Stochastic parameters for scalar fields in de Sitter spacetime

The stochastic effective theory approach, often called stochastic inflation, is widely used in cosmology to describe scalar field dynamics during inflation. The existing formulations are, however, more qualitative than quantitative because the connection to the underlying quantum field theory (QFT) has not been properly established. A concrete sign of this is that the QFT parameters depend on the renormalisation scale, and therefore the relation between the QFT and stochastic theory must have explicit scale dependence that cancels it. In this paper we achieve that by determining the parameters of the second-order stochastic effective theory of light scalar fields in de Sitter to linear order in the self-coupling constant $\lambda$. This is done by computing equal-time two-point correlators to one-loop order both in QFT using dimensional regularisation and the $\overline{\rm MS}$ renormalisation scheme and the equal-time four-point correlator to leading order in both theories, and demanding that the results obtained in the two theories agree. With these parameters, the effective theory is valid when $m\lesssim H$ and $\lambda^2\ll m^4/H^4$, and therefore it is applicable in cases where neither perturbation theory nor any previously proposed stochastic effective theories are.


I. INTRODUCTION
The mathematical framework that one uses to describe the dynamics of inflation is scalar quantum field theory (QFT) in de Sitter spacetime [1][2][3][4][5][6][7][8][9]. Specifically, the long-distance behaviour of scalar fields is directly related to inflationary observables [10]. Spectator fields that existed during inflation can become observationally relevant in the present day and thus warrant further study. Examples of their application include curvature and isocurvature perturbations [11][12][13], dark matter generation [14][15][16][17] and primordial black hole abundance [18,19], electroweak vacuum decay [20][21][22][23][24] and gravitational-wave background anisotropy [25]. Scalar QFT cannot be solved exactly so one has to turn to approximate methods. For example, when one considers a quartic self-interacting theory, with coupling λ, of a scalar field with non-zero mass m, the standard approach is to perform a perturbative expansion in small λ about the free field solution. However, such a perturbative expansion is only valid in the regime λ ≪ m 4 /H 4 , where H is the Hubble parameter, because the expansion fails to converge beyond this limit [26][27][28][29]. This is an infrared (IR) problem and so is particularly prevalent when one is interested in the long-distance dynamics of the fields, and so one must look for alternative methods [30][31][32][33][34][35][36][37][38][39][40][41].
The focus of this paper will be one such method: the stochastic effective theory of the long-distance behaviour of scalar fields in de Sitter. The premise is that, for sufficiently light fields m ≲ H, long wavelength field modes are stretched by the expanding spacetime to such a degree that they can be considered classical. The remaining short wavelength modes remain quantum, but their contribution to the long-wavelength dynamics can be summarised by a statistical noise contribution. This method was pioneered by Starobinsky and Yokoyama [42,43], who derived stochastic equations from the slow-roll "overdamped" (OD) equations of motion that govern the inflaton using a cut-off method to separate the long and short wavelength modes. Stochastic inflation has since become a powerful tool for performing computations relating to the inflaton . However, due to the nature of the slow-roll equations, it is limited to the regime m ≪ H and λ ≪ m 2 /H 2 . These conditions are satisfied if the scalar field in question is the inflaton existing in slow-roll, but there are plenty of scenarios where one might wish to go beyond this regime, particularly if one is interested in studying spectator fields [16,17,19,72,73].
This can be achieved with a second-order stochastic effective theory [48,53,57,74,75], in which case the stochastic dynamics takes place in phase space, and no slow-roll assumption is needed. However, in the existing literature on both first-order and second-order stochastic theories, the parameters of this stochastic effective theory have not been computed beyond tree-level. In particular, the mass parameter m 2 S of the stochastic theory has been taken to be equal to the renormalised mass parameter m 2 R of the quantum field theory. Obviously, this cannot be accurate because m 2 R depends on the arbitrary renormalisation scale M , and there is nothing in the stochastic theory that can cancel this dependence. This introduces a relative error of order O(λH 2 /m 2 ) in the stochastic theory, limiting its range of validity and meaning that it cannot be used to compute precise quantative predictions.
In this paper we extend the approach used in Refs. [74,75], to compute the parameters of the stochastic effective theory to full one-loop order in perturbation theory. This calculation does not suffer from the infrared problem to the same extent as direct perturbative calculations of observables, and it gives a relation between the two theories that cancels the renormalisation scale dependence explicitly. Instead of using a cut-off approach to derive our stochastic equations, we consider second-order stochastic equations that resemble the classical equations of motion for a scalar field in de Sitter, but with general stochastic parameters: the stochastic mass m S , stochastic coupling λ S and the noise contributions σ 2 ij . One can apply standard techniques to solve these equations to compute stochastic correlation functions, which can be compared with perturbative QFT correlators to determine what the stochastic parameters should be for the stochastic theory to be promoted to an effective theory of scalar fields in de Sitter. This method was first introduced for free fields in Ref. [74] before being extended to include self-interactions in Ref. [75].
In Ref. [75], the matching between the stochastic theory and perturbative QFT was done by comparing the equivalent 2-point functions. On the perturbative QFT side, this required us to renormalise the theory because the O(λ) correction to the Feynman propagator is ultraviolet (UV) divergent. In Ref. [75], we only considered the correction term to leading order such that λ ≪ m 4 /H 4 , which meant we could neglect a detailed discussion of renormalisation schemes. We found that the stochastic parameters required to reproduce such a term were simply equal to those found in Ref. [74] for free fields. The result was a stochastic effective theory that was valid in the regime m ≲ H and λ ≪ m 2 /H 2 . However, this doesn't give a full account of the O(λ) contributions to the stochastic parameters. In this paper, we will perform a more detailed analysis of the UV renormalisation on the QFT side, which can then be matched by stochastic results to give full expressions for the stochastic parameters to O(λ). Additionally, we will also consider the connected 4-point function in both approximations such that we can compute the relation between the stochastic coupling λ S and its QFT counterpart λ. Thus, we will extend the regime of our stochastic theory to m ≲ H and λ 2 ≪ m 4 /H 4 .
We start by giving a full account of perturbative QFT in Sec. II. We introduce the free field 2-point functions in Sec. II A before outlining the in-in path integral formalism in Sec. II B. We then perform the UV renormalisation of the 2-point function using the MS scheme of dimensional regularisation in Sec. II C, with further details of the calculation in Appendix A. This amounts to a mass redefinition in order to absorb the UV divergent terms. As a result of dimensional regularisation, the mass parameter becomes dependent on the renormalisation scale M . To round out the QFT, we compute the connected 4point function in Sec. II D. In Sec. III, we move onto the second-order stochastic theory.
This follows the same procedure as in Ref. [75], except now the matching procedure is performed using the fully UV-renormalised QFT results. We also include the computation of the stochastic connected 4-point function in Sec. III A 3, which is compared with its QFT counterpart to find an expression for λ S . Finally, we perform a comparison between results from perturbative QFT, OD stochastic theory and second-order stochastic theory, including both the new results from this paper and the old results from Ref. [75]. Additionally, we discuss how the M -dependence of the stochastic parameters becomes important. We conclude with some remarks in Sec. V.

