Anomalous scaling of passive scalar fields advected by the Navier-Stokes velocity ensemble: Effects of strong compressibility and large-scale anisotropy
AAnomalous scaling of passive scalar fieldsadvected by the Navier-Stokes velocity ensemble:Effects of strong compressibility and large-scale anisotropy
N. V. Antonov and M. M. Kostenko ∗ Chair of High Energy Physics and Elementary ParticlesDepartment of Theoretical Physics, Faculty of PhysicsSaint Petersburg State University, Ulyanovskaja 1Saint Petersburg–Petrodvorez, 198904 Russia
The field theoretic renormalization group and the operator product expansion are applied to twomodels of passive scalar quantities (the density and the tracer fields) advected by a random turbulentvelocity field. The latter is governed by the Navier–Stokes equation for compressible fluid, subject toexternal random force with the covariance ∝ δ ( t − t (cid:48) ) k − d − y , where d is the dimension of space and y is an arbitrary exponent. The original stochastic problems are reformulated as multiplicativelyrenormalizable field theoretic models; the corresponding renormalization group equations possessinfrared attractive fixed points. It is shown that various correlation functions of the scalar field,its powers and gradients, demonstrate anomalous scaling behavior in the inertial-convective rangealready for small values of y . The corresponding anomalous exponents, identified with scaling(critical) dimensions of certain composite fields (“operators” in the quantum-field terminology), canbe systematically calculated as series in y . The practical calculation is performed in the leadingone-loop approximation, including exponents in anisotropic contributions. It should be emphasizedthat, in contrast to Gaussian ensembles with finite correlation time, the model and the perturbationtheory presented here are manifestly Galilean covariant. The validity of the one-loop approximationand comparison with Gaussian models are briefly discussed. PACS numbers: 47.27.eb, 47.27.ef, 05.10.CcKeywords: fully developed turbulence, passive advection, anomalous scaling, renormalization group, operatorproduct expansion, composite fields, compressibility, anisotropy
I. INTRODUCTION
In a few past decades, intermittent interest has beenattracted to the problem of intermittency and anomalousscaling in fluid turbulence; see e.g.
Refs. [1]–[9] and theliterature cited therein. The phenomenon manifests it-self in singular (arguably power-like) behavior of variousstatistical quantities as functions of the integral turbu-lence scales, with infinite sets of independent anomalousexponents [1]. In spite of considerable success, the prob-lem remains essentially open: no regular calculationalscheme, based on an underlying dynamical model andreliable perturbation expansion (like the famous ε ex-pansion for critical exponents) was ever constructed forthe anomalous exponents of the turbulent velocity field.Both the natural experiments and numerical simu-lations suggest that deviations from the classical Kol-mogorov theory are even more strongly pronounced forpassively advected scalar fields (like the temperature orthe density of a pollutant) than for the velocity field itself[2]–[6]. At the same time, various simplified models, de-scribing passive advection by “synthetic” velocity fieldswith given statistics, appear easier tractable theoreticallyand allow analytical results to be derived [7]. Therefore,the problem of passive advection, being of practical im-portance in itself, may also be viewed as a starting point ∗ [email protected], [email protected] in studying intermittency and anomalous scaling in fluidturbulence on the whole.The most remarkable progress was achieved for theKraichnan’s rapid-change model [8], where the advectingvelocity field is taken Gaussian, not correlated in time,and having a power-like correlation function of the form ∼ δ ( t − t (cid:48) ) /k d + ξ , where d is the dimension of space, k isthe wave number and ξ is an arbitrary exponent. There,for the first time, the existence of anomalous scaling wasfirmly established on the basis of a microscopic model [8];the corresponding anomalous exponents were calculatedin controlled approximations [9] and, eventually, within asystematic perturbation expansion in a formal small pa-rameter ξ [10]. Detailed review of the theoretical researchon the passive scalar problem and the bibliography canbe found in Ref. [7].In the original Kraichnan’s model, the velocity ensem-ble was taken Gaussian, decorrelated in time, isotropic,and the fluid was implied to be incompressible. Morerealistic models should take into account finite correla-tion time and non-Gaussianity of the velocity ensemble,anisotropy of the experimental set-up, compressibility ofthe fluid, etc. ; see the discussion in [2, 3]. Here, the twokey issue arise: formulation of more realistic models andthe possibility to treat them (more or less) analytically.A most efficient way to study anomalous scaling is pro-vided by the field theoretic renormalization group (RG)combined with the operator product expansion (OPE);see the books [11, 12] for the detailed exposition of these a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t techniques and the references. In the RG+OPE sce-nario for the anomalous scaling in turbulence, proposedin [13], the singular dependence on the integral scalesemerges as a consequence of the existence in the cor-responding models of composite fields (“composite op-erators” in the quantum-field terminology) with negative scaling dimensions; termed as “dangerous operators;” formore detailed explanations and the references, see [12–15]. For Kraichnan’s model, the anomalous exponentscan be identified with the scaling dimensions (“criticaldimensions” in the terminology of the theory of criticalstate) of certain individual Galilean-invariant compositeoperators [10]. This allows one to give a self-consistentderivation of the anomalous scaling, to construct a sys-tematic perturbation expansion for the anomalous expo-nents in ξ , and to calculate the exponents in the second[10] and in the third [16] orders. The RG approach can begeneralized to the case of finite correlation time [17] andto the non-Gaussian advecting velocity field, governedby the stochastic Navier–Stokes equation [18]. A generaloverview of the RG approach to Kraichnan’s model andits descendants and more references can be found in [19].Numerous studies were devoted to the effects of com-pressibility on the intermittency and anomalous scaling[20]–[30]. Analysis of simplified models suggests thatcompressibility strongly affects the passively advectedfields. In particular, in contrast to the incompressiblecase, the diffusion can be depleted by the advection of apurely potential flow [22] and the phase transition froma turbulent to a certain purely chaotic state takes placewhen the degree of compressibility increases [25]. Itwas also shown that the anomalous exponents becomenon-universal due to dependence on the compressibil-ity parameter, such that the anomalous scaling is en-hanced, while the hierarchy of anisotropic contributionsis suppressed [26]–[30]. For passive vector ( e.g. mag-netic) fields, the issues of anomalous scaling, hierarchyof anisotropic contributions and the dependence on com-pressibility were discussed e.g. in [31]–[38].An important advantage of Kraichnan’s model is thepossibility to easily model compressibility [20]–[26]. Gen-eralization to the case of a Gaussian ensemble with finitecorrelation time is also possible [29, 30, 36]. However,synthetic models with non-vanishing correlation time suf-fer from the lack of Galilean symmetry, which may leadto “interesting pathologies” (quoting Ref. [3]). In the RGapproach, one of such a pathology manifests itself as anultraviolet (UV) divergence in the vertex [29], which inmore realistic models is forbidden by Galilean invariance,and for the incompressible Gaussian model is absent be-cause of rather technical reasons [17]. Thus it is desirableto describe the advecting velocity field by the correspond-ing Navier–Stokes equations [39] with a random stirringforce. However, this appeared to be a difficult task.In Refs. [40, 41], the leading-order correction in theMach number Ma to the incompressible scaling regimewas studied; generalization to all orders of the expan-sion in Ma was derived in [42]. The corrections are small for very small Ma and not very small momenta k , butbecome arbitrarily large (IR relevant in the sense of Wil-son) and destroy the incompressible scaling regime if Mais fixed and the momenta become small enough. Thus theoriginal incompressible regime becomes unstable, and acrossover to another unknown regime occurs. The case ofstrong compressibility was studied in Refs. [43]–[45]. Theresults are rather controversial, but all of those studiessupport the existence of a stationary resulting “compress-ible” regime, different from the original incompressibleone.In the present paper, we adopt the approach ofRef. [45], where the standard field theoretic RG was ap-plied to the problem of stirred hydrodynamics of a com-pressible fluid, and the resulting stationary scaling regimewas associated with the IR attractive fixed point of thecorresponding multiplicatively renormalizable field the-oretic model. That approach was later applied to theproblem of mass distribution in the self-gravitating mat-ter within the framework of a continuous stochastic for-mulation of the Vlasov–Poisson model [46]. The problemof anomalous scaling of the velocity field in that modelremains open, as for its incompressible predecessors, butthe passive scalar advection by such an ensemble can betreated analytically. This is the aim of the present work.The plan of the paper is as follows.In section II we revisit the RG approach to the stochas-tic Navier–Stokes equation for a compressible fluid, fol-lowing mostly Ref. [45], and introduce the basic notions(field theoretic formulation, canonical