Higher-order regularity in local and nonlocal quantum gravity Nicol` o Burzill` a, * Breno L. Giacchini, † Tib´ erio de Paula Netto, ‡ and Leonardo Modesto § Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China (Dated: June 2, 2021) In the present work we investigate the Newtonian limit of higher-derivative gravity theories with more than four derivatives in the action, including the non-analytic logarithmic terms resulting from one-loop quantum corrections. The first part of the paper deals with the occurrence of curvature singularities of the metric in the classical models. It is shown that in the case of local theories, even though the curvature scalars of the metric are regular, invariants involving derivatives of curvatures can still diverge. Indeed, we prove that if the action contains 2n + 6 derivatives of the metric in both the scalar and the spin-2 sectors, then all the curvature-derivative invariants with at most 2n covariant derivatives of the curvatures are regular, while there exist scalars with 2n + 2 derivatives that are singular. The regularity of all these invariants can be achieved in some classes of nonlocal gravity theories. In the second part of the paper, we show that the leading logarithmic quantum corrections do not change the regularity of the Newtonian limit. Finally, we also consider the infrared limit of these solutions and verify the universality of the leading quantum correction to the potential in all the theories investigated in the paper. 1. INTRODUCTION There has been an increasing interest in higher-derivative theories of gravity in recent years, especially those with more than four derivatives and weak nonlocalities. Such models are obtained by extending the Einstein–Hilbert ac- tion with curvature-squared terms such as RF 0 () R and C μναβ F 2 () C μναβ , where F 0,2 () are analytic functions of the d’Alembertian and C μναβ denotes the Weyl tensor. Among the motivations for this, we can mention the possibility of con- ciliating unitarity and renormalizability in the framework of perturbative quantum gravity. From one side, higher derivatives improve the behaviour of the propagator in the ultraviolet (UV) regime, which favours renormalizability. In fact, if F 0,2 are non-zero constants one has the fourth-derivative gravity, which is renormalizable [1]; while if F 0,2 () are non-trivial polynomials, the theory can be made super-renormalizable [2]. We remark that, even if the metric tensor is not quantised, the renormalization of mat- ter fields in curved space-time requires the introduction of at least four derivatives in the gravitational action [3]. It is well known, however, that local higher-derivative models usually contain ghost-like massive poles in the propagator, which vi- olate unitarity. In this approach, unitarity can be recovered either by projecting out the ghost-like poles of the propagator from the spectrum [4–9], or avoiding them ab initio by means of nonlocality [10–13]. The former possibility includes the Lee–Wick gravity, which requires an action with at least six derivatives of the metric so that all the ghost-like poles of the propagator can be complex [4, 5]. In the work [4] it was shown that these models are unitary at tree-level; while the generalisation of the Lee–Wick quantisation prescription [14–16] proposed in * [email protected]† [email protected]‡ [email protected]§ [email protected]Refs. [6, 7] guarantees unitarity at any perturbative order [8]. It is worthwhile to mention that the same procedure can be applied to real ghost modes [9], e.g., in the context of the sim- plest fourth-derivative gravity or to the additional ghost modes of higher-order theories. In what concerns the nonlocal higher-derivative gravity the- ories, we note that if the analytic functions F 0,2 () are not polynomials and have the form F s () = e H s () - 1 , (1.1) where H 0,2 (z) is an entire function, the propagator only con- tains the massless pole of the graviton. Therefore, these mod- els are automatically tree-level ghost-free; they can also be (super-)renormalizable, depending on the choice of the func- tions H 0,2 (z) [12, 13]. For a discussion on solutions and stabil- ity issues in nonlocal gravity models, see, e.g., Refs. [17–22]. In the present work, we investigate general properties of the Newtonian limit of a generic higher-derivative gravitational theory, including the leading logarithm quantum corrections, with particular focus on the occurrence of curvature singular- ities and on the far infra-red (IR) behaviour. Accordingly, the gravity effective action of our interest has the general structure Γ= - 1 κ 2 Z d 4 x p |g| n 2R + C μναβ F 2 () C μναβ - 1 3 R F 0 () R o , (1.2) where C μναβ denotes the Weyl tensor, κ 2 = 32πG and the form factors F s have the form F s () = F s () + β s log (/μ 2 s ), s = 0, 2. (1.3) Here, β s are constants, μ s are renormalization group invariant scales and F s () are analytic functions of the d’Alembertian. As discussed above, the choice of F s () corresponds to the definition of the higher-derivative sector of the classical ac- tion. In the formula (1.2) we do not write terms that are irrele- vant to the weak-field limit, such as the cosmological constant, superficial terms, and O(R 3 )-structures. arXiv:2012.11829v3 [gr-qc] 1 Jun 2021
Transcript
Higher-order regularity in local and nonlocal quantum gravity
Nicolo Burzilla,* Breno L. Giacchini,† Tiberio de Paula Netto,‡ and Leonardo Modesto§
Department of Physics, Southern University of Science and Technology,Shenzhen 518055, China
(Dated: June 2, 2021)
In the present work we investigate the Newtonian limit of higher-derivative gravity theories with more thanfour derivatives in the action, including the non-analytic logarithmic terms resulting from one-loop quantumcorrections. The first part of the paper deals with the occurrence of curvature singularities of the metric in theclassical models. It is shown that in the case of local theories, even though the curvature scalars of the metricare regular, invariants involving derivatives of curvatures can still diverge. Indeed, we prove that if the actioncontains 2n+6 derivatives of the metric in both the scalar and the spin-2 sectors, then all the curvature-derivativeinvariants with at most 2n covariant derivatives of the curvatures are regular, while there exist scalars with 2n+2derivatives that are singular. The regularity of all these invariants can be achieved in some classes of nonlocalgravity theories. In the second part of the paper, we show that the leading logarithmic quantum corrections donot change the regularity of the Newtonian limit. Finally, we also consider the infrared limit of these solutionsand verify the universality of the leading quantum correction to the potential in all the theories investigated inthe paper.
1. INTRODUCTION
There has been an increasing interest in higher-derivativetheories of gravity in recent years, especially those withmore than four derivatives and weak nonlocalities. Suchmodels are obtained by extending the Einstein–Hilbert ac-tion with curvature-squared terms such as R F0(�) R andCµναβ F2(�) Cµναβ, where F0,2(�) are analytic functions of thed’Alembertian and Cµναβ denotes the Weyl tensor. Among themotivations for this, we can mention the possibility of con-ciliating unitarity and renormalizability in the framework ofperturbative quantum gravity.
From one side, higher derivatives improve the behaviour ofthe propagator in the ultraviolet (UV) regime, which favoursrenormalizability. In fact, if F0,2 are non-zero constants onehas the fourth-derivative gravity, which is renormalizable [1];while if F0,2(�) are non-trivial polynomials, the theory canbe made super-renormalizable [2]. We remark that, even ifthe metric tensor is not quantised, the renormalization of mat-ter fields in curved space-time requires the introduction of atleast four derivatives in the gravitational action [3]. It is wellknown, however, that local higher-derivative models usuallycontain ghost-like massive poles in the propagator, which vi-olate unitarity. In this approach, unitarity can be recoveredeither by projecting out the ghost-like poles of the propagatorfrom the spectrum [4–9], or avoiding them ab initio by meansof nonlocality [10–13].
The former possibility includes the Lee–Wick gravity,which requires an action with at least six derivatives of themetric so that all the ghost-like poles of the propagator canbe complex [4, 5]. In the work [4] it was shown that thesemodels are unitary at tree-level; while the generalisation ofthe Lee–Wick quantisation prescription [14–16] proposed in
Refs. [6, 7] guarantees unitarity at any perturbative order [8].It is worthwhile to mention that the same procedure can beapplied to real ghost modes [9], e.g., in the context of the sim-plest fourth-derivative gravity or to the additional ghost modesof higher-order theories.
In what concerns the nonlocal higher-derivative gravity the-ories, we note that if the analytic functions F0,2(�) are notpolynomials and have the form
Fs(�) =eHs(�) − 1�
, (1.1)
where H0,2(z) is an entire function, the propagator only con-tains the massless pole of the graviton. Therefore, these mod-els are automatically tree-level ghost-free; they can also be(super-)renormalizable, depending on the choice of the func-tions H0,2(z) [12, 13]. For a discussion on solutions and stabil-ity issues in nonlocal gravity models, see, e.g., Refs. [17–22].
In the present work, we investigate general properties of theNewtonian limit of a generic higher-derivative gravitationaltheory, including the leading logarithm quantum corrections,with particular focus on the occurrence of curvature singular-ities and on the far infra-red (IR) behaviour. Accordingly, thegravity effective action of our interest has the general structure
Γ = −1κ2
∫d4x
√|g|
{2R + Cµναβ F2(�) Cµναβ
− 13 R F0(�) R
},
(1.2)
where Cµναβ denotes the Weyl tensor, κ2 = 32πG and the formfactors Fs have the form
Fs(�) = Fs(�) + βs log (�/µ2s), s = 0, 2. (1.3)
Here, βs are constants, µs are renormalization group invariantscales and Fs(�) are analytic functions of the d’Alembertian.As discussed above, the choice of Fs(�) corresponds to thedefinition of the higher-derivative sector of the classical ac-tion. In the formula (1.2) we do not write terms that are irrele-vant to the weak-field limit, such as the cosmological constant,superficial terms, and O(R3)-structures.
