Effective convergence of the 2PI-1/N expansion for nonequilibrium quantum fields
aa r X i v : . [ h e p - ph ] S e p Effective convergence of the 2PI- /N expansion for nonequilibrium quantum fields Gert Aarts and Nathan Laurie
Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
Anders Tranberg
Department of Physical Sciences, University of Oulu, P.O. Box 3000, FI-90014 Oulu, Finland (Dated: September 19, 2008)The 1 /N expansion of the two-particle irreducible effective action offers a powerful approach tostudy quantum field dynamics far from equilibrium. We investigate the effective convergence of the1 /N expansion in the O ( N ) model by comparing results obtained numerically in 1 + 1 dimensionsat leading, next-to-leading and next-to-next-to-leading order in 1 /N as well as in the weak couplinglimit. A comparison in classical statistical field theory, where exact numerical results are available,is made as well. We focus on early-time dynamics and quasi-particle properties far from equilibriumand observe rapid effective convergence already for moderate values of 1 /N or the coupling. PACS numbers:
Introduction –
The need to understand the dynamicsin heavy ion physics and the quark gluon plasma as wellas highly nonlinear phenomena in the (post-)inflationaryuniverse has lead to substantial development in the studyof quantum field evolution far from equilibrium. One par-ticular approach, firmly based on functional methods infield theory, employs the two-particle irreducible (2PI)effective action [1]. The nonperturbative 2PI-1 /N ex-pansion [2, 3], where N denotes the number of matterfield components, has in particular proven fruitful. Inrecent years this approach has been applied to a varietyof nonequilibrium problems in 3 + 1 dimensions, relatedto inflationary preheating [4, 5], effective prethermaliza-tion [6], fermion dynamics [7], transport coefficients [8]and kinetic theory [9, 10], slow-roll dynamics [11], topo-logical defects [12], nonthermal fixed points [13], expand-ing backgrounds [14], nonrelativistic cold atoms [15], etc.Besides these applications, theoretical aspects are undercontinuous investigation: renormalization in equilibriumis by now well understood [16, 17, 18] and practical imple-mentations of renormalization out of equilibrium are be-ing developed [19]. Moreover, progress in adapting thesemethods to gauge theories is steady [20, 21, 22].Applications of 2PI effective action techniques arebased on truncations, employing either a weak couplingor a large N expansion. So far, truncations stop at rel-atively low order: as far as we are aware all studies infield theory use next-to-leading order (NLO) truncationsin the coupling or 1 /N [30]. The reason for this situationis clear: beyond NLO the complexity of the equationsand the numerical effort required to solve them increasesdramatically [23]. In this paper we present the first re-sults beyond NLO in field theory.Remarkably, the lowest order truncations beyond meanfield theory include already many of the physical pro-cesses necessary to describe quantum field dynamics bothfar and close to equilibrium and are capable of capturingeffective memory loss, universality of late time evolution, and thermalization. The natural question to ask is thenhow accurate a truncation at a given order describes thedynamics in the full theory. There are several ways thiscan be investigated. In the case of a systematic expan-sion, it should be possible to compare different ordersof the expansion, shedding light on the effective conver-gence. When restricting to LO and NLO truncationsonly, the applicability of this approach is limited. Thereason is that in LO mean field approximations scatteringis absent and there is no notion of equilibration and ther-malization, as there is at NLO. It is therefore necessary toconsider the next-to-next-to-leading order (N LO) con-tribution as well. In the first part of this paper we studythis problem in the O ( N ) model and compare the dy-namics obtained at LO, NLO and (part of) N LO in the1 /N expansion in quantum field theory. In cases wherean exact solution is available, a direct comparison canbe carried out, and this approach has been