dimensions, renor-malizability and RG equations), needed for the furtheranalysis of the passive advection. The RG equations pos-sess an IR attractive fixed point, which implies existenceof a scaling regime in the inertial and energy-containingranges. The one-loop explicit expressions for the renor-malization constants and the RG functions (anomalousdimensions and β functions), calculated in [45], are pre-sented. The corresponding scaling dimensions of the fre-quency and the velocity are known exactly and coincidewith their analogs for the incompressible case. Anothernontrivial fixed point is unstable (it is a saddle point)and corresponds to the incompressible fluid.In section III we introduce the diffusion-advectionstochastic equations for the two types of passive scalarfield: the tracer (temperature, entropy or concentrationof a pollutant) and the density of a conserved quantity( e.g. density of a pollutant). We present the field theo-retic formulation of these models and show that they aremultiplicatively renormalizable. Then the RG equationscan be derived in a standard fashion. The renormal-ization constants and the RG functions are calculated inthe leading (one-loop) approximation, which is consistentwith the accuracy of the results derived in [45]. The full-scale models, involving the velocity field and the scalarfield, possess an IR attractive fixed point. Thus the ex-istence of a scaling regime in the IR range is established.Exact expressions for the scaling dimensions of the scalarfields are obtained.In section IV we calculate, in the leading order of theexpansion in y (one-loop approximation), critical dimen-sions of the composite operators built of the scalar fieldand its spatial derivatives, including some tensor opera-tors. In the next section, those dimensions are identifiedwith various anomalous exponents.In section V we apply the OPE to the analysis of theinertial-range behavior of various correlation functions:the correlation functions of the scalar fields and theirpowers for the density case and of the structure functionsfor the tracer case. We show that, for the density case,leading terms of the inertial-range behavior are deter-mined by the contributions of the operators built solelyof the scalar fields. Their critical dimensions are nega-tive, which leads to strong dependence on the integralscale and to the anomalous scaling, with the anomalousexponents identified with those dimensions.For the tracer case, more interesting quantities are thestructure functions that involve differences of the valuesof the scalar field at different points. Their anomalous be-haviour is determined by the scalar operators built of thegradients of the scalar field, whose negative dimensionsare identified with the corresponding anomalous expo-nents.In the presence of anisotropy, introduced into the sys-tem at large scales, contributions of the tensor operatorsin the OPE’s come into play: l th rank tensor operatorsdetermine the contribution in the correlation functionswith nontrivial angular dependence described by the l thorder spherical harmonics. Like for the Kraichnan model,those anisotropic contributions organize a kind of hierar-chy, related to the degree of anisotropy: they becomeless important as l grows, so that the leading term ofthe inertial-range asymptotic behavior is given by theisotropic contribution ( l = 0) in agreement with Kol-mogorov’s hypothesis of the local isotropy restoration.This issue is discussed for the pair correlation functionin the both models and for the structure functions of ar-bitrary order for the tracer.Section VI is reserved for the discussion, comparisonwith the Gaussian models and the conclusion. II. RG ANALYSIS OF THE STOCHASTIC NSEQUATION WITH STRONG COMPRESSIBILITYA. Description of the model
The Navier–Stokes equation for a viscid compressiblefluid has the form [39] ρ ∇ t v i = ν [ δ ik ∂ − ∂ i ∂ k ] v k + µ ∂ i ∂ k v k − ∂ i p + η i , (2.1)where ∇ t = ∂ t + v k ∂ k (2.2)is the Lagrangian (Galilean covariant) derivative, ∂ t = ∂/∂t , ∂ i = ∂/∂x i , and ∂ = ∂ i ∂ i is the Laplace operator. Equation (2.1) is obtained by combining the momen-tum balance equation ∂ t ( ρv i ) + ∂ k Π ik = η i , (2.3)whereΠ ik = ρv i v k + δ ik p − ν ( ∂ i v k + ∂ k v i ) − δ ik ( µ − ν ) ∂ l v l (2.4)is the stress tensor, with the continuity equation ∂ t ρ + ∂ i ( ρv i ) = 0 . (2.5)In those equations, v i is the velocity, ρ is the massdensity, p is the pressure, and η i is the density of theexternal force (per unit volume). All these quantities de-pend on x = { t, x } with x = { x i } , i = 1 . . . d , where d is an arbitrary (for generality) dimensionality of space.The constants ν and µ are two independent molecu-lar viscosity coefficients; in the viscous terms in (2.1) weexplicitly separated the transverse and the longitudinalparts. Summations over repeated vector indices are al-ways implied.The problem (2.1), (2.5) should be augmented by theequation of state, p = p ( ρ ). It will be taken in the sim-plest form of the linear relation( p − ¯ p ) = c ( ρ − ¯ ρ ) (2.6)between the deviations of the pressure and the densityfrom their mean values. The constant c has the meaningof the (adiabatic) speed of sound.Following [45], we divide equation (2.1) with ρ and inthe viscous terms replace ρ with its mean value. Thisapproximation (which is needed to obtain a renormaliz-able local field theoretic model) is implicitly justified bythe analysis of Ref. [42]; we also note that the viscosityplays a little role in the inertial range. We retain the samenotation ν and µ for the resulting constant kinematicviscosity coefficients. Then the equations (2.1), (2.5) canbe rewritten in the form ∇ t v i = ν [ δ ik ∂ − ∂ i ∂ k ] v k + µ ∂ i ∂ k v k − ∂ i φ + f i , (2.7) ∇ t φ = − c ∂ i v i , (2.8)where we have introduced the new scalar field φ = c ln( ρ/ ¯ ρ ) (2.9)and f i = f i ( x ) is the density of the external force (perunit mass).In the stochastic formulation of the problem, the exter-nal force becomes a random field that models the energyinput into the system from the large-scale stirring. Thedetails of its statistics are believed to be unessential, soit is taken to be Gaussian with zero mean, white in-time(this is required by the Galilean symmetry), and involv-ing some typical IR scale L (the integral scale). On theother hand, for the use of the standard RG technique it isimportant that its correlation function have a power-lawtail at large wave numbers. More detailed discussion canbe found in [14, 15, 48]. In the present case one chosesthe correlation function in the form [45] (cid:104) f i ( x ) f j ( x (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) (cid:90) k>m d k (2 π ) d D fij ( k ) exp { i kx } , (2.10)where D fij ( k ) = D k − d − y (cid:110) P ⊥ ij ( k ) + αP (cid:107) ij ( k ) (cid:111) . (2.11)Here P ⊥ ij ( k ) = δ ij − k i k j /k and P (cid:107) ij ( k ) = k i k j /k are thetransverse and the longitudinal projectors, respectively, k = | k | is the wave number, D and α are positive ampli-tudes. It is convenient to write D = g ν : the parameter g plays the role of the coupling constant (formal expan-sion parameter in the ordinary perturbation theory). Therelation g ∼ Λ y sets in the typical UV momentum scale(reciprocal of the dissipation length scale). The parame-ter m = L − provides IR regularization; its precise formis unessential and the sharp cut-off is the simplest choice for calculational reasons. The exponent 0 < y ≤ ε = 4 − d in theRG theory of critical behavior [11, 12]: it provides UVregularization (so that the UV divergences have the formof the poles in y ) and the coordinates of fixed points andvarious scaling dimensions are calculated as series in y .The most realistic (physical) value is given by the limit y →
4, when the functions in (2.11) can be viewed (withthe proper choice of the amplitude) as power-like mod-els of the function δ ( k ): it corresponds to the idealizedpicture of the energy input from infinitely large scales. B. Field theoretic formulation and Feynman rules
According to the general theorem [11, 12], the stochas-tic problem (2.7), (2.8), (2.10), (2.11), is equivalent tothe field theoretic model of the doubled set of fieldsΦ = { v (cid:48) i , φ (cid:48) , v i , φ } and the action functional S (Φ) = 12 v (cid:48) i D fik v (cid:48) k + v (cid:48) i (cid:8) −∇ t v i + ν [ δ ik ∂ − ∂ i ∂ k ] v k + u ν ∂ i ∂ k v k − ∂ i φ (cid:9) ++ φ (cid:48) (cid:2) −∇ t φ + v ν ∂ φ − c ( ∂ i v i ) (cid:3) , (2.12)where D f is the correlation function (2.10), (2.11), andall the needed summations over the vector indices andintegrations over x = { t, x } are implied, for example, v (cid:48) i ∇ t v i = (cid:90) dt (cid:90) d x v (cid:48) i ( x )[ ∂ t + v k ( x ) ∂ k ] v i ( x ) . (2.13)In (2.12) we passed to the new dimensionless parameter u = µ /ν > v ν φ (cid:48) ∂ φ with another positive dimensionless parameter v . Thisterm is not forbidden by the symmetry or dimensionalityconsiderations, so it will necessarily appear in the renor-malization procedure. From the physics viewpoints, itcorresponds to some redefinition of the relation betweenthe velocity and momentum [39]. From a more techni-cal point of view, it is needed to insure multiplicativerenormalizability of the model (2.12), which allows oneto easily derive the RG equations. One can insist onstudying the original model (2.7), (2.8) without such aterm. Then the RG equations must be solved with theinitial condition v = 0. In renormalized variables, thiscorresponds to a general situation with a nonzero valueof the corresponding renormalized parameter v . Sincethe IR attractive fixed point is unique (see below), thespecific initial condition is unessential.The field theoretic formulation means that variouscorrelation functions and response (Green) functions ofthe original stochastic problem are represented by func-tional averages over the full set of fields with weightexp S (Φ), and in this sense they can be viewed as the Green functions of the