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The leading logarithmic quantum corrections are related tothe quantities βs, which depend on the field content of a givenquantum field theory and can be calculated (for the resultsof standard matter fields, see, e.g., [23]). Regarding quan-tum gravity contributions, in the fourth-derivative gravity thebeta functions of the curvature-squared terms are unambigu-ous [24, 25], while those in general relativity are gauge- andparametrization-dependent—an issue which can be solved byusing the Vilkovisky–DeWitt formalism [26, 27]1. In the lo-cal super-renormalizable models the beta functions are gauge-independent and, if the degree of the polynomial Fs(�) is atleast three, they are one-loop exact [2]. However, since herewe aim for general results, we leave these parameters arbi-trary.
It is useful to recall some results for classical theories(i.e., when βs ≡ 0) in the Newtonian limit. For the fourth-derivative gravity, the (modified) Newtonian potential is finiteat r = 0 [1], but the curvature invariants still have singulari-ties [32]. The situation is completely different when Fs is anon-trivial polynomial; in this case, the Newtonian-limit met-ric is not only finite [33, 34], but all the invariants built onlywith curvature and metric tensors are regular [35]. Also, for aplethora of choices of the entire functions Hs(z) within thenonlocal models (1.1) one meets the same situation of thepolynomial gravity; for a detailed discussion, see Ref. [36].One of the present work goals is to investigate if the insertionof the logarithmic terms in (1.3) either improves or spoils theaforementioned results.
Actually, the generalisation carried out in this paper isthreefold. First, in what concerns the conditions for theregularity of the curvature invariants, here we also considerthe scalars build with derivatives of the curvatures; and it isproven that these quantities can still diverge in local higher-derivative gravity models. Then, a general characterization oftheories that have a regular Newtonian limit is presented, in-cluding the case of non-analytic form factors such as (1.3).The quantum corrections are treated in two different ways,namely, as the full resummation of the one-loop 1-particle ir-reducible dressed propagator, like in (1.2), and as the first or-der correction to the 2-point correlation function, which com-prises a perturbative expansion on βs. Finally, the IR limitis also discussed, and it is shown that it has a universal be-haviour related to the quantum logarithmic corrections. Dueto the difference between the case of fourth-derivative gravityand the other higher-derivative models, we hereby only dis-cuss the latter one, addressing the former case in the parallelwork [37].
The paper is organized as follows. In Sec. 2 we brieflyreview the Newtonian limit of the higher-derivative gravitymodel (1.2), while in Sec. 3 we present a theorem on theconditions that the metric potentials should fulfil to regularisethe scalars involving derivatives of the curvatures. In Sec. 4we give two explicit examples of classical theories that sat-
1See [28–30] for a discussion about the dependence on the fieldparametrization in quantum general relativity, and [31] for a recent appli-cation of the Vilkovisky–DeWitt formalism in effective quantum gravity.
isfy the assumptions of the theorem, namely, the polynomial-derivative gravity (with simple poles in the propagator) anda nonlocal gravity model. The discussion is extended in theSec. 5, where we characterise a large family of local and non-local gravity models satisfying the conditions of the theorem;this analysis also includes non-analytic quantum corrections.Finally, in Sec. 6 we derive results considering a perturbativeexpansion of the metric potentials in the quantum-correctionparameter βs. Some general results are presented, especiallyin the IR limit; and the quantum correction to the Newtonianpotential for two specific models are explicitly evaluated: thepolynomial-derivative gravity with simple poles in the propa-gator, and the simplest nonlocal ghost-free gravity. The resultsare summarised in Sec. 7, where we also draw our conclu-sions.
2. NEWTONIAN LIMIT
In the weak-field approximation, we consider metric fluc-tuations around Minkowski space-time,
gµν = ηµν + hµν (2.1)
and expand the action (1.2) up to second order on the field hµν.The quadratic part of the action (1.2) reads
Γ(2) = −1
2κ2
∫d4x hµν Hµν,αβ hαβ, (2.2)
where
Hµν,αβ = f2(�) [δµν,αβ� −(ηα(µ∂ν)∂β + ηβ(µ∂ν)∂α
)]
−13
[ f2(�) + 2 f0(�)] [ηµνηαβ� −(ηµν∂α∂β + ηαβ∂µ∂ν
)]
+23
[ f2(�) − f0(�)]∂µ∂ν∂α∂β
�,
(2.3)
and the functions fs(z) are defined as
fs(�) = 1 + Fs(�)�. (2.4)
The interaction between gravity and matter is introducedvia the matter action
S m = −12
∫d4x T µν hµν, (2.5)
where T µν is the energy-momentum tensor in the flat space-time, such that the principle of least action
δ(Γ(2) + S m) = 0 (2.6)
yields the equations of motion,
Hµν,αβ hαβ = −κ2
2T µν. (2.7)
As our interest is in the Newtonian limit, we shall considerthe metric associated with a point-like mass in rest, whoseenergy-momentum tensor reads
Tµν = ρ δ0µ δ
0ν with ρ(~r) = Mδ(3)(~r). (2.8)
3
In isotropic Cartesian coordinates we have the line element
where ϕ = ϕ(r) and ψ = ψ(r) are the Newtonian-limit poten-tials and r =
√x2 + y2 + z2 . Writing the metric potentials in
the form
ϕ =13
(2χ2 + χ0) , ψ =13
(χ2 − χ0) , (2.10)
it is possible to show that the auxiliary potentials χ0,2 are thesolutions of [35]
fs(−∆)∆χs = κs ρ , (2.11)
with
κs = 2πG (3s − 2) . (2.12)
One of the benefits of working with these auxiliary poten-tials is the explicit separation of the contributions owed to thescalar and spin-2 degrees of freedom. In fact, to the New-tonian limit, the relevant part of the propagator associatedto (2.3) is given by
Gµναβ(k) =P(2)µναβ
k2 f2(k2)−
P(0−s)µναβ
2k2 f0(k2), (2.13)
where P(2) and P(0−s) are the spin-2 and spin-0 projectors [38],k2 = kµkµ and we used Euclidean signature. As the massivepoles of the propagator are defined by the zeros of the func-tions f0,2(z), the potential χs only depends on the spin-s sec-tor of the theory. Also, the overall structure of the equationsdefining these potentials is essentially the same, see (2.11),which makes it possible to derive general results based oncertain particular characteristics of the functions fs(z), e.g.,by means of the effective source formalism of Sec. 5 (seealso [36]). Once the expressions for both χ0,2 are obtained,the potentials ϕ and ψ can be recovered as a linear combina-tion of them, through (2.10). In this sense, one can work withthe spin-s potentials without loss of generality.
3. REGULARITY CONDITIONS IN THE NEWTONIANLIMIT
It is widely known that the regularity of a given metric doesnot imply in the absence of curvature singularities. For ex-ample, consider the following curvature invariants associatedto (2.9),
R2µναβ = 4
[(∆ψ)2 + (∂i∂ jϕ)2 + (∂i∂ jψ)2], (3.1)
R2µν = (∆ϕ)2 + 5(∆ψ)2 − 2∆ϕ∆ψ + (∂i∂ jϕ)2
+ (∂i∂ jψ)2 − 2∂i∂ jϕ∂i∂ jψ,(3.2)
C2µναβ = 2(∂i∂ jχ2)2 −
23
(∆χ2)2 and R = 2∆χ0. (3.3)
In view of (2.10) one can say that the absence of singularitiesin the scalars above is related to the regularity of the quanti-ties2
∆χs = χ′′s +2χ′s
r, (3.4)
∂i∂ jχs1∂i∂ jχs2 = χ′′s1χ′′s2
+2r2 χ
′s1χ′s2, (3.5)
where we changed to spherical coordinates in the spatial sec-tor. Therefore, the existence of the limits
limr→0
χ′′s (r) and limr→0
χ′s(r)r
(3.6)
is a sufficient condition for avoiding curvature singularities atr = 0 in the invariants (3.1)–(3.3). These conditions dependon the derivatives of the potentials χs, not only on their finite-ness; this is why there are curvature singularities in the fourth-derivative gravity although the potentials are bounded [1, 32].
It turns out that the existence of the limits (3.6) also en-sures the regularity of the higher-order scalars of the type Rn,formed by the contraction of an arbitrary number n of curva-ture tensors. In fact, by dimensional arguments, such objectdepends solely on combinations of products of χ′′s and χ′s/r.
Nevertheless, if one aims to regularise not only the invari-ants of the type Rn, but also those involving derivatives of thecurvatures, the potentials should fulfil additional conditions.This can be readily seen from the evaluation of �nR (for anarbitrary n), which, according to (3.3), is given by
�nR = 2∆n+1χ0. (3.7)
For a generic central function π(r) one has
∆n+1π(r) = π(2n+2)(r) +2(n + 1)
rπ(2n+1)(r). (3.8)
Therefore, to regularise the scalar �nR it suffices to have apotential χ0(r) of class C2n+2 such that there exists the limit
limr→0
χ(2n+1)0 (r)
r. (3.9)
(Notice that the condition (3.6) is the particular case n = 0.)In addition, if χ0(r) is of class C2n+2, but (3.9) diverges, then�nR is not regular.
It is straightforward to verify that if the same conditionshold also for χ2, then the invariants of type �nRk, for any in-teger k > 1, are finite at r = 0. Indeed, the scalars Rk arebuilt only with combinations of products of χ′′s and χ′s r−1, thus�nRk is formed by sums of terms ∆n[(χ′′s1
)i(χ′s2r−1) j] for some
powers i, j such that i + j = k. Provided that the derivativesof odd order of both potentials χ0 and χ2 vanish at least up to(including) the (2n + 1)-th order, the Taylor representation of(χ′′s1
)i(χ′s2r−1) j has no term with odd power r2`−1 for ` 6 n.
2Throughout this work we use the prime and superscript notation to de-note differentiation with respect to r.
4
Then, [(χ′′s1)i(χ′s2
r−1) j](2n−1) ∼ r and it follows from (3.8) that�nRk is regular too, for any k > 1.
