Light bending in $F\left[g(\square)R\right]$ extended gravity theories
Breno L. Giacchini Ilya L. Shapiro
gr-qc 53B50, 83D05, 81T20












remarkTheorem[section]
propoProposition[section]
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    ln                   
    Δαβχγδεϵζικλ∇∂ϱρσωφτþθϑυΓΔΛΣΩΘΘΥ effective action 
    
    
    
    
    Light bending in  F[g()R] 
    extended gravity theories
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    Breno L. Giacchini^a,
      and  Ilya L. Shapiro^b,c,d
    
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    (a)  Centro Brasileiro de Pesquisas Físicas
    
    
    Rua Dr. Xavier Sigaud 150, Urca, 22290-180, Rio de Janeiro, RJ, Brazil
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    (b)  Departamento de Física, ICE, Universidade Federal de Juiz de Fora
    
    
    Campus Universitário - Juiz de Fora, 36036-900, MG, Brazil
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    (c)  Tomsk State Pedagogical University, Tomsk, 634041, Russia
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    (d)  Tomsk State University, Tomsk, 634050, Russia
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     E-mails:
     breno@cbpf.br,
       
    shapiro@fisica.ufjf.br
    
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    Abstract. We show that
    in the weak field limit the light deflection alone cannot distinguish
    between different R + F[g()R] models of gravity, where F
    and g are arbitrary functions. This does not imply, however, that
    in all these theories an observer will see the same deflection angle.
    Owed to the need to calibrate the Newton constant, the deflection
    angle may be model-dependent after all necessary types of
    measurements are taken into account.
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     MSC:53B50,  
    83D05,  
                 
    81T20   
    
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    PACS: 
    04.20.-q,     
    04.50.Kd 	 
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    Keywords:  Higher derivative gravity, light bending,
    Brans-Dicke theory, non-local gravity, scalar-tensor theories
    
    
    
    
    § INTRODUCTION
    
    
    Modified gravity theories attract a lot of attention nowadays, this
    concerns in particular the models constructed with the scalar
    curvature and its covariant derivatives. In this Letter we discuss the
    deflection of light in the linear regime in a generic model of this
    type. This issue has already been discussed in the literature using
    a variety of approaches applied to specific
    models <cit.>. In the various references one can
    meet different conclusions. In some works the statement is that
    the bending of light occurs exactly in the same way as in general
    relativity (GR), while in other works it is claimed that the light
    bending effect can be used to distinguish between some of such
    theories and GR. The aim of the present work is to present a
    general discussion on the subject and clarify to which extent the
    bending of light, taken alone or together with other observables,
    can distinguish among the theories which depend on
    an arbitrary function F(Y), where Y=g()R, and g is a
    function of the d'Alembert operator.
    
    The action of our interest
    is given by the expression
    𝒮_ gen = ∫ d^4 x √(-g){
    -2κ^2 R + F[g()R]} ,
    
    where ^2 = 32 π G and G is Newton's
    constant.
    The integrand of the gravitational action should be supplemented by
    the Lagrange matter density ℒ_M.
    
    Our interest will be concentrated on the light bending by a weak
    gravitational field, which means that the background is considered to
    be  a small deviation from the flat Minkowski space. This condition
    immediately provides significant simplification, as the non-linear
    in curvature terms are expected to have small effect on the bending
    of light. Then the influence of a non-flat background can be safely
    regarded as at least a second-order effect. Therefore, without
    losing generality, one can restrict consideration by the actions
    which are at most quadratic in scalar curvature, but may be
    non-polynomial in derivatives. In this way one arrives at the
    simplified action of the form
    𝒮 = ∫ d^4 x √(-g)[
    -2κ^2 R
    + R f() R
    + ℒ_M ]  ,
    
    where f() is a function of  d'Alembert operator, which is
    not necessarily analytic.
    
    The theory (<ref>)
    includes many
    interesting particular cases. Let us give a short
    list of the models of this type which have been discussed in the
    literature.
     1mm
    
    1)   Taking  f(x)= const.gives the well-known R+R^2 gravity, which is
    the basis of the
    successful  inflationary model of Starobinsky
    <cit.>. In this case the bending of light is exactly
    like in GR, as has been proved in the paper by Accioly et al.<cit.>. Our present consideration can be seen as a
    generalization of this work to the general model (<ref>).
    Let us note that this statement concerns only the prediction of light
    deflection  without taking other observables into account,
    as will be discussed in what follows.
     1mm
    