successfullyapplied using classical statistical field theory instead ofquantum field theory [25, 26]. In the classical limit, thenonperturbative solution can be constructed numericallyby direct integration of the field equations of motion,sampling initial conditions from a given probability dis-tribution. We use this approach to further quantify therole of truncations in the second part of the paper. We consider a real N -component scalarquantum field with a λ ( φ a φ a ) / (4! N ) interaction ( a =1 , . . . , N ). In order to allow for an economical 1 /N ex-pansion, an auxiliary field χ is introduced to split thefour-point interaction, which results in the action S [ φ, χ ] = − Z C dx Z d x h ∂ µ φ a ∂ µ φ a + 12 m φ a φ a − N λ χ + 12 χφ a φ a i . (1)The theory is formulated along the Schwinger-Keldyshcontour C , as is appropriate for initial value problems. Inthe symmetric phase ( h φ a i = 0), the 2PI effective actiondepends on the one-point function ¯ χ = h χ i and the two-point functions G ab ( x, y ) = G ( x, y ) δ ab = h T C φ a ( x ) φ b ( y ) i , (2) D ( x, y ) = h T C χ ( x ) χ ( y ) i − h χ ( x ) ih χ ( y ) i . (3)In terms of those, the action is parametrized as [3, 27]Γ[ G, D, ¯ χ ] = S [0 , ¯ χ ] + i G − + i G − ( G − G )+ i D − + i D − ( D − D ) + Γ [ G, D ] , (4)where G − = i ( (cid:3) + m + ¯ χ ) and D − = 3 N/ ( iλ ) denotethe free inverse propagators. Extremizing the effectiveaction gives equations for ¯ χ , G and D . The latter takethe standard form G − = G − − Σ and D − = D − − Π,with self energies Σ = 2 iδ Γ /δG and Π = 2 iδ Γ /δD .In order to solve an initial value problem, the prop-agators and self energies are decomposed in statistical( F ) and spectral ( ρ ) components. For instance, the ba-sic two-point function is written as G ( x, y ) = F ( x, y ) − ( i/ C ( x − y ) ρ ( x, y ), and the causal equations for F and ρ take the form (cid:2) (cid:3) x + M ( x ) (cid:3) F ( x, y ) = − Z x dz Σ ρ ( x, z ) F ( z, y )+ Z y dz Σ F ( x, z ) ρ ( z, y ) , (5) (cid:2) (cid:3) x + M ( x ) (cid:3) ρ ( x, y ) = − Z x y dz Σ ρ ( x, z ) ρ ( z, y ) . Here we used the notation R x y dz = R x y dz R d z . Theeffective mass parameter is given by M ( x ) = m +¯ χ = m + λ ( N + 2) / (6 N ) F ( x, x ). After decomposing D ( x, y ) = λ/ (3 N ) (cid:2) iδ C ( x − y )+ ˆ D F ( x, y ) − ( i/ C ( x − y ) ˆ D ρ ( x, y ) (cid:3) , similar equations can be found for ˆ D F andˆ D ρ [3]. /N expansion – In the 2PI-1 /N expansion thenontrivial contribution Γ [ G, D ] is written as Γ =Γ NLO2 + Γ N LO2 + . . . Powercounting is straightforward: aclosed loop of G propagators yields a factor N , whereas a D propagator contributes 1 /N . This yields one diagramat NLO ( ∼ N ) and two diagrams at N LO ( ∼ /N ), seeFig. 1. Cutting a G/D line gives the self energy Σ / Π. AtNLO the self energies have no internal vertices and aretherefore easily evaluated in real space, where they aregiven by the product of two propagators. For example,the statistical component of the NLO self energy is givenby the expressionΣ
NLO F ( x, y ) = − g (cid:20) F ( x, y ) ˆ D F ( x, y ) − ρ ( x, y ) ˆ D ρ ( x, y ) (cid:21) , (6)where g = λ/ (3 N ). At N LO, the number of terms in-creases substantially, due to the possibility that any line (2) (3) (4)
FIG. 1: 2PI − /N expansion: NLO (2 loops) and N LO (3and 4 loops) contributions. The full/dashed line denotes the
G/D propagator. can be of the F or ρ type. Moreover, self energies havetwo or four internal vertices, greatly increasing the com-plexity of the expressions. Let us first consider the 3-loopdiagram at N LO. The corresponding self energy has twointernal vertices and five propagators. The complete ex-pression can be found in Ref. [23] and has O (30) terms.Here we give two terms to illustrate the structure,Σ N LO F ( x, y ) = − g Z x dz Z y z dw ρ ( x, z ) ˆ D F ( x, w ) ρ ( y, w ) (cid:20) ˆ D F ( y, z ) F ( z, w ) + 14 ˆ D ρ ( y, z ) ρ ( z, w ) (cid:21) + . . . (7)The two internal vertices result in two nested memoryintegrals. The second diagram at N LO yields self ener-gies with four nested memory integrals. In contrast toNLO, at N LO the evaluation of self energies completelydominates the numerical effort.
