Quantum backreaction of O(N)-symmetric scalar fields and de Sitter spacetimes at the renormalization point: renormalization schemes and the screening of the cosmological constant

We consider a theory of N self-interacting quantum scalar fields with quartic O(N)-symmetric potential, with a coupling constant λ, in a generic curved spacetime. We analyze the renormalization process of the Semiclassical Einstein Equations at leading order in the 1/N expansion for different renormailzation schemes, namely: the traditional one that sets the geometry of the spacetime to be Minkowski at the renormalization point, and new schemes (originally proposed in [1, 2]) which set the geometry to be that of a fixed de Sitter spacetime. In particular, we study the quantum backreaction for fields in de Sitter spacetimes with masses much smaller than the expansion rate H. We find that the scheme that uses the classical de Sitter background solution at the renormalization point, stands out as the most appropriate to study the quantum effects on de Sitter spacetimes. Adopting such scheme we obtain the backreaction is suppressed by H/M pl with no logarithmic enhancement factor of lnλ, giving only a small screening of the classical cosmological constant due to the backreaction of such quantum fields. We point out the use of the new schemes can also be more appropriate than the traditional one to study quantum effects in other spacetimes relevant for cosmology.


I. INTRODUCTION
The motivations for studying quantum fields in de Sitter (dS) spacetime are diverse. The understanding of the predictionS and the robustness of the inflationary models usually requires to assess the importance of quantum effects for light scalar fields on a (quasi) de Sitter background geometry. For a pedagogical introduction to the main issues and the relevance of the IR behavior of quantum fields in inflationary cosmology see Ref. [3,4]. In the base cosmological model, known as ΛCDM model, the accelerated expansion of the universe can be described by adjusting the value of the so-called cosmological constant, Λ, for which there is no fundamental explanation nor understanding of the inferred particular value [5][6][7][8]. Being Λ a constant, if the classical predictions of the model are extrapolated to future times, the geometry of the universe would approach to that of dS. There are arguments in the literature indicating the adjusted value is too small compared to the theoretical estimates. An interesting concept that aims at overcoming this discrepancy is that of the "screening of the cosmological constant", which is based on the expectation that large infrared quantum effects produce an effective reduction (or screening) of the classical value [9][10][11][12][13]. dS stands out among other possible curved geometries for its symmetries, which in quantity equal those of the flat Minkowski spacetime. Therefore, it should be a good starting point for exploration of field theories in more generic backgrounds with non constant curvatures, such as Friedman Robertson Walker spacetimes. However, there are certain characteristics of dS that hinder the development of computational methods as powerful as those known for Minkowski. One of them is the breakdown of the standard perturbation theory for light quantum fields; such as for scalar field models with (self-) interaction potentials, which are widely used in cosmology [14].
In this paper we assess the impact of choosing the ultraviolet (UV) renormalization scheme in the physical understanding and robustness of the non-perturbative results. The approach we use is based on the work done previously in [1,2]. This consists in using the two-particle irreducible effective action (2PI-EA) method [33] to derive finite (renormalized) self-consistent equations for the mean fields, the two point functions, and the metric tensor g µν , which are non-perturbative in the (self-)coupling of the scalar fields. In Refs. [1,2] the study was done for the more difficult case of only one scalar field, where there is no parameter controlling the truncation of the diagrammatic expansion. Here we focus on the large N limit of an O(N )-symmetric scalar field theory.
In Sec. II we summarize the nonperturbative formalism we consider, following [34], and we derive the effective action in the large N limit. In Sec. III we critically study the renormalization process for the so-called gap equation, from which a nonperturbative dynamical mass is obtained. We consider three different renormalization schemes, namely: the minimal subtraction (MS) scheme, the Minkowski renormalization (MR) scheme and the de Sitter renormalization (dSR) scheme. A sketch of the derivation of the renormalized Semiclassical Einstein Equations (SEE) is provided in Sec. IV, which closely follows previous studies for N = 1 field in the Hartree approximation [2] (see also [1]). The renormalized SEE for a generic metric g µν are obtained in the dSR scheme, using a fixed dS metric with curvature R 0 . These equations reduce to the traditional renormalized SEE when R 0 → 0. In Sec. V we specialize the results for a dS background metric, for which dS self-consistent solutions are studied in VI. In this study we show that, given the role of dS curvature in generating a nonperturbative dynamical mass, the introduction of the dSR scheme is crucial in understanding the infrared effects for light quantum fields.
Our results agree with those found in Ref. [24], using nonperturbative renormalization group techniques, on that the expected large infrared corrections to the curvature of the spacetime show up as a manifestation of the breakdown of perturbation theory (rather than an instability). The corrections are screened when nonperturbative (mainly, the dynamical generation of a mass) effects are accounted for. Indeed, the infrared corrections to the dS curvature turn out to be controlled by the ratio H 2 /M 2 pl , which is small by assumption in the semiclassical regime. In particular, in contradiction with the results reported in [19], we find no logarithmic enhancement factor of ln λ in the renormalized stress energy tensor (we clarify the reason of this disagreement in Sec. VII).
Everywhere we set c = = 1 and adopt the mostly plus sign convention.