field theoretic model with action(2.12). The model corresponds to standard Feynman di-agrammatic techniques with two vertices − v (cid:48) ( v∂ ) v and − φ (cid:48) ( v∂ ) φ and the free (bare) propagators, determinedby the quadratic part of the action functional; in thefrequency–momentum ( ω – k ) representation, they havethe forms: (cid:104) vv (cid:48) (cid:105) = (cid:104) v (cid:48) v (cid:105) ∗ = P ⊥ (cid:15) − + P (cid:107) (cid:15) R − , (cid:104) vv (cid:105) = P ⊥ d f | (cid:15) | + P (cid:107) αd f (cid:12)(cid:12)(cid:12) (cid:15) R (cid:12)(cid:12)(cid:12) , (cid:104) φv (cid:48) (cid:105) = (cid:104) v (cid:48) φ (cid:105) ∗ = − i c k R , (cid:104) vφ (cid:48) (cid:105) = (cid:104) φ (cid:48) v (cid:105) ∗ = i k R , (cid:104) φφ (cid:48) (cid:105) = (cid:104) φ (cid:48) φ (cid:105) ∗ = (cid:15) R , (cid:104) φφ (cid:105) = αc k d f | R | , (cid:104) vφ (cid:105) = (cid:104) φv (cid:105) ∗ = i αc d f (cid:15) k | R | , (cid:104) φ (cid:48) φ (cid:48) (cid:105) = (cid:104) v (cid:48) φ (cid:48) (cid:105) = (cid:104) v (cid:48) v (cid:48) (cid:105) = 0 , (2.14)where we have denoted (cid:15) = − i ω + ν k , (cid:15) = − i ω + u ν k ,(cid:15) = − i ω + v ν k , R = (cid:15) (cid:15) + c k ,d f = g ν k − d − y (2.15)and omitted the vector indices of the fields and the pro-jectors.In the limit c → ∞ , the propagators (cid:104) vv (cid:48) (cid:105) and (cid:104) vv (cid:105) become purely transverse, while the mixed propagator (cid:104) vφ (cid:105) vanishes. Then the scalar field φ decouples from v, v (cid:48) (it does not enter the vertex in (2.7)), and we arriveat the well-known Feynman rules for the incompressiblefluid [12, 14, 15]. C. UV divergences, renormalization, andmultiplicative renormalizability
It is well known that the analysis of UV divergencesis based on the analysis of canonical dimensions; see e.g. [11, 12]. Dynamical models like (2.12) have two in-dependent scales: the time scale T and the length scale L . Thus the canonical dimension of any quantity F (afield or a parameter) is described by two numbers, thefrequency dimension d ωF and the momentum dimension d kF , defined such that [ F ] ∼ [ T ] − d ωF [ L ] − d kF . The obviousconsequences of the definition are the relations d kk = − d k x = 1 , d ωk = d ω x = 0 ,d kω = d kt = 0 , d ωω = − d ωt = 1 . (2.16)The other dimensions are found from the requirementthat each term of the action functional be dimension-less (with respect to the momentum and the frequencydimensions separately). Then one introduces the totalcanonical dimension d F = d kF + 2 d ωF , (2.17)which plays in the theory of renormalization of dynam-ical models the same part as the conventional canonicaldimension does in static problems. The canonical di-mensions for the model (2.12) are given in table 1, in-cluding renormalized parameters (without the subscript“o”), which will appear a bit later.The choice (2.17) for the total canonical dimension de-serves a more careful explanation. It means that all theviscosity or diffusion coefficients in the model are pro-nounced dimensionless (with respect to the new totaldimension), and the time and the space variables aremeasured in the same units; cf. [11, 12]. Experiencedreader recalls the c = 1 system of units in relativisticphysics, where all the distances are measured in the timeunits (light years). Here, we relate the dimensions by eq.(2.17) because the dispersion law for diffusion modes is ω ∼ k . However, our model involves another dispersionlaw, ω ∼ k , related to the sound modes. If we decide toset the speed of sound c dimensionless, we would haveto set d F = d kF + d ωF .A similar alternative exists in the so-called model H ofequilibrium dynamical critical behavior, where the mo-tion of the fluid is taken into account and several disper-sion laws are simultaneously present; see e.g. p. 552 inthe monograph [12]. The choice (2.17) means that we areinterested in the asymptotic behavior of the Green func-tions where ω ∼ k →
0; the RG treatment will modifyit to the Kolmogorov law ω ∼ k / → d F = d kF + d ωF would mean that we were interested in the asymptoticbehavior of the (same) Green functions for ω ∼ k → g ∼ Λ y becomes dimen-sionless) at y = 0, so that the UV divergences have theform of poles in y in the Green functions. The totalcanonical dimension of any 1-irreducible Green functionΓ (the formal index of UV divergence) is δ Γ = d + 2 − (cid:88) Φ N Φ d Φ , (2.18)where N Φ are the numbers of the fields entering into thefunction Γ, d Φ are their total canonical dimensions, andthe summation over all types of the fields Φ is implied.Superficial UV divergences, whose removal requires coun-terterms, can be present only in the functions Γ with anon-negative integer δ Γ . The counterterm is a polyno-mial in frequencies and momenta of degree δ Γ , with theconvention that ω ∼ k .For the model (2.12), dimensional analysis should beaugmented by the following additional considerations[45]:(i) All the 1-irreducible Green functions without theresponse fields ( N v (cid:48) = N φ (cid:48) = 0) involve closed circuitsof retarded propagators, vanish identically, and thereforerequire no counterterms [12].(ii) If for some reason a number of external momentaoccurs as an overall factor in all the diagrams of a givenGreen function, the real index of divergence δ (cid:48) Γ is smallerthan δ Γ by the corresponding number of unities [12, 15].In the model (2.12) the field φ enters the vertex φ (cid:48) ( v∂ ) φ only in the form of spatial derivative, which reduces thereal index of divergence: δ (cid:48) Γ = δ Γ − N φ . (2.19)The field φ enters the counterterms only in the form ofthe derivative ∂φ . In particular, for the 1-irreduciblefunction (cid:104) φ (cid:48) φ (cid:105) − ir one obtains δ Γ = 2, δ (cid:48) Γ = 0. Thus thecounterterm φ (cid:48) ∂ t φ , allowed by dimensional analysis, is infact forbidden, and the only possible structure is φ (cid:48) ∂ φ .(iii) Galilean invariance of the model (2.12) requiresthat the contributions of the counterterms be also invari-ant. In particular, this means that the covariant deriva-tive (2.2) enters the counterterms as a whole. As a con-sequence, the counterterm required for the 1-irreduciblefunction (cid:104) φ (cid:48) vφ (cid:105) − ir with δ Γ = 1, δ (cid:48) Γ = 0, necessarily hasthe form φ (cid:48) ( v∂ ) φ and appears in the combination φ (cid:48) ∇ t φ with the counterterm φ (cid:48) ∂ t φ discussed above. Hence, it isalso forbidden.Similarly, the divergences in the functions (cid:104) v (cid:48) v (cid:105) − ir with δ Γ = 2 and (cid:104) v (cid:48) vv (cid:105) − ir with δ Γ = 1 can be elimi- TABLE I. Canonical dimensions of the fields and parameters in the models (2.12), (3.4), (3.5), (3.8).
F v (cid:48) v φ (cid:48) φ θ (cid:48) θ m , µ , Λ ν , ν c , c g u , v w , u , v , w , g , αd kF d + 1 − d + 2 − d − − y d ωF − − / − / d F d − d − d + 1 − y nated by the two counterterms: v (cid:48) ∂ v and the combina-tion v (cid:48) ∇ t v . In fact, the latter is also forbidden by the generalized Galilean invariance with the time-dependenttransformation velocity parameter w ( t ) [47, 50]: v w ( x ) = v ( x w ) − w ( t ) , Φ w ( x ) = Φ( x w ) ,x = { t, x } , x w = { t, x + u ( t ) } , u ( t ) = (cid:90) t w ( t (cid:48) ) dt (cid:48) . (2.20)Here Φ denotes the three fields v (cid:48) , φ (cid:48) , φ . The action func-tional is not invariant with respect to such a transfor-mation: S (Φ w ) = S (Φ) + v (cid:48) ∂ t w . One can show, how-ever, that the generating functional of the 1-irreducibleGreen functions transforms in the identical way, Γ(Φ w ) =Γ(Φ) + v (cid:48) ∂ t w . Since in general Γ(Φ) = S (Φ) plus the dia-grams with loops (which contain all the UV divergences),the counterterms appear invariant under (2.20). This ex-cludes the counterterm v (cid:48) ∇ t v , invariant with respect toconventional Galilean transformation with a constant w ,but not invariant with respect to (2.20). More detaileddiscussion of the uses of the generalized Galilean trans-formation, especially for composite fields, can be foundin [12, 15, 50].(iv) Expressions (2.14) show that the propagators (cid:104) v (cid:48) φ (cid:105) and (cid:104) vφ (cid:105) contain the factor c , while (cid:104) v (cid:48) φ (cid:105) con- tains c . These factors appear as external numerical fac-tors in any diagram involving these propagators, and itsreal index of divergence reduces by the correspondingnumber of unities. In particular, any diagram of the 1-irreducible function with N φ (cid:48) > N φ contains the factor c N φ (cid:48) − N φ )0 . It then follows that the counterterm to the1-irreducible function (cid:104) φ (cid:48) v (cid:105) − ir with δ Γ = 3 necessarilyreduces to c φ (cid:48) ( ∂v ), while the structures φ (cid:48) ∂ ( ∂v ) etc. are forbidden. Another consequence is finiteness of thefunction (cid:104) φ (cid:48) vv (cid:105) − ir with δ Γ = 2. Each diagram of thisfunction contains the factor c , which forbids the coun-terterms like φ (cid:48) ( ∂v )( ∂v ) etc. , while the remaining struc-ture c φ (cid:48) v is forbidden by the Galilean symmetry.Using all these considerations one can check that allthe UV divergences in the model (2.12) are removed bythe counterterms of the form v (cid:48) i ∂ v i , v (cid:48) i ∂ i ∂ k v k , v (cid:48) i ∂ i φ, c φ (cid:48) ∂ i v i , φ (cid:48) ∂ φ. (2.21)All these structures are present in the extended actionfunctional (2.12) with v >
0, so the model is multiplica-tively renormalizable.Like for the incompressible case [49], for d = 2 a newUV divergence arises in the function (cid:104) v (cid:48) v (cid:48) (cid:105) − ir , and anew counterterm v (cid:48) ∂ v (cid:48) should be included. This caserequires special treatment, and in the following we as-sume d >