Quantum dynamics far from equilibrium –
We con-sider a spatially homogeneous system and solve the dy-namical equations numerically in 1 + 1 dimensions, bydiscretizing the system on a lattice with spatial lat-tice spacing a and temporal lattice spacing a t , using a t /a = 0 .
2. Initial conditions are determined by aGaussian density matrix, resulting in initial correla-tion functions F (0 , k ) = F ( a t , a t ; k ) = [ n ( ω k ) +1 / /ω k , F ( a t , k ) = F (0 , k )(1 − a t ω k / n is the initial (nonequilibrium) particle number and ω k = (cid:0) k + M (cid:1) / with M the initial mass, determinedselfconsistently from the mean field mass gap equation.Initial conditions for the spectral function are determinedby the equal time commutation relations.We consider LO mean field dynamics, obtained byputting the nonlocal memory integrals on the right-handside of Eq. (5) to zero. The only nonequilibrium aspectis in the time dependent mass M ( x ). The NLO ap-proximation is solved without further approximation. AtN LO the numerical effort differs substantially betweenthe 3-loop and the 4-loop diagram in Fig. 1, due to 2 and4 internal nested memory integrals respectively. There-fore we restrict ourselves to the first diagram at N LOand refer to this as N LO ′ . In principle, a subtle can-cellation between the 3- and 4-loop diagram could occur.However, since the second N LO diagram is (naively) -101
NLONNLO ’ mt -101 F ( t , ; k = ) N=2 N=4N=10 N=20
FIG. 2: Unequal time correlation function F ( t, k = ) for N = 2 , , ,
20 in the quantum theory. The dashed/full linesshow results for the 2PI-1 /N expansion at NLO/N LO ′ . suppressed by two powers of the coupling constant, thisis found not to be the case. We come back to this below.A similar comparison between 2- and 3-loop truncations,appearing at the same order in a coupling expansion, wasmade in Ref. [24].The unequal-time two-point function at zero momen-tum is shown in Fig. 2. The coupling is λ/m = 30,where m is the renormalized mass in vacuum. The latticespacing is am = 0 . Lm = 32. The mem-ory kernel is preserved completely. At LO (not shown)there is no damping in the unequal-time correlation func-tion. Beyond LO, we observe an underdamped oscilla-tion, the signal of a well-defined quasiparticle. Increasing N shows convergence of the two truncations considered.We can extract the quasiparticle properties far from equi-librium by fitting these curves to an Ansatz of the form Ae − γt cos M t , where M is the quasiparticle mass and γ its width (divided by 2). The resulting values are shownin Fig. 3 as a function of 1 /N . We observe effective con-vergence when the value of N is increased, as expectedfor a controlled expansion. For N &
10, the NLO andN LO ′ results are practically indistinguishable. In 1 + 1dimensions the perturbative onshell width from 2 → /N , as expected from naive powercounting. The quasi-particle mass converges to the value determined by thenonthermal fixed point of the mean field equations in thelarge N limit, which can be computed analytically [25].For the parameters used here we find M/m = 1 .