To establish more general results, let us define the set I2nof all the scalars constructed with curvature tensors and theirderivatives, with the restriction that the maximum number ofderivatives of curvatures is 2n. For example, I0 = {Rk; k > 1},while �Rk and (∇µRαβ)2 belong to I2. Accordingly, it is clearthat I2n ⊃ I2(n−1) ⊃ · · · ⊃ I0. It is also useful to introducethe definition of order of regularity of a function, as follows.
Definition. Given a function π : [0,∞)→ R and an integerp > 0, we shall say that p is the order of regularity of π if:
(i) π(r) is at least 2p-times differentiable on [0,∞) andπ(2p)(r) is continuous.
(ii) If p > 1, the first p odd-order derivatives of π(r) vanishas r → 0, namely
0 6 n 6 p − 1 =⇒ limr→0
π(2n+1)(r) = 0.
If these conditions hold we shall also say that the function π(r)is p-regular.
In terms of this definition, a continuous function which isregular at r = 0 is 0-regular, while the limits in Eq. (3.6)characterise the 1-regularity of a function at least twice con-tinuously differentiable3. Having Taylor’s theorem in mind,one can say that a real function π(r) is p-regular if the first podd-order coefficients of its Taylor polynomial around r = 0are zero. In this sense, an analytic function π(r) is∞-regular ifand only if it is an even function. Moreover, the condition (ii)of the definition is equivalent to say that π(2n+1)(r) −→
r→00 at
least linearly.According to the discussion presented here, if the potentials
χ0,2 are (n + 1)-regular then there exist regular scalars with 2nderivatives of the curvatures. A stronger result is stated asthe following theorem, whose proof we postpone to the Ap-pendix.
Theorem. Given an integer n > 0, a sufficient condition forthe regularity of all the elements in I2n is that the potentialsχ0 and χ2 are (n + 1)-regular.
Most of the discussions in the literature on higher-derivativegravity has been focused on the regularisation of the invariantsin I0 (see, e.g., [35, 36, 39] and references therein). One ofthe goals in the present work is to extend the characterisationof regular models in the Newtonian limit beyond the simplest0-regularity. In this spirit, in the next section we characterisethe local classical higher-derivative gravity models for whichthe set I2n (for a given n > 0) only contains non-singularscalars; while in the following sections we extend considera-tions to the cases involving leading logarithmic quantum cor-rections as well as classical nonlocal gravity models.
3The definition of “finiteness” (the standard notion of regularity) isequivalent to “0-regularity”. Therefore, throughout this work, we shall simplysay “regularity” instead of “0-regularity” without ambiguity in interpretation.This is in contrast to the definition of “regularity” adopted in [35, 36, 39],which coincides to what here we call “1-regularity”.
4. HIGHER-ORDER REGULARITY IN CLASSICALPOLYNOMIAL-DERIVATIVE GRAVITY MODELS
Summary of the section: we show that if a local gravita-tional model has 2(N + 1) derivatives in the spin-s sector,then the potential χs(r) is (N − 1)-regular, but it is notN-regular. According to last section’s theorem, it means thatall invariants containing up to 2(N − 2) covariant derivativesof the curvature tensors are singularity-free at r = 0. As ananticipation of Sec. 5, we also give an example of a nonlocaltheory for which the potentials are∞-regular.
Given a function f (z), the solution of (2.11) can be reducedto a quadrature by means of the three-dimensional Fourier orthe Laplace transform methods (see, e.g., [33–35, 39–41]). Inthe first case, it is possible to integrate over the angular coor-dinates of the three-vector ~k , the result is:
χ(r) = −κM2π2r
∫ ∞
0
dkk
sin(kr)f (k2)
, k = |~k|. (4.1)
Notice that we dropped the s-label for the sake of simplicity.In this section we assume that f (z) is a real polynomial
of degree4 N > 1, which corresponds to the local super-renormalizable models of Ref. [2]. Comparing Eqs. (1.2) and(2.4) we see that, for this choice, the gravitational action con-tains 2(N + 1) derivatives of the metric tensor. For simplicity,here we restrict considerations to the case in which the equa-tion f (z) = 0 has N simple roots z = −m2
i , with i = 1, ...,N.We allow, however, the occurrence of complex roots, whichcan only appear in conjugate pairings owed to the fundamen-tal theorem of algebra5. Moreover, in order to avoid tachyonsin the spectrum it is assumed that Re(m2
i ) > 0. The scenariowith complex roots is of greatest interest from the physicalviewpoint. Indeed, such models correspond to the class ofLee–Wick gravity [4, 5] in which the conflict between unitar-ity and renormalizability is solved if the ghost degrees of free-dom are quantized a la Lee–Wick with the prescription givenin [6, 7], but without introducing any extra fictitious scale.
Under these conditions, and recalling that f (0) = 1, thepolynomial f (z) can be factored as
f (z) =
N∏i=1
z + m2i
m2i
. (4.2)
To solve the integral (4.1) we can start applying the partialfraction decomposition,
1f (z)
=
N∏i=1
m2i
z + m2i
=
N∑i=1
Cim2
i
z + m2i
, (4.3)
4The case N = 1 is not considered here for it corresponds to the strictlyrenormalizable fourth-derivative gravity [1], which contains singularities al-ready in I0.
5Remember that here we consider that f (z) is a real polynomial so thatthe action is also real and polynomial in derivatives.
5
where
Ci =
N∏j=1j,i
m2j
m2j − m2
i
. (4.4)
Thus,
χ(r) = −κM2π2r
N∑i=1
Ci m2i
∫ ∞
0
dkk
sin(kr)k2 + m2
i
. (4.5)
Performing an analytic continuation to the complex plane viak 7→ z ∈ C, it is possible to define a closed contour C on theupper half-plane Π+ = {z ∈ C : Im (z) > 0} with an inden-tation around the origin, such that the values of the integralsin (4.5) over the real line are related to the poles inside C bymeans of the Cauchy’s residue theorem. The result is [33, 34]
χ(r) = −κM4πr
(1 −
N∑i=1
Ci e−mir). (4.6)
Some general comments about this solution are in order.Even though the coefficients Ci might be complex, the massesappear only in complex conjugate pairs, and the combinationin (4.6) guarantees that the potential is a real-valued func-tion [34]. From (4.6) it is possible to show that the presenceof complex poles in the propagator yields oscillatory contri-butions to the Newtonian potentials, which are damped byYukawa factors [42]. In Ref. [35] the solution (4.6) was gen-eralized to address the case of degenerate poles of arbitraryorder—the outcome is that the solution gets new additionalterms in the form of products of modified Bessel and powerfunctions. Furthermore, it was proved that the potential (4.6)is 1-regular if N > 1 (that is, in theories with at least sixth-derivatives) [35].
In what follows, we refine this result, showing that the po-tential is, actually, (N−1)-regular and that it cannot be regularto an order higher than this. As mentioned above, in the ex-plicit proof in this section we only deal with the case of simplepoles in the propagator; the most general case is postponed tothe next section.
To show that all the first N odd-order Taylor coefficients ofthe potential (4.6) are null, let us start by writing the seriesexplicitly, namely
χ(r) = −κM4π
[(1 −
N∑i=1
Ci
)1r
+
∞∑k=0
(−1)k+1
(k + 1)!
( N∑i=1
Ci mk+1i
)rk
].
(4.7)
Then, we use the relations that come from the partial fractiondecomposition (4.3) to show that the divergent and the afore-mentioned odd-power terms vanish.
Writing the r.h.s. of (4.3) as a single fraction we get
N∑i=1
Cim2
i
z + m2i
=1∏
i(z + m2i )
N−1∑`=0
α` z` (4.8)
where
αN−` ≡
N∑i=1
Cim2i B`−1,i (4.9)
and B`,i is defined as the sum of all the combinations of prod-ucts of distinct quantities m2
j with j , i, taken ` by `. Forexample,
B0,i ≡ 1, B1,i =
N∑j=1j,i
m2j ,
B2,i =
N∑j,k=1
j,k,i, j,k
m2jm
2k , · · · , BN−1,i =
N∏j=1j,i
m2j .
It is also useful to define the related quantities
B` =
N∑k1,··· ,k`=1ki,k j∀i, j
∏i=1
m2ki, ` = 1, · · · ,N, (4.10)
which is the sum of all the combinations of products of distinctquantities m2
i , taken ` by `; and B0 ≡ 1. We have immediatelythe recursive formula
B`,i = B` − m2i B`−1,i, (4.11)
that can be applied ` times to express B`,i in terms of the quan-tities B` solely, namely
B`,i =∑j=0
(−1) j m2 ji B`− j. (4.12)
Therefore, Eq. (4.9) can be rewritten as
αN−`−1 =∑j=0
(−1) j B`− j
N∑i=1
Cim2( j+1)i , (4.13)
for ` = 0, · · · ,N − 2, while ` = N − 1 implies
α0 =
N∏i=1
m2i
N∑j=1
C j. (4.14)
On the other hand, the comparison of Eqs. (4.3) and (4.8)yields
α0 =
N∏i=1
m2i =⇒
N∑i=1
Ci = 1 . (4.15)
The relation (4.15) is responsible for the finiteness of the po-tential at r = 0, see (4.7) [33–35]. In the general polynomialmodel with simple poles considered here, it has the explicitform ∑
i
∏j,i
m2j
m2j − m2
i
= 1. (4.16)
6
Although (4.16) can be proven to hold for any set of distinctnumbers {m2
i } [34], it is not necessary to work with the expres-sion of the coefficients C j in terms of mi to verify the cancel-lation of the singularity, since (4.15) is merely a consequenceof the partial fraction decomposition, as showed above.