2. Then the renormalized action functionalhas the form S R (Φ) = 12 v (cid:48) i D fik v (cid:48) k + v (cid:48) i (cid:8) −∇ t v i + Z ν [ δ ik ∂ − ∂ i ∂ k ] v k + Z uν∂ i ∂ k v k − Z ∂ i φ (cid:9) ++ φ (cid:48) (cid:2) −∇ t φ + Z vν∂ φ − Z c ( ∂ i v i ) (cid:3) . (2.22)Here g, ν, u, v, c are renormalized counterparts of theoriginal (bare) parameters (with the subscript “o”), thefunction D f is expressed in renormalized parameters us-ing the relation g ν = gµ y ν , the reference scale (or the“normalization mass”) µ is an additional free parameterof the renormalized theory; the renormalization constants Z i depend only on the completely dimensionless parame-ters g, u, v, α, d, y . The renormalized action (2.22) is ob-tained from the original one (2.12) by the renormalizationof the fields φ → Z φ φ , φ (cid:48) → Z φ (cid:48) φ (cid:48) and the parameters g = gµ y Z g , ν = νZ ν , u = uZ u ,v = vZ v , c = cZ c . (2.23) The renormalization constants in (2.22) and (2.23) arerelated as Z ν = Z , Z u = Z Z − ,Z v = Z Z − , Z φ = Z − φ (cid:48) = Z ,Z c = ( Z Z ) / , Z g = Z − ν . (2.24)The last relation follows from the absence of renormaliza-tion of the non-local term of the random force in (2.22);for the same reason the parameters m, α from the correla-tion function (2.10) are not renormalized: Z m = Z α = 1.No renormalization of the fields v, v (cid:48) is needed: Z v = Z v (cid:48) = 1 due to the absence of renormalization of theterm v (cid:48) ∇ t v .The renormalization constants are found from the re-quirement that the Green functions of the renormalizedmodel (2.22), when expressed in renormalized variables,be UV finite (in our case, be finite at y → Z = 1+ only polesin y .” The calculation in the first order in g (one-loopapproximation) gives [45] Z = 1 + ˆ gy A, Z = 1 + ˆ guy B, Z = 1 + ˆ gy ( d − dv ( v + 1) − α ˆ gy ( u − v )2 duv ( u + v ) ,Z = 1 + ˆ gy ( d − d ( u + 1)( v + 1) , Z = 1 , (2.25)with corrections of order ˆ g and higher. Here we passedto the new coupling constantˆ g = gS d / (2 π ) d , (2.26)where S d = 2 π d/ / Γ( d/
2) (2.27) is the surface area of the unit sphere in d -dimensionalspace and Γ( · · · ) is Euler’s Gamma function, and denoted A = d ( d − u − d + d − u − d ( d + 3)4 d ( d + 2)(1 + u ) + α (1 − u )2 du (1 + u ) , B = (1 − d ) ( d − u + ( d + 4) u + 12 d ( d + 2)(1 + u ) . (2.28)One important technical remark follows. The renor-malization constants in the MS scheme do not dependon the dimensional parameter c . On the other hand,all the propagators (2.14), and hence all the Feynmandiagrams, have a well-defined limit for c →
0. Thus inthe calculation of the constants Z – Z one can simplyset c = 0 in (2.14) and (2.15). Then the propagators (cid:104) φv (cid:48) (cid:105) , (cid:104) vφ (cid:105) , (cid:104) φφ (cid:105) vanish, while the form of the othersdrastically simplifies. In the calculation of the constant Z in front of the term c φ (cid:48) ( ∂v ) it is sufficient to takeinto account the diagrams with one and only one propa-gator (cid:104) φv (cid:48) (cid:105) or (cid:104) vφ (cid:105) . Then the needed c appears as anexternal factor, and in the remaining expression one canset c = 0.To avoid possible misunderstanding, we stress that weare interested in the model with finite and arbitrary c ,and that more involved calculation with the full-scalepropagators (2.14) would give the same results (2.25),(2.28) for the renormalization constants. In this respect,the parameter c is similar to τ ∝ T − T c , deviation of thetemperature from its critical value, in models of criticalbehavior: in the MS scheme, the renormalization con-stants do not depend on it and can be calculated directlyat the critical point τ = 0.The simple expression Z = 1 results from the can-cellation of nontrivial contributions from three Feynmandiagrams; we see no reason to expect that it is valid toall orders in g . D. RG equations and RG functions
Let us recall a simple derivation of the RG equations;detailed discussion can be found in [11, 12]. The RGequations are written for the renormalized correlationfunctions G R = (cid:104) Φ · · · Φ (cid:105) R , which differ from the orig-inal (unrenormalized) ones G = (cid:104) Φ · · · Φ (cid:105) only by nor-malization and choice of the parameters, and thus canbe equally used for the analysis of the critical behavior.The relation S R (Φ , e, µ ) = S ( Z Φ Φ , e ) between the func-tionals (2.12) and (2.22) results in the relations G ( e , . . . ) = Z N φ φ Z N φ (cid:48) φ (cid:48) G R ( e, µ, . . . ) (2.29)between the correlation functions. Here, as usual, N φ and N φ (cid:48) are the numbers of corresponding fields enter-ing into G (we recall that in our model Z v = Z v (cid:48) = 1); e = { ν , g , u , v } is the full set of bare parametersand e = { ν, g, u, v } are their renormalized counterparts;the ellipsis stands for the other arguments (times, coor-dinates, momenta etc. ).We use (cid:101) D µ to denote the differential operation µ∂ µ forfixed e and operate on both sides of the equation (2.29)with it. This gives the basic RG differential equation: {D RG + N φ γ φ + N φ (cid:48) γ φ (cid:48) } G R ( e, µ, . . . ) = 0 , (2.30)where D RG is the operation (cid:101) D µ expressed in the renor-malized variables: D RG = D µ + β g ∂ g + β u ∂ u + β v ∂ v − γ ν D ν − γ c D c . (2.31)Here we have written D x ≡ x∂ x for any variable x . Theanomalous dimension γ F of a certain quantity F (a fieldor a parameter) is defined as γ F = Z − F (cid:101) D µ Z F = (cid:101) D µ ln Z F , (2.32)and the β functions for the three dimensionless couplingconstants g , u and v are β g = (cid:101) D µ g = g [ − y − γ g ] ,β u = (cid:101) D µ u = − uγ u ,β v = (cid:101) D µ v = − vγ v , (2.33)where the second equalities result from the definitionsand the relations (2.29).From the relations (2.24) we obtain β g = g [ − y + 3 γ ] ,β u = u [ γ − γ ] ,β v = v [ γ − γ ] , (2.34)and for the anomalous dimensions we have γ φ = − γ φ (cid:48) = γ , γ c = ( γ + γ ) / , γ ν = γ ,γ v = γ v (cid:48) = γ α = γ m = 0 . (2.35)The relations in the second line follow from the absenceof renormalization of the corresponding fields and param-eters; see the remarks below equation (2.24).In the MS scheme all the renormalization constantshave the form Z F = 1 + ∞ (cid:88) n =1 z ( n ) y − n , (2.36)where the coefficients z ( n ) do not depend on y . Thenfrom the definition and the expressions (2.33) it followsthat the corresponding anomalous dimension is deter-mined solely by the first-order coefficient: γ F = −D g z (1) , (2.37)see e.g. the discussion [11, 12]. Then in the one-loopapproximation from the explicit expressions (2.25) onefinds: γ = − A ˆ g, γ = − B ˆ g/u,γ = ˆ g ( d − dv ( v + 1) + α ˆ g ( u − v )2 duv ( u + v ) ,γ = ˆ g (1 − d )2 d ( u + 1)( v + 1) , γ = 0 (2.38)with A and B from (2.28) and the corrections of order ˆ g and higher. E. The IR attractive fixed point
It is well known that possible IR asymptotic regimesof a renormalizable field theoretic model are associatedwith IR attractive fixed points of the corresponding RGequations. The coordinates g ∗ of the fixed points arefound from the equations β i ( g ∗ ) = 0 , (2.39)where g = { g i } is the full set of coupling constants and β i are the corresponding β functions. The type of a fixedpoint is determined by the matrixΩ ij = ∂β i /∂g j | g = g ∗ . (2.40)For the IR stable fixed points the matrix Ω is positive, i.e. , the real parts of all its eigenvalues are positive.In our model, g = { ˆ g, u, w } , and the β functions aregiven be the relations (2.33) and the explicit one-loopexpressions (2.38). We do not include the dimensionlessparameter α into the list of coupling constants, becauseit is not renormalized ( α = α and Z α = 1) and thecorresponding function β α = − αγ α vanishes identically.Thus the equation β α = 0 imposes no restriction on thevalue of α , and it remains a free parameter.Analysis of the expressions (2.33), (2.38) and (2.28)shows that in the physical range of parameters(ˆ g, u, v, α >
0) there is only one IR attractive fixed pointwith the coordinatesˆ g ∗ = 4 dy d − , u ∗ = v ∗ = 1 , (2.41)with possible higher-order corrections in y .Let us briefly explain derivation of (2.41). Any fixedpoint with ˆ g ∗ = 0 cannot be IR attractive, because oneof the eigenvalues of the matrix Ω coincides with thediagonal element ∂ g β g = − y <
0. For ˆ g ∗ (cid:54) = 0 fromthe equation β g = 0 we immediately find the relation γ ∗ = γ ∗ ν = y/
3, valid to all orders in y (here and below γ ∗ F = γ F ( g ∗ ) for any F is the value of the anomalous di-mension at the fixed point in question). Substituting thisrelation into the equation β u = 0 gives the equation for u ∗ with the only positive solution u ∗ = 1. Substitutingit into the equation β g = 0 gives the value of ˆ g ∗ (it is im-portant here that the functions β g and β u in the one-loopapproximation do not depend on v ). Finally, substitutingthe known values of ˆ g ∗ and u ∗ into the relation β v = 0gives the equation for v ∗ with the only positive solution v ∗ = 1. Now it is easy to see that the matrix (2.40) atthe fixed point (2.41) is triangular, so that its eigenval-ues coincide with the diagonal elements and are easilycalculated from the explicit expressions (2.38). They arepositive for all y > α > d > g, u, v > β functions vanish for g = 0 and that the functions β u and β v are negative for u = 0 and v = 0, respectively: β u | u =0 = − ˆ g ( d − d ( d + 2) , β v | v =0 = − ˆ g (cid:26) ( d − d + 1 du (cid:27) . It then follows that the IR asymptotic behavior of theGreen functions in our model can be governed only bythe fixed point (2.41): even if some other fixed pointsexist in the unphysical range, they cannot be reached bythe RG flow.Changing to the new variable f = 1 /u one can find an-other fixed point with f ∗ = 0 and ˆ g ∗ = 4( d +2) y/ d − u → ∞ corresponds to the purelytransverse velocity field, while the scalar field decouples.The point is unstable (it is a saddle point) in agreementwith the analysis of Refs. [40–42] which shows that theleading-order correction in the Mach number to the in-compressible scaling regime destroys its stability (in theRG terminology, it is relevant in the sense of Wilson). F. IR behavior and the critical dimensions