48 when N → ∞ . Classical statistical field theory –
In order to furtherinvestigate the applicability of the 1 /N expansion and inparticular the role of the 4-loop diagram at N LO, we γ / m mean fieldNLONNLO ’ M / m FIG. 3: Masses and widths extracted from the unequal-timecorrelation functions in Fig. 2 as a function of 1 /N . In themean field approximation, the width is zero for all values of N . turn to classical statistical field theory, where the non-perturbative evolution can be computed by numericallysolving the classical field equations, sampling initial con-ditions from the Gaussian ensemble. In order to carry outthe comparison, we also take the classical limit in the setof 2PI equations. In this limit statistical ( F ) two-pointfunctions dominate with respect to spectral ( ρ ) functions.There are several ways this can be motivated, using forinstance classical statistical diagrams or arguing for clas-sicality when occupation numbers are large [26, 28, 29].The result is that terms are dropped that are subleadingwhen F is taken to be much larger than ρ . For example,both in Eq. (6) at NLO and in Eq. (7) at N LO, the lastterms are dropped with respect to the first ones. We havecarried out this procedure for all terms appearing in selfenergies at N LO ′ in Ref. [23].Since the 4-loop diagram at N LO is suppressed bythe coupling constant, its effect is expected to diminishat weaker coupling. In order to verify this, we use a rela-tively small value of N = 4, where NLO and N LO ′ differat large coupling, both in the quantum and the classicaltheory. A comparison between NLO, N LO ′ , and the ex-act numerical result (MC for Monte Carlo) is shown inFig. 4. In the numerical integration of the classical equa-tions of motion, we have used a sample of 2 . × initialconditions. In the top left corner, the coupling constantis λ/m = 30. Damping at N LO ′ is less than at NLO,similar to the situation in the quantum theory. Decreas-ing the coupling constant, we observe that the agreementbetween the two 2PI truncations improves. Moreover, forweaker coupling, we note that the MC results lie betweenthe curves at NLO and N LO ′ , signalling a well converg-ing expansion. For even smaller coupling (not shown),we found that all curves lie on top of each other, even for N = 4. Since the N LO approximation is numerically
MCNLONNLO ’ mt -101 F ( t , ; k = ) λ/ m =30 λ/ m =30/4 λ/ m =30/8 λ/ m =30/2 N=4
FIG. 4: Unequal time correlation function F ( t, k = ) for N = 4 in the classical theory. Included are the NLO andN LO ′ results in the classical limit of the 2PI-1 /N expansion,and the exact result obtained by numerical integration of theclassical equations of motion (MC). The coupling constant is λ/m = 30 /n , with n = 1 , , , substantially more intensive than the NLO one, we haveinvestigated whether the memory kernel at N LO couldbe truncated earlier than at NLO, saving considerablecomputer resources. However, we found this not to bethe case: in the results shown here it was necessary topreserve memory kernels over the entire history, also forthe N LO contribution. At this moment this prevents usfrom going to larger times and studying the approach toequilibrium beyond NLO.