Besides, we have N − 1 relations of the type
αN−`−1 = 0, ` = 0, · · ·N − 2. (4.17)
Hence, the system (4.13) can be easily solved for the quan-tities
∑i Cim2n
i with n = 1, · · · ,N, iteratively starting from` = 0 and successively applying (4.17) up to ` = N − 1, whenwe use (4.15). The result is:
N∑i=1
Ci m2ni =
{0, if n = 1, · · · ,N − 1,(−1)N−1 ∏N
i=1 m2i , if n = N.
(4.18)Comparison with (4.7) then revels that the potential is (N−1)-regular, but not N-regular.
It was already known that all the polynomial models withat least six derivatives have 1-regular potentials [35]. Here weshowed that higher-order regularity can be achieved as oneincreases the number of derivatives in the action. In nonlocalghost-free models, depending on the choice of the entire func-tion H(z) the potentials can be ∞-regular. This is the case ofthe family of ghost-free gravities defined by the function
f (k2) = e(k2/m2)N, (4.19)
where m is a mass parameter and N ∈ N. In fact, in thesetheories the potential is an analytic even function [43, 44],i.e., it can be expressed as a power series in r2, ensuring theregularity of all the curvature and curvature-derivative localinvariants. For example, for N = 1 the explicit solution for(4.1) is [45] (see also [13, 46])
χ(r) = −κM4πr
erf(mr
2
)= −
κMm4π3/2 e−
m2r24 M
(1, 3
2 ,m2r2
4
), (4.20)
where M(a, b, z) = 1F1(a; b; z) is the Kummer’s confluent hy-pergeometric function. Further discussion on the regularity inmore general nonlocal models is carried out in the next sec-tion.
5. EFFECTIVE SOURCES AND REGULARITY
Summary of the section: we relate the order of regularity ofthe potentials χ(r) to the behaviour of the propagator in theUV. This is done through the effective source formalism, andallows a characterization of higher-derivative models with aregular Newtonian limit. The consideration is very generaland applies to local and nonlocal models, including somenon-analytic form factors, such as the logarithmic quantumcorrections.
The inclusion of the logarithmic quantum correction in thefunctions fs makes the task of evaluating the Newtonian po-tentials (4.1) much more involved than in the case of ana-lytic form factors. From one side, the pole structure of the
integrand becomes considerably richer than its purely classi-cal counterpart. To our best knowledge, only for the fourth-derivative gravity the detailed analysis of the structure of thesepoles, taking into account the one-loop logarithmic correc-tions, has been carried out [48]. Moreover, still in this simplestcase it seems that knowing the pole’s position is not very help-ful in solving the corresponding integral (4.1) using Cauchy’stheorem, and other methods should be applied [37].
Even if the classical theory is ghost-free at tree-level, onemay think that quantum corrections can introduce new ghostdegrees of freedom in the propagator (see, e.g., [47]). Weremark, however, that the logarithmic or other quantum cor-rections are only perturbative, and cannot affect the spectrumof the theory in their validity regime. This is strictly relatedto the perturbative unitarity based on the Cutkosky cuttingrules, which are also based on the Landau’s singularities of theloop amplitudes. Indeed, the singularities in the amplitudes inweakly non-local theories, defined by the form factors (1.1),are exactly the same of the local ones, as proved in [49–52].This is also evident when looking at the zeros of the denom-inator of the propagator, namely, considering the followingequation,
z[1 + βe−H(z)z log (z)] = 0 ,
which is the sum of all one-loop one-particle irreducibleamplitudes. The zeros appear for |βe−H(z)z log (z)| ∼ 1,which is in contradiction with the one-loop approximation|βe−H(z)z log (z)| < 1. The case of local higher-derivative grav-ity models can be analysed in a similar way; all in all, thisdiscussion is in agreement with the two explicit analytic so-lutions that we will present in Sec. 6, in which only the polesof the classical propagator contribute to the Newtonian poten-tial (see, for instance, Eqs. (6.23) and (6.33)). Ultimately, thepresence (and the stability) of a ghost originated from quan-tum corrections can only be decided with non-perturbativeresults—see [53–57] for a discussion related to this issue inthe context of the fourth-derivative gravity. Let us mentionthat the logarithmic quantum corrections produce also an IRsingularity in the propagator; this well-known fact is responsi-ble precisely to the asymptotic 1/r3 behaviour of the one-loopcorrection to the Newtonian potential [58–65], which we willdiscuss in more details in Sec. 6 (similar reasoning applies inquantum electrodynamics see, e.g., [66]).
Owed to the mentioned difficulties to obtain explicit solu-tions for the metric potentials with nonlocal form factors (in-cluding those with logarithmic corrections), our strategy toobtain general results concerning the occurrence of singular-ities in the Newtonian limit is to follow the approach used inRef. [36] in terms of effective sources6. While in [36] the rela-tion between the regularity of the source and the 1-regularityof the potential was derived under the assumption that thefunction f (z) is analytic, here we shall extend the consider-ations to non-analytic form factors. Moreover, as in [36] the
6See, e.g., [13, 45, 67–73] for applications of the formalism to specificmodels.
7
main concern was related to the regularity of the curvature in-variants of the type Rn, it was sufficient to prove the finitenessof the effective source. Here, in order to verify whether theconditions obtained in Sec. 3 for the regularity of invariantsconstructed with derivatives of curvatures are also satisfied,we should take into account the source’s differentiability.
The basic idea of the method is to rewrite Eq. (2.11) as astandard Poisson equation,
∆χ = κ ρeff , (5.1)
with the modified source
ρeff(r) =M
2π2
∫ ∞
0dk
k sin(kr)r f (k2)
. (5.2)
In this way, the effect of a non-constant, continuous functionf (z) on the Newtonian potential can be treated as the smearingof the original δ-source (2.8)—and the regularity propertiesof the potential χ can be deduced from those of the effectivesource ρeff.
Indeed, if the source in (5.1) is bounded and integrable, thenχ(r) is continuously differentiable. If, in addition, ρeff(r) is lo-cally Lipschitz continuous, the potential χ(r) is twice contin-uously differentiable (see, e.g., [74]). In particular, the exis-tence of ρeff(0) and χ′′(0) implies that
limr→0
χ′(r)r
< ∞,
as it can be directly verified by writing the Laplacian in spher-ical coordinates,
χ′′(r) +2rχ′(r) = κ ρeff(r) , (5.3)
and applying the limit r → 0 in both sides of this equation.Hence, under these circumstances, the finiteness of the sourcemeans that the Newtonian potential χ(r) satisfies the condi-tions (3.6), and all the curvature invariants of type Rn are reg-ular. In what follows, we investigate the conditions for thefiniteness and higher-order regularity of the effective source.
Assuming that the propagator does not have tachyonicpoles, f (z) does not change sign for z ∈ (0,∞). Moreover,putting f (0) = 1 (see Eq. (2.4)), the function
g(r, k) =k sin(kr)r f (k2)
, (5.4)
in the integrand of (5.2), is bounded on any compact. The inte-grability of (5.4), thus, depends on its behaviour as k → ∞.It holds, however, that if there exists k0 > 0 such that f (k2)grows at least as fast as k4 for k > k0, then
k > k0 =⇒ |g(r, k)| 6ck2 (5.5)
for some constant c. This means that g(r, k) is integrable, evenfor r = 0, provided f (z) grows as z2 or faster7 for sufficiently
7 It is actually possible to refine this condition to faster than z3/2+ε forε > 0.
large arguments, as it can be proved using the Weierstrass test.Under these circumstances, ρeff(r) is integrable and finite forr > 0, showing that the δ-singularity of the original source isregularised by the higher derivatives [36].
5.1. 1-regularity of the potential
So far, we have established conditions for the finiteness ofρeff. It is also possible to prove that, if those conditions hold,r = 0 is the global maximum of the effective source [36],which is an intuitive idea. The shape of this maximum can, inprinciple, depend on the gravity model. Now we show that iff (k2) grows faster than k4+ε (for an arbitrary ε > 0 and suf-ficiently large k) then this maximum is “smooth” in the sensethat the effective source is at least 1-regular,
limr→0
ρ′eff(r) = 0. (5.6)
In general, one cannot expect the effective source, viewed as afunction on R3, to be of class C∞ because the original sourceis a δ-function. However, it is useful to recall that if the sourceis differentiable and ρ′eff
(r) is bounded, as it is here, then ρeff islocally Lipschitz—and the Newtonian potential is 1-regular.
To prove Eq. (5.6), notice that (see (5.20) below)∣∣∣∣∣ ∂∂rg(r, k)
∣∣∣∣∣ =k
r f (k2)
∣∣∣∣∣k cos(kr) −sin(kr)
r
∣∣∣∣∣ 6 k3
2 f (k2). (5.7)
Therefore, if for k large f (k2) grows at least as fast as k4+ε (foran ε > 0), the integral ∫ ∞
0dk
∂
∂rg(r, k) (5.8)
converges uniformly for r > 0. Under these conditions ρeff isdifferentiable and we can apply differentiation under the inte-gral sign in (5.2). Furthermore, since in the limit r → 0 thefunction ∂
∂r g(r, k) converges uniformly to 0 on any compact,the limit on r can be interchanged with the integral in (5.8).This gives (5.6)—provided that f (k2) asymptotically growsfaster than k4+ε. In particular, because ρ′eff
(r) is bounded, theeffective source is Lipschitz continuous and the potential χ is1-regular.