It follows from the solution of the RG equation (2.30)that when an IR fixed point is present, the leading termof the IR asymptotic behavior of the Green function G R satisfies the equation (2.30) with the replacement g → g ∗ for the full set of the couplings; see e.g. the monograph[12]. In our case this gives (cid:40) D µ − γ ∗ ν D ν − γ ∗ c D c + (cid:88) Φ N Φ γ ∗ Φ (cid:41) G R = 0 . (2.42)We recall that D x ≡ x∂ x for any variable x , γ ∗ F is thefixed-point value of the anomalous dimension γ F , andthe summation over all types of the fields Φ is implied.In the one-loop approximation, from (2.38) and (2.41) weobtain γ ∗ ν = y/ , γ ∗ φ = − γ ∗ φ (cid:48) = − y/ O ( y ) ,γ ∗ c = − y/
12 + O ( y ) . (2.43)Canonical scale invariance is expressed by the twoequations (cid:40)(cid:88) F d kF D F − d kG (cid:41) G R = 0 , (cid:40)(cid:88) F d ωF D F − d ωG (cid:41) G R = 0 , (2.44)with the summations over all the arguments of the func- tion G R . From table 1 we obtain (cid:40) −D x + D µ + D m − D ν − D c − (cid:88) Φ N Φ d k Φ (cid:41) G R = 0 , (cid:40) −D t + D ν + D c − (cid:88) Φ N Φ d ω Φ (cid:41) G R = 0 , (2.45)where the dimensions d k,ω Φ of the fields are also given inthe table. Each of the equations (2.42), (2.45) describesthe scaling with dilatation of the variables whose deriva-tives enter the differential operator. One is interested inthe scaling with fixed “IR irrelevant” parameters µ and ν ; see [12, 14, 15]. In order to derive the correspondingscaling equation one has to combine (2.42), (2.45) suchthat the derivatives with respect to these parameters beeliminated; this gives: (cid:40) −D x + ∆ t D t + ∆ c D c + ∆ m D m − (cid:88) Φ N Φ ∆ Φ (cid:41) G R = 0(2.46)with∆ F = d kF + ∆ ω d ωF + γ ∗ F , ∆ ω = − ∆ t = 2 − γ ∗ ν . (2.47)Here ∆ F is the critical dimension of the quantity F (fol-lowing [12, 14, 15] we use this term to distinguish it fromcanonical dimensions), while ∆ t and ∆ ω are the criticaldimensions of the time and the frequency.From table 1 and expressions (2.43) we obtain∆ v = 1 − y/ , ∆ v (cid:48) = d − ∆ v , ∆ ω = 2 − y/ , ∆ m = 1(2.48)(these results are exact due to γ ∗ ν = y/ γ ∗ v,v (cid:48) ,m = 0)and∆ φ = d − ∆ φ (cid:48) = 2 − y/ O ( y ) , ∆ c = 1 − y/ O ( y ) . (2.49)We note that the analogs of the expressions (2.48), (2.49)in Ref. [45] contain a few misprints.Surprisingly enough, all the results (2.41), (2.43),(2.48), (2.49) are independent on α (and some of themdo not depend on d ). They are valid for all α >
0, butthe case α → ∞ (purely potential random force) requiresspecial attention. To study this limit, one should pass tothe new couplings g (cid:48) = gα , b = 1 /α and then set b = 0at finite g (cid:48) . This gives β g (cid:48) = − yg (cid:48) , β u = g (cid:48) ( u − du (1 + u ) , β v = g (cid:48) ( v − u ) du ( u + v ) . (2.50)The system (2.50) has no IR attractive fixed point, be-cause from β g (cid:48) = 0 it necessarily follows that g (cid:48) = 0, andsuch a point cannot be IR attractive due to ∂ g (cid:48) β g (cid:48) = − y <
0. In principle, the needed fixed point with g (cid:48)∗ ∼ y / can appear on the two-loop level, if the term oforder ( g (cid:48) ) appears in β g (cid:48) . Then the results (2.48) remainvalid, while (2.49) should be revised.0 III. PASSIVE SCALAR FIELDS:RENORMALIZATION, RG FUNCTIONS ANDFIXED POINTA. The models and their field theoretic formulation
There are two main types of diffusion-advection prob-lems for the compressible velocity field [39]. Passive ad-vection of a density field θ ( x ) ≡ θ ( t, x ) (say, the densityof a pollutant) is described by the equation ∂ t θ + ∂ i ( v i θ ) = κ ∂ θ + f, (3.1)while the advection of a “tracer” (temperature, specificentropy, or concentration of the impurity particles) is de-scribed by ∂ t θ + ( v i ∂ i ) θ = κ ∂ θ + f. (3.2)Here ∂ t ≡ ∂/∂t , ∂ i ≡ ∂/∂x i , κ is the molecular diffusiv-ity coefficient, ∂ = ∂ i ∂ i is the Laplace operator, v ( x ) isthe velocity field, and f ≡ f ( x ) is a Gaussian noise withzero mean and given covariance, (cid:104) f ( x ) f ( x (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) C ( r /L ) , r = x − x (cid:48) . (3.3)Here C ( r /L ) is some function finite at ( r /L ) → r /L ) → ∞ . In the following, wedo not distinguish the integral scale L , related to thenoise, and its analog L = m − in the correlation func-tion of the stirring force (2.11). Without loss of general-ity, one can set C (0) = 1 (the coefficient can be absorbedby rescaling of θ and f ). The noise mimics the effectsof initial and/or boundary conditions: it maintains thesteady state of the system and serves as the source ofthe large-scale anisotropy. (The latter term means thatthe anisotropy is introduced at scales of order L , whilethe statistics of the velocity field remains isotropic. Thecase of anisotropic velocity statistics is discussed, withinthe RG+OPE approach, in Refs. [51].) In more real-istic formulations, the noise can arise from an imposedlinear gradient of the (temperature) field. It turns out,however, that the specific form of the random stirring isunimportant, and in the following we use the artificialnoise with the correlation function (3.3).In the absence of the noise, equation (3.1) has the formof a continuity equation (conservation law); θ being thedensity of a corresponding conserved quantity. For (3.2),the conserved quantity is the auxiliary (response) field θ (cid:48) , which appears in the field-theoretic formulation of theproblem; see below. If the function in (3.3) is chosen suchthat its Fourier transform C ( k ) vanishes at k = 0, thefields θ or θ (cid:48) remain to be conserved in the statisticalsense in the presence of the external stirring.The models (3.1) and (3.2) were thoroughly studied forthe case of Kraichnan’s rapid-change model [21]–[28]; thecase of Gaussian velocity statistics with finite correlationtime was studied in [29, 30].The stochastic problem (3.1), (3.3) is equivalent tothe field theoretic model of the full set of fields Φ ≡ { θ (cid:48) , θ, v (cid:48) , v, φ (cid:48) , φ } with the action functional S Φ (Φ) = S θ ( θ (cid:48) , θ, v ) + S ( v (cid:48) , v, φ (cid:48) , φ ) , (3.4)where S θ ( θ (cid:48) , θ, v ) = 12 θ (cid:48) D f θ (cid:48) + θ (cid:48) (cid:8) − ∂ t θ − ∂ i ( v i θ ) + κ ∂ θ (cid:9) (3.5)is the De Dominicis–Janssen action for the stochasticproblem (3.1), (3.3) at fixed v , while the second termis given by (2.12) and the represents the velocity statis-tics; D f is the correlation function (3.3), and, as usual,all the required integrations and summations over thevector indices are implied.In addition to (2.14), the diagrammatic technique inthe full problem involves two propagators (cid:104) θθ (cid:48) (cid:105) = (cid:104) θ (cid:48) θ (cid:105) ∗ = 1 − i ω + κ k , (cid:104) θθ (cid:105) = C ( k ) ω + κ k , (3.6)and the new vertex − θ (cid:48) ∂ i ( v i θ ) = V i θ (cid:48) v i θ . In the momen-tum representation, the vertex factor V i in the diagramshas the form V i ( k ) = i k i , (3.7)where k is the momentum argument of the field θ (cid:48) (usingintegration by parts, the derivative at the vertex can bemoved onto the field θ (cid:48) ).The problem (3.2), (3.3) corresponds to the action(3.4), where the part S θ is given by S θ ( θ (cid:48) , θ, v ) = 12 θ (cid:48) D f θ (cid:48) + θ (cid:48) (cid:8) − ∂ t θ − ( v i ∂ i ) θ + κ ∂ θ (cid:9) . (3.8)The propagators are given by the same expressions (3.6),while the vertex factor (3.7) is replaced with V i ( k ) = − i k i , (3.9)where k is the momentum argument of the field θ . B. UV renormalization and all that
Canonical dimensions of the new fields and parametersthat appear in the models (3.4), (3.5), (3.8) are given intable 1, where we introduced a new dimensionless param-eter w = κ /ν with ν from (2.1).Now in the expression (2.18) for the formal index of UVdivergence the summation runs over the full set of fieldsΦ ≡ { θ (cid:48) , θ, v (cid:48) , v, φ (cid:48) , φ } . The rules (i)-(iv) from section II Cshould be extended and augmented as follows:(i) All the 1-irreducible Green functions without theresponse fields v (cid:48) , φ (cid:48) , θ (cid:48) vanish identically and require nocounterterms.1(ii) In the model (3.8), the field θ enters the vertex − θ (cid:48) ( v i ∂ i ) θ only in the form of derivative. Then the ex-pression (2.19) for the real index of divergence should bemodified as δ (cid:48) Γ = δ Γ − N φ − N θ . (3.10)In the model (3.5), the derivative at the vertex − θ (cid:48) ∂ i ( v i θ )can be moved onto the field θ (cid:48) using integration by parts,and the real index becomes δ (cid:48) Γ = δ Γ − N φ − N θ (cid:48) . (3.11)Since the field θ in model (3.8) and θ (cid:48) in model (3.8) canenter the counterterms only in the form of spatial deriva-tives, the counterterm θ (cid:48) ∂ t θ to the 1-irreducible Greenfunction (cid:104) θ (cid:48) θ (cid:105) − ir with δ Γ = 2, δ (cid:48) Γ = 1 is forbidden forthe both models.(iii) Another consequence of (ii) is that the countert-erms to the 1-irreducible function (cid:104) θ (cid:48) vθ (cid:105) − ir with δ Γ = 1, δ (cid:48) Γ = 0 necessarily reduce to the form θ (cid:48) ∂ i ( v i θ ) for themodel (3.5) and θ (cid:48) ( v i ∂ i ) θ for the model (3.8). Galileansymmetry requires, however, that these monomials enterthe counterterms in the form of invariant combinations θ (cid:48) [ ∂ t θ + ∂ i ( v i θ )] and θ (cid:48) ∇ t θ . Hence, they are also forbid-den.