II. QUANTUM FIELD THEORY IN DE SITTER SPACETIME
We begin by considering a perturbative approach to scalar QFT in de Sitter spacetime.
We will consider a spectator scalar field ϕ(t, x) with a scalar potential V (ϕ) in a de Sitter background parametersed by the scale factor where H = 1 a da dt is the Hubble parameter, which is constant, and t and η are physical and conformal time respectively. They are related by The action for scalar fields in de Sitter is given by Introducing the field momentum π(t, x), the equations of motion are given by where primes and dots denote derivatives with respect to ϕ and t respectively. Throughout this paper, we will consider a field with mass m and a quartic self-interaction parameterised by λ such that the potential is V (ϕ) = 1 2 m 2 ϕ 2 + 1 4 λϕ 4 . Additionally, we include a nonminimial coupling to gravity ξ, which is absorbed into the mass term such that m 2 = m 2 0 + 12ξH 2 .

A. Free scalar QFT
We will first consider free fields such that λ = 0. Following standard procedures of second quantisation [1][2][3][4], the mode functions in the Bunch-Davies vacuum are given by where k = |k|, ν = 9/4 − m 2 /H 2 and H  The quantities we are interested in are correlation functions: expectation values of the Bunch-Davies vacuum state. The most basic of these is the scalar field two-point function, which is computed in k-space as We can perform the Fourier transform to obtain the 2-pt function in coordinate space as Computing this integral [2,3,[76][77][78] results in the positive (+) and negative (-) frequency Wightman functions in the Bunch-Davies vacuum where Γ(z) is the Euler-Gamma function, 2 F 1 (a, b, c; z) is the hypergeometric function and α = 3/2 − ν, β = 3/2 + ν. The iϵ prescription indicates the pole about which we perform our contour integration in the complex plane. Note that I have used conformal time η here, defined in Eq. (2), for convenience. From the equation of motion (4), the Wightman functions obey the equation Physical correlators must be invariant under the de Sitter group. This means that the behaviour of such correlators can be written purely in terms of a de Sitter invariant combination of the spacetime coordinates. The quantity in question is We can write the Wightman functions in terms of the de Sitter invariant by expanding about small ϵ to give where we have introduced the Heaviside function θ(z). Henceforth, I will drop the iϵ prescription. From the Wightman functions, we can build our other scalar 2-point correlators.
For convenience, we define Then, we define various other 2-point functions in Table I.
where T (T ) is the (anti-)time ordered operator. It obeys the equation One can study the leading IR behaviour of the Feynman propagator by considering the asymptotic expansion about large y. The leading terms in such an expansion are 1 The leading IR behaviour of the spacelike (equal-time) Feynman propagator is then given

