Gravitational scattering upto third post-Newtonian approximation for conservative dynamics: Scalar-Tensor theories

We compute the scattering angle $\chi$ for hyperboliclike encounters in massless Scalar-Tensor (ST) theories up to third post-Newtonian (PN) order for the conservative part of the dynamics. To calculate the gauge-invariant scattering angle as a function of energy and orbital angular momentum, we use the approach of Effective-One-Body formalism as introduced in [Phys.Rev.D 96 (2017) 6, 064021]. We then compute the nonlocal-in-time contribution to the scattering angle by using the strategy of order-reduction of nonlocal dynamics introduced for small-eccentricity orbits.


I. INTRODUCTION
The observation of gravitational wave signals, with the first observation by the LIGO-Virgo collaboration in 2015 [1], opened a new era to probe the dynamics of the strongfield gravity regime.The third generation of detectors [2][3][4], along with the next generation of telescopes, such as the Einstein telescope [5] and the Cosmic Explorer [6] will be crucial for probing the strong-field dynamics of gravity by constraining the parameters of the alternative theories of gravity.
Amongst the theories alternative to Einstein's General Relativity (GR), the simplest theory is the addition of the massless scalar field to GR known as the scalar-tensor (ST) theory.The ST theories have been extensively studied and tested [7][8][9][10][11][12].Besides arising naturally in the UV complete theories of gravity, the addition of the scalar field is also equivalent to f (R)-theories of gravity [13].The two-body problem for ST theories have been extensively studied within the post-Newtonian (PN) approximation for both the dynamics and waveform generation in Refs.[14][15][16][17][18][19][20][21].
The detection of the gravitational wave signals relies on a large bank of (semi-) analytical accurate waveform templates to match filter against the data observed in the detectors.Therefore, the two-body PN dynamics in ST theories have been mapped within the effective-one-body (EOB) formalism [22][23][24][25][26] to incorporate the corrections due to massless scalar-tensor theories in the EOB approach based waveform models [27][28][29].These results were obtained for the elliptic motions of the compact binaries.
The EOB description of the unbound, scattering states of the binary systems was introduced in Ref. [30].Recently, the approach was used to compute the scattering angle within the PN approximation in GR [31].The main aim of this paper is to compute the scattering angle in ST theories upto 3PN order using the EOB Hamiltonian in ST theories.* tj317@cam.ac.uk The paper is organised as follows.In Sec.II, we give a brief reminder of ST theories and the EOB formalism in ST theories.Then, in Sec.III we derive the scattering angle for local part of the dynamics in ST theories at the 3PN order and derive the scattering angle for nonlocal part of the dynamics at 3PN order using order-reduction approach in Sec.IV.Finally, in Sec.V we sum the local and nonlocal contribution at 3PN in large-j expansion.

II. BRIEF SCALAR-TENSOR THEORY AND EOB REMINDER
We consider mono-scalar massless ST theories described by the minimal coupling of the scalar field to the metric in the Einstein Frame, and its action reads where g µν is the Einstein metric, R is the Ricci scalar, ϕ is the scalar field, Ψ collectively denotes the matter fields, g ≡ det(g µν ) and G is the bare Newton's constant.
Here, we adopt the conventions and notations of Damour, Jaranowski and Schäfer (DJS, hereafter) [7,9].In Einstein Frame, the field equations for ST theories are derived in [7].The coupling of the scalar field with the matter fields gives rise to the dynamics of the scalar field where the coupling between the matter and the scalar field is measured by the parameter in the equations of motion.The scalar field is nonminimally coupled to the metric in Jordan Frame (physical frame) where gµν is the metric in Jordan frame, and hence the Einstein-Frame mass is defined as where m(ϕ) is the scalar-field dependent mass.The Jordan-Frame parameters of Ref. [16,18,19] that encompass the scalar field effect upto the third PN order are converted in DJS conventions, i.e. the Einstein-Frame parameters (see, Table 1 of Ref. [24]).The mass function m(ϕ) is used to define the Einstein Frame parameters following Refs.[7,9] i.e. ) Finally, before proceeding to the computations of the scattering angle for ST theories, we briefly review the EOB formalism in ST theories.In the description of EOB, the relation between the real and EOB Hamiltonian is, where ν = µ/M is the symmetric mass ratio.The reduced-mass effective Hamiltonian ( Ĥeff ) is given by where pr , pφ (with the magnitude j = pφ ) are the dimensionless radial and angular momenta, and r is the dimensionless radial separation.The dimensionless variables are defined as, Hereafter, the superscript hat will be used to denote the dimensionless variables.The three EOB potentials (A, B, Q e ) upto 3PN in gauge choice of Ref. [32] (also known as Damour, Jaranowski and Schäfer gauge) formally read where the ν-dependent coefficients a i , b i and q i take into account both GR and ST corrections which are separated as q 3 = q GR 3 + q 3,ST . (2.16) The GR coefficients are known analytically up to 6PN [32][33][34][35][36], except for some unknown coefficients proportional to ν 2 .As for the ST theories, the nonlocal-in-time contributions start at the 3PN-order, the 3PN coefficients can be decomposed as a 4,ST = a I 4,ST + a II 4,ST , (2.17) where the superscripts I and II denote the local and nonlocal contributions, respectively.These coefficients can be further decomposed as, q I 3,ST = q loc 3,ST + q log 3,ST ln(u) . (2.22) These corrections to the EOB potentials in the ST theories upto 3PN order have been derived in [22][23][24][25][26].

