Nonperturbative renormalization group for scalar fields
in de Sitter space: Beyond the local potential approximation
Nonperturbative renormalization group techniques have recently proven a powerful tool to tackle the nontrivial infrared dynamics of light scalar fields in de Sitter space. In the present article, we develop the formalism beyond the local potential approximation employed in earlier works. In particular, we consider the derivative expansion, a systematic expansion in powers of field derivatives, appropriate for long wavelength modes, that we generalize to the relevant case of a curved metric with Lorentzian signature. The method is illustrated with a detailed discussion of the so-called local potential approximation prime which, on top of the full effective potential, includes a running (but field-independent) field renormalization. We explicitly compute the associated anomalous dimension for O() theories. We find that it can take large values along the flow, leading to sizable differences as compared to the local potential approximation. However, it does not prevent the phenomenon of gravitationally induced dimensional reduction pointed out in previous studies. We show that, as a consequence, the effective potential at the end of the flow is unchanged as compared to the local potential approximation, the main effect of the running anomalous dimension being merely to slow down the flow.
The inflationary paradigm provides a consistent picture of the early Universe with both observational and theoretical successes. It has also brought much attention to the topic of quantum fields in curved space-time and, more specifically, to the case of de Sitter space. Of particular interest is the case of light scalar fields (in units of the space-time curvature), whose quantum fluctuations undergo a dramatic amplification on superhorizon scales due to the accelerated expansion. This can be viewed as a tremendous particle production from the classical gravitational field Mottola:1984ar (). If this is responsible for the observed power spectrum of primordial density fluctuations in inflationary cosmology Parentani:2004ta (), it also results in a nontrivial infrared dynamics Starobinsky:1994bd (); Tsamis:2005hd (); Weinberg:2005vy (). In particular, perturbative (loop) contributions are typically plagued by infrared and secular divergences which require resummation techniques or genuine nonperturbative approaches Starobinsky:1994bd (); Mazzitelli:1988ib (); Burgess:2009bs (); Rajaraman:2010xd (); Serreau:2011fu (); Beneke:2012kn (); Akhmedov:2011pj (); Gautier:2013aoa (); Boyanovsky:2012qs (); Serreau:2013psa (); Serreau:2013koa (); Nacir:2013xca (); Nacir:2016fzi (). Among those, the nonperturbative renormalization group (NPRG; see Refs. Berges:2000ew (); Delamotte:2007pf () for reviews), recently adapted to the case of scalar fields in de Sitter space Kaya:2013bga (); Serreau:2013eoa (); Guilleux:2015pma (), provides a promising tool to study the onset of gravitational effects as one progressively integrates modes from subhorizon to superhorizon scales.
Nontrivial results have been obtained in this context for O() scalar field theories using the simple local potential approximation (LPA) Serreau:2013eoa (); Guilleux:2015pma (), where one only retains the full functional flow of the effective potential but neglects that of other (e.g. derivative) terms Morris:1994ki (). In particular, it has been shown that the large superhorizon fluctuations induce a dimensional reduction of the renormalization group (RG) flow, resulting in an effective zero-dimensional theory. The effective potential can be expressed in terms of a normal integral—as opposed to a functional one—which can be put in exact correspondence with the late time stationary probability distribution function of the stochastic approach Starobinsky:1994bd (). This dimensional reduction also nicely explains the phenomenon of radiative symmetry restoration discussed in Ref. Ratra:1984yq () for the case and in Refs. Mazzitelli:1988ib (); Serreau:2011fu () for . The NPRG analysis shows that such gravitationally induced symmetry restoration occurs for any value of and in any space-time dimension, in agreement with the results of Ref. Lazzari:2013boa () in the stochastic approach. The NPRG approach has recently been applied to the study of symmetry restoration in scalar quantum electrodynamics in de Sitter space, with similar results Gonzalez:2016jrn ().
It is, therefore, of interest to investigate possible corrections to the LPA, which is the simplest, yet nontrivial, approximation in the context of NPRG methods. A typical extension is the derivative expansion, where one includes the running of the kinetic term and of higher derivative terms in the effective action Morris:1994ie (). This is suitable for the study of physical quantities primarily sensitive to long wavelength field configurations, such as critical exponents at a continuous phase transition in statistical physics Berges:2000ew (); Delamotte:2007pf (). Other approximation schemes can be based on a functional expansion of the effective action in powers of the fields. This generates an infinite tower of coupled flow equations for vertex functions which has to be truncated in one way or another. In the present work, we undertake the study of NPRG methods in de Sitter space beyond the LPA, focusing on the derivative expansion, having in mind the dynamics of long wavelength fluctuations. Our aim is twofold: First, establishing a consistent formulation of the derivative expansion in de Sitter space; second, putting the formalism at work in the simplest Ansatz beyond the LPA.
