Two interacting scalars system in curved spacetime -- vacuum stability from the curved spacetime Effective Field Theory (cEFT) perspective

In this article we investigated the influence of the gravity induced higher dimensional operators on the issue of vacuum stability in a model containing two interacting scalar fields. As a framework we used the curved spacetime Effective Field Theory (cEFT) applied to the aforementioned system in which one of the scalars is heavy. After integrating out the heavy scalar we used the standard Euclidean approach to the obtained cEFT. Apart from analyzing the influence of standard operators like the non-minimal coupling to gravity and the dimension six contribution to the scalar field potential, we also investigated the rarely discussed dimension six contribution to the kinetic term and the new gravity induced contribution to the scalar quartic self-interaction.


Introduction
The Effective Field Theory (EFT) approach turns out to be an immensely useful framework to parametrize and analyze the effects of newly observed (or yet not observed, but expected) phenomena in many branches of physics. In the case of particle physics it is most often invoked to describe possible influence of the beyond Standard Model (BSM) particles on observed quantities that can be measured in experiments (see, for example, [1][2][3] and references therein). Recently, there has been a renewed influx of efforts at writing down the systematic way of producing EFT from a set of minimal assumptions about new particles sector [4][5][6][7]. In an effort to extend this approach to curved spacetime the formalism of the curved spacetime EFT has been recently developed in [8].
In the current article we applied this formalism to investigate the problem of the vacuum stability in the presence of non-trivial gravitational field. The problem of the SM vacuum stability is one of the open questions in modern theoretical cosmology. For many years it has been investigated using both the flat spacetime approach [9][10][11] (see also citations therein) and taking into account the gravity effect in the case when Higgs was coupled to gravity [12][13][14][15][16][17]. More recently, there was a renewed surge of interest in better understanding the vacuum stability in curved spacetime in the case when scalar field is coupled non-minimally to gravity [18][19][20][21] (for some interesting discussion pertaining to the influence of the higher dimensional operators which couple kinetic terms of the Higgs and inflaton fields see [22]). The aforementioned papers follow mainly calculational route pioneered by classical papers [23][24][25], however there are also some alternative approaches like for example [26][27][28]. At this point it is prudent to note that beside the Coleman-De Luccia bounces in the case of curved spacetime there are also other forms of instantons, the most noticeable Hawking-Moss one [29]. Some more discussion on their properties and interpretation may be found, for example, in [30,31].
Modern particle physics lacks full understanding of the non-gravitational interactions that may be relevant at the high energy scale related to the question of vacuum stability. Most importantly we do not know what is the precise description of dark matter or inflationary sector and how they interact with ordinary matter. Although there are many models for dark matter or inflation that are in agreement with observations we do not have sufficient data to prove which one is actually realized in Nature. In this context we may consider the requirement of vacuum stability, or at least metastability with the lifetime longer than the age of the Universe, as a non-trivial constraint on the beyond Standard Model physics. Having this in mind it seems reasonable to use the framework that was developed precisely for this type of problems, namely the Effective Field Theory or more accurately its curved spacetime counterpart.
As was mentioned above, the main purpose of this article is to show that the application of cEFT to the problem of vacuum stability is, first of all, possible and, secondly, may lead to a discovery of new interesting phenomena. To this end, we employed a model containing two scalar fields interacting through a quartic term (Higgs portal type of the interaction). Assuming that one of the fields is heavy we integrated it out and obtained an effective theory for the light scalar. The novel feature of our approach manifests itself in the gravity induced operators that do not have its counterparts in flat spacetime. From this we proceeded by analyzing contributions of these operators to the lifetime of false vacuum. To better understand the mechanism that leads to the obtained results we made use of two observables. One of them is the Ricci scalar and is related to geometry, while the second one is the Euclidean counterpart of the energy density and is related to the matter content of the discussed model.
The rest of the article is structured as follows. Section 2 contains a description of the scalar sector and derivation of cEFT. It also contains derivation of equations of motion and description of the numerical procedure used in solving them. In Section 3 we described and discussed the obtained results. Section 4 contains the summary of our findings.