B. The in-in formalism
Having laid the foundations with free fields, we now turn our attention to the more interesting situation where we include interactions such that λ ̸ = 0. The addition of the interaction means we cannot straightforwardly compute the scalar correlators; the equations of motion (4) cannot be solved analytically. Instead, we will consider a perturbative approach where one performs a small-λ expansion about the free solution that we computed in the last section.
We will use the in-in 2 path integral formalism, where we begin in some initial vacuum state -the in state |0 − ⟩ -evolve our system to some intermediate state before evolving back 1 Note that for m > √ 2H, there are other terms in the asymptotic expansion that dominate more than the second term in Eq. (15). For this paper, we are interested in light fields m ≲ H so this will never be the case. 2 Also known as Schwinger-Keldysh or closed time path to the in state [79][80][81][82]. The in-in generating functional is defined by where J ± are external sources that source the evolution from/to |0 − ⟩ respectively, and we take the normalisation Z[J, J] = 1. In the path integral representation, we introduce two auxiliary fields, ϕ + and ϕ − , to differentiate the contributions from the two paths. The in-in generating functional is then given by All scalar correlators can be built from the in-in generating functional via The Bunch-Davies vacuum states considered in Sec. II A, |0⟩, are really the in state |0 − ⟩.
Henceforth, I will drop the subscript '-' on the vacuum state.
The free in-in generating functional is given by 3 such that all the free field 2-point functions given in Table I can be computed using the in-in formalism via Eq. (19). To include quartic self-interactions, we write the generating functional as Applying Eq. (19) to (21), we can compute scalar correlators for this self-interacting theory.

C. Two-point QFT correlation functions to O(λ)
The quantity of most interest for this work is the time-ordered 2-point function. Eq. (13) gives us this quantity for free fields; now, we will add corrections from interactions in a perturbative manner. By expanding Eq. (21) to leading order in small coupling λ, the generating functional to O(λ) is given by Using Eq. (19), the time-ordered 2-point function to O(λ) is The first line is just the free Feynman propagator (13) while the second line gives the con- which is yet to be computed, and we have used the fact that i∆ F (z, z) = where the bare mass is m 2 B = m 2 0,B + 12ξ B H 2 . We infer that the 2-point function to O(λ) can be computed simply by replacing the mass m in the propagator by the Hubble-rate dependent effective mass i.e., It is important to note that the effective mass m 2 eff (H) is finite, but both the bare mass m 2 B and the field variance φ 2 are ultraviolet divergent. Therefore, the calculation requires regularisation and renormalisation. In order to make our effective theory directly applicable to particle physics theories, we want to use dimensional regularisation and the MS renormalisation scheme, which is the convention in particle physics.
In dimensional regularisation, one takes the number of spacetime dimensions to be d = 4 − ϵ. When one then takes the limit d → 4, the UV divergence appears as a 1/ϵ pole. For our purposes, we need the full expression for the field variance, including finite terms, and we are not aware of such a calculation in the literature. Therefore we present it in full detail in Appendix A. The result is is the polygamma function, γ E is the Euler-Mascheroni constant, and µ is an arbitrary energy scale. The next step is renormalisation, which involves absorbing the divergence into the mass parameter. In the MS scheme of dimensional regularisation, the renormalised mass is given by where M is the renormalisation scale. Explicitly, we must renormalise both the scalar mass and the non-minimal coupling respectively as We see that it is crucial to include the non-minimal coupling term for the renormalisation counterterms to be independent of H. Finally, we can now express the effective mass in Eq. (26) in terms of the MS renormalied mass m 2 R as Note that the explicit dependence on the renormalisation scale M cancels the implicit dependence through m 2 R , and therefore the effective mass m 2 eff (H) is renormalisation scale independent, as it must be.
For comparison with the stochastic approach, we will be interested in the long-distance behaviour of the 2-point function (26). Focussing on spacelike separations, the leading term in the asymptotic expansion about long distances is where the exponent 4 is In principle, we could extend the calculation to higher orders in λ, albeit with increasing levels of complexity. However, there is a problem that stems from the IR limit hidden amongst our results. To see this, we expand the 2-point function (31) to leading order in small-mass m 2 ≪ H 2 to give One can see that both corrections to the amplitude and exponent are of relative order λH 4 m 4 R in this small-mass expansion. In order for the sum to converge at higher orders in λ, we . This is not a priori true since our perturbation theory takes λ to be the small parameter about which we expand. Thus, perturbative QFT is limited to the following region in the parameter space: λ ≪ 1 and λ ≪ m 4 /H 4 . Therefore, to leading order in λH 4 /m 4 , the long-distance behaviour of the spacelike field correlator is 5 This is as far as perturbative QFT will take us for 2-point correlation functions. In order to go beyond this, we must employ alternative methods, such as the stochastic effective theory of scalar fields in de Sitter.