To this aim, we shall investigate the so-called local potential approximation prime (LPA’), where one includes a running, but field-independent renormalization of the standard kinetic term. This allows for relatively simpler calculations as compared to the full (field-dependent) next-to-leading order in the derivative expansion. In the context of statistical physics, the LPA’ is able to capture the main qualitative features of the phase structure of O() scalar theories in flat Euclidean space. In particular, it correctly describes the existence of a nontrivial critical regime for the two-dimensional XY model (), which corresponds to the Berezinsky-Kosterlitz-Thouless (BKT) transition Berezinsky:1970fr (); Kosterlitz:1973xp (); Kosterlitz:1974sm (); Berges:2000ew (); Delamotte:2007pf (). For the present purposes, this approximation is enough to illustrate the approach. We shall see that, due to the phenomenon of dimensional reduction described above, the inclusion of the running field renormalization factor alters the RG trajectories but not the end result of the flow as far as the effective potential is concerned. For instance, symmetry restoration happens later along the flow—i.e., deeper in the infrared—but the final effective potential exactly agrees with that of the LPA and, thus, with the stochastic approach.
The paper is organized as follows. In Sec. II, we present the general NPRG setup in de Sitter space-time. We discuss the issues of a proper formulation of the derivative expansion in general curved space-time and present a consistent approach in de Sitter space-time. We apply the formalism to the LPA’ for O() scalar theories in Secs. III and IV. We present a detailed calculation of the running anomalous dimension and we discuss the infrared and ultraviolet limits. The resulting flow equations in the infrared regime and the consequences of the gravitational dimensional reduction are discussed in Sec. V and we conclude in Sec. VI. Some technical details are presented in Appendixes A, B, and C and we present a discussion of the LPA’ flow equations in Minkowski space-time with particular emphasis on the Lorentz violating effects from the regulator in Appendix D.
Ii General setting
ii.1 NPRG in de Sitter space
The basic setup of the NPRG approach in de Sitter space has been developed in Refs. Kaya:2013bga (); Serreau:2013eoa (); Guilleux:2015pma (). We briefly review it here. We consider the expanding Poincaré patch of a de Sitter space-time with Lorentzian signature in dimensions. In terms of the comoving spatial coordinates the line element reads
where the cosmological time and the conformal time are related by , with the expansion rate. Unless explicitly stated, we set in the following. We shall be interested in the case of scalar field theories and, to fix the ideas, we shall explicitly consider theories with O() symmetry described by the classical action
where is the invariant integration measure, with the determinant of the metric tensor. The potential is a function of the O() invariant , and a summation over repeated space-time or O() indices is understood. Note that the potential includes possible couplings to the (constant) space-time curvature.
The NPRG approach consists in deforming the classical action with a space-time-dependent mass term, , with
where the infrared regulator acts as a large mass term for (quantum) fluctuations on sizes larger than and essentially vanishes for short wavelength modes, thereby suppressing the contribution from the former to the path integral. The idea is then to progressively integrate out the potentially dangerous infrared fluctuations by continuously changing the scale from a ultraviolet scale111The scale can be taken to infinity for renormalizable theories. , where the (bare) theory is defined, down to . One defines a regularized effective action which interpolates between the bare action, and the standard effective action, i.e., the generating functional of one-particle-irreducible vertex functions, . It satisfies an exact (functional) flow equation Wetterich:1992yh (); Morris:1993qb ()
where a dot denotes a derivative with respect to the RG time . The functional trace and convolution product refer to a given integration measure. We use the covariant measure defined above. Accordingly, we define the covariant two-point vertex function
where we have defined the covariant functional derivative as
The two-point function (5) relates to the exact propagator of the regulated theory as
where the inversion refers to the convolution product .
There are two technical points to be emphasized here. The first one is that we are primarily interested in computing the correlation functions of the theory in a given initial state specified in the infinite past; see below. Such an initial-value problem is most conveniently formulated in the in-in, or closed-time-path formalism Schwinger:1960qe (); Berges:2004yj (), where time integrations are to be understood along Schwinger’s closed time contour , which goes from infinite past to infinite future and back. In the present context, this amounts, e.g., to the replacements and ; see Ref. Parentani:2012tx () for details.222Discussions of NPRG methods for nonequilibrium systems can be found in Refs. Gasenzer:2008zz (); Canet:2011wf (); Berges:2012ty ().