    2)  f(x) is a polynomial function of order N. The bending of
    light for the particular case of a linear function f() has
    been explored in the recent work <cit.>. The Ricci-squared
    term has also been included into consideration in <cit.>, so the
    theory (<ref>) represents only the part of the model which
    is related to scalar curvature. Here we generalize the corresponding
    results of <cit.> (see also <cit.>) to arbitrary
    polynomial or non-polynomial functions.
     1mm
    
    3)  The logarithmic form factors ln( /μ^2) are
    typical for the quantum corrections coming from the loops of massless
    fields, and correspond to the Minimal Subtraction scheme running
    of effective constants. At higher loops there may be higher powers of
    the same logarithms. In case of massive fields the expressions for the
    gravitational form factors are more complicated <cit.>, but
    qualitatively are like  ln( [+m^2]/μ^2) for the
    quantum field of mass m. At low energy this expression becomes
    a constant, which corresponds to the gravitational  version of
    Appelquist and Carazzone decoupling theorem.
     1mm
    
    4)  The functions f_-1()=μ^-1 and
    f_-2()=μ^-2 have been introduced in <cit.>
    in order to explain how the renormalization group running of the
    Einstein-Hilbert and cosmological terms can look like. The main
    source of importance of these terms for cosmological applications
    is that under the global scaling the Rf_-1()R-term behaves
    exactly like the Einstein-Hilbert term, and the Rf_-2()R-term
    behaves like the cosmological term. At the same time, since both
    terms are non-local, they do not reproduce the Einstein-Hilbert and
    cosmological terms exactly, and this leads to their fruitful use in
    cosmology, as suggested in <cit.> for
    the R(^-1R)-type actions and in <cit.> for the
    R^-2R-term (see further references therein). In view of
    this phenomenological success, it would be interesting to know
    what is the effect of these terms for the light bending.
    
    In what follows we
    show that
    for the modified
    actions (<ref>) in the weak field limit the deflection of light is not directly
    affected by the terms quadratic in scalar curvature. This does not
    mean, however, that the predictions for the light bending is the same
    in all the models (<ref>), since the experimentally
    measured Keplerian mass of astronomical bodies can differ from
    theory to theory.
    
    The paper is organized as follows. In Sec. <ref> the bending
    of light is obtained on the basis of tree-level amplitudes, for the
    cases of GR and the models (<ref>). The purely classical,
    geometric optics approach is used in Sec. <ref>, where we also
    discuss the correspondence with the Brans-Dicke models and the
    role of conformal transformations for the light bending and other
    observables. Finally, in  Sec. <ref> we draw our conclusions.
    
    
    
    
    
    § TREE-LEVEL APPROACH
    
    
    The interaction between light and a weak gravitational field can be
    explored using the scattering approach by means of the tree-level
    Feynman diagrams <cit.>. Loop
    corrections may be included as well into this formalism
    <cit.>. In the present work our
    interest is concentrated on the classical behaviour of the theories
    (<ref>), and this includes the non-local form factors which
    can be attributed to the dressed propagator of gravitational modes.
    Hence in this framework we only need to deal with the tree-level
    diagrams.
    For the sake of consistency we shall start by
    briefly reviewing the use of the same method within GR, before going
    on in applying it to 
    extended models (<ref>).
    
    
    
    
     §.§ Light bending in general relativity
    
    
    The calculation for GR can be found in some books (see,
    e.g., <cit.>), but our derivation is carried out in a
    slightly different way, such that consequent generalization to the
    more complicated model (<ref>) is straightforward.
    
    The starting point is
    the metric
    written as a fluctuation around the Minkowski space, i.e.,
    g_μν = η_μν + κ h_μν, with
    η_μν = diag(+1,-1,-1,-1). Since gravity couples
    to matter via the energy-momentum tensor, in the first order in
    κ the interaction is described by
    ℒ_int =  -  κ/2  h^μν T_μν ,
    LT
    where T^μν is the matter energy-momentum tensor in
    flat space-time, which defines the interaction vertex. For the
    electromagnetic field we have
    
    T_em^μν =   -  η_αβ F^μαF^νβ + 1/4η^μν  F_αβF^αβ,
    
    with F_μν = ∂_μ A_ν - ∂_ν A_μ.
    We will also need the interaction with a minimal
    massive scalar,
    
    T_scal^μν = ∂^μϕ ∂^νϕ
    - 1/2η^μν(   ∂_αϕ ∂^αϕ - M^2 ϕ^2 )  .
    