It remains to deal with the limiting situation of ε = 0, inwhich f (k2) ∼ k4 asymptotically, like in the sixth-derivativegravity. In this case, it is possible to define an integrable func-tion which serves as an upper bound like in the r.h.s. of (5.7),but it depends on r. In fact, for k sufficiently large, it holds∣∣∣∣∣ ∂∂r
g(r, k)∣∣∣∣∣ 6 2k2
r f (k2)∼
crk2 , (5.9)
for some constant c. Thence, the integral (5.8) converges uni-formly on intervals which do not contain r = 0 as a limitingpoint, but limr→0 ρ
′eff
(r) cannot be evaluated by interchangingthe limit r → 0 and the integral. Owed to this, these sourcesmay have a spike in r = 0 but, if they are still locally Lipschitz,χ is 1-regular.
8
For instance, the effective source for the classical sixth-derivative gravity (see Sec. 4) with a pair of simple poles withmasses m1 and m2 can be read off from the general result ofRef. [36],
ρeff(r) =Mm2
1m22
4π(m22 − m2
1)
(e−m1r − e−m2r
r
). (5.10)
It is straightforward to verify that ρ′eff(r) is bounded (which
is enough to guarantee the 1-regularity of the potential) eventhough it does not vanish at r = 0,
ρ′eff(0) = −Mm2
1m22
8π. (5.11)
To prove that this qualitative result remains unchanged inthe more general case with logarithmic quantum correctionsand/or classical nonlocalities, one can change variables kr 7→u (r , 0) in (5.8), which becomes∫ ∞
0dk
∂
∂rg(r, k) =
∫ ∞
0du
u (u cos u − sin u)r4 f (u2/r2)
. (5.12)
Having assumed that f (k2) ∼ k4 asymptotically, there exists asmall enough r0 such that 0 < r < r0 yields∣∣∣∣∣ ∫ ∞
0dk
∂
∂rg(r, k)
∣∣∣∣∣ 6 ∣∣∣∣∣ ∫ 1
0du
u (u cos u − sin u)r4 f (u2/r2)
∣∣∣∣∣+ c
∣∣∣∣∣ ∫ ∞
1du
u cos u − sin uu3
∣∣∣∣∣, (5.13)
for some constant c. Since the first integrand on the r.h.s. isbounded for (r, u) ∈ (0, r0] × (0, 1], the corresponding inte-gral is bounded for r < r0. It is easy to prove that the re-maining integral, over an unbounded interval, is finite. There-fore, limr→0 ρ
′eff
(r) is bounded even for a non-analytic functionf (k2) which asymptotically grows as k4.
The reasoning above proves that if the functions f (k2)asymptotically grow at least as fast as k4, then the potentialsχ are 1-regular, and the curvature invariants in I0 are regular.This comprehends a large class of local and nonlocal theories.In particular, all polynomial-derivative models with more thanfour derivatives in both scalar and spin-2 sectors have a regu-lar Newtonian limit, and this feature is not changed if the loga-rithmic quantum corrections are taken into account. Also, theone-loop quantum corrections do not spoil the regularity forthe weakly nonlocal models for which f (k2) tend to a polyno-mial of the type kn (n > 4) in the UV, as well for the familyof models for which the classical f (k2) is an exponential func-tion (4.19). This result is a generalisation of [36] to the casein which the effective sources can be non-analytic functions.
Finally, it is useful to mention that in the case of the fourth-derivative gravity, f (k2) ∼ k2 in the UV and ρeff(r) divergesas r → 0, which means that at least one of the conditionsin (3.6) are violated. Indeed, the effective source for the clas-sical model was calculated in [36], and it was shown that itbehaves like r−1 for small r. When the logarithmic one-loopquantum corrections are taken into account,
f (z) = 1 + z [α + β log (z/µ2)], (5.14)
and the asymptotic behaviour becomes f (k2) ∼ k2 log k2.Then, combining the result of Ref. [37] with Eq. (5.3) we ob-tain, in the small-r approximation,
ρeff(r) ∼r→0
M4πr
1α − 2β log (µr)
, (5.15)
which diverges slightly more slowly than in the classicalmodel, but not enough to regularise the potential beyond the0-regularity [37].
The effective source also diverges for nonlocal gravity the-ories defined by the form factors (1.1) that tend to a constantin the UV. One such example is the Kuz’min form factor [11],
H(k2) = λ[γ + Γ(0, k2/m2) + log (k2/m2)
], (5.16)
in the case with λ = 1. Here and in the following γ denotes theEuler-Mascheroni constant, Γ(0, z) is the incomplete gammafunction, m is a mass parameter and λ ∈ N. For large momen-tum it satisfies
limk→∞
eH(k2) ≈ eλγ(
km
)2λ, (5.17)
whence, in the classical theory, f (k2) ∼ k2 if λ = 1 and the as-sociated potential is only 0-regular. This can be viewed in thenumerical evaluation of χ(r) and χ′(r) using (4.1), displayedin the upper panels of Fig. 1. Notice that for λ = 1 the poten-tial is finite, but its first derivative does not vanish as r → 0,indicating the singularity of the source (and of the curvatureinvariants). As in the local fourth-derivative gravity [37], theleading logarithmic quantum correction does not change thisoutcome. On the other hand, for λ > 2 the corresponding po-tential is at least 1-regular, regardless of the logarithmic cor-rections, as discussed above.
5.2. Higher-order regularity
The main argument used above to prove the 1-regularityof the potential involved the first derivative of the effectivesource. It is possible to generalise the previous discussion tohigher-order derivatives and investigate the higher-order reg-ularity of the Newtonian potentials.
Accordingly, we first observe that since the function g(r, k)in (5.4) is even and analytic in r, and it can be expressed as aseries,
g(r, k) =k2
f (k2)
∞∑`=0
(−1)`
(2` + 1)!(kr)2` , (5.18)
whence,
limr→0
∂n
∂rn g(r, k) =
0 , if n is odd,(−1)n/2
(n+1)kn+2
f (k2) , if n is even. (5.19)
Furthermore, the derivatives with respect to r are bounded (fora fixed k). This can be seen by noticing that since
∂n+1
∂rn+1 g(r, k) = −n + 1
r∂n
∂rn g(r, k) +kn+2
r f (k2)sin
[kr +
(n + 1)π2
],
9
λ = 1
λ = 2
λ = 3
λ = 4
0 2 4 6 8 10
-0.8
-0.6
-0.4
-0.2
0.0
r
χ(r)
λ = 1
λ = 2
λ = 3
λ = 4
0 2 4 6 8 100.00
0.05
0.10
0.15
0.20
0.25
0.30
r
χ'(r)
λ = 2
λ = 3
λ = 4
0 2 4 6 8 10
-0.04
-0.03
-0.02
-0.01
0.00
0.01
r
χ(3) (r)
λ = 3
λ = 4
0 1 2 3 4 5 6 7 8-0.01
0.00
0.01
0.02
0.03
r
χ(5) (r)
FIG. 1. Numerical evaluation of χ(r) and its first odd-order deriva-tives for the Kuz’min form factor for λ ∈ {1, 2, 3, 4} in (5.16).
the extrema of ∂n
∂rn g(r, k) are limited by kn+2
(n+1) f (k2) . Taking (5.19)and the analyticity of g(r, k) in r into account, we have∣∣∣∣∣ ∂n
∂rn g(r, k)∣∣∣∣∣ 6 kn+2
(n + 1) f (k2), (5.20)
which generalises (5.7).Regarding the upper bound defined by (5.20) as a function
of k it follows that if f (k2) grows at least as fast as kn+4, thenthe improper integral ∫ ∞
0dk
∂n
∂rn g(r, k)
converges uniformly. As in the previous subsection, underthese circumstances the source ρeff can be differentiated n-times, namely,
ρ(n)eff
(r) =M
2π2
∫ ∞
0dk
∂n
∂rn g(r, k) (5.21)
and the limit r → 0 can be interchanged with the integral inthe expression above. In particular, the odd derivatives of thesource vanish at r = 0 because of (5.19).
This result can be reformulated as: if the function f (k2)asymptotically grows at least as fast as k4+2N for an integerN > 0, then the effective source ρeff(r) is (at least) 2N timesdifferentiable and ρ(n)
eff(0) = 0 for all odd n 6 2N—in other
words, ρeff(r) is N-regular. As a corollary, we notice that iff (k2) asymptotically grows faster than any polynomial, thenthe effective source is an analytic function of r and is ∞-regular.
Of course, the converse of the collorary is not true, as inRef. [36] it was shown by explicit calculation that ρeff(r) isalso analytic if f (k2) is a polynomial. For example, for the
polynomial model with N > 2 simple poles considered inSec. 4 we have f (k2) ∼ k2N and the analytic function [36]
ρeff(r) =M
4πr
N∑i=1
Ci m2i e−mir, (5.22)
with Ci defined in (4.4). The finiteness at r = 0 follows fromthe result (4.18) with n = 0, which holds for any N > 2. Theremaining identities can be used to explicitly show that forN > 3 the source (5.22) is (N − 2)-regular.
As this example already suggests, and like in Sec. 5 5.1,the higher-order regularity properties of the source can be ex-tended to the Newtonian potential χ(r). Namely, if the sourceρeff(r) is of class C2N , the potential is C2N+2, and it is straight-forward to verify that the p-regularity of the source impliesthat the potential is (p + 1)-regular. The conclusion is that ifthe function f (k2) asymptotically behaves as k4+2N for someN > 0, then the associated potential is (N + 1)-regular and allthe curvature invariants in I2N are regular.
To close this section, let us return to the example of theclassical Kuz’min form factor (5.16). Since in the UV wehave f (k2) ∼ k2λ, according to the discussion above the po-tential χ must be (λ − 1)-regular. In Fig. 1 we display thenumerical evaluation of the first odd-order derivatives of thepotential χ(r) for λ ∈ {1, 2, 3, 4}, which verifies our result andshows that the potential cannot have an order of regularityhigher than λ − 1, just like for the local (polynomial) formfactors. Indeed, for a form factor with a certain λ, we see thatχ(2λ−1)(0) , 0. Again, the leading logarithmic quantum cor-rections do not modify the regularity order of the potential.A similar consideration applies to the more general nonlocalform factors proposed in Refs. [12, 13], which also tend to apolynomial in the UV.