II. THE 2PI EFFECTIVE ACTION AT LEADING ORDER IN 1/N
We consider an O(N )-symmetric scalar field theory in de Sitter spacetime with action, where i, j are the index of the N scalar fields φ i of the theory (where the sum convention is used), m B is the bare mass of the fields, ξ B is the bare coupling to the curvature R, λ B is the bare parameter controlling the fields (self-)interaction, and δ ij is the identity matrix in N dimensions.
A systematic 1/N expansion can be obtained in the framework of the so-called two-particle irreducible effective action (2PI EA). The definition of the 2PI EA along with the corresponding functional integral can be found in both papers and textbooks (see for instance [4,[33][34][35]). In this section we briefly summarize the main relevant aspects of the formalism for the model we are considering and the results obtained in [34] using the 1/N expansion.
We work in the "closed-time-path" (CTP) formalism [33] and introduce a set of indexes a, b which can be either + or − depending on the time branch. The starting point to obtain the 2PI EA is the introduction of a local source J(x) as well as a non-local one K(x, x ) in the generating functional Z[J, K]. The 2PI EA is the double Legendre transform with respect to that sources and is a functional of the mean fieldsφ i a and the propagators G ij ab . The result can be written as [34]: where Γ 2 [φ i , G, g µν ] is −i times the 2PI vacuum Feynman diagrams with propagator A ab given by and with vertex defined by S F int [ϕ i , g µν ], which is the interaction action obtained after recollecting the cubic and higher orders inφ i that emerged when expanding Following [34], the leading order in the 1/N expansion in the unbroken symmetry case ( φ i = 0) result: where c abcde is equal to ±1, if a = b = c = d = e = ±, and zero otherwise.
By setting to zero the variation of this 2PI EA with respect to the scalar field and the propagators, we obtain: whereφ 2 =φ iφj δ ij /N and [G 1 ] = 2G(x, x) = 2G ij (x, x)δ ij /N is the coincidence limit of the Hadamard propagator, which is a divergent quantity. Therefore, the two equations above contain divergences. In this paper we use dimensional regularization together with an adiabatic expansion. In the next section we analyze the renormalization process.

III. RENORMALIZATION IN CURVED SPACETIMES AND RENORMALIZATION SCHEMES
In this section we describe the renormalization process of Eqs. (6) and (7) in three different renormalization schemes, which we refer to as: the minimal subtraction (MS) scheme, the Minkowski renormalization (MR) scheme and the de Sitter renormalization (dSR) schemes.