III. SCALAR-TENSOR SCATTERING ANGLE : LOCAL CONTRIBUTIONS
In this section, we derive the contribution to the scattering angle for encounters of two non-spinning bodies for the local part of the conservative dynamics up to third PN order in Scalar-Tensor (ST) theories.As the nonlocal-in-time (tail) effects start only at the 3PN-level in ST theories, the scattering angle upto 3PN can be separated as a sum of functions, where χ loc and χ nonloc are local, nonlocal contributions to the scattering angle, respectively.
To compute the scattering angle we use the EOBderived integral expression for the scattering angle χ [31].The action in EOB takes the form, where E eff is the EOB energy and T eff is the EOB metric coordinate time.Using the equation of motion of orbit, and the Hamilton's equations, the scattering angle is where u = 1/r, u (max) = 1/r min is the distance of the closest approach of two bodies, and Ē is the dimensionless energy variable defined as [31] Ē A. PN-expanded χ loc for Scalar-Tensor theories Let us now first compute the radial momentum pr as a function of u = 1/r, orbital angular momentum and energy which would then be used to compute the explicit integral of Eq. (3.4).This is obtained by iteratively solving in p2 r the EOB energy conservation law, which yields, with η ∼ 1/c as a PN-order marker1 .This kind of formal PN expansion of pr generates a sequence of divergent integral in the limit [0, u (max) ].However, it was shown in [37] that the correct value of such a PN expanded integral is obtained by first using the Newtonian limit of u (max) as the upper limit of the integral and then taking the Hadamard partie finie of the integrals.The upper limit of the integral, u max , is the positive root closest to zero of the Eq.(3.7) and at the Newtonian level it reads The integrals of Eq. (3.4) for ST theories are evaluated using this standard technique except a logarithmic integral arising at 3PN order.To simplify the expressions of the scattering angle, we introduce an auxiliary variable and a function The scattering angle for local contribution 1 2 χ loc upto 3PN order can be decomposed as a sum of contributions from each PN order, i.e.
where, Here, for simplicity we do not substitute the values of the ST corrections a i,ST , b i,ST and q i,ST .The explicit expressions of the corrections have been derived in Refs.[22][23][24][25][26].
Finally, the last contribution, I χ , to the 3PN scattering angle is, Since this integral can not be solved using the standard techniques as above, we simplify the integral by using suitable integration by parts as where the last term is now a convergent integral defined as The integral of Eq. (3.18) can not be expressed in terms of the elementary functions.However, after suitable change of variables, the integral can be computed in large j-expansion (small ǫ-expansion) at fixed p ∞ .We follow the approach of Ref. [31] to compute the j-expansion of the integral.
Here, we display the first three contributions to integral in j-expansion, where The higher order contributions can be computed following the same approach.

B. Final result of the 3PN scattering angle in large j-expansion
The result presented in Eq. (3.14) of the scattering angle at the 3PN-order is fully explicit except the integral I χ of Eq. (3.16).To compute this integral, we expressed it into a simpler integral of Eq. (3.17).Then at the end of the last subsection, we computed large j-expansion of this remaining part, I χ , of the integral I χ .
Let us now insert the results of Eq. (3.17) in the large j-expansion of the scattering angle at the 3PN order.As the contributions to both I χ and I χ start at 1/j 4 order in their large j-expansion, we only show the large j-expansion of the exactly known part of χ (3PN) loc /2 upto 1/j 4 , i.e.In Eq. (3.20), we computed the integral I χ in the large j-expansion, and its first term reads, Now, combining Eqs.(3.22) and (3.23) gives the large jexpansion of the total scattering angle at the 3PN order for local part of the dynamics in ST theories.