(iv) From the straightforward analysis of the Feynmandiagrams it follows that, for any 1-irreducible function, N θ (cid:48) − N θ = 2 N , where N is the total number of barepropagators (cid:104) θθ (cid:105) entering the diagram. Clearly, no di-agram with N < N θ (cid:48) − N θ is an even non-negative integer for anynontrivial Green function. This fact, a consequence of thelinearity of the original stochastic equations (3.1), (3.2)in the field θ , appears crucial for the renormalizability ofthe models (3.5) and (3.8). Indeed, the total canonicaldimension d θ = − N θ , while (3.10) does not depend on N θ .Without the restriction N θ ≤ N θ (cid:48) , we would face theinfinity of superficially divergent functions (cid:104) θ (cid:48) θ . . . θ (cid:105) − ir ,and hence the lack of renormalizability.Finally, we are left with the only superficially diver-gent 1-irreducible Green function (cid:104) θ (cid:48) θ (cid:105) − ir with the onlycounterterm θ (cid:48) ∂ θ . It is naturally reproduced as mul-tiplicative renormalization of the diffusion coefficient, κ = κZ κ . No renormalization of the fields θ (cid:48) , θ isneeded: Z θ (cid:48) = Z θ = 1. The renormalized analog of theaction functional (3.5) has the form S R Φ (Φ) = S Rθ ( θ (cid:48) , θ, v ) + S R ( v (cid:48) , v, φ (cid:48) , φ ) (3.12)with S R from (2.22) and S Rθ ( θ (cid:48) , θ, v ) = 12 θ (cid:48) D f θ (cid:48) + θ (cid:48) (cid:8) − ∂ t θ − ∂ i ( v i θ ) + κZ κ ∂ θ (cid:9) , (3.13)and similarly for (3.8): S Rθ ( θ (cid:48) , θ, v ) = 12 θ (cid:48) D f θ (cid:48) + θ (cid:48) (cid:8) − ∂ t θ − ( v i ∂ i ) θ + κZ κ ∂ θ (cid:9) . (3.14)It remains to note that, if the term with D f is omit-ted, the models (3.5) and (3.8) can be mapped onto eachother by means of the interchange θ ( t, x ) ↔ θ (cid:48) ( t, x ) andthe reflection t → − t . In particular, this means that therenormalization constants Z κ in (3.13) and (3.14) coin-cide to all orders of the perturbation theory, because thecorrelator D f does not appear in the relevant diagrams;see the next subsection. C. Explicit leading-order results. Fixed points andscaling dimensions
Let us turn to the explicit calculation of the renor-malization constant Z κ in the leading one-loop order;for definiteness, consider the case of the density field(3.13). The constant is found from the requirement thatthe 1-irreducible Green function (cid:104) θ (cid:48) θ (cid:105) − ir be UV finite(that is, finite at y →
0) when expressed in renormalized parameters. The corresponding Dyson equation in thefrequency–momentum representation reads: (cid:104) θ (cid:48) θ (cid:105) − ir ( ω, p ) = +i ω − κ p + Σ θ (cid:48) θ ( ω, p ) , (3.15)where the “self-energy operator” Σ θ (cid:48) θ is given by the in-finite sum of 1-irreducible graphs. In the one-loop ap-proximation it has the form:Σ θ (cid:48) θ = X X + X + … X X X X X + + … X X (3.16)where the wavy line denotes the bare propagator (cid:104) vv (cid:105) from (2.14), the solid line with a slash denotes the barepropagator (cid:104) θθ (cid:48) (cid:105) from (3.6), the slashed end correspond-ing to the field θ (cid:48) . The dots with three attached fields θ (cid:48) , θ , v denote the vertex (3.7).In the leading-order approximation, the renormaliza-tion constant in the bare term of (3.15) is taken only in2the first order in g , that is, κ = κZ κ (cid:39) κ (1 + z (1) g/y ),while in the diagram (3.16) all Z ’s are replaced with uni-ties. Furthermore, we only need to know the divergentpart of (3.16), which is proportional to p (see the pre-ceding subsection). Thus we can set ω = 0 in (3.15) andkeep in the expansion in p of the resulting integrand onlythe p term. Like for the original NS model, its diver-gent part is independent on c ∼ c and can be calculateddirectly at c = 0; see the discussion in subsec. II C. Thenthe expression for (3.16) becomes:Σ θ (cid:48) θ = i p s (cid:90) dω (cid:48) π (cid:90) k>m d k (2 π ) d i( p + k ) l D sl ( ω (cid:48) , k ) − i ω (cid:48) + wν | p + k | , (3.17) where D sl ( ω (cid:48) , k ) = gµ y ν (cid:40) P ⊥ sl ( k )( ω (cid:48) ) + ν k + αP (cid:107) sl ( k )( ω (cid:48) ) + u ν k (cid:41) (3.18)is the velocity correlation function from (2.14) with theproper substitutions, including c = 0.Integrations over the frequency are easily performed,for example, (cid:90) dω (cid:48) π − i ω (cid:48) + wν | p + k | ω (cid:48) ) + u ν k = 12 uν k ( uk + w | p + k | ) . (3.19)In the terms containing the factor p s p l one can immedi-ately set p = 0 in (3.19), while in the exceptional termwith p s k l P (cid:107) sl ( k ) = p s k s one should expand (3.19) up to the linear term in p :1 uk + w | p + k | = 1( u + w ) k (cid:26) − w ( u + w ) ( pk ) k (cid:27) . With the aid of the formulas (cid:90) d k k i f ( k ) = 0 , (cid:90) d k k i k s k f ( k ) = δ is d (cid:90) d k f ( k ) , (cid:90) d k k i k s k l k p k f ( k ) = δ is δ lp + δ il δ sp + δ ip δ sl d ( d + 2) (cid:90) d k f ( k ) , (3.20)where f ( k ) is any function depending only on k = | k | , allthe resulting integrals are reduced to the scalar integral J ( m ) = (cid:90) k>m d k k d + y = S d m − y y (3.21)with S d from (2.27).Collecting all the terms givesΣ θ (cid:48) θ = − ˆ g dy (cid:16) µm (cid:17) y (cid:26) ( d − w ) + αu ( u + w ) − αwu ( u + w ) (cid:27) (3.22)with ˆ g defined in (2.26). Then the renormalization con-stant, needed to cancel the pole in y in (3.15), in the MSscheme should be chosen as Z κ = 1 − ˆ g dwy (cid:26) ( d − w ) + α ( u − w ) u ( u + w ) (cid:27) , (3.23)while the corresponding anomalous dimension is γ κ = ˆ g dw (cid:26) ( d − w ) + α ( u − w ) u ( u + w ) (cid:27) , (3.24)with the corrections of the order ˆ g and higher. The function β w = (cid:101) D µ w for the new dimensionlessparameter w has the form β w = − wγ w = w [ γ ν − γ κ ] , (3.25)cf. equation (2.33). Substituting the one-loop expres-sions (2.41), (3.24) and the exact relation (2.43) into theequation β w = 0 gives, after some simple algebra, theequation( w − d − w + 1)( w + 2) + 2 α ] = 0 , (3.26)with the only positive solution w ∗ = 1.The corresponding new eigenvalue of the matrix (2.40)coincides with the diagonal element ∂β w /∂w | g = g ∗ = y [3( d −
1) + α ] / d − > , because the functions (2.33) do not depend on w . Weconclude that the fixed point with the coordinates (2.41)and w ∗ = 1 is IR attractive in the full space of couplings g, u, v, w and governs the IR asymptotic behavior of thefull-scale models (3.5), (3.8).The critical dimensions of the fields θ , θ (cid:48) are obtainedfrom the data in table 1 and the expression (2.47) for ∆ ω :∆ θ = − y/ , ∆ θ (cid:48) = d + 1 − y/ . (3.27)3These expressions are exact due to the absence of renor-malization of the fields θ and θ (cid:48) . IV. COMPOSITE FIELDS AND THEIRDIMENSIONS
The key role in the following will be played by certaincomposite fields (“composite operators” in the quantum-field terminology). A local composite operator is a mono-mial or polynomial constructed from the primary fieldsΦ( x ) and their finite-order derivatives at a single space-time point x = { t, x } . In the Green functions with suchobjects, new UV divergences arise due to coincidence ofthe field arguments. They are removed by additionalrenormalization procedure. As a rule, operators mixin renormalization: renormalized operators are given bycertain finite linear combinations of the original mono-mials. However, in the following only a simpler situationwill be encountered, when the original operator F ( x ) andthe renormalized one F R ( x ) are related by multiplicativerenormalization F ( x ) = Z F F R ( x ) with the renormaliza-tion constant of the form (2.36). Then the critical di-mension of the operator is given by the same expression(2.47) and, in general, differs from the simple sum of thedimensions of the fields and derivatives that enter theoperator.The total canonical dimension of any 1-irreducibleGreen function Γ with one operator F ( x ) and arbitrarynumber of primary fields (the formal index of UV diver-gence) is given by δ Γ = d F − (cid:88) Φ N Φ d Φ , (4.1)where N Φ are the numbers of the fields entering into Γ, d Φ are their total canonical dimensions, d F is the canonicaldimension of the operator, and the summation over alltypes of the fields is implied. Superficial UV divergencescan be present only in the functions Γ with a non-negativeinteger δ Γ . A. Renormalization of the composite fields θ n .Explicit leading-order results Let us begin with the simplest case of the operators F ( x ) = θ n ( x ) in the density model. Then d F = − n in(4.1). Due to the linearity of the stochastic equation (3.1)in θ , the number of fields θ in any 1-irreducible functionwith the operator F ( x ) cannot exceed their number inthe operator itself. This is easily seen from the fact thatthe chains of the propagators (cid:104) θ (cid:48) θ (cid:105) , (cid:104) θθ (cid:105) in any dia-gram cannot branch; cf. item (iv) in sec. II C. Then theanalysis of expression (4.1) shows that the superficial di-vergence can only be present in the 1-irreducible functionwith N θ = n and N Φ = 0 for the fields Φ other than θ .For this function δ Γ = 0, the divergence is logarithmic,and the corresponding counterterm has the form θ n ( x ).Hence, our operators are multiplicatively renormalizable: F ( x ) = Z n F R ( x ) with certain renormalization constantsof the form (2.36).Now we turn to the calculation of the constants Z n in the leading (one-loop) approximation. Let Γ( x ; θ ) bethe generating functional of the 1-irreducible Green func-tions with one composite operator F ( x ) and any numberof fields θ . Here x = { t, x } is the argument of the op-erator and θ is the functional argument, the “classicalanalog” of the random field θ . We are interested in the θ n term of the expansion of Γ( x ; θ ) in θ ( x ), which wedenote Γ n ( x ; θ ). It can be written asΓ n ( x ; θ ) = (cid:90) dx · · · (cid:90) dx n θ ( x ) · · · θ ( x n ) (cid:104) F ( x ) θ ( x ) · · · θ ( x n ) (cid:105) − ir . (4.2)In the one-loop approximation the function (4.2) is rep-resented diagramatically as follows:Γ n ( x ; θ ) = F ( x ) + 12 X X + X + … X X X X X + + … X X (4.3)The first term is the tree (loopless) approximation, andthe thick dot with the two attached lines in the dia-gram denotes the operator vertex, that is, the variationalderivative V ( x ; x , x ) = δ F ( x ) /δθ ( x ) δθ ( x ) . (4.4)In the present case, the vertex V ( x ; x , x ) = n ( n − θ n − ( x ) δ ( x − x ) δ ( x − x ) (4.5) contains ( n −