Summary –
We have investigated convergence proper-ties of the 2PI-1 /N expansion for nonequilibrium quan-tum fields. We included contributions at N LO, andfound that the expansion is convergent even at large cou-pling as N is increased. At fixed N , we found convergenceas the coupling strength is reduced. These results confirmthat the 2PI-1 /N expansion truncated at NLO alreadygives quantitatively accurate results for moderate valuesof N or coupling strength: At any coupling N = 10 worksvery well, whereas for the often used N = 4, a trade-offin terms of a smaller coupling is required for precisioncalculations. We expect this behaviour to persist in 3 + 1dimensions, but a direct confirmation would be welcome.The numerical work was conducted on the MurskaCluster at the Finnish Center for Computational SciencesCSC. G.A. and N.L. are supported by STFC. A.T. is sup-ported by Academy of Finland Grant 114371. [1] For a review, see J. Berges, AIP Conf. Proc. (2005)3 [hep-ph/0409233].[2] J. Berges, Nucl. Phys. A , 847 (2002)[hep-ph/0105311]. [3] G. Aarts, D. Ahrensmeier, R. Baier, J. Berges and J. Ser-reau, Phys. Rev. D , 045008 (2002) [hep-ph/0201308].[4] J. Berges and J. Serreau, Phys. Rev. Lett. (2003)111601 [hep-ph/0208070].[5] A. Arrizabalaga, J. Smit and A. Tranberg, JHEP (2004) 017 [hep-ph/0409177].[6] J. Berges, S. Borsanyi and C. Wetterich, Phys. Rev. Lett. , 142002 (2004) [hep-ph/0403234].[7] J. Berges, S. Bors´anyi and J. Serreau, Nucl. Phys. B (2003) 51 [hep-ph/0212404].[8] G. Aarts and J. M. Mart´ınez Resco, Phys. Rev. D (2003) 085009 [hep-ph/0303216].[9] S. Juchem, W. Cassing and C. Greiner, Nucl. Phys. A (2004) 92 [nucl-th/0401046].[10] J. Berges and S. Borsanyi, Phys. Rev. D , 045022(2006) [hep-ph/0512155].[11] G. Aarts and A. Tranberg, Phys. Lett. B (2007)65 [hep-ph/0701205]; Phys. Rev. D (2008) 123521[0712.1120 [hep-ph]].[12] A. Rajantie and A. Tranberg, JHEP (2006) 020[hep-ph/0607292].[13] J. Berges, A. Rothkopf and J. Schmidt, Phys. Rev. Lett. (2008) 041603 [0803.0131 [hep-ph]].[14] A. Tranberg, arXiv:0806.3158 [hep-ph].[15] T. Gasenzer, J. Berges, M. G. Schmidt and M. Seco,Phys. Rev. A (2005) 063604 [cond-mat/0507480].[16] H. van Hees and J. Knoll, Phys. Rev. D , 025010 (2002)[hep-ph/0107200]; ibid. (2003) 160 [hep-ph/0301201]; Nucl. Phys. A (2004)149 [hep-ph/0312085].[18] J. Berges, S. Borsanyi, U. Reinosa and J. Serreau, Phys.Rev. D (2005) 105004 [hep-ph/0409123]; Annals Phys. (2005) 344 [hep-ph/0503240].[19] S. Borsanyi and U. Reinosa, arXiv:0809.0496 [hep-th].[20] A. Arrizabalaga and J. Smit, Phys. Rev. D (2002)065014 [hep-ph/0207044].[21] U. Reinosa and J. Serreau, JHEP (2006) 028[hep-th/0605023]; JHEP (2007) 097 [0708.0971[hep-th]].[22] S. Borsanyi and U. Reinosa, Phys. Lett. B (2008) 88[0709.2316 [hep-ph]].[23] G. Aarts and A. Tranberg, Phys. Rev. D (2006)025004 [hep-th/0604156].[24] A. Arrizabalaga, J. Smit and A. Tranberg, Phys. Rev. D (2005) 025014 [hep-ph/0503287].[25] G. Aarts, G. F. Bonini and C. Wetterich, Phys. Rev. D , 025012 (2001) [hep-ph/0007357].[26] G. Aarts and J. Berges, Phys. Rev. Lett. , 041603(2002) [hep-ph/0107129].[27] J. M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev.D (1974) 2428.[28] G. Aarts and J. Smit, Nucl. Phys. B (1998) 451[hep-ph/9707342].[29] F. Cooper, A. Khare and H. Rose, Phys. Lett. B (2001) 463 [hep-ph/0106113].[30] Mean field (leading order 1 /N/N