Action and equations of motion
We begin our investigations by writing down the action for the gravity-matter system. For the description of the gravity sector we use the standard Einstein-Hilbert action where R is the Ricci scalar and G is the Newton constant. In what follows, we will be using the (+ + +) sign convention of [32], which includes the mostly plus convention for the metric tensor (−, +, +, +). Our matter sector will be composed of two scalar fields, namely H that may represent the Higgs doublet and X that may represent the real scalar singlet of heavy dark matter. The U V action for this sector is where the potential for the complex H field may be given by V |H| 2 = 1 2 m 2 H |H| 2 + λ H 4! |H| 4 and d µ is a covariant derivative that may in general contain gauge fields parts. For the case where X represents the heavy scalar singlet with mass m 2 X > 0 (we assume the following mass hierarchy: m 2 X ≫ |m 2 H |), d µ reduces to the standard covariant derivative in curved spacetime ∇ µ . For now we may disregard gauge fields of the Standard Model and therefore we may set H to be a real scalar field, which also implies d µ H = ∇ µ H. In (2.2) we also included the non-minimal coupling of both scalars to gravity, the strength of these couplings is controlled by parameters ξ H and ξ X . The next step is to obtain cEFT for the scalar H that will be valid at energy and curvature scales smaller than m X . To this end, we will follow [8] and integrate out the heavy Z 2 -symmetric real scalar singlet X. Having done this we may write the action functional for our cEFT as where we defined the curvature dependent coefficients in the following manner: As we may see, the coefficients (2.4)-(2.8) were calculated up to terms of order O(R 2 ), where R 2 = R 2 , R µν R µν , K ≡ R µνρσ R µνρσ and R µν is the Ricci tensor and R µνρσ is the Riemann tensor. Yet, as was pointed out in [8], in the case when the Ricci scalar is non-zero the dominant contributions will come from terms linear in R. Moreover, in what follows we will not discuss the curvature contribution to dimension six operators, that is we will set c GdHdH = 0 and consider c dHdH and c 6 independent of the Ricci scalar. This is justified by the fact that curvature dependent parts of these coefficients are suppressed by additional powers of m 2 X as compared to the terms proportional only to λ HX . To sum up, the action functional for both gravity and matter parts of the investigated system is In the above we specified the H field to be a real scalar h (this may represent the real component of the Higgs doublet) and fixed its potential to be V ( Although this potential contains the term h 3 which is not present in the Higgs sector, we may think of it as a part of the λ ef f (h) term that allows us to capture the essential (from the point of view of vacuum stability) feature of the Higgs effective potential, namely the presence of the second minimum at large energies. To be precise, we do not imply that this type of potential is accurate at capturing quantitative behavior of the Higgs vacuum, like for example the numeric value of lifetime, but it is good enough to capture at least some qualitative behaviors like an increase or decrease of lifetime due to higher order operators. Moreover, in (2.9) we rewrite the action in such a way that various coefficients denoted c i are pure numbers which is signalled by the absence of tildes above them.