The second, related point concerns the issue of the de Sitter isometries. In general, one chooses a regulator function which respects as many symmetries of the problem at hand as possible, in order to ensure that the resulting RG flow respects the latter down to . In the present case, it is not clear how to construct a proper regulator which respects all the de Sitter isometries.333This is, in fact, a general issue for space-times with Lorentzian signature, in relation with causality; see, e.g., Canet:2011wf (). Here, we follow Refs. Kaya:2013bga (); Serreau:2013eoa (); Guilleux:2015pma () and choose an infrared regulator of the form
which preserves a large subset of de Sitter isometries444In the expanding Poincaré patch this corresponds to the subgroup of space and time translations on the hyperboloid and spatial rotations. The former are generated by the spacelike and timelike Killing vector fields and , where is the cosmological time whereas and , with , are the physical and comoving spatial coordinates, respectively Eling:2006xg (); Busch:2012ne (). Together with the generators of spatial rotations , the subgroup algebra is (9) (10) (11) and all the other commutators vanish. Busch:2012ne (); Adamek:2013vw (); Parentani:2012tx (). Here, in the second line, we have introduced physical coordinates and momentum variables, and . The important point here is that we regulate physical momenta (as opposed to comoving ones). Interestingly, this induces an effective regulation of the time variable because of the way momentum and time are tight together by the gravitational redshift.
As already mentioned, the flow equation (4) is exact. However, such a nonlinear functional partial differential equation is in general not solvable exactly and one has to use approximations. The simplest local potential Ansatz (LPA) has been discussed at length in previous works Kaya:2013bga (); Serreau:2013eoa (); Guilleux:2015pma () and produces interesting physical results. The purpose of the present work is to explore the possible formulation and applications of approximations beyond the LPA, in particular, the derivative expansion that we now discuss.
ii.2 Derivative expansion and LPA’
The derivative expansion has been widely used in statistical physics applications of the NPRG Morris:1994ie (); Delamotte:2007pf (). It is a systematic expansion in powers of derivatives of the field, which aims at capturing the dynamics of long wavelength excitations, relevant, e.g., for computing critical exponents. It appears that this very idea is not completely straightforward in a general curved space-time because of possible couplings to the Riemann tensor and its contractions. In general, one expects the typical variation of the field to be related to those of the curvature and a sensible derivative expansion is likely to count gradients of both quantities on an equal footing.555 One could instead count derivatives of the metric tensor (see, e.g., Ref. Shapiro:2015ova ()) but this seems to us unphysical since the latter can be made arbitrarily large by (in)appropriate choices of coordinates. At first nontrivial order in gradients, this includes the standard term , but also terms of the form666A term is also possible but not independent since . , , and . Furthermore, all the coefficients of the gradient terms (including the zeroth-order local potential term) should be seen as functions of the field and all scalars made of the Riemann tensor, such as , , etc. For instance, keeping the full curvaturedependence in the potential is crucial in order to correctly capture the physics of superhorizon modes. In de Sitter space, this leads to nontrivial effects such as dynamical mass generation with Starobinsky:1994bd (). Finally, to make matters even more intricate, we mention that the non-commuting covariant derivatives make the separation between different orders in gradient ambiguous. For instance, the apparently fourth-order term
where is the Laplace-Beltrami operator, contains second-order contributions. In brief, a derivative expansion is far from being trivial (if implementable at all) in a general space-time.
Of course, matters simplify in space-times with a large number of symmetries. In particular, in the maximally symmetric, de Sitter space, the Riemann tensor is covariantly conserved, , and all scalar contractions are constant, e.g., , , etc. One can thus devise a systematic expansion in powers of, e.g., , with coefficients depending on the field only.777For instance, one has . For a theory with a single scalar field, one writes
Yet, this is not the end of the story…
The flow of the local potential can be obtained by writing Eq. (4) with the Ansatz (13) for constant field configuration. To derive the flow equation for the kinetic term , we consider the two-point vertex function (5) for a constant field configuration, namely, at second order in gradients,
where we denote the covariant Dirac distribution on the time contour as
In Minkowski space-time, we could now exploit the translation invariance in both the spatial and the temporal directions, valid for constant fields, and diagonalize the operator by going to -dimensional Fourier space, i.e., and . The RG flows of the various renormalization factors of the derivative expansion can then be obtained as the coefficients the momentum expansion of around . For instance, one would have .