    
    In the context of GR, gravitational interaction is related to the exchange of
    only one particle, the graviton, whose propagator can be written as
    
    D_μν,αβ^GR (k)
     =  P^(2)_μν,/k^2
    - P^(0-s)_μν,/2k^2
    in the momentum-space representation. Here P^(2)_μν,
    and P^(0-s)_μν,αβ are the spin-2 and spin-0
    projectors (see, e.g., <cit.>) defined through
    
    P^(2)_μν,αβ    =    12( θ_μαθ_νβ
    + θ_μβθ_να)
    - 13θ_μνθ_αβ,
    
    
    P^(0-s)_μν,αβ    =    13θ_μνθ_αβ,
    
    where the longitudinal and transverse vector-space projectors are
    ω_μν = k_μ k_ν/k^2,
      and  θ_μν = η_μν - ω_μν.
    
    Let us note that the other parts of the gravitational propagator are
    irrelevant, because it is going to be directly contracted
    with the matter sources in the tree-level approximation. For the same
    reason we do not need to worry about the gauge-fixing dependence
    for the quantum gravitational field. After going on-shell this
    dependence will disappear anyway.
    
    At the tree-level we can model the massive source which produces
    the gravitational field by a massive scalar field. Therefore we have
    to evaluate the exchange of graviton between such a scalar and a
    photon.
    The leading contribution to the amplitude ℳ associated
    to this process is of the order κ^2 and is related via the
    LSZ reduction formula to the function
    
    G_4 = -^2/4∬ d^4 z_1 d^4 z_2
     ⟨ 0 |
    T ϕ(x) ϕ(x') A_μ(y) A_ν(y')
    T_scal^(z_1)
    T_em^ρσ (z_2) h_αβ(z_1)
    h_ρσ (z_2) | 0 ⟩  _c  .
    G4
    
    Using Wick's theorem, the term
    h_αβ(z_1) h_ρσ (z_2) yields the graviton
    propagator (<ref>). The energy-momentum
    tensor of the electromagnetic field satisfies the conservation law
    _ T^_em = 0 and is traceless,
    T^_ em = 0. Consequently, the terms  proportional
    to _μν in the graviton propagator do not contribute to
    the expression (<ref>), and hence to the scattering amplitude.
    We can thus replace the projectors þ_μν by the metric
    η_μν in the graviton propagator.
    From the physical perspective the situation is such that photons
    interact only with the spin-2 sector of the propagator.
    
    
     
    
    The lowest order contribution to the scattering matrix comes from
    the tree diagram in Fig. <ref>,
    in which one photon with initial momentum p and
    one scalar with momentum q interchange one graviton, resulting
    in a final state with momenta p^' and q^'. The
    corresponding scattering amplitude is
    ℳ   =   
    V_μν^(ϕ) (q,q^')
     D^μν,_GR (k)   V_ ρ^(A)(p,p^')
     ^ρ (p)  ^* (p^')  ,
    
    where we included the polarization vectors of the photons. The
    vertex functions are
    
    V_μν^(ϕ) (p,p^') = -i/2[
    p_μ p^'_ν +  p^'_μ p_ν
    + η_μν (M^2 - p · p^' ) ]  
    for the scalar, and
    
    V_ μν^(A) (p,p^')     =     -iκ/2[ p · p^' ( η_μαη_νβ
    + η_μβη_να  - η_μνη_αβ )
    +  η_αβ p^'_μ p_ν
    + η_μν ( p_α p^'_β + p_β p^'_α )
    
           - ( η_βν p^'_μ p_α
    + η_αμ p^'_β p_ν
    + η_αν p^'_μ p_β
    + η_βμ p^'_α p_ν ) ]   
    for the photon.
    
    As discussed above, in the amplitude (<ref>) one can
    replace the propagator D^μν,αβ_GR(k) by
    the expression
    1/2k^2( η_μαη_νβ
    + η_μβη_να).
    
    Further simplification is achieved by recalling that we are
    interested in the deflection of a light ray passing nearby a very
    massive object (such as a star or a galaxy), described by the scalar
    field. In this regime it is appropriate to work in the large-mass
    approximation for the scalar field, that means q=0,
    q^'=-k,
    M ≫|k| and M ≫|p|,
    and consider that the photon undergoes an elastic scattering
    with |p| = |p^'|. For a
    vanishing momentum transfer the scalar-scalar-graviton vertex
    simplifies to
    V_μν^(ϕ) = - i κ M^2 δ_μ^0δ_ν^0.
    For the remaining vertex, it is useful to work in the Lorentz gauge
    for which ϵ^0=0 and
    ϵ·p = 0. 
    It is then immediate to verify that Eq. (<ref>)
    simplifies to
    ℳ =  -  κ^2 M^2 E^2/|k|^2 ϵ (p) ·ϵ^* (p^')   .
    