6. PERTURBATIVE SOLUTION OF THE POTENTIAL
Summary of the section: here we focus on the quantumlogarithmic corrections to the Newtonian potentials, treatedas the first order correction to the 2-point correlationfunction. This differs from the approach employed in theprevious section, which considered the full resummation ofthe one-loop 1-particle irreducible dressed propagator. Weevaluate the explicit form for the correction at first order in βfor two specific models: the polynomial gravity with simplepoles and one ghost-free nonlocal gravity model. Generalresults are also obtained concerning the UV and the IRbehaviours of the quantum-corrected potentials.
Another approach to obtain a solution for the potential withlogarithmic quantum corrections is to solve the differentialequation (2.11) using perturbation theory in β. This relies onthe assumption that the scale related to the one-loop quantumcorrection term is much smaller than the classical counterpartbecause the former is of order O(~). Thence, we shall rewriteEq. (2.4) splitting its classical and quantum parts,
f (z) = fc(z) + β z log (z/µ2), (6.1)
10
FIG. 2. Contour of integration used to evaluate (6.11), poles andbranch cut defined by (6.10).
and look for solutions of the potentials in the form
χ = χc + χq + O(β2), (6.2)
where χc is the O(β0) classical potential and χq is the O(β)one-loop correction.
Substituting (6.1) and (6.2) into (2.11) gives the equationsat each order in β,
fc(−∆)∆χc = κMδ(~r), (6.3)
fc(−∆)∆χq = β log (−∆/µ2)∆2χc. (6.4)
Hence, in the three-dimensional Fourier space we have thetransformed potentials
χc(k) = −κM
k2 fc(k2), (6.5)
χq(k) = −βk2
fc(k2)log
(k2
µ2
)χc(k), (6.6)
which after the integration over the angular coordinates yield
χc(r) = −κM2π2r
∫ ∞
0dk
sin(kr)k fc(k2)
(6.7)
and
χq(r) =βκMπ2r
∫ ∞
0dk
k sin(kr) log (k/µ)[fc(k2)
]2 . (6.8)
As mentioned above, the potential χc given by (6.7) co-incides with the one without logarithm quantum corrections,whose explicit solution for different types of classical gravitymodels can be found, e.g., in [1, 13, 33–35, 42–45]. In par-ticular, the case of polynomial-derivative gravity with simplepoles has been discussed in detail in Sec. 4. Therefore, ourmain concern here involves the integral
I ≡∫ ∞
0dx
x sin(xr) log (x/µ)[fc(x2)
]2 , (6.9)
which appears in Eq. (6.8). In what follows, we shall evaluateit explicitly for the two models discussed in Sec. 4: the lo-cal polynomial gravity and the exponential ghost-free gravity.Subsequently, we present general results regarding the regu-larity of χq and its behaviour in the IR regime.
6.1. Polynomial gravity
The first example we consider is the case in which fc is apolynomial function with only simple roots, given by Eq. (4.2)with N > 1. Differently from the general formula (4.1),in (6.9) the logarithm function appears in the numerator be-cause of the perturbative expansion. Hence, we can follow themethod developed in [37] and apply Cauchy’s residue theo-rem in the context of the pole structure of the classical theory.
Let us define the function
h(z) =zeirz log (z/µ)[
fc(z2)]2 , −
π
26 arg z <
3π2
(6.10)
which has poles at z = ±imi (i = 1, · · · ,N). The branch cutdefined in (6.10) corresponds to the negative part of the imag-inary axis, therefore it is possible to construct the orientedsimple closed path C depicted in Fig. 2, for which Im (z) > 0.Notice that C has an indentation around z = 0, where log z isnot defined. Since there is only a finite number of poles, wecan take R > maxi{|mi|} and ε < mini{|mi|}. Only the poles atz = +imi are inside C, then�
Cdz h(z) = 2πi
∑i
Res (h(z), imi). (6.11)
On the other hand,�C
dz h(z) =
∫ R
ε
dxxeirx log (x/µ)[
fc(x2)]2 +
∫CR
dz h(z)
+
∫ −ε
−Rdx
xeirx( log (|x|/µ) + iπ)[fc(x2)
]2 +
∫Cε
dz h(z).(6.12)
Utilizing Jordan’s lemma, it follows that the integral along thesemicircular arc CR vanishes when R → ∞; similar consider-ation shows that the integral along Cε also vanish in the limitε → 0. Thus, making the substitution x 7→ −x in the third in-tegral in the r.h.s. of formula (6.12) and comparing with (6.11)one has
I = πRe[∑
i
Res (h(z), imi)]
+π
2
∫ ∞
0dx
x cos(rx)[fc(x2)
]2 . (6.13)
In order to evaluate the remaining integral in (6.13), weemploy the partial fraction decomposition, which now gives(c.f. (4.3))
1[fc(z2)
]2 =
N∑i=1
2∑j=1
Ai, j
m2i
z2 + m2i
j
, (6.14)
with the coefficients
Ai,1 = 2m2i
∑k,i
m4k
(m2i − m2
k)3
∏`,i,k
m2`
m2i − m2
`
2
(6.15)
11
and
Ai,2 =∏j,i
m2j
m2j − m2
i
2
. (6.16)
Therefore, using (6.14) we get∫ ∞
0dx
x cos(rx)[fc(x2)
]2 =
N∑i=1
m4i Ai,2
∫ ∞
0dx
x cos(rx)(x2 + m2
i )2
+ m2i Ai,1
∫ ∞
0dx
x cos(rx)x2 + m2
i
. (6.17)
The above integrals have the result (for Re (m2i ) > 0) [75]∫ ∞
0dx
2m2i x cos(rx)
(x2 + m2i )2
= 1 + mir sinh(mir) Chi(mir)
− mir cosh(mir) Shi(mir),
(6.18)
and
∫ ∞
0dx
x cos(rx)x2 + m2
i
= sinh(mir) Shi(mir)
− cosh(mir) Chi(mir),(6.19)
where we define the hyperbolic integrals
Chi(z) = γ + log z +
∫ z
0dt
cosh t − 1t
, (6.20)
Shi(z) =
∫ z
0dt
sinh tt
. (6.21)
On the other hand, the real part of the residues in (6.13) is given by
Re[∑
i
Res (h(z), imi)]
=
N∑i=1
m2i e−mir
4
[2Ai,1 log
(mi
µ
)+ Ai,2mir log
(mi
µ
)− Ai,2
]. (6.22)
Collecting (6.17), (6.18), (6.19) and (6.22) we get the expression for the quantum correction to the potential,
Taking into account the series representations of the functions above, the potential (6.23) can be written as
χq(r) =βκM4π
∞∑k=0
{2
(2k)!
[H2k − γ − log(µr)
]Gk r2k−1 +
1(2k + 1)!
Ek r2k}
(6.24)
where
Hk =
k∑n=1
1n
(6.25)
is the k-th harmonic number, and
Gk =
N∑i=1
m2(k+1)i
(Ai,1 − kAi,2
), (6.26)
Ek =
N∑i=1
m2k+3i
{Ai,2 − log
(miµ
) [2Ai,1 − (2k + 1)Ai,2
]}.
(6.27)
With a procedure similar to the one of Sec. 4 one can provethat
k 6 2N − 2 =⇒ Gk = 0, (6.28)
whereas
G2N−1 = −
N∏i=1
m4i , 0. (6.29)
Due to these relations, the non-analytic part of (6.23) has thegeneral structure
r2k−1 log (µr) with k > 2N − 1. (6.30)
Moreover, the potential χq is (2N − 2)-regular and the one-loop quantum correction does not spoil the regularity of theclassical solution. In fact, in Sec. 4 we proved that χc is (N −1)-regular. The explicit calculations of this section are also intotal agreement with the general discussion of Sec. 5; we shallreturn to this issue in Sec. 6 6.3.
Finally, we point out that even though the potential χq as-sociated to the fourth-derivative gravity [37] can be obtainedfrom the general expression (6.23) by setting N = 1, in sucha case relations (6.28) only hold for G0. In that case we have
12
A1,1 = 0 and A1,2 = 1, so that χq(r) is finite (0-regular) butdoes not satisfy the 1-regularity conditions (3.6) as the lead-ing non-analytic term is already r log (µr) (see [37] for furtherdiscussion on this model).
6.2. Nonlocal ghost-free gravity
As a further example of a form factor that admits a compactexpression for the quantum correction χq(r), let us considerthe simplest case of nonlocal ghost-free gravity, for which
fc(k2) = ek2/m2. (6.31)
In this case, the integral (6.9) is given by
I =
∫ ∞
0dx xe−2x2/m2
sin(xr) log (x/µ), (6.32)
whose solution can be obtained by taking a derivative (withrespect to a) of the integral representation of Kummer’s func-tion [76, 77],
M(
32 − a, 3
2 ,−z2
4b
)=
2b32−a
z Γ(
32 − a
) ∫ ∞
0dx x1−2ae−bx2
sin(zx),
defined for Re (a) < 3/2, Re (b) > 0, and using the propertyM(a, b, z) = ezM(b − a, b,−z). The final result for (6.8) reads
χq(r) =βκMm3
8(2π)3/2 e−m2r2
8
[2 − γ + 2 log
( m8µ
)−∂
∂aM
(a, 3
2 ,m2r2
8
) ∣∣∣∣a=0
].
