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Self-force of a scalar charge in the space-time of extreme charged anti-dilatonic wormhole

The self-interaction for a static scalar charge in the space-time of extreme charged anti-dilatonic wormhole is calculated. We assume that the scalar charge is the source of massless scalar field with minimal coupling of the scalar field to the curvature of spacetime

(see equation (7) below). This Green's function can be expressed in terms of independent solutions of the corresponding homogeneous equation. As a rule, the Wronskian of these solutions coincides, up to a normalization constant, with the coefficient in front of the delta-function in the Green's function equation. In the case under consideration, this is not so. To solve this problem, we used the properties of the delta-function (see (9)). This enables us to reduce the problem to the standard case.
The paper is organized as follows. In section II we describe the gravitational background. In section III we obtained the unrenormalized expression for self potential of the static scalar charge on the considered background. Section IV describes the procedure of renormalization of the self potential and the final result.
Throughout this paper, we use units c = G = 1.

II. BACKGROUND
Let us choose the line element of space-time in the following form [79,80] This metric describes extreme charged anti-dilatonic wormhole, for which following relations are valid where Q -electric charge of the wormhole, M -mass of the wormhole and Q φ -dilatonic charge of the wormhole.

III. SELF POTENTIAL OF THE STATIC SCALAR CHARGE
Let us consider a scalar field φ with scalar source j. The corresponding field equation has the form where g (4) is the determinant of the metric g µν , q is the scalar charge and τ is its proper time. The world line of the charge is given byx µ (τ ). For a particle at rest in static space-time (1) field equation (4) can be rewritten as follows where we take into account dτ /dt = √ −g tt = e −α(r) for the static charge. Due to spherical symmetry of the problem under consideration we represent the potential in the following form where Y lm (Ω) is the spherical functions of argument Ω = (θ, ϕ), cos γ ≡ cos θ cosθ + sin θ sinθ cos(ϕ −φ). The radial part, g l , satisfies the equation Introducing a new function G l (r,r) = e α(r) g l (r,r) and taking into account δ(r,r) e α(r) = δ(r,r) e α(r) (9) one can obtain the following equation for G l (r,r) Two independent solutions of corresponding homogeneous equation will be denoted by Ψ 1 (r) and Ψ 2 (r) Ψ 1 (r) is chosen to be the solution which is equal to zero at r → +∞ and divergent at r → −∞. Ψ 2 (r) is chosen to be the solution which is equal to zero at r → −∞ and divergent at r → +∞. That is, Then one can represent the solution of equation (10) in the following form Normalization of Ψ is achieved by integrating (10) with respect to r from (r − ǫ) to (r + ǫ) and letting ǫ → 0. This results in the Wronskian condition We consider the radial equation (11) in domains r > 0 and r < 0. We may easily construct independent solutions of this equation for these two domains separately where P l and Q l are the Legendre polynomials of the first and second kind. Asymptotically The Wronskian of these solutions has the following form (see Ref. [81]) The solutions over all space can be written as follows where α 1,2 ± , β 1,2 ± are constants. Applying the boundary conditions (12) in (18) and using (16) we get Then the solutions (18) reduce to the following form Substituting these expressions into the Wronskian and taking into account (17) we get , r < 0 The Wronskian condition (14) implies the constraints on the coefficients: The matching conditions of solutions Ψ 1 (r), Ψ 2 (r) at r = 0 are the following Using these conditions one can obtain the following relations [70] β 1 + φ 2 These relations can be rewritten as follows Taking into account (15) and using the relations (see Ref. [81]) the Wronskians in (26) can be calculated as: Then, we can rewrite the equations (26) as Using these relations and (13,15,20,22) we obtain for r >r > 0 where z = ir |Q| andz = ir |Q| .
To find ∞ l=0 (2l + 1)Q l (z)Q l (z) we use the integral representation for the Legendre function of the second kind [81] Q l (z) = 1 2 and get Therefore for r >r > 0 and θ =θ, ϕ =φ

IV. RENORMALIZATION AND RESULT
The procedure of the self-force evaluation requires the renormalization of a scalar potential φ(x;x) which is diverged in the limit x →x (see, for example, papers [82,83]). This renormalization is achieved by subtracting the DeWittSchwinger counterterm φ DS (x;x) from φ(x;x) and then letting x →x For a scalar charge at rest in static curved space-time the DeWittSchwinger counterterm φ DS (x;x), which must be subtracted, has the following form [84] φ DS (x i ; where [85,86] Γ i jk are the Christoffel symbols calculated at pointx. The DeWittSchwinger counterterm φ DS (x;x) in the limit θ =θ, ϕ =φ can be easily calculated using the metric (1) Using the expression (35) for φ(r, r) we obtain the renormalized expression for φ in domain r > 0 φ ren (r) = lim r→r [φ(r,r) − φ DS (r,r)] = − q|Q|e −α(r) π(r 2 + Q 2 ) .
φ ren in domain r < 0 coincides with this expression because of the symmetry r ↔ −r of the problem. The limiting case r → ∞ gives us the following result φ ren (r) ≈ − q|Q| πr 2 .
The only nonzero component of the self-force is F r (r) = − q 2 g rr ∂φ ren (r) ∂r = − q 2 e −3α(r) r |Q| − 1 2 arctan( r |Q| ) Thus, we have obtained an analytic expression (42) for the self-force on a static scalar charge in an extreme charged anti-dilatonic wormhole spacetime (1).