    Here E = p_0 = p_0^' is the energy of the photon. The
    polarization factor can be dropped in the limit of vanishing
    momentum transfer (see <cit.> for further discussion
    on helicity effects such as the flip in the case of more general,
    rotating sources). Also, we shall insert the normalization factors
    √(2E_i) for each external particle. Recalling that
    ^2 = 32 π G, the scattering amplitude then reads
    𝒱 (k) = - 8 π G M E/|k|^2  ,
    
    whose Fourier transform gives the interaction potential felt by the photon:
    
    
    V(r) = - 2 G M E/|r|  .
    
    As expected this potential is twice the standard Newtonian potential,
    which is felt by non-relativistic particles <cit.>.
    This factor two, indeed, took an important role in the development of
    GR, since it was missing in the first light bending prediction made by
    Einstein based solely on the equivalence principle <cit.>.
    With the potential (<ref>) above and using classical scattering
    theory it is possible to compute the deflection angle undergone by a
    light ray owed to the weak field produced by a massive
    object[Let us note that one may also evaluate the
    unpolarized small-angle scattering cross section from (<ref>)
    as in Refs. <cit.>.
    However, as discussed in <cit.>, this quantum cross section
    cannot be generally applied to the deflection by astronomical bodies.].
    
    
    
    
     §.§ Light bending in 
    extended theories
    
    
    The approach described above can be easily adapted to extended
    gravity theories, in which the propagator of the gravitational
    interaction has a richer structure. For example, in the case of the
    action (<ref>) with f(x) being a polynomial function
    of degree N, there can be N+1 scalar particles of mass
    μ_i ≠ 0 (i=0,1,⋯,N) along with the usual (massless)
    graviton. It is possible for the propagator to have
    degenerate poles, complex poles, or even to have solely the pole
    at k^2=0. The crucial point is that all those possibilities stem
    from modifications in the scalar sector of the propagator, while the
    spin-2 mode propagates in the same way as in GR. Namely, the
    propagator of the extended theory (<ref>) can be
    written as
    
    D_μν,αβ^ext. (k)
       =   P^(2)_μν,αβ/k^2
    - P^(0-s)_μν,αβ/2k^2
    [1 - 3 κ^2 k^2 f(-k^2)]  .
    
    In order to gain extra degrees of freedom in the tensor sector one
    needs to include an R_ρf̃() R^ρ-term
    into the action <cit.>.  In the present work we do not
    consider this extension and deal only with the
    model (<ref>).
    
    It is easy to see that, regardless of additional (compared to GR)
    degrees of freedom
    mediating gravitational interaction in the theory (<ref>),
    the lowest order vertex associated to the interaction between
    photons and gravitational field is exactly the same as in GR. This
    happens because the extra modes are all in the scalar sector, which
    interacts through the trace of the energy-momentum tensor, as
    discussed in Sec. <ref>. In other words, photons interact only
    with the spin-2 (graviton) sector of the propagator, which is the
    same as GR, in spite of the Rf()R-terms in the action. This is
    the main reason of why the light bending alone cannot distinguish
    between original GR and this type of extensions.
    One can write for the Feynman amplitude
    ℳ    =    
    V_μν^(ϕ) (q,q^')  D_ext.^μν, (k)
      V_αβρσ^(A)(p,p^')  ^ρ (p)
     ^*σ (p^') 
        =    
    V_μν^(ϕ)(q,q^')
    ( η^μαη^νβ
    + η^μβη^να/2k^2)
     V_αβρσ^(A)(p,p^') ^ρ (p)
     ^* (p^')
    
       =   
    V_μν^(ϕ) (q,q^')  
    D_GR^μν,αβ (k)
     V_αβρσ^(A)(p,p^')
     ϵ^ρ (p) ϵ^*σ (p^') ,
    
    that is the same as in GR (<ref>). All in all, light bending
    alone cannot distinguish among any two theories of the
    type (<ref>). This general result is in agreement with
    the calculations
    of <cit.>
    for particular models by means of other methods, which will be also
    discussed  in Sec. <ref>.
    