IV. NON-LOCAL CONTRIBUTIONS TO THE SCATTERING ANGLE
In this section, we compute the leading order (LO) nonlocal contributions to the scattering angle using the order-reduction approach of Ref. [36] for bound orbits.This approach has been recently used to derive the nonlocal contributions to the EOB metric potentials for bound orbits in ST theories [25,26].Here, we will use this approach for hyperboliclike orbits in ST theories following Refs.[31,38].
As the tail contribution to the Hamiltonian starts at 3PN order in ST theory one can compute the scattering angle χ nonloc by considering the Hamiltonian where H N is the Newtonian-order Hamiltonian and H tail is the LO tail contribution [19], as all the other PN contributions upto 3PN order have been already considered in local scattering angle computation.
For the computation of the scattering angle using the general Hamilton-Jacobi derived equation, the function radial momentum is first computed by solving for p2 r the energy conservation law, in M = 1 (µ = ν) units.At LO in tail, the solution of the equation in pr is where p0 r is the Newtonian contribution.Inserting the solution in Eq. (4.2), the nonlocal contribution to the scattering angle reads where The LO tail contribution to the Hamiltonian in ST theories reads [19] s,i (t + τ ) , (4.7) where Pf is the Hadamard partie finie function with the Hadamard partie finie scale s and I (2) s,i is the second timederivative of the scalar dipole moment, I s,i .In the centerof-mass (COM) frame, the scalar dipole moment is where s A , s B are the sensitivities of two bodies.Thus the LO potential W tail is, s,i (t)I In Refs.[25,26], it is shown that the scalar dipole moment in action-angle variables and using the Kepler's equations for ellipticlike orbits is a periodic function, and hence can be decomposed into Fourier series.Here, we are considering hyperboliclike motions, therefore the Cartesian coordinates are parameterised as and the hyperbolic Kepler equation is where e is the eccentricity, a the semi-major axis, and n = 1 ā3/2 ; ā = −a .( Similar to the ellipticlike orbits, the scalar dipole moment for hyperboliclike motions can also be decomposed into Fourier series, i.e. Ĩs,i = dt I s,i e iωt , where Ĩs,i (ω) is the Fourier-transform of the scalar dipole moment.
Inserting the Fourier transformation of the scalar dipole moment (and τ = G AB M (t ′ − t)) 3 in Eq. (4.9) yields, The partie finie integral of the last term in the above equation is, where γ Euler is Euler's Gamma.Inserting Eq. ( 4.16) into Eq.(4.15) gives the Fourier-domain formula for the potential, where Ĩs,i (−ω) = Ĩ * s,i (ω).To compute the explicit expression of Eq. (4.17) of potential W rmtail in terms of Ē and j, we insert the Fourier transform of scalar dipole moment.For this, we evaluate the Fourier transforms of (x, y) for hyperbolic orbits, i.e x = dt e iωt x(t) , y = dt e iωt y(t) .  (1 p (q) , ( we find the Fourier transform as x = πa ω p q H (1)  p (q) − H where We then consider the Fourier transform of (x, y) in large-j limit which is equivalent to large-e limit e = 1 + 2 Ēj 2 .The large-e limit of Eqs.(4.20)-(4.21)yields, where q = iu.The Hankel functions evaluated at purely imaginary arguments are related to modified Bessel functions K ν as (see Eq. (9.6.4) of [39]) 1 (ix) .4.17) and then taking j-derivative of potential W tail , the explicit expression of the scattering angle in large-e limit yields, where we recall that ŝ = s/(G AB M ) is the dimensionless regularisation scale defining the nearzone-farzone separation.Here, the subscript "±" denotes the symmetric and anti-symmetric parts of the ST parameters, e.g.

V. SUMMING THE LOCAL AND NON-LOCAL CONTRIBUTIONS TO χ3PN IN LARGE-J EXPANSION
In Sec.III, we first computed the local scattering angle upto 3PN order and then in Sec.IV we separately computed the nonlocal contributions at the 3PN order.The results at 1PN and 2PN levels were given in fully explicit and exact form.However, the results at the 3PN order were obtained in the large-j expansion for both the local contribution (due to the logarithmic term I χ ) and the nonlocal contribution.On combining the two separate 3PN order contributions to the scattering angle at 3PN, we find where we use the notations of Refs.[24,25], with X A,B ≡ m 0 A,B /M .As the scattering angle is gaugeinvariant, the arbitrary scale ŝ has been cancelled between the two contributions as expected.

VI. CONCLUSIONS
Building upon the results of [22][23][24][25][26] for the corrections in the EOB metric coefficients (A, B, Q e ) for massless scalar-tensor theory for the conservative part of the dynamics, we determined the scattering angle for hyperboliclike orbits upto the 3PN order for both the localin-time and the nonlocal-in-time part of the dynamics.First, we compute the scattering angle for the local part of the dynamics by: (i) deriving the radial momentum as a function of u, orbital angular momentum and energy by iteratively solving the EOB energy conservation law; (ii) calculating the scattering angle using the standard techniques of Ref. [37] for solving divergent integrals arising in the PN-expansion of the radial momentum except the integral I χ at the 3PN order; and (iii) computing the inte-gral I χ by using the appropriate integration by parts and expanding in large-j the remaining integral after change of variables [31].We then computed the total contribution to the 3PN order scattering angle in the large-j expansion.
Then, we computed the nonlocal-in-time contribution by using the approach introduced in Ref. [36] for GR of order-reducing (time-localisation) the Hamiltonian in small-eccentricity case for hyperboliclike encounters [31,38].Finally, we substituted the ST corrections of the metric potentials (A, B, Q e ) and sum both the local and nonlocal contributions in the large-j expansion at 3PN order.As a test of our results, we checked that that the scattering angle coincides with the scattering angle of GR (see Ref. [31] for GR results) in the GR limit as expected.This paper must be seen as a first step to compute the gauge-invariant scattering angle within the PN expansion for massless scalar-tensor theories.In future work we will address radiation reaction contributions to scattering.

3
Here, the dimensionless variable τ = τ G AB and t = T G AB M .