A. Minimal subtraction scheme
In the MS scheme, we split each bare parameter (m 2 B , ξ B , λ B ) into the MS scheme parameter (m 2 , ξ, λ) which define the finite part, and the divergent contribution (δm 2 , δξ, δλ), which contain only divergences and no finite part, In order to renormalize the equations we rewrite them as, defining the physical mass m ph as the solution to the following gap equation: (10) We now use the well-known Schwinger-DeWitt expansion for [G 1 ] to split the propagator into its divergent and finite terms [20], where = d − 4, which is factored out in the first term making explicit the divergence as d → 4, T F is a finite function that depends on the spacetime (here {R} denotes the curvature tensors), where a scaleμ with units of mass must be included in order for the physical quantities to have the correct units along the dimensional regularization procedure.
Introducing Eq. (11) into Eq. (10) and demanding that the divergent terms cancel out with the contributions of the counterterms, we obtain a finite equation for the physical mass (i.e., the renormalized gap equation): where the function F (m 2 ph , {R}) is defined by and has the following properties: In Appendix A we provide the expression for the counterterms.
Therefore, the gap equation in the MS scheme depends on the mass scaleμ, which is an arbitrary scale with no obvious physical interpretation. A way to define renormalized parameters with a physical meaning is to use the effective potential, which can be obtained by integrating the following equation with respect toφ, following Eq. (9): In this way, a natural definition for the renormalized parameters (m R , ξ R λ R ) is to set them to be equal to the corresponding derivatives of the effective potential as a function ofφ and R evaluated atφ = 0 and R = 0 (that is, as defined in Minkowski space) and more generally at R = R 0 . This is the option we adopt next.

B. Minkowski renormalization scheme
Choosing Minkowski geometry at the renormalization point, corresponds to setting R 0 to zero. Therefore, from the effective potential V ef f the renormalized parameters are obtained from (12) as follows: And the result is: From these equations, we can find the following useful relations between the MR parameters defined above and the MS parameters, where we introduced λ * R to simplify the notation. Then, we can write the equation for m 2 ph , using solely one set of parameters, (24) One can easily check that, in the free field limit (λ R → 0, and therefore λ * R → 0) the physical mass reduces to the renormalized mass, m 2 ph → m 2 R .

C. De Sitter renormalization schemes
Now we set the geometry of the spacetime at the renormalization point to be that of a fixed de Sitter spacetime, corresponding to R = R 0 = constant. As above, the renormalized parameters (m 2 R , ξ R , λ R ) are defined in terms of the effective potential, Therefore, the generalization of the expressions (19), (20) and (21) that relate these parameters to the MS parameters are: As for the MR case, one can derive the following relations between these renormalized parameters and the MS ones: where the function J(R 0 , m 2 R , ξ R ) is defined by the last equality and goes to zero when R 0 → 0.
Then, using the equations (28), (29) and (30), the new expression for the physical mass m 2 ph is found, (33) This result reduces to the previous one (in the MR scheme, given in (24)), when R 0 → 0. Finally, the resulting counterterms are given by: Before proceeding any further, we present the expression of the function F dS (m 2 , R) which is the one defined in (13) evaluated in the de Sitter spacetime. To see a more detailed derivation we encourage the reader to read [1,2], is the DiGamma function and we define ν 4 ≡ 9/4 − 12(m 2 + ξR)/R 2 . One important property of the function g is that, in the infrared limit (39)

IV. RENORMALIZATION OF THE SEMICLASSICAL EINSTEIN EQUATIONS
We are halfway to our goal; the procedure below corresponds to the other half. The equations obtained above from the 2PI EA, describe the dynamics ofφ and G for a generic metric g µν . In order to assess the effect of the quantum fields on the spacetime geometry, we need to set to zero the variation of the 2PI EA action including the gravitational part with respect to g µν . This is equivalent to compute the expectation value of the stress-energy tensor T µν and use it as a source in the Semiclassical Einstein Equations (SEE). The resulting SEE are given by [20] When the dimension is set to d = 4, the Gauss-Bonet theorem implies that these tensors are not all independent from each other, and hence we have that [20] The stress energy tensor T µν in the large N approximation, can be obtained from the 2PI EA in Eq. (5). The computation is described in [34] [2], and the result for a generic metric is where and the index B only states that the parameters there involved are the bare ones.
Let us now separate T µν B into (46) where T µν R depends only on the renormalized parameters, but contains divergences coming from G 1 and its derivatives. As is well known, these divergent contributions can be properly isolated by computing the adiabatic expansion of T µν R up to the fourth order. The sum of such contributions are a tensor we call T µν ad4 , which is given by [20,34]: [ [ The renormalization process follows closely that described in Ref. [2] for N = 1 in the Hartree approximation. To proceed we need to use the counterterms for (m B , ξ B , λ B ) obtained as described above. In what follows we write the results in terms of the renormalized parameters (m R , ξ R , λ R ) in the dSR scheme. The expressions in the other schemes can be found using the relations derived in the previous section. Then, by separating the full expression of the fourth adiabatic order of the T µν given in (44) (which we call T µν ad4 ) into its divergent and finite terms, we can write T µν ad4 = T µν div ad4 + T µν con ad4 . After performing carefully the limit when → 0 of Γ(− /2 − a)x /2 , the convergent part results [2]: and the divergent terms can be absorbed into the following redefinition of the gravitational constants on the LHS of the SEE: Hence, we can now write a finite expression for the SEE: 1R H µν + α 2R H µν + α 3R H µν = T µν ren + T µν con ad4 , (57) where [ T µν − T µν ad4 ] = T µν ren .
Notice the above renormalization procedure only uses dS spacetime at the renormalization point. This is the main difference with respect to the traditional renormalization procedure for which a Minkowski spacetime is used. The metric g µν involved in both sides of the SEE (in the geometric tensors and in the stress energy tensor), which is the solution of the SEE, is unspecified. The traditional equations are recovered in the MR scheme (i.e., when R 0 → 0).