    
    
    
     §.§ Light massive modes and rescaling of potential
    
    
    
    Despite the equality of the interaction potential in the whole class
    of theories (<ref>), the statement that all these theories
    predict the same deflection angle for light may be misleading.
    The possible effect is due to the difference between the product
    of the Newton's constant G and the mass M of the body
    which enters into the formulas for the deflection, and the one
    which comes from other measurements and/or observations.
    Indeed, G is measured in the experiments, e.g., the Cavendish-type
    ones. The measurement of the mass M can be achieved
    by detailed investigation of the orbits of astronomic bodies. As a
    result, in order to predict a deflection angle one has to take into
    account not only the interaction potential with relativistic particles,
    but also the non-relativistic potential resulting from the gravitational
    scattering of massive particles. This quantity can be computed, as
    in (<ref>), as the Fourier transform of the
    non-relativistic scattering amplitude
    ℳ   =    V_μν^(ϕ_1) 
    D^μν,αβ  V_αβ^(ϕ_2),
    
    where ϕ_1 and ϕ_2 are scalar fields with masses M and
    m. As the energy-momentum tensor for massive scalar fields has
    a non-vanishing trace, the non-relativistic potential depends on the
    form of the function f(x) and hence on the massive parameters of
    the model.
    
    For example, the non-relativistic potential for a polynomial f(x)
    acquires Yukawa-like correction terms for each massive simple pole
    in the
    propagator <cit.>,
    
    V(r)
       =   
    -  G M m/|r|( 1  + 1/3∑_i=0^N∏_j = 0 
     j ≠ i^N
    μ_j^2/μ_j^2 - μ_i^2 e^- |r|μ_i).
    
    If the quantities μ_i
    are real and are much
    larger than a typical scale r_lab^-1 of experimental
    measurements of G, then the Yukawa terms are suppressed and the
    potential reduces to the Newtonian one. On the other hand, if 
    the masses of the extra degrees of freedom satisfy
    μ_i ≪ r_lab^-1 for a given experimental/observational
    system, then the exponentials can be approximated by the unit, and
    it is possible to show that the potential is roughly 4/3 of Newton's one <cit.>. In this case our laboratory
    measurements would actually measure an effective
    G_eff = 4/3 G, thus the predicted bending angle
    for a light-ray passing close to the solar limb would be 3/4 of the
    general relativity's prediction.
    
    This simple example shows that the function f(x) can have an
    indirect influence on the prediction of deflection angles. The effect
    is indirect because it does not concern the bending of light itself,
    but the measured value of the product GM, which appears in
    the expression for the bending angle.
    In the realistic astrophysical
    situations all this concerns only the models with very light,
    albeit massive, gravitational degrees of freedom.
    
    
    
    
    
    
    § GEOMETRIC OPTICS AND NULL-GEODESIC STRUCTURE
    
    
    Solving the wave equation
    may be viewed as a cleaner way of evaluating the bending of
    light, based on the assumption that the source of the gravitational
    field is classical, not quantum.
    As discussed
    in <cit.> 
    this procedure avoids possible subtleties
    which appear when using the calculations via Feynman diagrams.
    For this reason, in this section we describe how the higher-derivative
    and non-local extensions (<ref>) manifest in the bending
    of light using classical, geometric arguments.
    
    The analogy between the propagation of massless particles in curved
    space-time and geometrical optics in a medium can be regarded as one
    of the most straightforward methods for describing the gravitational
    deflection of light. One can consult, e.g., Refs. <cit.>
    for the detailed description of this method. Let us give a brief
    survey of the main results.  For a static and spherically symmetric
    source the metric is assumed  to have an isotropic form,
    
    ds^2
       =   
    (1 + 2 Φ) dt^2
    - (1 - 2Ψ) [ dr^2 + r^2 ( dθ^2
    + sin^2θ  d ϕ^2) ],
    
    where Φ=Φ(r) and Ψ=Ψ(r) only. It is possible to
    demonstrate that in this case light propagation in the geometric
    optics limit is equivalent to the propagation in the flat space in
    a medium with the effective local refractive index
    
    n(r)
       =   √(1 - 2Ψ(r)/1 + 2Φ(r)).
    
    The deflection angle can be evaluated with Snell-Descartes law
    (see <cit.> for an explicit calculation). This
    method is also equivalent to the use of Maxwell equations in a
    space-time which represents a small fluctuation around Minkowski one <cit.>.
    
    In the case of GR, to the first order in G, the
    metric (<ref>) associated to a point source of mass
    M is given by the potentials
    Φ (r)    =   Ψ (r)   =  -  G M/r  .
    
    Due to the smallness of these
    quantities, Eq. (<ref>) yields the
    effective refractive index in GR,
    
    n_GR(r)
       =   
    1  + 2GM/r +  O(G^2) .
    
    
    Let us derive the effective refractive index for
    extended models of gravity (<ref>) with
    an arbitrary function f(x).
    To this end we rewrite the potentials Φ and Ψ in
    terms of the new functions Φ_* and Ψ_*,
    Φ (r)
       =    V(r) + Φ_* (r)
    ,   Ψ (r) = V(r) + Ψ_* (r)
    ,   with  Δ V = ^2 ρ/8 .
    
    In the last expression ρ is the static density of matter sourcing
    the field. It is clear that for GR the equations of motion must yield
    Φ_* = Ψ_*≡ 0. The solution in the
    form (<ref>) for the field generated by a static mass
    distribution can be obtained from the trace and the (0,0)-component
    of the equations of motion, which read
          
    4[1-3h(-Δ)] Δ (Φ - 2 Ψ)
     = κ^2 ρ ,
    
          
    4[ h(-Δ)-1] ΔΦ - 8 h(-Δ) ΔΨ
    =  - κ^2 ρ ,
    
          
    h() = [ 1 + 2 κ^2 f() ] .
    
    Let us remember that in the static case  boils down to -.
    Using the Ansatz (<ref>) it is
    straightforward to show that the functions Φ_* and Ψ_*
    satisfy the equations
    1-3h(-Δ)/1 - h(-Δ) ΔΦ_*
    = ^2 ρ/8,
        1-3h(-)/1 - h(-) Ψ_*
    = - ^2  ρ/8,
    
    from which we assume Φ_*(r) = - Ψ_*(r).
    
    Taking the point-like mass source
    ρ(r) = M δ^(3)(r), the effective
    refractive index is
    
    n_ext(r)
        =    √(1 - 2V - 2Ψ_*/1 + 2V + 2Φ_*)
        =    
    1 - ( V - Φ_* )
     - ( V + Φ_* ) + O(G^2)
    
        =     1 + 2 GMr + O(G^2) .
    
    The above expression is
    the same one which stems from GR (<ref>).  Once again, we
    see that the light deflection alone cannot be used to distinguish between
    the two theories. This matches and generalizes the previous results of <cit.> in the context of the R+R^2 gravity and
    of <cit.> for the sixth-order gravity. A quick inspection
    of (<ref>) reveals that this happens because of the cancellation
    of the R-extended contribution, which appears with exactly the same
    value, but with an opposite sign, in the h_00 and h_11
    components. However, as it was discussed in the previous section,
    in order to define the light bending one has to establish the quantity
    GM from a qualitatively different measurement.
    
    To better explain this conclusion one may write the
    metric (<ref>) in the PPN-inspired
    form <cit.>, with
    Φ = - G_effM/r ,     Ψ = -  G_eff M/r,
    
    being G_eff the quantity measured in a given experiment.
    Let us stress that it is not generally possible to cast theories of the
    form (<ref>) into the PPN formulation <cit.>.
    This happens because the non-relativistic limit in these theories may be
    non-Newtonian, since the potential is not proportional to r^-1 only
    but may have an additional non-trivial dependence on r, as
    discussed in <cit.>.
    Indeed, the quantities G_eff and γ are actually
    functions of r. In the phenomenologically relevant models of this
    sort one can thus expect a scale for which it is possible to observe
    deviations from the Newtonian mechanics, such as anomalous
    precession of elliptical orbits <cit.>. The force law
    can be investigated in such a regime and the free parameters of the
    model determined, including the gravitational constant G.
    Nevertheless, it may happen that these effects are not observable at
    a certain scale r̅ for which G_eff(r̅) and
    (r̅) are roughly constant and the laws of mechanics are
     close to the ones of Newton. In this scenario one can work with the
     metric in the PPN form above
     (see, e.g., <cit.>). Then, from the relations
          Φ = V + Φ_* = V ( 1 + Φ_*/V)
    
          Ψ = V - Φ_* =  V (1+Φ_*/V)
    
    in the vicinity of r̅ one reads off
    
    G_eff=( 1 + Φ_*/V) G
        and    γ = V-Φ_*/V+Φ_*.
    
    
    Assuming that  Φ_*(r̅) ≪ V(r̅) in a certain
    region, it follows that
    (r̅)  ≈   1 - 2 Φ_*(r̅)/V (r̅).
    