Clearly, this step is not as simple in a general curved space-time because of the lack of space and/or time translation symmetries. But even in the maximally symmetric de Sitter space, which possesses translational invariances both in the (cosmological) time and in the (comoving) spatial directions, the problem remains tricky because the corresponding Killing fields do not commute and thus cannot be diagonalized simultaneously.
Exploiting spatial translation and rotation invariance in comoving coordinates and introducing the Fourier transform
where we have noted the Laplacian at fixed . At first sight, an easy way to extract the flow of the coefficients of the derivative expansion seems to consider an expansion in around , similar to the flat-space case. This is, however, too naive, again because of the noncommutation of the generators of space and time translations or, equivalently, because of the redshift of physical momenta encoded in the term . Indeed, it is easy to check that higher-order derivative operators , with , all contribute a term. For instance,
More generally, all operators contain terms with . It follows that the expansion of in powers of does not coincide with the derivative expansion: each coefficient of the former actually mixes an infinite number of terms from the latter.
The bottom line of the above discussion is that one should expand the function on a basis of eigenfunctions of the operator , which is clearly not the case of the spatial plane waves underlying the Fourier decomposition (17). As a practical constraint, it is also desirable that the relevant set of eigenfunctions be simple enough so that one can perform actual calculations. In this spirit, we propose to extract the coefficients of the derivative expansion by considering the Fourier representation (17) at . The relevant operator is then , with , which is diagonalized by the functions , with . We have
with . Accordingly, we introduce the following Fourier-like transform along the time contour:
and we shall perform an expansion in powers of in order to isolate the coefficients of the derivative expansion. Indeed, we have
from which we get the flows of the desired functions, e.g.,
Some comments are in order:
Second, the transform (21) on the closed time contour can be expressed as a standard Fourier transform along the real time axis in terms of the cosmological time and properly rescaled quantities. Indeed, using the identity and introducing the function888 It is this rescaled function which depends only on in the limit of subhorizon physical momenta and time difference . Indeed, in this limit, the rescaled propagator tends to the Minkowski propagator Serreau:2013psa (), which only depends on time through the difference . It is easy to show that this rescaled propagator is related to the vertex function (27) through , where we have used the fact that the regulator function vanishes for momenta . It follows that only depends on time through the time difference in the subhorizon limit.
Moreover, introducing the statistical () and spectral components of any nonlocal999Local terms, proportional to and its derivatives, can easily be treated as in Eq. (22). two-point function on the contour as Berges:2004yj ()
where, in the second equality, we further introduced the retarded two-point function
We see that the transform (21) is given by the Fourier transform (in cosmological time) of the retarded component of the two-point function evaluated at a complex frequency:
The approach described above reduces to straightforward prescriptions in the flat space-time, Minkowski limit . Making factors explicit, we replace by and by in the above discussions such that in Eqs. (20) and (24). Similarly, in Eq. (32), so that
with the -dimensional Fourier transform of the function introduced in (27). Note that, in the limit , the latter only depends on times through the difference and is time-independent for any , reflecting the fact that the generators of space and time translations now commute.
A drawback of our approach is that it implicitly assumes that the regulator, hence the flow, respects the de Sitter symmetries and, in particular, the symmetric role of timelike and spacelike gradients in the operator . However, as we have emphasized previously, our regulator only respects a subgroup of the de Sitter isometries and treats time and space differently. This implies that the various derivative terms compatible with the symmetries (see, e.g., Ref. Busch:2012ne ()) of the regulated theory should receive a priori different renormalization factors. For instance, at second order, the time and space gradient terms would receive independent renormalization factors: . If the method outlined above allows one to consistently extract the flow of , the discussion below Eq. (18) shows that it is not clear how to unambiguously extract that of . Nevertheless, the whole approach—with the present regulator—makes sense only if the explicit symmetry breaking induced by the regulator is small, in which case it is enough to compute the flow of following the above procedure. We test these ideas in Appendix D in the case of Minkowski space, where we can explicitly compute both and . Of course, a possible way out would be to construct a fully de Sitter invariant regulator. We postpone a detailed study of these issues to a later work and, from now on, we simply assume that such symmetry breaking effects can be neglected.