(6.33)
The solution above is analytic because Kummer’s confluenthypergeometric function,
M(a, b, z) = 1F1(a; b; z) =
∞∑n=0
(a)nzn
(b)nn!, (6.34)
is entire for b = 3/2. Here, (p)n = p(p + 1) · · · (p + n − 1)is the Pochhammer symbol. The last term inside the bracketsin (6.33), actually, has a simple power series representation:from (a)n = a(n−1)!+O(a2), we get lima→0 ∂a(a)n = (n−1)! ,so that
∂
∂aM (a, b, x)
∣∣∣∣a=0
=
∞∑n=1
xn
n (b)n. (6.35)
Therefore, since the functional dependence of Eq. (6.33) in-volves only r2, the quantum correction to the potential is ∞-regular (see discussion in Sec. 3), just like its classical coun-terpart, given by Eq. (4.20). Finally, the analyticity of thepotential χq in this example can be qualitatively explained asthe limiting scenario of the polynomial gravity (discussed inSec. 6 6.1) when the number of derivatives in the action tendsto infinity; thence, the logarithmic terms which occur in (6.30)hide in the infinity.
6.3. Regularity in general higher-derivative gravity
The procedure to evaluate χq can be cumbersome for moregeneral higher-derivative models defined by other nonlocalform factors, or by polynomial functions fc which contain de-generate roots. However, some general properties of the one-loop quantum correction to the potential in these models canbe derived without the need to work out the explicit solution.
In what concerns the small-r behaviour, this follows fromthe observation that χq is sourced by a smeared distribution,see Eq. (6.4). In fact, its integral representation (6.8) is verysimilar to the one of the source (5.2), both integrands beingregular as k → 0. Therefore, we can apply the same formalismof Sec. 5 to the potential χq, which is the solution of
∆χq = κ β ρeff , (6.36)
where
ρeff(r) =M
2π2
∫ ∞
0dk
k3 sin(kr) log (k2/µ2)
r[fc(k2)
]2 . (6.37)
Rewriting the integrand of (6.37) as (c.f. (5.2))
gq(r, k) =k sin(kr)r φ(k2)
with φ(k2) =
[fc(k2)
]2
k2 log(k2/µ2) , (6.38)
one can follow all the discussion of Sec. 5 concerning thefiniteness and higher-order regularity of the potentials byanalysing the function φ(k2) instead of f (k2). Notice that eventhough φ(k2) diverges as k → 0 (while in Sec. 5 it is assumedthat limk→0 f (k2) = 1), both integrands gq and g vanish ask → 0. In this sense, for most of the cases, here the behaviourfor small k is actually improved with respect to the one ofSec. 5. Regarding the behaviour for large k, if fc(k2) ∼ k2n
asymptotically, then φ(k2) ∼ k2(2n−1)/ log k. This shows thatfor n > 1 it happens that φ grows faster8 than fc.
The analysis of Sec. 5, mutatis mutandis, allows us to con-clude that χq is finite (0-regular) if the classical action containsat least four derivatives of the metric—or, in the case of non-local theories, if the associated fc(k2) asymptotically growsat least as fast as k2. Moreover, if χc is p-regular, then χq is(2p)-regular. The explicit examples involving the polynomialmodels considered above perfectly agree with this general re-sult. In short, for the higher-derivative models considered inthis work, the perturbative quantum correction to the potentialis at least as regular as the classical part, and in most of thecases it is regular at a higher order.
6.4. Infrared limit
Regarding the far-IR limit, it is not difficult to see that thelarge-r leading quantum corrections to the classical mechan-ics’ Newtonian potential are captured by the O(β) correction
8On the other hand, for n = 1 (as in the fourth-derivative gravity) thesituation is the opposite—see [37] for further discussion on this particularmodel. Still, the quantum correction χq is finite at r = 0, like the classicalpotential.
13
(6.8). This can be verified by using (6.1) into (4.1) and makingthe change of integration variable kr 7→ u, which yields
χ(r) = −κM2π2r
∫ ∞
0du
sin u
u[fc(u2/r2) +
2βu2
r2 log (u/µr)] . (6.39)
Therefore, for large r,
χ(r) ∼r→∞−κM2π2r
∫ ∞
0du
sin uu fc(u2/r2)
+κMβ
π2r3
∫ ∞
0du
u sin u log (u/µr)fc(u2/r2)
.
(6.40)
The last integral in (6.40) is just (6.8) in the new vari-ables. Noticing that fc(u2/r2) → 1 for large values of r (seeEq. (2.4)), the first term in (6.40) gives the classical Newtonpotential,
χc(r) ∼r→∞−κM4πr
, (6.41)
while the quantum part tends to
χq(r) ∼r→∞
βκMπ2r3
[ ∫ ∞
0du u sin u log u
− log(µr)∫ ∞
0du u sin u
].
(6.42)
Using the exponential regularization to the distributional in-tegrals above, the result of the first integral is −π/2 and theother one, a δ-function, from which we obtain
χq(r) ∼r→∞−βκM2πr3 , (6.43)
for any classical higher-derivative model defined by analyticform factors. This result is in agreement with the common loreabout the effective theory of quantum gravity, which statesthat the details of the underlying “true” quantum theory ofgravity are unimportant for the behaviour of the low-energyregime [58–65].
Of course, the particular cases presented in the previoussubsections exemplify this general result. Indeed, by meansof the explicit solutions obtained, Eqs. (6.23) and (6.33), thelarge-r limit (6.43) of the quantum corrections to the potentialcan be directly verified.
7. SUMMARY AND CONCLUSION
The present work can be regarded as a generalisation ofprevious results concerning the possibility of avoiding space-time singularities in higher-derivative theories of gravity (see,e.g., [33–36, 39, 40, 42–44, 70, 71]). Due to the difficulties inobtaining exact solutions for the full non-linear theory, mostof the results in the literature are derived in the Newtonianlimit. Here, the considerations were still restricted to the lin-earized version of the model, but we made two generalisa-tions.
Instead of solely considering the curvature invariants madeonly by curvature tensors, we also discussed the regularityproperties of scalars containing derivatives of the curvatures.In this vein, the main result was a relation between the num-ber of derivatives in the action and the maximum number ofderivatives in the regular scalars: all the curvature-derivativeinvariants with at most 2n derivatives of curvatures are reg-ular if the local gravity action has at least 2n + 6 deriva-tives in both scalar and spin-2 sectors (moreover, there arescalars with 2n + 2 derivatives which are singular). The reg-ularity of all the local curvature invariants can be achievedin some classes of nonlocal gravity, namely, those definedby a form factor that grows faster than any polynomial, inthe UV. So far, the known solutions that are “infinitely reg-ular” are the Nicolini–Smailagic–Spallucci metric [67, 68]and similar generalisations [70]. Other known solutions withsingularity-free Kretschmann invariant may have higher-orderdivergences. One example is Dymnikova’s metric [78], forwhich a direct evaluation of �2R reveals a divergence at r = 0.This is in accordance with the analysis of the present papersince, in that solution, the Taylor expansion of the metric com-ponents has the first non-zero odd-order coefficient at O(r5).
Furthermore, in our analysis we also allowed for the possi-bility of some universal non-analytic form factors associatedwith quantum corrections. The conclusion is that the logarith-mic corrections do not change the regularity of the Newtonianlimit, inasmuch as they are sub-leading with respect to theclassical part of the form factor in all the super-renormalizablemodels.
Since the main set of theories considered in this paper havea classical propagator at least as strong as k−6, the quantum-correction k4 log k2 has a more prominent role in the far-IRregime, where it gives the leading correction to Newton’s po-tential, proportional to βr−3. We showed that this qualitativebehaviour is not affected by the specific classical action (thevalues of the quantities β are model-dependent, though). Thisresult supports the hypothesis of the universality of the effec-tive approach to quantum gravity in the IR [58–65]. Also, itis worth mentioning that we evaluated the quantum correctionto the potential to linear order in β for two models, viz. thepolynomial-derivative gravity with simple poles in the propa-gator and one case of nonlocal ghost-free gravity. Such com-putations can be viewed as related to the first-order correctionto the 2-point correlation function.
Last but not least, we would like to stress the correctnessof our result in the ultraviolet regime, regardless the linearapproximation. Indeed, the asymptotic freedom of the the-ory [79] at short distances guarantees the stability of the po-tential (or the metric) under nonlinear corrections.
ACKNOWLEDGMENTS
This work was supported by the Basic Research Program ofthe Science, Technology and Innovation Commission of Shen-zhen Municipality (grant no. JCYJ20180302174206969).
14
Appendix A: Proof of the Theorem of Sec. 3
Here we present the main steps of the proof of the theo-rem stated in Sec. 3, namely, that given an n > 0 the suffi-cient condition for the regularity of all the elements in I2n isthat the potentials χ0,2 are (n + 1)-regular. We choose to workin the isotropic coordinate system, with spherical coordinates(r, θ, φ) for the spatial sector, i.e., the flat-space metric reads
Since the components of the metric (A.1) do not have thesame dimension, let us define the balanced component of acovariant rank-` tensor T in isotropic spherical coordinates,denoted by ]Tµ1,··· ,µ` [, as follows: if Tµ1,··· ,µ` is a componentof T such that s indices (0 6 s 6 `) are angular indices (θor φ), then ]Tµ1,··· ,µ` [ ≡ r−s Tµ1,··· ,µ` . This definition is moti-vated from the fact that in an invariant S q
2n all the indices ofthe building blocks (A.4) are contracted (possibly with otherbuilding block) and the contraction of a pair of angular indicesinvolves ηθθ or ηφφ, which are proportional to r−2. Therefore,in the balanced component we distribute the factor r−2 comingfrom the angular components of the inverse metric between itstwo indices.