    On the other hand, if the condition  Φ_*(r̅) ≪ V(r̅)does not hold, the parameter  may be very different from the
    value γ_GR = 1 of GR. Then the bending angle
    can vary from theory to theory. For instance[The
    derivation of this result can be found in textbooks on GR, for
    the terms of Snell-Descartes law see <cit.>.], from (<ref>) the deflection θ undergone
    by a light ray with impact parameter b is
    θ = 4GMb^-1. In terms of G_eff this result
    has the form
    þ = 4G_effM/( 1 + Φ_*/V ) b = 
    2(1+)  G_effM/b ,
    
    with γ defined in (<ref>).
    
    Let us consider a simple example. In the case of a constant function
    f() ≡ (6α)^-2 (that is, R+R^2 gravity) with a
    point-like mass as a source one has (m ≡ακ^-1)
    Φ_*(r) = - GM e^-mr/3r,
        γ = 3 - e^-mr/3 + e^-mr.
    
    If m is sufficiently large, the Yukawa term has a range
    shorter than the scale of the recent torsion-balance experiments.
    Then for all practical purposes the quantity G_eff
    measured in laboratory is equal to G and γ = 1. On
    the other hand, if the scalar is very light, with a range much larger
    than the Solar System scales, then the measured solar mass in
    terms of Newtonian mechanics is such that
    
    (GM)_eff = (1 + 1/3  e^-mr̅)
    GM  ≈ 4GM/3,
    
    and ≈ 1/2. This example shows that one cannot assume
    Φ_* ≪ V in general. Inserting =1/2 into (<ref>) one
    finds the bending angle þ = 3/4θ_GR, in
    agreement with <cit.> (see also the example briefly
    discussed in Sec. <ref>). The PPN parameter γ
    in (<ref>) was derived in <cit.>, where it
    was also argued that its correct value should be, instead, =1 in
    view of the fact that Φ + Ψ = 2 V as in (<ref>). As we
    have shown above there is no conflict between Eqs. (<ref>)
    and (<ref>) after one takes into account the observable
    quantities in (<ref>). The result γ = 1 was found
    in <cit.> under the assumption that the free parameter
    m is such that the theory does not violate known experimental
    results. This is in agreement with our discussion, i.e., considering
    that m is such that it may affect the Newtonian force law only
    at sub-micrometre scale. Finally, the deflection of light in this model
    was investigated also in
    Refs. <cit.>
    with the conclusion that it cannot be used to distinguish between GR
    and R+R^2 at the leading order in G (see <cit.> for
    calculations at the next order). According to the discussion in the
    present work this assertion is partially correct, since any prediction
    is based on the quantity GM and the measured (GM)_eff
    can differ from GR and quadratic gravity.
    
    
    
    
    
     §.§ Relation to conformal transformation of the background
    
    
    The description of the bending of light in terms of the effective
    refractive index is closely related to the fact that light rays follow
    null-geodesics associated to the metric g_μν. In fact,
    the statement that the refractive index for the extended
    model (<ref>) equals general relativity's one to the
    first order in G can be translated to the geodesic structure of
    the manifolds in both theories in terms of conformal transformations.
    This association has been carried out, for example,
    in <cit.> for the model R+R^2, and
    in <cit.> for the more general
    model which is a polynomial function of d'Alembertian. Using the
    metric (<ref>) parametrised as (<ref>)
    and with the result Ψ_* = - Φ_* obtained above it is cursory
    to verify that
    
    g_μν^ext = 
    ( 1 + 2 Φ_* ) g_μν^GR ,
    
    were only the terms to order G were kept. Here
    
    g_μν^GR(r)  = η_μν + 2 Vδ_μν
    is GR's weak-field metric associated to the same matter configuration
    in the de Donder gauge.
    
    Since null-geodesics are conformal invariant, up to O(G)
    the null-geodesics of GR are also null-geodesics in the extended
    model (<ref>). As a result, light rays undergo the same
    gravitational deflection in both theories. On the other hand, time- and
    space-like geodesics are not conformally mapped to geodesics, and
    massive particles ought to deflect in a different manner.
    As mentioned before, the reasoning presented in this section
    parallels the description in terms of Maxwell equations on a
    curved background.  The conformal transformation
    g_μν^GR↦ g_μν^ext
    does not affect the equations of motion of the electromagnetic
    field, as we discussed above (see also <cit.>).
    