D. Four-point QFT correlation functions to O(λ)
To round out our discussion of perturbative QFT, we will briefly consider the 4-pt functions. For this paper, we will largely consider them as a tool for computing stochastic parameters and so won't go into a huge amount of detail. However, it is important to recognise that these objects are computationally challenging and our work has raised some questions surrounding this, which we will touch on at the end of this section.
Using the Schwinger-Keldysh formalism outlined in Sec. II B, we can combine Eq. (19) with (20) and (21) to obtain the time-ordered 4-pt scalar correlation function to O(λ) as The first line after the equality sign is the free part, composed of a combination of Feynman propagators. The next three lines are a similar combination but this time occur at O(λ), 5 Note that it was this 2-point function that was used in Ref. [75]. A key extension in this work is that we now consider Eq. (31) when we match the perturbative QFT and stochastic 2-point functions.
This introduces the M -dependence into the stochastic theory, which is an important UV effect even at long-distances.
thus containing the O(λ) piece of the 2-pt function (23), multiplied by the free Feynman propagator. The final lines indicate the new contribution to the 4-pt function that first appears at O(λ). These terms make up the connected 4-point function, which we will focus on here. Explicitly, this is related to the 2-point functions by where the subscript 'C ' stands for 'connected'. To compute this quantity, we must perform the z-integral in Eq. (35). Since the integrand is composed of a series of hypergeometric functions, doing an analytic calculation is extremely difficult. Moreover, attempts at a numerical computation have proved fruitless due to the existence of poles in the integrand.
Unfortunately, unlike its 2-pt counterpart, the 4-pt integral cannot be solved by a mass redefinition. We can make some progress by moving from position to momentum k-space, as outlined in Ref. [41,83,84], where computations simplify and the pole structure is no longer a problem. For this purpose, we will focus on equal-time 4-pt functions. The Fourier transform goes as where δ (3) (k) = (2π) 3 δ (3) (k) and we use the shorthand notation for the connected 4-pt func- , and the 'tilde' indicates equivalent quantities in k-space. Note that we will use conformal time in the following calculations, as defined in Eq. (2). The equal-time connected 4-pt function in k-space is given bỹ where the Wightman function in k-space is given by Eq. (6) and the Feynman propagator is found by using its definition in Table I. Using the time-ordering, one can simplify the integral such thatG where This integral is hard to solve in general. Analytic solutions are difficult because the integrand is a product of Hankel functions whilst the oscillatory behaviour of the integrand in the limit η z → −∞ make numerical computations challenging. However, for the purposes of this paper, we are only interested in the long-distance behaviour of the correlator. Assuming that that all four momenta k i ∀i ∈ {1, 2, 3, 4} are of the same order of magnitude, which we denote by k, we can therefore assume that −kη ≪ 1. We then separate the integral at some Then, the 4-point function can be written as Considering the kη ≪ −1 limit of the Wightman functions, one observes that the first term in Eq. (41) is proportional to the power law k −3−4ν R . We show this more explicitly in Appendix B.
For the second term, we can make use of the approximation −kη z ≪ 1 to write the Wightman functions as Substituting Eq. (43) into the second term of (41), one can compute the integral to givẽ where the first term includes contributions from the the first term of Eq. (41) and from the lower limit of the second integral, which are both of the same order O(k −3−4ν R ). This is actually the leading contribution in the IR limit for light fields, over the second term in to obtain the equal-time connected 4-pt function in coordinate space