2) fields θ . (We recall that δθ ( x ) /δθ ( x (cid:48) ) = δ ( x − x (cid:48) ) ≡ δ ( t − t (cid:48) ) δ ( x − x (cid:48) ).) Two more fields areattached to the plain vertices θ (cid:48) ∂ ( vθ ) at the bottom ofthe diagram.Since the divergence is logarithmic, one can set all theexternal frequencies and momenta equal to zero. Thenall θ ’s acquire the common argument x and the diagrambecomes proportional to the operator θ n ( x ) with the co-efficient, given by the “core” of the diagram: (cid:90) dω π (cid:90) d k (2 π ) d k s k l ω + w ν k D sl ( ω, k ) , (4.6)where the first factor in the integrand comes from thevertices (3.7), the second one comes from the propagators (cid:104) θ (cid:48) θ (cid:105) in (3.6) with the replacement κ → wν , and the4last factor is the velocity propagator from (3.18). Notethat only the second term from D sl gives nonvanishingcontribution to (4.6). Integration over the frequency is easily performed using the formula (cid:90) dω π ω + a )( ω + b ) = 12 ab ( a + b ) , (4.7)and after the contraction of the tensor indices the integralover the momentum reduces to (3.21). Collecting all thefactors givesΓ n ( x ; θ ) = θ n ( x ) (cid:26) n ( n − α ˆ g wu ( u + w ) (cid:16) µm (cid:17) y y (cid:27) , (4.8)with ˆ g defined in (2.26) and up to a finite part and higher-order corrections.The renormalization constant Z n is found from the re-quirement that the renormalized analog Γ Rn = Z − n Γ n ofthe function (4.2) be UV finite in terms of renormalizedparameters (mind the minus sign in the exponent). Inour approximation, it is sufficient to replace θ n → Z − n θ n only in the first term of the expression (4.8) and then tochoose Z n to cancel the pole in the second term. In theMS scheme this gives Z n = 1 + n ( n − α ˆ g wu ( u + w ) 1 y . (4.9)Then for the corresponding anomalous dimension eq.(2.37) gives γ n = − n ( n − α ˆ g wu ( u + w ) , (4.10)with the higher-order corrections in ˆ g .For the critical dimensions of the operators θ n fromthe expression (2.47) one obtains∆[ θ n ] = n ∆ θ + γ ∗ n , (4.11)and substituting the fixed-point values (2.41) and w ∗ = 1into (4.10) finally gives∆[ θ n ] = − n + ny − n ( n − α dy d − , (4.12)with the higher-order corrections in y . These dimensionsare negative (“dangerous” in the terminology of [12–15])and decrease as n grows. One can argue that danger-ous operators can always appear in a field theoretic onlyas infinite families with the spectrum of dimensions notbounded from below.Now let us turn to the same operators θ n in thetracer model. From the expression (4.1) and the lin-earity of the stochastic equation (3.2) it follows that,like for the density case, the superficial UV diver-gences can only be present in the 1-irreducible function (cid:104) θ n ( x ) θ ( x ) · · · θ ( x n ) (cid:105) − ir . Clearly, at least one of the ex-ternal tails of the field θ is attached to a vertex θ (cid:48) ( v∂ ) θ : it is impossible to construct a nontrivial diagram of thedesired type with all the external tails attached only tothe vertex (4.5) of the operator F ( x ). Therefore at leastone derivative ∂ , acting on a tail θ , appears as an exter-nal factor in the diagram. Consequently, its real index ofdivergence δ (cid:48) Γ is necessarily negative, and the diagram isin fact UV convergent; cf. item (iii) in sec. II C.This means that the operators θ n are in fact UV finite, Z n = 1, and their scaling dimensions are given by theexpression ∆[ θ n ] = n ∆ θ = − n + ny/ y . B. Renormalization of the composite fields ( ∂θ ) n inthe tracer model. Explicit leading-order results In the tracer model, of special importance are tensoroperators, constructed solely of the gradients of the pas-sive scalar field. Such operators with the lowest canoni-cal dimension contain the minimal number of derivatives(one derivative per each field) and have the form F ( n,l ) i ...i l = ∂ i θ · · · ∂ i l θ ( ∂ i θ∂ i θ ) s + . . . . (4.14)Here l is the number of the free vector indices (the rankof the tensor) and n = l + 2 s is the total number of thefields θ entering into the operator. The ellipsis standsfor the subtractions with Kronecker’s delta symbols thatmake the operator irreducible (so that contraction withrespect to any pair of the free tensor indices vanish), forexample, F (2 , ij = ∂ i θ∂ j θ − δ ij d ( ∂ k θ∂ k θ ) . (4.15)For all these operators d F = 0, and the real indexof divergence is δ (cid:48) Γ = δ Γ − N θ with δ Γ from (4.1). In-deed, now one derivative ∂ appears as an external fac-tor in a diagram for any external tail θ , no matter is itattached to the ordinary vertex θ (cid:48) ( v∂ ) θ or to the ver-tex (4.5) for the operator (4.14). Like for the operators5 θ n , the number of the fields θ in any 1-irreducible func-tion cannot exceed their number in the operator itself: N θ ≤ n , cf. the discussion in sec. IV A. It then followsthat superficial UV divergences can only be present inthe 1-irreducible functions (cid:104) F ( n,l ) ( x ) θ ( x ) . . . θ ( x k ) (cid:105) − ir with k ≤ n . For such functions δ (cid:48) Γ = 0 and δ Γ = k , sothat the corresponding counterterm can only involve themonomials F ( k,p ) from (4.14) with certain values of therank p . We conclude that the family of the operators(4.14) is closed with respect to renormalization in thesense that F ( n,l ) = Z ( n,l )( k,p ) F ( k,p ) R with a certain ma-trix of renormalization constants. Since Z ( n,l )( k,p ) = 0for k > n , this matrix is block-triangular with the diag-onal sub-blocks corresponding to n = k , and so is thecorresponding matrix ∆ F in (2.47).We are interested presumably in the scaling dimen-sions, associated with the operators (4.14). They aregiven by the eigenvalues of the matrix ∆ F , which arecompletely defined by its diagonal sub-blocks. A sim-ple analysis shows that the corresponding diagrams donot involve the propagator (cid:104) θθ (cid:105) from (3.6); this is againa consequence of the linearity of the original stochasticequation (3.2). Hence, the diagonal blocks can be calcu-lated directly in the model without the random noise in(3.2), because the correlation function of the noise (3.3)enters the diagrams only via the propagator (cid:104) θθ (cid:105) . Thefunction (3.3) is the only source of the anisotropy in theproblem. Without the noise, the model becomes SO ( d )covariant, and the irreducible tensor operators with dif- ferent ranks cannot mix in renormalization. This meansthat the diagonal sub-blocks of the matrix ∆ F are in factdiagonal, and their diagonal elements coincide with theeigenvalues of the full matrix ∆ F .We finally conclude that, as long as the scaling dimen-sions are concerned, the operators (4.14) can be treatedas multiplicatively renormalizable, F ( n,l ) = Z ( n,l ) F ( n,l ) R with certain renormalization constants Z ( n,l ) , the diago-nal elements of the full matrix Z ( n,l )( k,p ) .For practical calculations, it is convenient to contractthe tensors (4.14) with an arbitrary constant vector λ = { λ i } . The resulting scalar operator has the form F ( n,l ) = ( λ i w i ) l ( w i w i ) s + . . . , w i ≡ ∂ i θ, (4.16)where the subtractions, denoted by the ellipsis, necessar-ily involve the factors of λ = λ i λ i . The counterterm to F ( n,l ) is proportional to the same operator, and in or-der to find the constant Z ( n,l ) , it is sufficient to retainonly the principal monomial, explicitly shown in (4.16),and to discard in the result all the terms with factors of λ . Then, using the chain rule, the vertex (4.4) for theoperator F ( n,l ) can be written in the form V ( x ; x , x ) = ∂ F ( n,l ) ∂w i ∂w j ∂ i δ ( x − x ) ∂ j δ ( x − x ) (4.17)up to irrelevant terms. The differentiation gives ∂ F ( n,l ) /∂w i ∂w j = 2 s ( w ) s − ( λw ) l (cid:2) δ ij w + 2( s − w i w j (cid:3) + l ( l − w ) s ( λw ) l − λ i λ j ++ 2 ls ( w ) s − ( λw ) l − ( w i λ j + w j λ i ) , (4.18)where w = w k w k and ( λw ) = λ k w k . Two more factors w p w r are attached to the bottom of the diagram, thederivatives coming from the vertices θ (cid:48) ( v∂ ) θ . The UVdivergence is logarithmic, and one can set all the externalfrequency and momentum equal to zero; then the core ofthe diagram takes on the form (cid:90) dω π (cid:90) k>m d k (2 π ) d k i k j D pr ( ω, k ) 1 ω + w ν k . (4.19) Here the first factor comes from the derivatives in (4.17), D pr from (3.18) is the velocity correlation function (2.14),and the last factor comes from the two propagators (cid:104) θ (cid:48) θ (cid:105) . The substitutions Z → c → k arereduced to the scalar integral (3.21) using the relations(3.20). Combining all the factors, contracting the tensorindices and expressing the result in n = l +2 s and l gives:Γ n ( x ; θ ) = F ( n,l ) ( x ) (cid:26) − ˆ g yd ( d + 2) (cid:16) µm (cid:17) y (cid:18) Q w (1 + w ) + α Q wu ( u + w ) (cid:19)(cid:27) , (4.20)where Q = − n ( n + d )( d −
1) + ( d + 1) l ( l + d − , Q = − n (3 n + d −
4) + l ( l + d − , (4.21)and ˆ g is defined in (2.26). Then the renormalization con- stant Z ( n,l ) in the MS scheme reads Z ( n,l ) = 1 − ˆ g yd ( d + 2) (cid:26) Q w (1 + w ) + α Q wu ( u + w ) (cid:27) , (4.22)6see the explanation in sec. IV A below eq. (4.8). Then forthe corresponding anomalous dimension eq. (2.37) gives γ ( n,l ) = ˆ g d ( d + 2) (cid:26) Q w (1 + w ) + α Q wu ( u + w ) (cid:27) , (4.23)with the higher-order corrections in ˆ g .Finally, for the scaling dimension, associated with theoperators (4.14), the general expression (2.47) gives∆ ( n,l ) = n + n ∆ θ + γ ∗ ( n,l ) = ny/ γ ∗ ( n,l ) . (4.24)Substituting the fixed-point values (2.41) and w ∗ = 1into (4.23) one finally obtains∆ ( n,l ) = ny/ y { Q + αQ } d − d + 2) , (4.25)with the higher-order corrections in y .In particular, for the scalar operator with l = 0 oneobtains:∆ ( n, = − yn { ( n − d −
1) + α (3 n + d − } d − d + 2) . (4.26)Again, we meet an infinite family of dangerous opera-tors with the spectrum of dimensions not bounded frombelow. For a fixed n , the dimension (4.25) increaseswith the rank l , so that for the maximum possiblerank l = n one always has ∆ ( n,n ) >