To describe the gravitational field we need to specify an ansatz for the metric. Since the standard procedure in the case of investigations of vacuum stability is to compute a solution to the Euclidean equations of motion in the case when the metric possesses a S 3 topology we may write the line element as (2.10) For this parametrization of the metric the non-zero components of the Einstein tensor are given by where the overdot is defined as dA(τ ) dτ ≡Ȧ. Meanwhile, the Ricci scalar is On the other hand, the Euclidean action for cEFT is given by (basically speaking the process of obtaining the Euclidean action in this case comes down to the fact that the exponential factor e iS cEF T in the path integral goes to e −S Euc cEF T ) For the above action the relevant Einstein equations could be rewritten as and M 2 P l ≡ 1 8πG is the reduced Planck mass. Meanwhile, the equation for the scale factor can be written asÄ Using the above formulas and (2.19) we may writë On the other hand, the equation of motion of the Higgs field is The system of equations (2.23) and (2.25) is the one that will be solved numerically to obtain profiles for the scale factor A(τ ) and scalar field h(τ ). To this end, we used the Maple software and its built-in ordinary differential equations (ODE) solvers [33]. But before this can be done we should make the field h, the Euclidean time τ and various constants appearing in the action (2.14) dimensionless. To this end, we use the following rescaling: where we decided that for a better readability of the paper we will use the same symbols for both dimensionfull and dimensionless quantities. Since in our analysis we will always be using dimensionless quantities this should not lead to any confusion. In the above, M represents some scale of mass dimension one. Usually in the literature this is set to be the reduced Planck mass M P l . Yet, in our case we have another energy/mass scale to consider, namely m X . This stems from the fact that our effective theory is valid for energies and curvatures smaller than the heavy particle mass. For this reason it is convenient to set M = m X . After discussing the rescaling of the variables and parameters needed for numerical computations let us present the details of the numerical setup. Let us start by pointing out that basic equations that we intend to solve, namely (2.24) and an appropriate linear combination of equations (2.15) and (2.16) that leads to (2.20) possess singularities at τ = 0 where we should have A(τ ) |τ =0 = 0. This implies the following set of boundary conditions: where v H is a constant. The first set of the above conditions implies that the scalar field is constant, but v H does not need to be the minimum of the potential, in fact it is not. The second set translates to the requirement that there is no conical singularity at τ = 0 and the Ricci scalar as given by (2.13) To obtain such a solution we treated v H as a free parameter. We chose some value for it and solved our system of equations. Then we repeated it until we found v H for which h goes to zero at large τ . This procedure of solving a system of ODEs is often called 'shooting' and its description can be found for example in [34]. As an additional point let us note that if h(τ ) goes sufficiently fast to zero, and provided that c 0 = 0, the spacetime becomes asymptotically flat. What we described above is the general algorithm for solving our problem. Let us now focus on some details of its implementation. Firstly, it is not feasible to start numerical integration from the singular point therefore we will start not at τ = 0 but at τ = ǫ, where ǫ = 10 −5 . To obtain the boundary condition at this point we look for the series solution to (2.23)-(2.25) at τ = ǫ. To this end, we use the following ansatz for h and A: (2.28) The form of the series was chosen in such a way that h and A fulfill boundary conditions (2.27). Moreover, the splitting of A was dictated by the need to isolate the asymptotic behavior for τ → ∞, namely A(τ ) |τ →∞ ∼ τ . This form turns out to be more convenient for numerical calculations. Having put the above ansatz into the equations we found solutions that satisfy them up to terms linear in τ . This implies that we introduce an error of the order of O(ǫ 2 ) to our numerical scheme regardless of the accuracy of numerical integration. Then, as our starting condition for integration we set h(ǫ) and Ap(ǫ), let us note that the value of h(ǫ) still depends on v H . Secondly, we cannot integrate up to τ = ∞, therefore need some stopping criteria. The obvious one is h(τ ) |τmax = 0 but this is insufficient. As a supplementary criterion we chooseḣ(τ ) |τmax < ǫ τ , where ǫ τ is a small parameter. We tested our code for a value of this parameter from within the range 10 −6 , 10 −8 and found no significant difference in the obtained results. For this reason we chose ǫ τ = 10 −6 . This set of conditions implies that after some Euclidean time h will decrease rapidly and then approach h(τ ) = 0 in a very gentle way, as so after reaching τ max it can be glued to the analytic solution h(τ ) = 0. To sum up, the described procedure gives us the value of τ max up to which we need to perform the integration. As far as Ap(τ ) is concerned, it turns out that the abovementioned conditions lead to satisfactory behavior of this function.

Results
After explaining our analytical and numerical setups in the previous section let us focus on the obtained results. As was mentioned above, we are looking for solutions of equations of motion that represent bounce (or soliton-like) behavior for the scalar field and the corresponding metric function. From the physical standpoint this type of solutions represents a field that is close to true vacuum at τ = 0 and then decays to false vacuum at large τ . One of the standard ways of obtaining the vacuum decay probability (in the context of the Euclidean approach used in this study) is to calculate the action on the solutions of equations of motion and then the vacuum decay probability (per unit volume) is given by In the above formula A represents a prefactor coming from a quantum correction to the action. In flat spacetime it may be either accounted for by dimensional arguments or calculated numerically. Let us also note that there is possibility that it could be calculated analytically with the help of the heat kernel method, but this is the problem for a separate study. As for the calculation of the prefactor in curved spacetime, there is an additional difficulty in accounting contributions of the fluctuation of the metric field. Returning to the case at hand, since the prefactor represents a higher order correction to e −S we will not discuss it any further. Let us now focus on the leading contribution to the vacuum decay that comes from the exponential factor. In this factor S represents a difference between the Euclidean action calculated at the bounce solution and the same action calculated at the solution representing false vacuum S = S euc cEF T (bounce) − S euc cEF T (f alse vacuum), where S euc cEF T is given by (2.14). At this point let us remark that in our study we will set the c 0 parameter in the potential of the scalar field to be equal to 0, what is equivalent to setting the cosmological constant to zero. Moreover, for our model the false vacuum solution is given by h(τ ) = 0, therefore it implies (with c 0 = Λ = 0) that R = 0. This means that S euc cEF T (f alse vacuum) = 0. This also implies the integration in S euc cEF T (bounce) over a finite interval τ ǫ 0, τ max . This is so because for τ larger than τ max we have by definition h(τ ) |τ >τmax = 0 and therefore R(τ ) |τ >τmax = 0.
As far as the boundary terms are concerned, we should supplement the action given by (2.14) with the Gibbons-Hawking-York boundary term [35][36][37] where κ(h) was defined in (2.18), h is the induced metric on the boundary hypersurface and K is the trace of its extrinsic curvature. For our action and the metric ansatz it may be explicitly written as In the formula above we integrated over the S 3 coordinates. Since for our setup the fields and metric configurations for both the false vacuum and the bounce configuration are identical outside τ = τ max , there is no contribution to the exponent in (3.1) from the hypersurface τ = ∞. As far as τ = 0 hypersurface is concerned, S GHY vanishes there due to the vanishing of the metric function A, therefore this also does not contribute to Γ. Below we present and discuss the results of our calculations. Figure 1a presents the influence of the non-minimal coupling term ξ H h 2 R on the vacuum stability. We restricted ourselves to the case ξ H 0 (and c HH 0), since for this κ(h) as defined by (2.18) is positive. If we want to interpret κ(h) as the effective Planck mass this restriction means that we demand that it is always positive. From this figure we infer that as |ξ H | increases S decreases, hence large negative ξ H destabilizes false vacuum. This is in agreement with the results obtained in [20].
In Figure 2a we plotted the profile of the scalar field h for various values of the nonminimal coupling parameter. Differences in the profiles are very small and for large |ξ H | they are the biggest in the region where the field changes its value most rapidly. This, in the conjunction with the fact that the exponent factor changes visibly with ξ h , leads to some interesting conclusions that we will discuss below.
Firstly, let us point out that from Figure 1b we may see that initially increasing |ξ H | leads to a decrease of the v H parameter (this parameter represents the value of the scalar field at τ = 0). After |ξ H | becomes bigger than approximately 0.13 this trend reverses and for large negative ξ H v H becomes bigger than for the minimally coupled case.
In Figure 3 we plotted the change in one of the observables mentioned in the Introduction, namely the energy density defined as ρ ≡ 1 κ(h) T τ τ , where T τ τ is the appropriate component of the energy-momentum tensor as defined in (2.19). From this plot we may see that for a minimally coupled scalar field ρ is always positive and its maximum value is at τ = 0. As we increase ξ H this maximum value starts to decrease and for sufficiently large negative ξ H ρ becomes negative with local minimum attained for some τ ǫ (0, τ max ). Meanwhile, for τ → τ max ρ approaches zero which is in agreement with our boundary condition h(τ ) |τ =τmax = 0. This together with the fact that the overall shape of the profiles of the scalar field does not differ significantly for various ξ h leads us to the conclusion that the dominant contribution to the action associated with the non-minimal coupling is through the derivative term that contributes to the energy-momentum tensor and therefore to the geometry itself. This is also visible in Figures 4a and 4b. In Figure 4b we plotted the second observable, namely the Ricci scalar for various ξ H . We may infer from it that for the minimally coupled scalar R is negative in the vicinity of τ = 0, then it has a local maximum at around τ ≈ 8.0 (this roughly corresponds to the bubble size as can be seen from Figure 2a), and then it goes to zero as expected due to the boundary conditions. Meanwhile, for large negative ξ H the Ricci scalar becomes positive at τ = 0 and then attains a local maximum and eventually decreases to zero. In our opinion this cannot be explained solely by the small difference in the value of the field at τ = 0 which will contribute to the equations of motion and the action through the potential term. Therefore, we conclude (as was stated before) that the actual effect must be due to the gradients of the field that influence geometry. Another interesting fact that can be inferred from Figure 4a is that the Ricci scalar is not a monotonic function of the Euclidean time. Moreover, from this figure we may also see that R changes its sign close to the bubble wall.
In Figure 4a we plotted the profile of the metric function Ap(τ ). Again we see that for the minimally coupled scalar the Ap(τ ) functions tend to a positive constant and for large negative ξ H tend to a negative one. As a side note, let us point that since the metric function A(τ ) grows up as τ + Ap(τ ), the obtained spacetime is an asymptotically flat one.
Let us at this point pause and tackle one more technical problem. We work within the framework of the effective field theory in curved spacetime. This implies that there is an energy scale µ > above which our theory is no longer valid. For our case this scale is proportional to the mass of the integrated out heavy particle X. Due to the employed rescaling (2.26) we may set the numerical value of µ > to be equal to 1. Translating this energy scale to the energy density ρ > ∼ µ 4 > we need the following constraints to be fulfilled: ρ ρ> < 1 and R µ 2 > < 1. As we may see from Figures 3 and 4b, despite the fact that the second minimum of our potential is roughly at h ∼ 1 ∼ µ > the abovementioned constraints are fulfilled. Now we will focus on the influence of the c HH Rh 4 term on the vacuum stability. The first interesting thing we may observe on the plot of Figure 5a is the fact that for small |c HH | an increase in |c HH | leads to a stabilization of the false vacuum. This is true for |c HH | less than approximately 0.2. Then this trend gets reversed and further increase in |c HH | leads to a decrease in the vacuum stability. From Figure 5b we may see that the behavior of v H (c HH ) is similar to the v H (ξ H ) case, namely for small |c HH | v H decreases then after crossing |c HH | ≈ 0.06 it starts to increase with an increase in |c HH |.
In Figure 6 we plotted the energy density ρ for various values of c HH . For small |c HH | ρ is positive everywhere and then for large |c HH | it becomes negative for the interior of the bubble of true vacuum. As for the Ricci scalar case we plotted its dependence on the c HH in Figure 7b. Similarly to the case of the dependence of the Ricci scalar on the non-minimal coupling ξ H we may see that for small |c HH | R is negative inside the bubble and for large |c HH | it becomes positive. Moreover, in this case R also changes its sign close to the bubble wall. The same c HH dependent distinction in the behavior of geometry is visible on the level of the Ap(τ ) function as depicted in Figure 7a. For small |c HH | it goes to a positive constant while for large |c HH | it tends to a negative one.
After discussing the influence of the gravity induced operators on the vacuum stability we will now focus on the non-geometric ones. To this end, we discuss the case of dimension six operators c 6 h 6 and much less often discussed c dHdH ∇ µ h 2 ∇ µ h 2 one. After inspecting (2.4) and (2.8) we may see that in our toy model these coefficients are related to each other at high energy. On the other hand, if we take into account running of these couplings their values in the low energy limit will differ, yet we will not discuss this running in this article. Having this in mind we plotted the influence of both operators on the vacuum stability in Figure 9a. From it we may see that an increase in these coefficients leads to an increase of the stability of false vacuum. Moreover, if both coefficients in front of these operators are positive, this stability is enchanted in comparison to the case c 6 = 0. On the other hand, in the case when c 6 is negative this vacuum stabilization is suppressed. In Figure 9b we plotted the dependence of v H on the same coefficients. The value of the scalar field h at τ = 0 decreases as we increase the c dHdH coefficient, this is also the case if we consider variation in the c 6 coefficient. This decrease is weaker in the case when c 6 has an opposite sign to the c dHdH coefficient.
Focusing a little bit more on these coefficients leads us to some additional interesting conclusions. In Figure 10a we plotted the action calculated for various c 6 c dHdH rates. Before we discuss this let us make one technical remark. Most of these rates lay beyond the allowed parameter space for our model. Nevertheless, they are physically reasonable in the sense that in the case when c 6 and c dHdH are treated as independent coefficients the rates represent physically allowed values for them. The above situation could be the case when the integrated heavy sector was more complicated than the one containing a single massive scalar field. Returning to Figure 10a, if the c 6 c dHdH ratio is negative and big enough, the c 6 h 6 operators start to dominate and lead to a decrease in the action value and therefore to a decrease in the false vacuum lifetime. In our case the negativity of the aforementioned ratio implies that the coefficient c 6 is negative. This implies that in a Lorentzian spacetime it will have an opposite sign to the quartic self-interaction term in the scalar field potential. This increase in vacuum instability is in agreement with what we would expect by analyzing only the potential for the scalar field. By doing this we could write λh 4 − c 6 h 6 = λ(h) ef f h 4 and conclude that the c 6 term will make λ(h) ef f negative for large enough fields therefore it would worsen the stability. For completeness, in Figure 10b we plotted the behavior of v H for various c 6 c dHdH rates. For small rates v H decreases for an increasing c dHdH and for sufficiently large and negative c 6 this tendency is reversed.

Summary
In the article we discussed a possibility of using the curved spacetime Effective Field Theory (cEFT) to tackle the problem of investigating the influence of gravity on vacuum stability. To this end, we firstly obtained cEFT for the light scalar by integrating out the heavy one. Then we focused on investigating the role of the dimension six operators that contribute both to the potential and kinetic parts of the scalar field action. We also discussed the influence of the novel dimension gravity induced operator that does not have its counterpart in the flat spacetime case. To aid our analysis we also used two observables, one of which was the Ricci scalar and was related to geometry and the second one was the Euclidean counterpart of energy density.
For the non-minimal coupling of the light scalar to gravity we found that the value of the Euclidean action calculated at the solutions to the equations of motion change monotonically with the change of ξ H . This type of behavior was already discussed in literature.
On the other hand, for the novel c HH Rh 4 type operator this change is not monotonic when we change the value of c HH . This is a nontrivial observation since both operators may be classified as non-minimal type contributions to the scalar field potential.
Let us also note that our investigation allows to observe some interesting features of the Ricci scalar not discussed so far in the context of vacuum stability. These were the non-monotonicity of R as a function of Euclidean time and the fact that the Ricci scalar changes its sign close to the bubble wall location.
As for the operators of dimension six that are also present in the flat spacetime case we discussed the standard contribution to the scalar field potential with coefficient c 6 and much less often discussed contribution to the kinetic term with coefficient c dHdH . In our cEFT approach they turn out to be linked to each other since both of them stem from the Higgs portal like coupling among scalar fields λ HX by the formula c 6 c dHdH = λ HX . From the analyzed cases we observed that if the abovementioned coefficients ratio is close to unity, the c 6 h 6 term will give a dominating contribution to the action.
In conclusion, we may state that applying cEFT to the case at hand turns out to be very productive and revealed some novel features in the problem of vacuum stability in curved spacetime.