In the following, as a proof of principle of the feasibility of the above program, we consider a somewhat simplified version of the derivative expansion, where one neglects the field dependence of the field renormalization factor . This so-called local potential approximation prime, or LPA’, corresponds to the Ansatz
This has already been considered in Ref. Serreau:2013eoa () where, however, the flow of was not explicitly computed. It is the purpose of the following section to compute the latter and its influence on the flow of the local potential. Because the left-hand side of Eq. (24) does not depend on anymore whereas the right-hand side does (this is part of the inconsistencies of any Ansatz), we have to specify a value of where to evaluate it. We follow the standard practice in this context and choose the minimum of the running potential . We thus define the running anomalous dimension as
Iii Anomalous dimension for a single field
iii.1 Flow equation
where the derivative only acts on the explicit regulator dependence on the right-hand side. This form of the equation makes it particularly simple to obtain the flow of the two-point vertex function:
We must compute the three- and four-point vertex functions using our preferred Ansatz. This is where the LPA’ Ansatz (34) greatly simplifies matters. The derivative term being quadratic in the field, does not contribute to the three- and higher-point vertices. We have
and the flow equation (III.1) becomes
After exploiting spatial homogeneity and isotropy, this yields, in comoving spatial Fourier space,
with and . This exact scaling relation is a consequence of the de Sitter isometries (in fact, of the subgroup mentioned previously) which precisely states how physical momenta get correlated by the gravitational redshift. It allows one to scale out the comoving momentum and to deal the time evolution for a physical momentum evolution. Accordingly, one introduces a closed contour in momentum; see Ref. Parentani:2012tx () for details. It is straightforward to show that the transform (21) writes
where and are respectively the statistical and spectral correlators defined as
It is useful to note the symmetry properties and .
To compute the derivative acting on these propagators, we consider the variation in the relation , which yields
By identifying the statistical and spectral parts on both sides, we get
for which .This simple form allows one to perform some of the integrals analytically. After some algebra, the resulting flow can be written as
where we have left the field dependence on both sides implicit for simplicity and where we defined the integrals
with the integration measure
Here, we have split the contributions with all momenta below the RG scale from those which involve modes above this scale. Specifically, the contributions and involve modes which, as a result of the gravitational redshift, have nontrivial correlations with modes . As we shall see, these contributions vanish in the Minkowski limit where the redshift is absent. They also give subdominant contributions in the infrared limit.
Another consequence of the gravitational redshift is the fact that a cutoff scale on physical momenta effectively restricts the range of time integration. This is visible on Eqs. (51)–(56) where the variables and originally arise from time integrations. To illustrate this further, consider the change of variables , with and for a given in the integral (51), where we have made the dimensionful factors explicit.101010Note that the mass dimension of is . The various powers of momenta in the definition (51) contribute for . Hence, one must include a factor on the right-hand side of Eq. (51). A similar analysis shows that one must include a factor in the defining equations of , whose mass dimension is . Introducing
this yields, taking the case for illustration,
where . The time integration is bounded by the time111111Note that for in Eq. (59). at which the physical momentum crosses the running scale . The expression (59) also makes clear how this effect of the gravitational redshift disappears in the flat-space limit , where . In this case, one gets
which is, indeed, the result one obtains by applying the present formalism directly in Minkowski space Guilleux:2015pma (); Maximethesis (). This discussion easily generalizes to the integrals with the further change of variable . In that case, the factor ; see Eq. (33). As for the integrals , which involve modes , the above analysis yields a time integral , which guarantees that in the flat-space limit.
The flow equation (50) is quite complicated and cannot be written in a simple form in general. However, it greatly simplifies in two opposite limits that we now discuss, where all dimensionful scales are either large or small in units of the curvature. The first one corresponds to the Minkowski limit discussed above while the second is the one of prime interest to us, where curvature effects become important. At this point, it is worth emphasizing that, in the LPA’, only the contribution on the second line of Eq. (III.1) depends on the variable , from which it follows that . Because our prescription is to evaluate Eq. (35) at the minimum of the potential, is identically zero in the symmetric phase. As a consequence, we will be interested in cases where the potential presents a spontaneous symmetry breaking shape along the RG flow. In particular we shall study the possible effect of the running anomalous dimension on the phenomenon of symmetry restoration in the deep infrared regime.
iii.2 Heavy UV regime: Minkowski flow
The integrals (51)–(56) can be computed analytically in the limit where both the RG scale and the curvature of the running potential are large in units of , which is equivalent to sending . We shall not reproduce this instructive but cumbersome calculation here. We refer the reader to Ref. Maximethesis () for details. Instead, we shall compare with a calculation of the LPA’ flow equations directly in Minkowski space, detailed in Appendix D. The limit of Eq. (50) yields
where and is the regulated potential curvature. Setting gives the flow of the curvature of the potential [see Eq. (23)]: