Microscopic identification of dissipative modes in relativistic field theories

We present an argument to support the existence of dissipative modes in relativistic field theories. In an O(N) $\varphi^4$ theory in spatial dimension $d\le 3$, a relaxation constant $\Gamma$ of a two-point function in an infrared region is shown to be finite within the two-particle irreducible (2PI) framework at the next-leading order (NLO) of 1/N expansion. This immediately implies that a slow dissipative mode with a dispersion $p_0\sim i\Gamma \p^2$ is microscopically identified in the two-point function. Contrary, NLO calculation in the one-particle irreducible (1PI) framework fails to yield a finite relaxation constant. Comparing the results in 1PI and 2PI frameworks, one concludes that dissipation emerges from multiple scattering of a particle with a heat bath, which is appropriately treated in the 2PI-NLO calculation through the resummation of secular terms to improve long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent $z$ for the two-point function as $z=2-\eta$. We also discuss possible improvement of the result.


Introduction
It is generally a nontrivial and difficult challenge to deduce from a microscopic theory hydrodynamic behavior at long timescales. As was clearly stated by Kadanoff and Martin [1], this is because "hydrodynamic equations only appear when the behavior is dominated by the secular effects of collisions" and thus "most straightforward techniques for determining the correlation functions cannot be successfully applied". In the present paper, we will demonstrate in a concrete model that the two-particle irreducible (2PI) framework, which appropriately deals with secular effects through resummation, is indeed able to describe the hydrodynamic behavior from the microscopic level, in particular, a dissipative mode in a two-point correlation function.
There are three different levels of descriptions for the long-time (and long-distance) behavior of a locally equilibrated system. At timescales much longer than the correlation time (the typical timescale of the system), hydrodynamics is established as the framework for the time evolution of conserved densities through macroscopic variables such as temperature, chemical potential, and flow velocity. Diffusion and viscosity constants are regarded as low-energy effective constants. When we study, e.g., the temperature or momentum dependence of the low-energy constants, we need to PTEP 2015, 053A02 Y. Saito et al. include fluctuations around hydrodynamic behavior by employing a "mesoscopic" effective theory. As a typical example, the mode-coupling theory (MCT) [2][3][4][5] separates the slow and fast degrees of freedom and describes the effective dynamics of the slow modes while treating the effects of the fast modes as stochastic noise and bare parameter constants. In particular, non-conserved variables such as an order-parameter field ϕ may be included in addition to hydrodynamic modes in MCT when they contribute to slow dynamics of interest. Only after the microscopic theory is solved will all the phenomenological parameters be fixed, which is the third-level description. However, such a microscopic approach, if applied naively, is suitable only for short-time description; it starts to lose its accuracy at timescales longer than the typical scattering time.
It should also be noticed that hydrodynamics applies at a scale much longer than the correlation length ξ and therefore breaks down as the system approaches a critical point, where ξ → ∞. MCT was originally devised to describe the slow dynamics near the critical point by including the orderparameter fluctuations ϕ as well as those of conserved densities, and it is successful in reproducing the singularity of the transport coefficients as the critical point is approached. Accordingly, MCT gives empirical values for the dynamic critical exponent z, which characterizes the dissipation time of a disturbance as τ ∼ ξ z . Such a process should be associated with a dissipative mode whose dispersion is given by p 0 ∝ i|p| z , with p 0 and p the mode energy and momentum.
Considering that MCT is just a phenomenological theory and that the dissipative mode (at the critical point) should couple in the two-point function of the order-parameter field ϕ, it is preferable if we can investigate the low-frequency behavior of the two-point function at the microscopic level. As far as we know, the 2PI approach [6][7][8] is the only framework that allows us to describe the two-point function at the microscopic level, making the secular effects from multiple scattering of particles tractable through resummation. This motivated us to work on the two-point function of ϕ in the 2PI framework.
Recently, the 2PI effective action has been actively applied to non-equilibrium phenomena. One of the merits of the 2PI framework is that it treats a condensate φ = ϕ and the two-point correlation function of fluctuations G = (ϕ − φ)(ϕ − φ) in a self-consistent way, where · indicates averaging with a density matrix, and that we can systematically resum higher-order diagrams for φ and G while preserving the conservation laws of the system [9][10][11]. Therefore, the 2PI framework is expected to be very useful in describing the long-time evolution, where resummation of secular effects is crucial.
In the present paper, we analyze the two-point function G of a microscopic theory nonperturbatively, and show how to identify the dissipative mode, which will play an important role in the long-time behavior of the system. Here we take a relativistic O(N ) ϕ 4 scalar theory as a microscopic theory. Another motivation for our study is to see how a dissipative mode appears in the microscopic two-point function of a relativistic field theory. The bare mode with the dispersion relation, p 2 0 = p 2 + m 2 , receives corrections by the interactions in perturbation theory, but is still propagating. On the other hand, we are interested here in the motion in the long-time limit, which corresponds to p 0 ∼ 0. It has not been fully revealed how dissipative modes emerge microscopically in relativistic field theories. We show that purely dissipative motion is possible in relativistic theory when we include multiple interactions among the modes, which can be efficiently taken into account in the 2PI framework.
After confirming the existence of a dissipative mode in the theory, we briefly extend our discussion to a system near a critical point. Indeed, there is a controversial situation concerning the dynamic critical exponent z between relativistic and nonrelativistic scalar theories. Analysis of the two-point 2 [12,13] predicts z close to 2, while a relativistic model gives different values for z depending on the calculation methods: Dynamic renormalization group [14,15] gives z ∼ 1 while numerical simulation [16] gives z ∼ 2. In fact, as is shown in the present paper, these analyses (except Ref. [16]) of the two-point function of ϕ give us the exponent z, not of a purely dissipative mode, but of the propagating modes; the latter motion is faster than the former dissipative modes. This mis-identification is the origin of the controversy. 1 Before closing this introduction, let us comment on several physical situations whose hydrodynamics are of interest. First, the hydrodynamic picture is widely applied to describing the long-time and long-distance behavior of various systems, from quark-gluon plasma (QGP) in heavy-ion collisions [24][25][26] and the Bose-Einstein condensate of a cold atomic gas [27], to astrophysical phenomena such as supernova explosions [28], and even to the universe itself [29]. In particular, the hydrodynamic picture is a very powerful tool in describing the space-time evolution of QGP, which is an expanding local equilibrium matter made of quarks and gluons. It is in general necessary to figure out the hydrodynamic behavior of QGP including diffusion in any circumstance, but the effects of diffusion are expected to become more relevant if the system is near or at the critical point. In fact, a critical point is conjectured in the T -μ phase diagram of quantum chromodynamics (QCD), where transition from the "hadron phase" (at low temperature T and low chemical potential μ) to the QGP phase (at high T ) is of the 2nd order, and heavy-ion collision experiments for the critical point survey have already started [30,31]. Locating the critical point on the QCD phase diagram is one of the main objectives in QGP physics now, and to this end we obviously need correct understanding of the dynamics near the critical point. There are indeed several works applying MCT to the critical dynamics of QCD [21][22][23][32][33][34]. However, it is still controversial in the sense that the hydrodynamic mode dominates the critical dynamics. Experimentally, it is crucial to identify the Rayleigh peak (heat-diffusion mode) separating from the Brillouin peaks (propagating sound modes), which is indeed performed in, e.g., inelastic photon scattering by liquid argon and neon [35]. Similar identification should be possible even in QCD matter around the critical point [36]. Second, note also that researchers are trying to describe non-equilibrium phenomena from microscopic theories by exploiting the 2PI framework. For example, the problem of how QGP can be formed within a short time after heavy-ion collisions [37][38][39][40] and also non-equilibrium phenomena in cold atomic systems [41] is being discussed. We hope that fundamental understanding on the dissipative modes in the present paper would be helpful for various phenomena, including QGP physics. This paper is organized as follows: In Sect. 2, we discuss how a dissipative mode emerges from a two-point correlation function using both one-particle irreducible (1PI) and 2PI effective actions. In particular, we focus on the way in which the effects of multiple scattering with a heat bath are included, and propose a criterion for the emergence of a dissipative mode. In Sect. 3, we check the criterion in an O(N )-symmetric ϕ 4 model within two different calculation methods: 1PI and 2PI frameworks in a 1/N expansion at the next-leading order (NLO). We will find that the two-point function in the 1PI-NLO calculation fails to describe the dissipative mode, but that the 2PI-NLO 1 There are previous studies within MCT treating dynamic critical exponents of nonrelativistic [5,[17][18][19][20] and relativistic [21][22][23] field theories. These works, except for Ref. [23], assumed the existence of dissipative modes ( p 0 ∼ −i|p| 2 ) and obtained the result z ∼ 2 at the leading order of 1/N or expansion. Reference [23] obtained a different result, z ∼ 1, for a relativistic model, claiming that as slow modes one should include a "momentum" variable canonically conjugate to a field variable in relativistic models. We will not address this problem since our scope in this paper is to find a microscopic description of dissipative modes. calculation can have a dissipative mode with a finite relaxation constant. In Sect. 4, assuming that the dissipative two-point correlation function obtained in the previous section is not modified at the critical point, we estimate the dynamic critical exponent z. We also discuss possible improvement of our result. The last section is devoted to a summary and a discussion.

Dissipative mode in the two-point correlation function
We consider a system slightly disturbed from equilibrium and how it undergoes relaxation at timescales much longer than the typical scattering time, by inspecting the two-point function. 2 Before proceeding to a concrete model analysis, let us first clarify the general properties of two-point functions in an infrared region.
The slowest motion of the system is governed by the hydrodynamic modes consisting of conserved density fluctuations as well as order-parameter fluctuations in the case of critical phenomena, and the Nambu-Goldstone modes in the case of the symmetry-broken phase. At a "mesoscopic" timescale, MCT describes the relaxation processes phenomenologically as an effective theory that involves other slow degrees of freedom in addition to the hydrodynamic modes. Nonlinear couplings among those modes renormalize, e.g., transport coefficients in hydrodynamics.
In the present paper we deal with a non-conserved "order-parameter" field ϕ in the symmetric phase, for a particular example. According to the phenomenological MCT, the bare two-point correlation function of the field ϕ with a mass m is given as [5,19,20,44] Here is the relaxation constant, which is easily understood after the inverse Fourier transform with respect to t: G pheno (t, p) ∼ e − ·(p 2 +m 2 )t . On the other hand, in a microscopic field theory, a retarded two-point function G R ( p 0 , p) is generally written as where R ( p 0 , p) is a retarded self-energy and m 0 is a bare mass. Henceforth, we indicate retarded and advanced quantities with indices R and A, respectively. The retarded function, G R , describes the evolution of a single mode excited from the system in thermal equilibrium, and the mode attenuates by scatterings with other modes that play the role of a heat bath. In the present and following sections, we deal with the system off the critical point (T = T c ) and extract the dissipative term, i −1 p 0 , from the self-energy. The system at the critical point will be considered in Sect. 4. Firstly, at p 0 = 0, the self-energy R ( p) has neither a pole, an essential singularity point, nor a branch point, except when p 0 = 0 is approached along the line | p 0 | = |p| with |p| → 0 [45,46]. Secondly, although a cut lies on the axis m p 0 = 0, the retarded self-energy is defined infinitesimally above this cut in the complex p 0 plane, and, therefore, R can be expanded around p 0 = 0 as long as one keeps | p 0 |/|p| 1. 4 2 When a system is in the vicinity of equilibrium, relaxation is described by the retarded two-point function evaluated in equilibrium (linear response theory) [42,43]. 3 We can obtain the two-point function for a conserved field by replacing with gp 2 in Eq. (1), where g is the transport coefficient related to the conserved quantities. 4 Expansions around p 0 = 0 are not always justified. For instance, it is known that two-point functions have essential singularities at p 0 = |p| [45,46], and thus cannot be written as the power series of p 0 around this point.

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Downloaded from https://academic.oup.com/ptep/article-abstract/2015/5/053A02/1531914 by guest on 28 July 2018 (This can be explicitly verified using the sunset self-energy diagrams calculated in Ref. [47].) Since the real and imaginary parts of R are respectively even and odd functions of p 0 [43,48,49], they can be expanded as Here it should be understood that the dimensions in O p n 0 and O p n are canceled with other dimensionful quantities such as temperature T and/or m 0 (and coupling constant λ in general), i.e., O ( p 0 /T ) n , . . . , etc. When we refer to an "infrared region", it is always meant that the energy and momentum are much less than these dimensionful parameters. Note that the coefficients c 0 , c 1 , and c 2 depend on λ and T . 5 With the expression Eq. (3) valid in the infrared region, the two-point function of the microscopic theory can be approximated as From Eq. (4) we see that the two-point function G R ( p 0 , p) of a microscopic theory develops a dissipative mode if the imaginary part of a self-energy is nonzero finite in the infrared limit: We propose to check the finiteness of this quantity as a criterion for the existence of a dissipative mode, and to identify c 1 as the inverse of the phenomenological relaxation constant −1 in Eq. (1). Later we will show that it is indeed nonzero finite at the NLO in the 1/N expansion in the 2PI formalism. According to the Cutkosky rules generalized at finite temperature [43,[50][51][52][53], one can relate m R to the sum of cut diagrams. The cut diagrams, in which internal lines are cut, can be interpreted as the scattering processes shown in Fig. 1, where an incoming (outgoing) particle with p = ( p 0 , p) disappears (emerges) through the scattering with particles in the heat bath. More precisely, shown in the left diagram of Fig. 1 is the process where a particle with momentum p scatters off j particles with momenta q 1 , . . . , q j in the heat bath, and turns to − j particles with momenta q j+1 , . . . , q in the final state. The right diagram shows the reverse process. Consider the soft limit of interest p → 0, where the energy-momentum conservation between the initial and final states, q 1 + · · · + q j = q j+1 + · · · + q , must be fulfilled. Clearly, the process in which there are no incoming particles from the heat bath (i.e., the initial total energy-momentum is zero) does not contribute to Eq. (5) because the condition for the final total energy-momentum q 1 + · · · + q = 0 can never be satisfied for massive particles in the heat bath. From this simple observation, one concludes that m R ( p 0 → 0, p → 0) = 0 at T = 0, i.e., there is no infrared dissipation in the vacuum. At T = 0, however, the imaginary part of the self-energy m R ( p 0 → 0, p → 0) receives, in general, contributions from the processes where the incoming particle scatters off particles in the heat bath and contributions from their reverse processes.
A more careful inspection reveals that the imaginary part of the self-energy m R ( p 0 , p) can be written in a rather intuitive form: where ρ ( j) and ρ ( − j) are, respectively, spectral functions for incoming and outgoing thermal particles, and will be expressed as a product of thermal distribution functions f p = e β| p 0 | − 1 −1 as follows: One finds that the product of f q denotes incoming particles from the heat bath, and that of (1 + f q ) denotes outgoing particles into the heat bath. The spectral functions, ρ ( j) and ρ ( − j) , specify kinematical windows for scattering states. Therefore, the quantity m R ( p 0 , p) is nonvanishing when the kinematical windows for the incoming and outgoing particles overlap with each other. This is a very useful point of view when we evaluate m R ( p 0 , p) in a concrete model. We will indeed see below that m R ( p 0 , p) in the O(N ) scalar model can be written in this form, and we can assess if it is zero or nonzero by checking the kinematical windows.
Note that we will also treat "composite" fields such as ϕ 2 , since the Schwinger-Dyson equation relates two-point functions to higher-point functions, which could have two (or more) fields at the same space-time point. A similar consideration to that discussed above should hold for those "composite" fields.

Relaxation constant at NLO in the 1/N expansion
In this paper, as a typical example of relativistic quantum field theories, we employ an O(N )symmetric ϕ 4 where N ) is an N -component real scalar field, m 2 0 and λ 0 are the bare mass and coupling constant, and we assume that d ≤ 3 so that the action is renormalizable. The partition function in the imaginary-time formalism is given by where β is an inverse temperature β = 1/T , and we have introduced one-and two-point source fields, J and K , respectively (for reviews of field theories at finite temperature, see Refs. [43,48,49]). For instance, the one-and two-point functions are obtained from the functional derivative of Z [J, K ], where δϕ = ϕ − ϕ and ρ eq denotes the density matrix of the canonical ensemble, ρ eq = e −β H . From this partition function Z [J, K ], one can introduce the 2PI effective action via a double Legendre transform in J and K , while the ordinary 1PI effective action is defined as the single Legendre transform in J with K ≡ 0. The advantage of the 2PI effective action is that it reorganizes the perturbative expansion series by resumming the higher-order terms self-consistently into the two-point correlation function. Details of the formalism of the 2PI effective action are presented in Ref. [8]. Because dissipation will result from multiple scatterings of the mode in the heat bath, one needs some sort of non-perturbative framework that can treat the processes involving infinitely many interactions. Thus, in the following calculation, we employ the 1/N expansion, which is a typical non-perturbative expansion. The leading-order (LO) contribution is the so-called tadpole diagram, which only brings in a shift of the mass m 2 0 → m 2 . Here we restrict ourselves to the symmetric phase: m 2 > 0. In the symmetric phase, the condensate of the order parameter is zero: ϕ = 0. Henceforth, we write δϕ as ϕ for brevity. The scattering processes among the particles appear at the NLO and in higher-order diagrams. The NLO diagram for the self-energy in the 2PI formalism that is relevant for the scattering processes in the symmetric phase 6 is found to be where iω n = 2πin/β with integer n is the Matsubara frequency. The straight line and the wiggly line in Eq. (10), respectively, denote the Matsubara Green function of the elementary field ϕ, i.e., G ∼ ϕϕ , and that of the composite field ϕ 2 , i.e., D ∼ ϕ 2 ϕ 2 . The Matsubara Green function D iω n , p satisfies the following diagrammatic equation in general: (11) The blob in Eq. (11) denotes a vertex function ϕ 2 ϕϕ . This vertex function has been studied with a self-consistent equation in Refs. [55,56]. For the purpose of evaluating the self-energy at NLO, however, we only need this vertex function at LO and then the equation reduces to a geometric series of the one-loop bubble diagram (iω n , p), which leads to the second line of Eq. (10): 7 At the end of the calculation we perform the analytic continuation from the imaginary to the real energy, iω n → p 0 + i ( is an infinitesimal positive number). Then the imaginary-time correlators are replaced by their retarded counterparts, which are explicitly defined as Here G ab R = G R δ ab in the symmetric phase, and the field indices are suppressed in ϕ 2 = ϕ a ϕ a and where By using all the information given above, one can find the explicit form of the imaginary part of the NLO self-energy Eq. (10) (see Appendix A for derivation): where n( p 0 ) = e βp 0 − 1 −1 , and the spectral functions for two-point correlators are defined as In particular, in the LO approximation for D R , Note that the expression Eq. (17) for m R at NLO is valid in both the 1PI and 2PI formalisms, except that G is the free two-point function in the 1PI formalism while it is the full self-consistent two-point function in the 2PI formalism. Note also that Eq. (17) can be rewritten in the form alluded to earlier in Eq. (6), so that the effects of the heat bath can be explicitly seen: and two other combinations for p 0 and q 0 should be obvious. Now one can judge the finiteness of the relaxation constant through the inspection of overlapping kinematical windows specified by the spectral functions ρ and ρ D .
In the infrared region, m R ( p) can be approximated as Substituting Eq. (20) into Eq. (5), we find the equation that determines c 1 , which we now identify as the inverse of the phenomenological relaxation constant −1 . The aim of the present paper is to show that is nonzero finite. For notational brevity, we define a constant γ by which is proportional to . In the following subsections, we will examine if γ is nonzero finite at the NLO of the 1/N expansion in the 1PI and 2PI formalisms. 8

1PI-NLO evaluation
First, we confirm that a dissipative mode does not emerge in the 1PI two-point function at the NLO. In the 1/N expansion in the 1PI formalism, the two-point function G R appearing in the diagrams is the free two-point function obtained at LO with the tadpole effect included in the self-energy, whose spectral function is The other function ρ D ( p 0 , p) in Eq. (21) is the spectral function for a "composite" ϕ 2 field. Because the λϕ 4 interaction couples the ϕ 2 field with two ϕ fields in either way of ϕ 2 ϕ → ϕ or ϕ 2 → ϕϕ  ( p 0 , p) has a nonzero value in the 1/N expansion at NLO in the 1PI framework: one is a space-like region s = p 2 0 − p 2 < 0, and the other is a two-particle continuum region s > (2m) 2 . (or their reverse processes), the spectral function ρ D ( p 0 , p) is nonzero in two kinematical regions: the space-like region s = p 2 0 − p 2 < 0 and two-particle continuum region s > (2m) 2 (Fig. 2). The supports of the spectral functions ρ( p 0 , p) and ρ D ( p 0 , p) do not overlap, and therefore the integral Eq. (21) for (1/γ ) 1PI at NLO vanishes. Therefore, we conclude that, in the 1PI-NLO calculation, dissipation is not seen in the infrared limit of the ϕ mode.
One obvious way to proceed is to go beyond the NLO in the 1PI framework. At the next-tonext-leading order (NNLO), the expression for the relaxation constant becomes more involved than Eq. (21), but it certainly gives a nonzero value because it is known that the spectral functions ρ and ρ D become nonzero except for p 0 = 0 on the p 0 -p plane once the so-called sunset diagram is included in the self-energy [47,49,57]. The sunset diagram appears as a sub-diagram in the NNLO-1PI calculation (see, e.g., Fig. 3, where the wavy line in red includes a sunset diagram). We will report progress on this approach in a separate paper.
Notice that the sunset diagram is already resummed at NLO in the 2PI effective action. Thus, in this paper, we take another direction. While keeping the NLO approximation, we exploit the 2PI effective action formalism, which self-consistently takes into account the processes relevant to dissipation in the self-energy of the two-point function G. Indeed, inclusion of the sunset diagram alone is known to give a finite [58,59], although it is not the full NLO calculation. Below, we develop a formalism that allows us to treat all the NLO diagrams including the sunset diagram.

2PI-NLO evaluation
Let us now confirm that the existence of a dissipative mode is accommodated in the 2PI-NLO calculation by showing that Eq. (21) is nonzero finite. As we have already emphasized several times, the 2PI effective action self-consistently determines two-point functions (and the condensate ϕ if O(N ) symmetry is broken). The self-consistent structure is seen in the expression Eq. (17) of m R ( p 0 , p) at NLO: It is given by the spectral functions ρ and ρ D , but they are in turn given as 10 imaginary parts of two-point functions. Therefore, the precise form of m R ( p 0 , p) is found only after the Schwinger-Dyson-type equation is solved self-consistently. We are going to show, however, from general properties of the spectral functions ρ and ρ D that m R ( p 0 , p) in 2PI-NLO has a term proportional to p 0 with a finite coefficient.
Recall that, in the 1PI-NLO calculation, the supports of the two spectral functions ρ and ρ D do not overlap in the (q 0 , q) space, giving a vanishing result for Eq. (21), but that 1PI-NNLO will give a nonzero finite result thanks to the sunset diagram as included in Fig. 3. The same contribution is included in the 2PI-NLO diagrams (written in terms of full two-point functions) and thus Eq. (21) is expected to become nonzero finite.
We will see this property below. Because the spectral functions are odd in the frequency q 0 and n (x) = n (−x) = −e x /(e x − 1) 2 , Eq. (21) yields is the surface area of the d-dimensional sphere. 9 Since the spectral functions ρ and ρ D are positive semi-definite for q 0 > 0 in general, and, as mentioned above, they have nonzero values except for q 0 = 0 in the integration domain in the 2PI-NLO calculation [47,49,57], we conclude at this point that Eq. (23) has a nonzero value, in contrast to the 1PI-NLO result.
In the following, we verify that Eq. (23) for 1/γ is not diverging, but finite. To this end, we first notice that the spectral functions would not have singularities. For example, the delta-function singularity of ρ coming from the one-particle pole in the vacuum turns into a peak with a finite width due to the interactions in the heat bath via resummation in the 2PI formalism. Therefore, we can reasonably assume that the integrand does not have any singularity.
Then, what remains to be checked is the behavior of the integrand in the infrared (IR) and ultraviolet (UV) regions. For that purpose, we change the variables from (q 0 , |q|) to (Q, θ), defined by q 0 = Q cos θ, |q| = Q sin θ : In terms of these new variables, the IR and UV limits correspond to Q → 0 and Q → ∞, respectively, while there is no restriction on the angle θ . Thus, we evaluate the integrand in these limiting regions, and then perform the integration over θ to check the finiteness of the integral. Consider the IR region, β Q 1, where the integrand of Eq. (24) is approximated as In the IR limit Q → 0, the spectral functions ρ and ρ D become zero as long as the mass is kept finite. 10 Thus, the integrand, Eq. (25), is bounded by Q a (a < d − 2) in the IR region and the 9 The spectral functions are assumed to be rotationally invariant in equilibrium without external fields. 10 However, we expect that the spectral function ρ will diverge in the IR limit if one considers the massless limit. As we will comment later, such a divergent behavior will affect the critical dynamics of the two-point functions. Estimation of this divergence is now under investigation. Q-integral is IR-finite, when the spatial dimension d satisfies d ≥ 1. Then, we consider the θ -integral. A divergence may arise in the vicinity of θ = π/2. In this case, q 0 /|q| 1 is satisfied, and, therefore, ρ = 2 m G R and ρ D = 2 m D R are proportional to the frequency q 0 = Q cos θ (see the argument above Eq. (3)). We find that the θ -integral is also finite for d > 0. Hence, we conclude that no divergence appears from the IR region at least for d ≥ 1, which includes the dimensions of interest, 1 ≤ d ≤ 3.

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Next, we do not expect any divergence either in the UV limit Q → ∞, because physically nonzero 1/γ should be a purely in-medium effect, as is seen in the integrand that contains n (q 0 ). As long as θ = π/2, the last factor in the integrand gives an exponential suppression ∼ e −β Q cos θ as Q → ∞. The remaining part gives at most a positive power of Q, as we will see later. Therefore, there is no UV divergence for θ = π/2.
The only concern is the integrand when the angle θ is close to π/2, namely, in a deep spacelike region q 0 = Q cos θ → 0, q = Q sin θ → ∞. No matter how large Q is, we can take the quantity β Q cos θ small enough by choosing θ very close to π/2. Then, we find a different estimate ∼1/(β Q cos θ) 2 for the last factor in the integrand. Let us define an angle α that satisfies β Q cos α 1, or, equivalently, π/2 − α 1 β Q . We estimate the UV behavior of Eq. (24) by putting α as the lower limit of the θ -integral (we suppress the lower limit of the Q-integral here because we are just studying the UV behavior of the integrand): Since ρ and ρ D are proportional to q 0 = Q cos θ when βq 0 is small enough, the integrand is regular in the limit β Q cos θ → 0. Let us now introduce two functions as follows: Then, the θ -dependence of the integrand can be neglected, and the right-hand side of Eq. (26) is evaluated as where we have used the approximations sin θ ∼ 1 and cos α ∼ π/2 − α, valid in the vicinity of θ = π/2. Using the condition π/2 − α 1 β Q , we can further put an upper limit on the integration shown in Eq. (28): Now the problem has reduced to checking the UV behavior of the quantity: If this product decays faster than 1/Q as Q → ∞, the integral is UV-finite. In order to identify the high-momentum behavior of the functions h 1 and h 2 , a naive dimensional analysis is sufficient. 12 where we have provided the overall mass dimensions by Q and explicitly shown the dependence on other dimensionful parameters T, m, and λ. As Q → ∞, h 1 and h 2 decay at least as fast as 1/Q 3 and 1/Q 4−d , respectively, if h 1 and h 2 do not diverge in this limit. This is indeed the case as long as we stay in the symmetric phase, as seen from a plausible argument shown below.
Recall that the self-energy given by Eq. (10) is valid in the symmetric phase. Let T c (m, λ) be the critical temperature below which the O(N ) symmetry is broken for a given mass m and a given coupling constant λ. Then, T = T c (m, λ) defines a surface in the three dimensional space (T /Q, m/Q, λ/Q 3−d ), which separates the broken and symmetric phases as shown in Fig. 4. Let us first consider the limit λ/Q 3−d → 0 with fixed Q and with some fixed T and m, as shown by dashed arrows in Fig. 4, before discussing the behavior of h 1 , h 2 in the limit of T /Q, m/Q, λ/Q 3−d → 0 with fixed ratios of T : m : λ 1/(3−d) . In this non-interacting limit, one can explicitly evaluate the spectral functions ρ and ρ D . Now ρ is given by the delta-function Eq. (22), so that ρ, and therefore h 1 , does not diverge in the region with which we are concerned, |q|/|q 0 | 1. Thus, h 1 depends on λ with a positive power in the space-like region. The other spectral function ρ D is given by a single bubble diagram (see Eqs. (12), (18)): In the limit λ → 0, the two-point function in the diagram can be replaced by a free two-point function: where is an infinitesimal positive number. After analytic calculation, we obtain the UV behavior of h 2 as follows: This inequality holds for d ≤ 3 and |q|/2T ≥ 1, the latter of which is always satisfied in the UV region. Therefore, h 2 decays at least as fast as 1/|q| 4−d ∝ 1/Q 4−d , which is consistent with the naive dimensional counting as shown in Eq. (31). It should also be noted that, due to the exponential part in the last line of Eq. (34), h 2 can be bounded by 1/Q 4−d even if we take the limit T → 0 or m → 0. Now let us come back to the relevant limit Q → ∞ with T , m, and λ fixed in the symmetric phase, as shown by solid arrows in Fig. 4. In this limit, we approach the origin of the three dimensional space T /Q, m/Q, λ/Q 3−d . The same limit can be achieved by making T, m, λ → 0 with fixed ratios T : m : λ 1/(3−d) for a finite Q. One can also approach the origin by taking the non-interacting limit λ → 0 first and then by setting T and m to zero. It is physically quite plausible that the two limits are the same as long as they are taken in the symmetric phase, where no singularity is expected.
In view of the above, we conclude that h 1 and h 2 decay as fast as Q −3 and Q d−4 , respectively. Then, the dimension of the integrand in Eq. (29) becomes Q 2d−8 and thus the integral is convergent for d ≤ 3, which are the dimensions of interest.
Hence, in case of the spatial dimension 1 ≤ d ≤ 3, Eq. (24) becomes nonzero and finite constant. Thus, we finally conclude that This means that a two-point correlation function obtained from the self-consistent equation in 2PI effective action has a dissipative mode in the IR region. Some comments are in order. First of all, we emphasize that multi-scattering processes in the heat bath are taken into account in the 2PI-NLO but not in the 1PI-NLO calculation. As a result the dissipative mode, which does not exist in the free theory, appears non-perturbatively in the 2PI-NLO calculation. Second, it should be noticed that the dissipative mode emerges irrespective of whether the microscopic field theory is relativistic or nonrelativistic. This fact implies the same scaling behavior of the dissipative mode for the relativistic and nonrelativistic field theory at the critical point. They get modified by scatterings in the heat bath to acquire damping widths as well as energy shifts. Therefore, the scaling behavior of the propagating modes for the relativistic and nonrelativistic field theories is in general different. Third, we have not attempted to obtain a numerical value of the relaxation constant, although it is in principle possible if one obtains the two-point functions around thermal equilibrium. Namely, if one solves the Kadanoff-Baym equation near equilibrium, then one can evaluate 1/γ defined in Eq. (21) with the resultant solutions. We leave this problem for our future work. Rather, in the next section, we discuss an important consequence deduced from the presence of the diffusion term, even without knowing its coefficient.

Implications for the dynamic exponent z
So far, we have considered the case where the effective mass m 2 ≡ m 2 0 − e R ( p = 0) is nonzero. Roughly speaking, the limit m → 0 corresponds to the critical point where the correlation length diverges, ξ ∼ 1/m → ∞. This divergence also affects the time evolution of a disturbed system towards equilibrium. Off the critical point m = 0, small disturbances die away exponentially with the relaxation time τ . As the critical point is approached, the relaxation time becomes enormously long, which is called critical slowing down, and we define the dynamic critical exponent z via τ ∼ ξ z [2,3,[60][61][62]. In this section, we first present a consequence on the dynamic critical exponent z from the dissipative mode identified in the previous section, assuming that the relaxation constant remains nonzero finite at the critical point. Then we remark on a possible modification of such a naive assumption.
In order to identify the dynamic critical exponent z, we exploit the dynamic scaling relation at the critical point. We assume that under a scale transformation the two-point function transforms anisotropically with respect to p 0 and p: where b is a dimensionless scale factor that is taken bigger than one: b > 1. The prefactor b 2−η is fixed by the static part of the two-point function characterized by an exponent η. Indeed, if one substitutes b = /|p| into Eq. (36) and sets p 0 = 0, one recovers the static scaling for the correlation function G static R (p) ∼ ( /|p|) 2−η , where an arbitrary dimensionful parameter is introduced to make the combination /|p| dimensionless. We take of the order of the critical temperature T c . Next, we will rewrite Eq. (36) into a more useful form to evaluate z. Since the mass dimension of G R is −2, we can write G R ( p 0 , p) as [5]: Thus, we can extract z from the different scaling properties of p 0 and p in the two-point function.
The self-consistent equation obtained in the 2PI framework has previously been utilized to evaluate the static critical exponents η [63] and ν [56] (see Refs. [63][64][65] for application of the 2PI framework to 2nd-order phase transitions). While the imaginary part of the self-energy m R ( p) vanishes in the static limit, the momentum dependence of the real part e R ( p) brings in the scaling form of the two-point function: Of course, this is the same form as identified in the prefactor in Eq. (36). Since this scaling function is simply obtained from e R ( p), we can substitute it into the two-point function Eq. (4) to find the expression at the critical point: This yields the dynamic critical exponent which depends on N and d through η [63,66,67]. In order to better understand the result Eq. (40), let us carefully see how it appears. For that purpose, it is instructive to go back to the 1PI-NLO calculation, which, however, does not produce a dissipative mode in the IR region. The absence of a dissipative mode in 1PI-NLO is deduced from the IR behavior of m R ( p); namely, in the IR region it starts from O p 2 0 , p 2 instead of O( p 0 ). Even though we do not have a dissipative mode, we are able to define a "dynamic exponent" from the dispersion relation in the IR region. However, this is an exponent for the "propagating mode" and must be distinguished from the exponent of a dissipative mode that governs the very long-time evolution of the system. Indeed, if one considers a two-point function in the , p 2 , one finds dispersions for "propagating modes" p 0 ∼ ±|p| as the leading contribution with a small correction from the self-energy of order O(1/N ). Thus, one will find an exponent z prop. = 1 + O(1/N ), where we have introduced z prop. to remind that it is not the exponent for a dissipative mode, but for a propagating mode. If one performs the same analysis in nonrelativistic field theories, one will find z prop. = 2 + O(1/N ) again for propagating modes whose leading dispersions are p 0 ∝ p 2 . Namely, the leading contribution z 0 in the exponent z prop. = z 0 + O(1/N ) depends on whether the system is relativistic or nonrelativistic, and this is one of the reasons why there was confusion in the literature. Now, coming back to the 2PI-NLO calculation where we do have a dissipative mode, we find that the presence of a dissipative term, −i −1 p 0 , modifies the dispersion, and the leading contribution to z is given by 2. Higher-order correction −η to z = 2 is from e R ( p) and it is of O(1/N ). Therefore, our result z = 2 − η in Eq. (40) looks similar to the one for nonrelativistic propagating modes z prop. = 2 + O(1/N ), but the origin is different.
Let us further proceed one more step. Our result Eq. (40) is an immediate consequence of the static scaling behavior under the assumption that the relaxation term −i −1 p 0 remains non-singular [68,69]. However, the relaxation constant may depend on p in general. It is explicitly shown in Eq. (12) of Ref. [17] that, as long as we stay in a region | p 0 |/|p| 1 (see footnote 4), the selfenergy at the critical point can be expanded as the power series of p 0 . Although the powers of p 0 are integers in this expansion, there appear ln p terms at the critical point. Resumming the ln p terms that are proportional to p 0 , one finds that the dissipative term depends on p with the anomalous dimension −c , such as −i −1 p 0 ∼ −i −1 0 ( /|p|) c p 0 with 0 and c being constants in the IR region. Then one obtains and accordingly the exponent z is modified to This is also clearly seen in the time dependence of the two-point function. Equation (41) leads to G R (t, p) ∼ exp{−t/τ (p)} where the relaxation time is given by τ (p) ∼ 1/ (p)p 2−η ∼ 1/p 2−η+c . Thus, in the IR region p → 0, the relaxation time diverges. We have extended the evaluation of c in Ref. [70]. Our result Eq. (42) with modified is consistent with "model A" in the classification for nonrelativistic systems based on the mode-coupling theory [5]. Model A contains only (multi-component) non-conserved fields without conserved fields, and the dynamic critical exponent is given by z = 2 + cη, where c is a numerical constant of order one [5,17,71]. This seems apparently to be the case since we are only looking at the fundamental field ϕ. But it is actually nontrivial because in the 2PI framework the energy-momentum tensor is conserved and the coupling between the non-conserved fields ϕ and conserved fields is relevant to the dynamic universality classification. However, without knowing the explicit numerical value for c , it is difficult at present to judge which model our result should be classified as.
There is an explicit numerical simulation for a particular case with N = 1 and d = 2 [16]. From numerical evaluation of the dynamic critical exponent z ∼ 2−η 0.87 , it is claimed that it belongs to the dynamic universality class, model C. However, since we have used the 1/N expansion, which is not valid for N = 1, we cannot directly compare ours to the numerical simulation result, which is, however, intriguing.

Summary
The long-time behavior of a system recovering equilibrium should be characterized by a dissipative hydrodynamic mode. Because of the nonlinear mixing among the modes, we expect that even the two-point function G of the order-parameter field ϕ carries information on this long-time dissipation.
We have shown, however, that the NLO calculation of G in the 1PI 1/N expansion only gives real and imaginary corrections to the propagating mode of δϕ, but does not generate the dissipative mode in G, and in particular that the imaginary part of the self-energy, m R ( p 0 , p)/ p 0 , vanishes in the IR limit, p 0 , p → 0. At the critical point, we find that the mode spectrum scales as p 0 ∝ p z prop. with z prop. = z 0 + O(1/N ), where z 0 = 1 and 2 for relativistic and nonrelativistic cases, respectively. Since there is no dissipative mode in the IR limit, this exponent is just associated with the propagating mode, whose dispersion depends on whether the system is relativistic or nonrelativistic.
In the 2PI framework, on the other hand, we resummed the multiple scattering effects in the self-consistent two-point function G although working at NLO in the 1/N expansion. Once selfconsistency is imposed on G, the spectral function ρ( p 0 , p) will become nonzero, except for p 0 = 0, even at NLO in the 1/N expansion. (Recall, e.g., the sunset diagram, which is included at this order.) We then argued that the dissipative mode inevitably emerges as an IR pole of the two-point function G by showing that the imaginary part of the self-energy m R ( p 0 , p)/ p 0 is generally nonzero finite in the IR limit.
At the critical point, this dissipative mode implies that we will have z ∼ 2 − η even in relativistic scalar theories, which is in contrast to the 1PI-NLO result. For more precise evaluation of the exponent z, one may need to compute the transport coefficient utilizing the 4PI framework. We leave this as our future work. Furthermore, it is not clear whether the dissipative mode that we found is the same as the dissipation of the hydrodynamic mode, such as heat. To this end, we apparently need a more detailed and systematic evaluation of the coupled two-point functions of ϕ and the energy-momentum densities, etc.  number). Thus, we find that Eq. (A2) becomes In order to obtain the retarded self-energy, we perform an analytic continuation iω n → p 0 + i : where we have used the following properties: Now, we take the imaginary part of Eq. (A4). Since the spectral functions are imaginary parts of correlation functions, we finally obtain Eq. (17): In the second line of Eq. (A7), we have changed the integration variables of the second term to q 0 → q 0 + p 0 and q → p + q. 19