III. SECOND-ORDER STOCHASTIC EFFECTIVE THEORY
We will now consider the second-order stochastic effective theory, which was introduced in Ref. [74,75]. We again consider a scalar field with a mass m S and quartic self-interaction λ S , where the subscript 'S' stands for 'stochastic', to differentiate from the QFT quantities introduced above. The stochastic equations are then given by with a white noise contribution The noise amplitudes σ 2 ij are left unspecified for the time being, other than the fact that they do not depend on the spacetime coordinates and that they are symmetric, preserving the reality of the noise. The form of the stochastic parameters m S , λ S and σ 2 ij will be determined by comparing stochastic correlators with their perturbative QFT counterparts. We will choose these quantities to be the same, promoting our stochastic theory from something general to an effective theory of QFT.
A. The second-order stochastic correlators

The spectral expansion
The time-evolution of the one-point probability distribution function (1PDF) P (ϕ, π; t) associated with the stochastic equations (48) is described by the Fokker-Planck equation where L F P is the Fokker-Planck operator. For a space of functions {f |(f, f ) < ∞} with the inner product we define the adjoint of the Fokker-Planck operator, L * F P , as Note that all integrals over ϕ and π have the limits (−∞, ∞) unless otherwise stated. Explicitly, The 1PDF can be written as a spectral expansion where Λ N and Ψ ( * ) N (ϕ, π) are the respective eigenvalues and (adjoint) eigenstates to the (adjoint) Fokker-Planck operator and c N are coefficients. We consider eigenstates that obey the biorthogonality and completeness relations and there exists an equilibrium state P eq (ϕ, π) = Ψ * 0 (ϕ, π)Ψ 0 (ϕ, π) obeying ∂ t P eq (ϕ, π) = 0. All sums of this form run from N = 0 to N = ∞.
From Eq. (54), making use of the relations (56), we find that the transfer matrix can be written with the spectral expansion as

B. Comparison with perturbative QFT
Now that we have developed the formalism for the stochastic theory, we will now promote it to an effective theory of the IR behaviour of scalar fields in de Sitter spacetime. To do this, we will compare the stochastic and QFT correlators and choose the stochastic parameters such that they match. This procedure was first outlined in Ref. [74,75].

Free stochastic parameters
We will begin by considering free fields, which was first done in Ref. [74]. It will prove convenient to change our field variables from (ϕ, π) to (q, p), with the transformation where α S = 3 2 − ν S and β S = 3 2 + ν S with ν S = 9 4 − m 2 S H 2 . All of the formalism introduced in the previous section can also be applied to (q, p) variables. In particular the Fokker-Planck operators are given by where the free part is given by and the interacting part is given by The (q, p) noise amplitudes are written in terms of their (ϕ, π) counterparts as Following the work of Ref. [74], we compute the stochastic free field correlator as and match it to the free Feynman propagator (15) to obtain an expression for the free stochastic parameters qp noise is chosen such that there is an analytic continuation from timelike to spacelike stochastic correlators, a behaviour prevalent in QFT.
However, the choice of σ 2 pp is arbitrary. In this paper, we focus on two possible choices: that the subleading term in the Feynman propagator is reproduced (σ 2 Q,pp = σ

2(N LO) pp
) or that the subleading term in the stochastic field 2-point function vanishes (σ 2 Q,pp = 0). We will see later that this choice doesn't impact physical results.
Since σ 2(0) qp = 0, the variables p and q separate and so we now use two indices (r, s) ∈ {0, ∞}, corresponding to p and q respectively, as opposed to just N . Thus, the free field eigenquations are given by where the Λ while the normalised eigenstates can be written in terms of the Hermite polynomials H n (x) as , (80a) 7 To take the limit, we have used the identity lim ϵ→0 Using the eigenequations with the biorthogonality conditions for (q, p), equivalent to Eq.
where for Eq. (82b) and (82c), r ′ ̸ = r and s ′ ̸ = s. One can then compute the spacelike stochastic 2-point correlator to O(λ S ) using Eq. (65) as ⟨f (q 1 , p 1 )g(q 2 , p 2 )⟩ = dq r dp r r ′ rs ′ s Ψ (0) 00 (q r , p r ) Ψ (0) * 00 (q r , p r ) 00 (q r , p r ) Ψ (0) * 00 (q r , p r ) We also wish to incorporate O(λ S ) effects into our stochastic parameters so we expand them about the free parameters (76) such that Note that because we are now considering an interacting theory, we have to use the renormalised mass m R , given in Eq. (30). Additionally, for simplicity, we will consider the case where σ 2(0) pp = 0 for this section, though we will compare with the other case later.
Using Eq. (65), we can compute the spacelike q − q, q − p, p − q and p − p stochastic 2-point functions. Converting these to (ϕ, π) variables using Eq. (70), we then write the spacelike stochastic field 2-point function to O(λ S ) as where the exponents are and we have used the free matched stochastic parameters (76).