V. RENORMALIZED SEMICLASSICAL EINSTEIN EQUATIONS IN DE SITTER
Let us now specialize these results for de Sitter spacetimes. In dS, the geometric quantities appearing on the LHS of the SEE are proportional to the metric g µν , with a proportionality factor that depends on R and the number of dimensions d: Moreover, for any other tensor of range two we have similar properties, for example, De Sitter invariance also implies that every scalar invariant is constant, and particularly [G 1 ] is independent of the spacetime point. Using this and (59) in (45), it is immediate to conclude that the tensor T µν in Eq. (44) is also proportional to g µν and given by Using the definition of m 2 ph in Eq. (10), we obtain where = d − 4. Notice we cannot yet take → 0 in the denominator, due to the fact that it is multiplied by bare parameters. In order to perform such a limit, first we need to remind the expressions (30), (31) and (32). After some algebra and after neglecting the O( ) terms, we obtain To compute the renormalized expectation value, T µν ren = T µν − T µν ad4 , we must evaluate T µν ad4 from Eq. (47) for the de Sitter spacetime. It is then when we use the expressions in (58). We now use that T µν ad4 = T µν div ad4 + T µν con ad4 , where Then, after subtracting the tensor T µν ad4 given in Eq. (47), and neglecting the terms that are O( ), the result can be written as (66) Therefore, the RHS of the SEE in de Sitter is given by (67) The LHS is simply The quadratic tensors that were introduced, (1) H µν , (2) H µν and H µν , vanish for d = 4. However, their presence was important for the renormalization procedure (and there is a finite remnant on the RHS of the SEE due to the well known trace anomaly). Then, as seen in (67) and in (68), we can factorize the metric g µν from both sides, obtaining a scalar and algebraic equation with a sole degree of freedom of the metric, R, (69) where we have divided both sides by N and defined a rescaled Planck mass M pl , that respects N k R = 8π/M 2 pl = 8πG N .

VI. DE SITTER SELF-CONSISTENT SOLUTIONS
We are now ready to study the backreaction effect on the spacetime curvature due to the computed quantum corrections. In Sec. II we found that a dynamical physical mass m ph is generated, which obeys Eq. (33) witĥ φ = 0. Notice that m ph depends not only on the curvature scalar R, but also on the parameters m R , ξ R λ R , and the curvature R 0 defined at the renormalization point. In Sec. III, the renormalized SEE were derived, which reduce to the algebraic equation for R given in (69).
Our goal in this section is to solve both (33), in the symmetric phase (φ = 0), and (69) self-consistently, for different renormalization schemes and values of the free parameters. We do this numerically. First, in Sec.VI A, we analyze the effects on the physical mass characterizing its departures from the renormalized value m R . Then, in Sec.VI A, we concentrate on the effects on the spacetime curvature R, which can be characterized as departures from the classical solution (R − R cl )/R cl and/or from its value at the renormalization point (R − R 0 )/R 0 . We focus on the infrared limit, namely m 2 ph + ξ R R H 2 , so that, after using Eqs. (37) and (39), Eq. (33) can be approximated by where we remind that κ = 11/6−2γ E . This is a quadratic equation for (m 2 where the functions A dS , B dS and C dS are defined as: So, the physical mass m 2 ph in the infrared limit can be expressed as First of all, we need to make sure that the parameters of the problem ensure that the quantity m 2 ph is real and positive, otherwise the de Sitter-invariant propagator solution would not be valid. Besides it is necessary to recall we are restricting the analysis to the infrared regime, meaning m 2 ph + ξ R R R 12 . We define the variable Recall the renormalized mass is, by definition, the physical mass evaluated at the renormalization point. So, this variable characterize how much the physical mass departures from the renormalized one when the values of the curvature are beyond that point (which is generically the case if R 0 is not the full solution of the problem including quantum effects -which is unknown beforehand-). Assuming ξ R 1, the condition m 2 ph + ξ R R R 12 can be replaced by Throughout this section we study three cases: the R 0 = 0 case that corresponds to a flat curvature or a Minskowski spacetime, the case of R 0 = 0.01M 2 pl which is a possible value for a curved spacetime in the semiclassical regime, and the case R 0 = R cl , where R cl is the solution to the background curvature with the quantum effects neglected.
From Figure 1 one can see that for the same values of the curvature R (shown on the common horizontal axes) the order of magnitude of the vertical axes change significantly depending on R 0 (the value of R at the renormalization point). For intermediate values of the parameters the obtained results are similar and lay between the corresponding curves. The departures characterized by ∆ m 2 are significantly larger for lighter fields, as can be seen by comparing the right panel (which corresponds to m 2 R = 10 −3 M 2 pl ) to the left one (where m 2 R = 10 −7 M 2 pl ). Notice the different ranges of the horizontal axes. This means that when the fields are light, the physical mass is less robust against corrections beyond the renormalization point. This result is an expected manifestation of the infrared sensitivity of light fields to the spacetime curvature.
From Figure 1 one can also see that ∆ m 2 is larger the larger the coupling constant λ R , which is another manifestation of the infrared sensitivity to quantum corrections. This means that when the interaction between the scalar fields intensifies, the physical mass becomes more sensitive to small changes of the curvature (as can be also seen analytically from (33)).  Figure 1: The variable ∆ m 2 defined in (73) as a function of R, for ξ R = 0 and varying λ R between two values: λ R = 0.1 (green dotted lines) and λ R = 1 (red solid lines). The first column corresponds to m 2 R = 10 −7 M 2 pl and the second one to m 2 R = 10 −3 M 2 pl . The first row is for R 0 = 0, the second one for R 0 = 0.01M 2 pl , and the last one for R 0 = 4Λ R . The grey area corresponds to values for which the restriction ∆ m 2 < ∆ c 2 is violated (see Eq. (74)).
An important result is shown on the third row of the Figure 1 for the case the curvature at the renormalization point is taken to be the classical background solution, R 0 = R cl = 4Λ R . Notice that when varying λ R the same phenomenon shows up. However, we find the values of the physical mass remain relatively close to the renormalized one, within a certain degree of precision, for a wide range of values of R. In other words, choosing the classic curvature scalar at the renormalization point, the physical mass turns out to be less sensible to curvature (middle) and R 0 = 4Λ R (bottom). ξ R was varied between the values ξ R = 0 (solid red lines) and ξ R = 0.01 (green dotted lines). The region for which ∆ m 2 < ∆ c 2 is shown as a grey-painted area.
variations. Hence, this case is ultimately the most appropriate to study the self-consistent solutions and to assess the quantum backreaction effects produced by the quantum fields.
Let us now study what happens to ∆ m 2 when ξ R varies. Figure 2 shows plots of ∆ m 2 as a function of R, where we fixed m 2 R = 10 −7 M 2 pl and λ R = 0.1. A similar conclusion to the previous cases can be drawn, when we vary m 2 R and λ R . The sensibility of ∆ m 2 against changes of R turns out to be minimal when setting R 0 = R cl . Therefore, once again this point seems to be the most appropriate to study the quantum effects on the curvature.

B. Backreaction solutions
Once the limits upon the physical mass in Eq. (74) are established, we can proceed to find the Λ R values, by solving self-consistently the system formed by the gap equation for m 2 ph (33), withφ = 0, and the SEE (69). To measure the departures of the scalar curvature from the classical one, we use the variable In what follows, we present plots of ∆ R as a function of Λ R , for different values of the parameters. It is necessary to take into account the upper limit on the physical mass obtained from the analysis of the previous section (see Eq. (74)). In this case, this limit shows up as a restriction for the possible values that R can take, which corresponds to a lower limit on R, we call R min . Although we take into account this restriction, it does not show up in any of the plots below because it is only relevant for other (not plotted) values of R.  The top plot of Figure 3 shows ∆ R vs Λ R , for the three R 0 cases we are considering and two values of the mass: m 2 R = 10 −7 M 2 pl and m 2 R = 10 −3 M 2 pl . For the heavier case, we obtain relatively larger variations with respect to the renormalization point. In particular, the quantity R − R cl is only negative when setting R 0 = 4Λ R . On the contrary, for the lighter mass case this quantity is in all cases negative, meaning that the curvature scalar that includes the quantum effects is smaller than the classical one. For the lighter case, the plot in the middle of the same figure shows results for two different values of λ R , for each R 0 case. No significant variations are obtained, whereas once again R − R cl turns out to be negative. From the plot on the bottom of Figure 3, one can see curves for two different values of ξ R (ξ = 0 and ξ R = 0.01). The values of R − R cl are negative and its absolute value increases with Λ R .
In the previous subsection, we arrived at the conclusion that, in order to study the quantum backreaction effects, the most appropriate choice for R 0 at the renormalization point is R 0 = R cl . Using this, in this section we obtain that the resulting curvature is in all cases smaller than the classical one (and this does not occur for the other studied R 0 values). As a general result what we found for the renormalization point R 0 = R cl is the so-called 'screening of the cosmological constant Λ R '. For the same value of Λ R , the curvature scalar R value is reduced due to the quantum effects of the scalar fields. In another words, for the same value of Λ R , the quantum effects screen R cl (ie., the resulting curvature scalar R is smaller than the classical one R cl = 4Λ R ).

VII. COMPARISON WITH PREVIOUS WORKS
Some of the results we have obtained can be compared with previous work. In particular, we consider here the results presented in [19] and [36]. Both papers consider the same scalar field theory as we are considering here (given by Eq. (1)) in the semiclassical large N approximation, and both found a screening phenomenon of the cosmological constant, but using alternate approaches. In [19] the same 2PI EA formalism in the large N limit is considered, while in [36] the analysis of the backreaction is done using the so called Wilsonian renormalization group framework. The main difference is in their conclusion on the size and parametric dependence of the quantum backreaction effects for light fields. While [19] concludes the backreaction is non pertubatively large, obtaining an unsuppressed effect proportional to a logarithmic enhancement factor log λ, the conclusion in [36] indicates there is no enhancement factor as λ → 0 and that the corrections are controlled in the semiclassical approximation by a factor of R/M 2 pl . As we have shown above, by computing the renormalized parameters, finding both the physical mass equations and the SEE as a function of these and numerically solving both of these equations we have ultimately found the screening. However, in [19] and [36] the results are shown in terms of the Minimal Subtraction (MS) parameters. In order to compare our result with theirs, we set m = 0, ξ = 0 and write our results (now expressed in terms of the dSR renormalized parameters (m R , ξ R , λ R ) and R 0 ), in terms of λ andμ. It is important to remember a couple of things. The fact that the MS mass m is set to zero is possible here, as long as m ph remains a real and positive. When setting R = R 0 (at the renormalization point), we obtain m R = m ph , as seen in (33). Therefore, by fixing the curvature R to be equal to the one at the renormalization point R 0 (i.e., by setting R = R 0 ) we can obtain the physical mass as a function of R 0 from the relations given in Eqns. (28), (29) and (30). The computations of m R and ξ R in this limit can be seen in Appendix B. After inserting them in our expression for T µν = T µν ren + T µν con ad4 and expanding in √ λ, we obtain where we have used that R 0 = R. Therefore, our result disagree with the one presented in [19], on that we do not find a term proportional to R 2 0 log(λ), which would generate a large backreaction effect for small values of λ. As one can see in (76), the contributions involving the coupling constant are suppressed by √ λ, and do not result in a large backreaction effect. Notice that Eq. (76) shows the next to leading order in √ λ is the one depending onμ, but the leading order is independent onμ. We see that, provided λ 1, the dominant contribution to the LHS of the SEE (see Eq. (69)) is positive, leading to the phenomenon of screening (i.e., R is smaller than R cl = 4Λ R ). All contributions are suppressed by R/M 2 pl . Therefore, our results agree with the ones presented in [36] and differ from those in [19].
As far as we understand the discrepancy is due to a mistake in the procedure followed in [19] to compute the SEE from the 2PI EA, after having evaluated the action for a dS metric. Indeed, the effective action (which in dS is given by the effective potential) obtained in [19] is compatible with ours. However, as done in [36], the correct procedure to obtain the SEE from the effective action evaluated in a dS metric, with curvature R = 12H 2 , is to perform the derivative ∂ H (H −d V ) = 0, with d = 4 and where V ef f is the (H−dependent) scalar field effective potential. In [19], however, it seems the factor H −d was not included in the derivation and this gives the extra term with the logarithmic factor.

VIII. CONCLUSIONS
The main subject of the present work has been the problem of the backreaction of quantum fields on the spacetime curvature through the SEE. We focus on the O(N ) theory defined in Eq. (1) in the symmetric phase of the field (i.e., for vanishing vacuum expectation values of the scalar fields), in the large N approximation.
A main result of this paper is a set of finite (renormalized) self-consistent equations for backreaction studies in a general background metric in the de Sitter renormalization (dSR) schemes defined in Sec. III C, namely: the renormalized gap-equation in Eq. (33), necessary to obtain finite equations for the fields and the two point functions (i.e., Eqs. (6) and (7)), and the renormalized SEE presented in Sec. IV. We emphasize the fact that these equations can be used as a starting point to study the quantum backreaction problem beyond dS spacetimes, since dS is used only as an alternative spacetime at the renormalization point. This choice generalize the traditional one that uses Minkowski geometry at the renormalization point. As happens for the light fields in dS spacetime considered in this paper, this generalization could be significantly useful when infrared effects are sensitive to the curvature of the spacetime. We point out the use of dSR schemes may be also useful for non-dS geometries, such as for more generic Friedman Robertson Walker spacetimes used in cosmology.
Another important result is the specific study of the quantum backreaction problem for dS spacetimes. This has allow us to explicitly illustrate the importance of the dSR schemes in the understanding of the physical results. More specifically, we have obtained a system of two equations (Eq. (33) and Eq.(69)) that can be solved numerically to assess the effects on the curvature due to the presence of quantum fields for different values of the renormalized parameters of the fields (i.e., the mass m R , the coupling constant λ R and the coupling to the curvature ξ R ) and the renormalized cosmological constant Λ R .
First, we have analyzed the impact of choosing the geometry at the renormalization point in the relative difference between the physical mass and the renormalized mass as the physical background geometry (characterized by the curvature scalar R in this case) changes. We have obtained the difference is minimal, for a wide range of values of R in the semiclassical approximation, when the dS geometry fixed at the renormalization point is the classical solution, R 0 = R cl = 4Λ R . This indicates the definition of the physical mass is less sensitive to curvature variations and changes in the interaction between the quantum fields and their coupling to the curvature. Hence, we conclude the choice of this dSR scheme (with R 0 = R cl ) is more convenient to study the quantum backreaction effects than choosing the plane geometry R 0 = 0 or another fixed value for R 0 .
Then, we have studied the relative difference between the curvature (that is affected by the quantum interac-tions between the fields) and its classical approximation (where the quantum effects are neglected). We have found that for light fields this difference is always negative, that is, that the curvature is smaller than the classical one. For the other mass values analyzed, this phenomenon has also been found, but only for the dSR scheme with R 0 = R cl case. Given the previous conclusion on this dSR scheme regarding the sensitivity to quantum physics, we consider this result can then be interpreted as an screening of the cosmological constant induced by quantum effects.

ACKNOWLEDGMENTS
We thank D. F. Mazzitelli and L. G. Trombetta for useful comments and discussions. DLN has been supported by CONICET, ANPCyT and UBA.