It is also useful to recall the definition of p-regularity (in-troduced in Sec. 3) of a function π(r), which is of class C2N
(with N > p > 0) and its first p odd-order derivatives van-ish as r → 0; we shall denote this property symbolically asP(π) = p. According to Taylor’s theorem, if N > 1 andN > p such a function can be written as
π(r) =
N−1∑`=0
c2` r2` +
N−1∑`=p
c2`+1 r2`+1 + q2N(r) r2N , (A.5)
where c` = π(`)(0)/`! and the remainder q2N(r) satisfieslimr→0 q2N(r) = 0. In particular, for small r,
π′(r) − r π′′(r) ∼
O(r0), if p = 0,O(r2), if p = 1,O(r3), if p > 2.
(A.6)
Let us now assume that the metric potentials are p-regular,with p > 1, and satisfy the conditions underlying (A.5). Itis straightforward to check that the components (A.2) of theRiemann tensor have the following small-r behaviour and reg-ularity properties:
]Rαβγδ[ ∼ r0, and P(]Rαβγδ[) = p − 1. (A.7)
Thus, the balanced components are (p − 1)-regular. Since allthe scalars S q
0 ∈ I0 (q ∈ N) have as building blocks the ob-jects in (A.7), it is clear that near the origin they tend to a con-stant value (they do not diverge) and that their first odd-orderterm occur at r2p−1, for any p > 1, which means that they are(p − 1)-regular—in short, P(S q
0) = p − 1∀q ∈ N. Hence, ifp = 1, the elements in I0 are 0-regular but not higher-orderregular. Finally, it is easy to see that if the metric potentialsare only 0-regular some components ]Rαβγδ[ diverge, whichmeans that the finiteness of the potentials is not enough toavoid curvature singularities.
The next step is consider terms with one covariant deriva-tive of the Riemann tensor. By direct calculation one can ver-ify that the non-zero components satisfy
]∇µRαβγδ[ ∼{
r0, if p = 1r1, if p > 2. (A.8)
In the evaluation of some of these components it is necessaryto use the identity (A.6), which only contain terms at least oforder r2 for p > 1. This is an important feature, as the leadingterms r0 or r would generate singular balanced components.
Thus, we can say that r]∇µRαβγδ[ is (p − 1)-regular, whichmeans that it is not 1-regular for p = 1. Of course, sincethe object in (A.8) has an odd number of indices, it is notpossible to make any scalar with only one of it. However,considering an even number of them we can build scalars suchas (∇µRαβγδ)2 ∈ I2. From Eq. (A.8) we see that these scalarsare regular for any p > 0, but they are only (p − 1)-regular,just like the scalars in I0.
Having established the regularity order and the behaviournear r = 0 for the quantities ]Rαβγδ[ and r]∇µRαβγδ[ we are inposition to extend considerations for a generic building blockwith any number of derivatives,
∇`r∇mθ ∇
nφRαβγδ. (A.9)
Given two natural numbers d and k, let us first assume thehypotheses:
Ia. the total number d = ` + m + n of derivatives is odd,and P(r ]∇`r∇
mθ ∇
nφRαβγδ[) = k for some k > 1, for any
combination of `,m, n such that ` + m + n = d;
Ib. there exists the limit limr→0 ]∇`r∇mθ ∇
nφRαβγδ[. Together
with Ia, this is equivalent to r ]∇`r∇mθ ∇
nφRαβγδ[ ∼ O(r2).
15
We shall refer to the hypotheses above in the concise notationI(d, k). Now we take one more covariant derivative of (A.9),analysing each case separately. Applying ∇φ we get:
∇φ∇`r∇
mθ ∇
nφRαβγδ = −`Γ
φφr∇
`−1r ∇
mθ ∇
n+1φ Rαβγδ
− nΓrφφ∇
`+1r ∇
mθ ∇
n−1φ Rαβγδ − nΓθφφ∇
`r∇
m+1θ ∇n−1
φ Rαβγδ
− mΓφφθ∇
`r∇
m−1θ ∇n+1
φ Rαβγδ − Γνφα∇`r∇
mθ ∇
nφRνβγδ
− Γνφβ∇`r∇
mθ ∇
nφRανγδ − Γνφγ∇
`r∇
mθ ∇
nφRαβνδ
− Γνφδ∇`r∇
mθ ∇
nφRαβγν.
(A.10)
Assuming Ia and Ib above and taking into account (A.3) itfollows, for example,
limr→0
]Γrφφ∇
`+1r ∇
mθ ∇
n−1φ Rαβγδ[ = const.
and
P(]Γrφφ∇
`+1r ∇
mθ ∇
n−1φ Rαβαβ[) = P(r ]∇`+1
r ∇mθ ∇
n−1φ Rαβγδ[) − 1
= k − 1.
All in all, it is straightforward to verify that analogous rela-tions hold for every term in (A.10), giving
]∇`r∇mθ ∇
n+1φ Rαβγδ[ ∼ O(r0), P(]∇`r∇
mθ ∇
n+1φ Rαβγδ[) = k − 1.
(A.11)Notice, however, that if in Ia we had allowed k = 0, thenin Ib we would have r ]∇`r∇
mθ ∇
nφRαβγδ[ ∼ O(r) and therefore
]∇`r∇mθ ∇
n+1φ Rαβγδ[ would not be regular.
Similar considerations can be applied to the terms∇`r∇
m+1θ ∇n
φRαβγδ and ∇`+1r ∇
mθ ∇
nφRαβγδ, with the same qualita-
tive result of (A.11). Thus, if Ia and Ib above hold, then
P(]∇µ∇`r∇mθ ∇
nφRαβγδ[) = k − 1 (A.12)
for any index µ. One can use this relation, for example, toinvestigate the building blocks ∇µ1∇µ2 Rαβγδ, with two deriva-tives. We already know that if the metric potentials are p-regular, then P(]r∇µRαβγδ[) = p − 1. Therefore, it is imme-diate to get P(]∇µ1∇µ2 Rαβγδ[) = p − 2, for p > 2. On theother hand, if p = 1 then ]∇µRαβγδ[ ∼ O(r0) (see (A.8)) andsome components diverge, e.g., ]∇φ∇φRαβαβ[ ∼ r−1. This ex-plains why �R diverges if p = 1 (compare with the examplein Sec. 3). If p > 1, then ]∇µ1∇µ2 Rαβγδ[ ∼ O(r0), but the oc-currence of the first odd power depends on p; for p = 2 itis already at linear order (thus this term is regular but not 1-regular). To sum up, only if p > 2 then all the scalars in I2are bounded at r = 0.
After having established the result of applying one covari-ant derivative to a regular object with odd number of deriva-tives, let us now assume:
II. the total number d = ` + m + n of derivatives is even,and P(]∇`r∇
mθ ∇
nφRαβγδ[) = k for some k > 1, for any
combination of `,m, n such that ` + m + n = d,
which we shall refer as II(d, k). Since the action of one co-variant derivative converts an odd polynomial into even, it is
more useful to consider the regularity order of the quantityr∇µ∇`r∇
mθ ∇
nφRαβγδ. As in the previous cases, it is necessary to
take one covariant derivative with respect to each coordinate.Let us consider
∇r∇`r∇
mθ ∇
nφRαβγδ = ∂r∇
`r∇
mθ ∇
nφRαβγδ
− mΓθrθ∇`r∇
mθ ∇
nφRαβγδ − nΓ
φrφ∇
`r∇
mθ ∇
nφRαβγδ
−∑ε∈A
[Γθrθ δ(ε, θ) + Γ
φrφ δ(ε, φ)
]∇`r∇
mθ ∇
nφRαβγδ
=
[∂r −
m + n + jωr
]∇`r∇
mθ ∇
nφRαβαβ,
(A.13)
whereA = {α, β, γ, δ} and jω is the number of angular indices(θ and φ) in Rαβγδ. Since ]∇`r∇
mθ ∇
nφRαβγδ[ is at least 1-regular,
its limit as r → 0 must be finite and it can be written as
]∇`r∇mθ ∇
nφRαβγδ[ = c0 + O(r2) (A.14)
for a constant c0. Also, recalling the definition of the balancedcomponents,
diverges, this singularity is precisely cancelled by the extraterm appearing in (A.13), so that
r ]∇`+1r ∇
mθ ∇
nφRαβγδ[ ∼ O(r2), P(r ]∇`+1
r ∇mθ ∇
nφRαβγδ[) = k.
(A.17)The behaviour of (A.17) can be verified also for the terms
involving an extra derivative ∇θ or ∇φ. In fact, the remainingcomponents can be dealt by commuting the derivatives, apply-ing the Bianchi identities and noticing that ∇m′
θ ∇n′φ Rαβγδ = 0
if the total number of indices θ (or φ) is odd (for any m′, n′).Therefore, if II holds, then,
r ]∇µ∇`r∇mθ ∇
nφRαβγδ[ ∼ O(r2), P(r]∇µ∇`r∇
mθ ∇
nφRαβγδ[) = k.
(A.18)It is useful to notice that had we allowed k = 0 in II, the onlychange in the result (A.18) is that r ]∇µ∇`r∇
mθ ∇
nφRαβγδ[ ∼ O(r).
Given the relations II(d, k) =⇒ I(d + 1, k) and I(d, k) =⇒
II(d+1, k−1), starting from Rαβγδ and∇µRαβγδ one can succes-sively apply covariant derivatives until one reaches k = 0. Theconsiderations above show that if the metric potentials χ0,2 are(p + 1)-regular, all the terms ]∇`r∇
mθ ∇
nφRαβγδ[ are bounded if
d = `+ m + n 6 2p + 1; therefore, all the invariants in I2p areregular, as the theorem stated.
16
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