Stochastic 4-point functions to O(λ S )
Using the perturbative eigenspectrum computed above, we can also compute the 4-point function to O(λ S ). We will focus on the connected piece to compare with its QFT counterpart (36) so we can use the free stochastic parameters from Eq. (76) and, for simplicity, we will make the choice σ 2 Q,pp = 0.
In general, the equal-space stochastic 4-point functions are given by Eq. (67). We will again switch from (ϕ, π) variables to (q, p) using Eq.

Stochastic parameters to O(λ)
We can now compute the O(λ) stochastic parameters by comparing the O(λ S ) stochastic 2-point and 4-point functions with their QFT counterparts. The procedure for matching our 2-point functions remains unchanged from the free case; we again have our three conditions (plus a choice) for the stochastic results to match perturbative QFT: (i) the leading exponents match, (ii) the leading prefactors match, (iii) the analytic continuation between spacelike and timelike correlators is preserved and (iv) either the subleading terms match or it vanishes in the stochastic correlators. We will consider both cases presented by (iv), though as stated these are not unique choices. For the following, I will consider the latter choice; the former is dealt with in Appendix C. We will see that the choice doesn't affect the physical results, as indeed it shouldn't.
However, we now have a fifth stochastic parameter to contend with -λ S -which can be matched to its QFT counterpart by considering the connected 4-point function. By equating the spacelike stochastic 4-point function (95), using the free stochastic parameters (76), to the spacelike quantum 4-point function (46), we find that the term O(|Ha(t)x| −9+6ν are equal for any value of m if Thus, at this order, the λ parameters in both quantum and stochastic theories are the same and we will drop the subscript S henceforth. We note that the second-order stochastic theory also gives us an expression of O |Ha(t)x| −6+4ν , which we know also appears in the QFT counterpart (46), denoted by the '...+'. Thus, the stochastic theory gives us a way of computing this term explicitly, which is difficult to do in perturbative QFT.
We can now turn our attention to the 2-point functions, and the other 4 O(λ) stochastic parameters. In order for the stochastic spacelike 2-point function (85) to reproduce the perturbative QFT 2-point function (31), the parameters must have the values Note that the mass parameter is now dependent on the renormalisation scale M . Using these parameters in our second-order stochastic equations (48) gives us a second-order stochastic theory of quartic self-interacting scalar QFT in de Sitter.
These expressions still have an IR problem, but it is milder than that of perturbative QFT. Expanding the O(λ) terms to leading order in m 2 /H 2 , we have We see that the sum will converge when λ ≪ m 2 /H 2 ; however, since we have corrected at this order, the error associated with the stochastic parameters is actually O λ 2 H 4 Converting our stochastic noise back to using the (ϕ, π) variables, our stochastic parameters are given by Thus, using the stochastic parameters (100) elevates the stochastic theory (48) to an IR effective theory of quartic self-interacting scalar QFT in de Sitter.

A. Regimes of validity
One of the strengths of the stochastic approach is that one can employ numerical techniques to compute correlation functions. For the second-order theory, one can use a matrix diagonalisation scheme to compute the eigenspectrum non-perturbatively. This method is outlined in Ref. [74]. Thus, the only limitation is the perturbative computation of the stochastic parameters. Conversely, the perturbative method outlined in Sec. II for QFT doesn't have an associated non-perturbative scheme; all results are purely perturbative.
Thus, the second-order stochastic theory can be used to go beyond perturbative QFT. Additionally, the second-order theory extends the regime of validity of the overdamped stochastic theory pioneered by Starobinsky and Yokoyama [42,43]. For more details on this method, see Ref. [72,74].
For the massive, quartic self-interacting scalar field theory considered here, the regime of these three approximations is OD stochastic: Second-order stochastic: We make a graphical comparison of these regimes in Fig. 1. For the purposes of making the boundaries obvious, we choose "≪ 1" to mean "< 0.2", though in reality we wouldn't expect these boundaries to be so clear cut.
The blue left hashed region represents the parameter space described by perturbative QFT. We can see that for light fields m ≲ H, this region is entirely covered by the secondorder stochastic theory. This is unsurprising given that the stochastic correlators were found directly from the 2-point functions of perturbative QFT. Beyond the light field limit, perturbative QFT continues to extend (though it is still limited to λ ≪ 1 -it is after all a perturbative theory!). This extension is not covered by either stochastic approaches as they both require light fields. The overdamped stochastic approach -the green, right hashed region -is resigned to nearmassless m ≪ H fields, but does go beyond perturbative QFT due to the non-perturbative methods available to it. Further, it is far simpler to compute stochastic correlation functions than their QFT counterparts, hence its popularity within its regime of validity.
The OD stochastic approach is encompassed by the second-order stochastic effective theory, as represented by the orange region in Fig. 1. However, the second-order stochastic effective theory goes further, also encompassing perturbative QFT entirely in the light field limit. We can also see that there is a large chunk of the parameter space, even for nearmassless fields, that is only covered by the second-order stochastic theory. The introduction of O(λ) corrections to the stochastic parameters means it goes beyond the OD approach, even in the limit m ≪ H, while the non-perturbative methods available mean that it can extend beyond perturbative QFT 8 . This suggests that the second-order stochastic effective theory can be used to probe hitherto untapped regions of the parameter space.