    The result (<ref>) turns out to be the linearised version 
    of the theorem proved in <cit.> on the relation of the
    model (<ref>), scalar-tensor theories and GR with extra 
    scalar fields[Indeed, it can be shown that the 
    higher-derivative theories whose Lagrangian is formed by functions
    of ^n R (for all n ∈ℕ) are dynamically equivalent
    to a (multi)scalar-tensor theory, which is then conformal equivalent
    to GR with scalar fields <cit.>.]. By evoking the same 
    arguments in terms of conformal transformations (see, e.g., 
    <cit.>), it follows that in the generalized Brans-Dicke (BD) 
    theory <cit.> in the form
    𝒮_BD = ∫ d^4 x √(-g)[
    ϕ R + ω g^μν∂_μϕ∂_νϕ
    - U(ϕ) + ℒ_M ]
    
    the non-trivial contribution to the bending of light also comes from
    the observable value (GM)_eff rather than to a new force
    acting on light rays.
    
    The PPN parameter γ for massive Brans-Dicke (mBD) theories,
    i.e., with ω≠ 0 and U^''(ϕ) ≠ 0, was
    computed in <cit.>, with the result
    _mBD = 2 + 3 - e^-mr/2 + 3 + e^-mr ,
    BD
    where m depends on ω and on the form of U. For a suitable
    potential U and in the limit ω→ 0, it is possible
    to show that γ_mBD above reduces to the expression
    in (<ref>), evaluated for the R+R^2 gravity. This is
    expected, since this theory is dynamically equivalent to a BD theory
    with ω=0 but with a non-trivial potential U <cit.>.
    Light deflection (or the PPN parameter γ) has been computed
    in the scalar-tensor formulation in <cit.>
    with the results consistent to those presented in the previous
    sections, as expected. We note that if m ≪ r^-1 (i.e., the scalar
    field has a range much larger than the experimental scale) then
    γ_mBD≈ (2ω+2)/(2ω+4), which gives
    γ_mBD≈ 1/2 for the quadratic equivalent model
    with ω = 0, in agreement to the example discussed in the last
    section.  It is important to notice, however, that this only occurs
    for a very light  massive field; in the case m ≫ r^-1 and = 0
    one has γ_mBD≈ 1. This situation is different
    from the massless BD theory, for which ω = 0 implies in
    _BD = 1/2 independently of the scale.
    
    
    
    
    
    
    § CONCLUSIONS
    
    
    The gravitational deflection of light is one of the main predictions
    of GR, which has a growing precision of measurement. It is quite
    reasonable to use this effect to test the models of modified gravity,
    including higher-derivative and non-local ones. In the literature one
    may find different conclusions on the bending of light within the
    simplest extension of GR, the R+R^2 gravity. A recurrent
    statement is that light bending cannot distinguish between this
    theory and GR in the linear
    regime <cit.>. This result directly follows from the fact
    that in quadratic gravity light is not subjected to an extra potential
    other than the Newtonian one. However, the situation may change
    if taking into account what are the true observable quantities in the
    theory. In this context  the predictions for the light bending can
    differ, as the Keplerian mass of the object responsible for the
    deflection is theory-dependent <cit.>. Similar results have
    been obtained for the bending of light in the sixth-order
    gravity <cit.>.
    
    In the present work we extended such considerations to the whole
    family of gravity theories based on the Einstein-Hilbert action
    with an extra term R f() R. Namely, we presented different
    arguments to show that in the weak field limit the deflection of photons is not
    directly affected by the terms quadratic
    in scalar curvature, regardless of the form of the function f(x).
    It turns out that this fact is closely related to the conformal invariance
    of Maxwell equations. It is worth noticing that in the weak-field
    approximation the result applies also to the general models of
    the form (<ref>).
    
    The form of f(x) can produce an indirect effect on the light
    bending through the redefinition of the (measured) Keplerian mass
    M_eff. In particular, the effective PPN parameter 
    was calculated as a quantity dependent on the function f(x),
    assuming that its spatial dependence can be neglected in some
    regimes. This derivation confirms that the predictions
    of light bending can differ  from theory to theory among the
    class (<ref>), even though they have the same
    interaction potential with light (written in terms of G).
    
    Finally, it is worth to point out that the introduction of the terms
    of the second order in Ricci and/or Riemann tensors—such as
    R_μν g() R^μν, for a local or non-local function
    g(x)—changes the propagator of the spin-2 sector, which yields a
    direct effect in the light bending <cit.>.
    More general results in this direction will be presented elsewhere.
    
    
    
    
    
    § ACKNOWLEDGEMENTS
    
    B.L.G. is thankful to CNPq for supporting his Ph.D. project.
    I.Sh. is grateful to CNPq, FAPEMIG and ICTP for partial support
    of his work.
    
    
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