B. Comparing approximations with O(λ) stochastic parameters
In Ref. [75], we performed a careful analysis of how the three approximations compare with each other. Now, we wish to update that analysis to incorporate the updated stochastic parameters. There are two important changes that we have made here that will effect the results. The first is that we have O(λ) corrections to the stochastic parameters, which were not included in the previous paper [75]. The second is that now we have done the renormalisation more carefully and so we have a renormalisation-scale dependent mass parameter m R (M ), which must be chosen when doing numerical computations.
We will revisit the two examples discussed in Sec. B1 and B2 of Ref. [75] by computing the exponent for the leading term in the long-distance behaviour of the scalar 2-point functions.
For QFT, this corresponds to the quantity given in Eq. , of their respective spectral expansions. They are computed numerically. Note that the result used in Ref. [75] for the OD stochastic theory is the same as the one here, other than the fact we now use m R (M ) instead of m.
We plot Λ 1 for all three approximations as a function of the coupling λ for fixed m 2 R /H 2 and scale M = a(t)H. The first example will be for m 2 R /H 2 = 0.1 (Fig. 2), where we are in a regime where the OD stochastic approach is valid, while the second will be for m 2 R /H 2 = 0.3 (Fig. 4), where we expect it to fail.
In both examples, we expect perturbative QFT to hold for small λ and fail as λ increases, since it is in this regime that λ → m 4 /H 4 .
These two approximations are given by the yellow dotted (OD stochastic) and blue dashed (perturbative QFT).
Using similar reasoning, the second-order stochastic theory should be expected to hold for small λ and begin to fail as we increase λ in both plots. This is a less severe failure as the breakdown is now for λ 2 → m 4 R /H 4 . We will also consider how the choice of the σ 2 pp noise amplitude affects our results. The red and green lines represent the choices σ 2 pp = 0 and σ 2 pp = σ 2(N LO) pp respectively. For both cases, we also consider the free (dot-dashed) and O(λ) (solid) stochastic parameters, given in Eq. (76) and (98) The first example is for m 2 R /H 2 = 0.1. This is chosen because the mass is sufficiently small such that the OD stochastic approach will be valid beyond perturbative QFT. Consider respectively. Thus, we find that this choice doesn't affect physical results once you include the O(λ) effects, as indeed they shouldn't.
However, for free stochastic parameters, this choice does affect the result; therefore, it is important to incorporate these new O(λ) effects for the theory to be reliable.
It is worth noting that the excellent agreement between the second-order and OD stochastic results is due to the choice of renormalisation scale M = a(t)H. One can see from Eq.
(100a) that the stochastic mass m S depends on the renomalisation scale as ∼ ln the stochastic mass m S parameter (100a), whereas the OD theory does not because it doesn't incorporate any UV renormalisation. Thus, the two will be different for such a choice. Fig.   3 shows the effect of a different choice up to M = 5a(t)H. One can see that there is very little change to Λ 1 for the second-order theory and that the change is much larger for the OD case. Thus, even in the regime where m R ≪ H, where the OD stochastic theory is deemed to be valid, it still has some error associated with UV renormalisation. For completeness, we also include the example where m 2 R /H 2 = 0.3 for M = a(t)H, where Λ 1 as a function of λ is given in Fig. 4. This plot is identical to that of Fig. 4 in Ref. (green) and σ 2 pp = 0 (red).
of σ 2 pp to be irrelevant. This consolidates the point that this choice doesn't affect physical results but only when we have included the relevant O(λ) effects.

V. CONCLUDING REMARKS
We have shown that the second-order stochastic theory is a valid effective theory of the long-distance behaviour of scalar fields in de Sitter, within the regime of validity m ≲ H and λ 2 ≪ m 4 /H 4 . This extends the work started in Ref. [74,75] to incorporate the full O(λ) correction to the stochastic parameters, which includes a complete discussion of the UV renormalisation at this order. Notably, we have found a stochastic theory that incorporates a dependence on the renormalisation scale M in such a way that physical results are Mindependent. This is not true for the widely used overdamped stochastic approach.
While this is the final installment in a trilogy of papers, there is plenty of study left in the second-order stochastic theory. Currently, this theory has only been tested on perturbative QFT; one could more rigorously test the effective theory by comparing it to other QFT approximations such as the 1/N approximation [85,86] or Monte-Carlo simulations [87].
However, the main outstanding question is whether one can derive the stochastic parameters from an underlying microscopic picture, as opposed to using the matching procedure discussed here, which relies on an alternative method -in this case, perturbative QFTbeing available. It is not clear to us how one should proceed in this direction.
In spite of these formal questions, the second-order effective theory already has uses in inflationary cosmology. Its numerical tools mean that it can generate novel results in this field; directions could include precision calculations of curvature and isocurvature perturbations or extensions to primordial black hole abundance computations. It is clear that this method has the potential to be an important tool in the arsenal of inflationary cosmologists.
Before we begin, it is useful to note that the calculation is straightforward with pointsplitting regularisation [77]. Taking the coincident point limit of the Feynman propagator (13) from the spacelike direction such that x = (t, x) and x ′ = (t, 0), the field variance becomes φ 2 P S :=i∆ F (x, 0) where ν B = 9 4 − m 2 B H 2 , ψ (0) (z) is the polygamma function and γ E is the Euler-Mascheroni constant. The subscript 'PS' indicates it is computed using point-splitting.
To carry out the same calculation in dimensional regularisation, we consider the k-space integral (7), which gives where d 3 k = d 3 k (2π) 3 . In dimensional regularisation, the number of space dimension becomes D = 3 − ϵ, and therefore we have where µ is introduced as the regularisation scale and the subscript 'DR' indicates that we have defined this integral via dimensional regularisation. This integral cannot be computed analytically, and therefore we split it in two pieces, where δ is an arbitrary energy scale 9 The sum of the two contributions will of course give a δ-independent result. The first line is ultraviolet divergent but can be easily computed in 9 Note that we cannot choose δ = 0 because dimensional regularisation would then give a zero result for the first line.
dimensional regularisation, whereas the second line is finite and can therefore be computed in three dimensions.
Computing the divergent integral in the first line of Eq. (A4) and taking the limit ϵ → 0, we obtain The remaining integral in Eq. (A5) is finite, and it can be computed using the point-splitting result in Eq. (A1). We first note that the point-splitting regularised variance can be written (A6) Next, we repeat the split to divergent and finite integrals in Eq. (A4), Comparing Eqs. (A5) and (A7), we can see that the difference between the point-splitting and dimensional regularisation results is simply ϵ + ln πµ 2 |x| 2 + γ E + 1 4π 2 a(t) 2 |x| 2 . (A8) Using the point-splitting result from Eq. (A1), we can therefore write the field variance in dimensional regularisation as While this integral can't be computed in general, we can get some information about the IR behaviour of the 4-pt function. Consider the split of the integral G (4) for some parameter Λ < 1. Focussing on the IR limit of the integral, K < K z < Λ, we can take the limit K z < 1 such that we can use the approximate form of the Wightman functions (43)∆ ± (η z , η, k) ≃ π 4Ha(η) 3/2 a(η z ) 3/2 Then, we can compute the IR region of the integral (B6) to find that the 4-pt function will have the following behaviour: The K −6ν is just the term that we found in Eq. (44) and comes from the K z → K limit.
The other two contributions come from the limit K z → Λ. Converting these to coordinate space via a Fourier transform, one finds that Since ν ≤ 3/2, it is immediately clear that the final term is subleading. However, for near-massless fields, ν ∼ 3/2, the first and second terms give a similar contribution. As one increases the mass of the field, the second term is in fact the leading contribution over the first term. So, it appears that the contribution computed in Sec. II D is subleading. However, for this work, we only care about the 4-point function as a comparison tool between the stochastic theory and QFT so we can use this subleading term is it appears in the stochastic 4-point function.