0. This hierar-chy, which is conveniently expressed by the inequality ∂ l ∆ ( n,l ) >
0, becomes more strongly pronounced when α grows: ∂ l ∂ α ∆ ( n,l ) >
0. All these properties will beimportant in the OPE analysis of sec. V.
C. More tensor operators
We will also need to know the critical dimensions of the l th rank irreducible tensor operators, built only of twofields θ and l spatial derivatives. An example is providedby the operator F i ...i l ( x ) = θ ( x ) ∂ i · · · ∂ i l θ ( x ) + . . . . (4.27)As earlier in (4.14), the ellipsis stands for the subtrac-tions with Kronecker’s delta symbols that make the op-erator irreducible. Of course, for any given l >
1, thereare several such operators with different placement of thederivatives: in the special case (4.27), all the derivativesact on the same field. However, all the other such op-erators differ from (4.27) by a total derivative, which iseasily seen from the relation F ( x ) ∂G ( x ) = − G ( x ) ∂F ( x ) + ∂ ( F ( x ) G ( x )) . (4.28) Thus the set of independent l th rank operators can bechosen as (4.27) and the operators having the forms ofderivatives, for example, for l = 2, as θ∂ i ∂ j θ + . . . and ∂ i ∂ j ( θθ ) + . . . . In the calculation of their critical di-mensions, it is sufficient to consider the SO ( d ) covariantmodel without the noise (3.3); see the discussion in thepreceding subsection. Then the operators with differentranks do not mix in renormalization. The analysis ofrenormalization also shows that the operator (4.27) canmix only with its own “family” of derivatives: the opera-tors with additional derivatives (like ∂ t or ∂ ) or with thefields θ (cid:48) , φ , φ (cid:48) , v (cid:48) have too high canonical dimensions d F ,the appearance of v is forbidden by Galilean symmetry,and extra θ ’s are forbidden by the linearity of the model.The same relation (4.28) also shows that for odd l ,the operator (4.27) itself reduces to a derivative (moreprecisely, to a linear combination of derivatives). In thefollowing, we will be interested only in the operators notreducible to derivatives, and thus, from now on, we willconsider only even values of l . Then (4.27) is nontrivialand it cannot admix to the derivatives from its family,although they can admix to (4.27). Thus the correspond-ing renormalization matrix Z F appears block triangular,and so is the matrix ∆ F . The eigenvalue, associated withthe nontrivial operator (4.27), coincides with the corre-sponding diagonal element of ∆ F . We conclude that inthe calculation of the critical dimension, associated withthe operator (4.27), the latter can be treated as if it weremultiplicatively renormalizable.Like in the preceding subsection, it is convenient tocontract the operator (4.27) with an arbitrary constantvector λ = { λ i } . The resulting scalar operator has theform F l = θ ( λ i ∂ i ) l θ + . . . , (4.29)where the terms, denoted by the ellipsis, necessarily in-volve the factors of λ . In order to find the correspondingrenormalization constant Z l , it is sufficient to keep onlythe principal monomial, explicitly shown in (4.16), and toretain in the result for the counterterm only terms of thesame form. Then the relevant part of the vertex factor(4.4) is V ( x ; x , x ) = δ ( x − x )( λ i ∂ i ) l δ ( x − x ) + { x ↔ x } . (4.30)The one-loop approximation for the functional (4.2)for the operator (4.29) has the same form (4.3). Let uschoose the external momentum p to flow into the diagramthrough the left lower vertex and to flow out throughthe right lower one. The external momentum flowingthrough operator’s vertex and all the external frequen-cies are set equal to zero: this is sufficient to find theneeded counterterm. Furthermore, we will put w = u = 1in the propagators from the very beginning, because weare eventually interested in the value of the anomalousdimension at the fixed point w ∗ = u ∗ = 1.Let us begin with the tracer case. Then the core of thediagram in (4.3) takes on the form7 p i p j (cid:90) dω π (cid:90) k>m d k (2 π ) d l ( λ q ) l gµ y ν k − d − y ω + ν k (cid:110) P ⊥ ij ( k ) + αP (cid:107) ij ( k ) (cid:111) ω + ν q . (4.31)Here the factor p i p j comes from the vertices (3.9), thefactor 2i l ( λ q ) l comes from the vertex (4.30) for even l (for the odd l the two terms in (4.30) would cancel eachother and instead of factor 2 one would get 0), the factorsdepending on k represent the velocity correlation func-tion from (2.14) with the proper substitutions, including c = 0 and w = u = 1. The last factor comes from thepropagators (cid:104) θ (cid:48) θ (cid:105) . The momentum k flows through thevelocity propagator, so that q = k + p .In the resulting expression we retain only terms of theform ( λ p ) l and drop all the other terms, containing λ or p . Thus we can replace p i p j (cid:110) P ⊥ ij + αP (cid:107) ij (cid:111) → ( α − pk ) /k . The integration over ω in (4.31) is easily performed using(4.7) and gives: gµ y ( α − l (cid:90) k>m d k (2 π ) d ( pk ) ( λ q ) l k − d − y q ( k + q ) . (4.32)Now we expand all the denominators in the integrandof (4.31) in p (dropping all the terms with p ):1 q (cid:39) k + 2( pk ) = 1 k ∞ (cid:88) s =0 ( − s ( pk ) s k s , k + q (cid:39) k + pk ) = 12 k ∞ (cid:88) m =0 ( − m ( pk ) m k m , (4.33)and expand the numerator using Newton’s binomial for-mula: ( λ q ) l = l (cid:88) n =0 C nl ( λ k ) n ( λ p ) l − n . (4.34)In the resulting three-fold series over n, m, s l (cid:88) n =0 C nl ( λ p ) l − n ∞ (cid:88) m,s =0 ( − m ( − s ( pk ) m + s +2 ( λ k ) n k s + m ) we only need to collect the terms proportional to ( λ p ) l ,which leads to the restriction n = s + m + 2 and henceto the finite double sum: s + m +2 ≤ l (cid:88) s,m =0 ( − m ( − s C s + m +2 l ( λ p ) l − m − s − ( pk ) m + s +2 ( λ k ) s + m +2 k s + m ) . (4.35)Substituting it to the (4.32) gives rise to the integrals J i ...i n ( m ) = (cid:90) k>m d k (2 π ) d k − d − y k i . . . k i n k n (4.36)with n = s + m + 2 ≥
2. They are easily found using theisotropy considerations, cf. (3.20): J i ...i n ( m ) = δ i i . . . δ i n − i n + all permutations d ( d + 2) . . . ( d + 2 n − J ( m )(4.37)with J ( m ) from (3.21). The sum over all possible per-mutations of 2 n tensor indices in the numerator of (4.37) involves (2 n − n )! / n n ! terms, but we have tokeep only the terms that give rise to the structure ( λ p ) n after the contraction with the vectors λ and p in (4.35).It is easy to grasp that there are only n ! such permuta-tions.Collecting all the factors gives for the core (4.31) of thediagram in (4.3) the following expression:i l ( λ p ) l ( α − g (cid:16) µm (cid:17) y y S l ( d ) , (4.38)8where ˆ g is defined in (2.26) and S l ( d ) = s + m +2 ≤ l (cid:88) s,m =0 ( − s + m s C s + m +2 l ( s + m + 2)! d ( d + 2) . . . ( d + 2( s + m ) + 2) . (4.39)For l = 0, the sums (4.35) and (4.39) contain no terms,so that S ( d ) = 0.For the functional (4.2) we then obtain (i p i → ∂ i )Γ ( x ) = F l ( x ) (cid:26) α −
1) ˆ g y (cid:16) µm (cid:17) y S l ( d ) (cid:27) (4.40)with the operator F l from (4.29); note the additionalfactor 1 / F l = Z l F Rl in the MS scheme we obtain Z l = 1 + ( α −
1) ˆ g y S l ( d ) , (4.41)and the corresponding anomalous dimension is: γ l ( g ) = − ( α −
1) ˆ g S l ( d ) . (4.42)The sum S l ( d ) in (4.39) can be reduced to a simplerone-fold sum for general l . Let us pass from s and m tothe new summation variables k = s + m and m and sub-stitute the explicit expression for the binomial coefficient C k +2 l = l ! / ( k + 2)!( l − k − S l ( d ) = l ! k +2 ≤ l (cid:88) k =0 (cid:40) k (cid:88) m =0 m (cid:41) ( − k ( l − k − d ( d + 2) . . . ( d + 2 k + 2) . (4.43)Now the internal summation over m is readily performedto give 2 − − k , so that, after changing the summationvariable k → k + 2, we obtain S l ( d ) = 2 N l ( d ) − M l ( d ) , (4.44)where N l ( d ) = l ! l (cid:88) k =2 ( − k − ( l − k )! d ( d + 2) . . . ( d + 2 k − , M l ( d ) = l ! l (cid:88) k =2 ( − k − ( l − k )! d ( d + 2) . . . ( d + 2 k − . (4.45)The first sum can be calculated explicitly for any l , cf.[28]: N l ( d ) = 4 l ( l − d ( d + 2 l − , (4.46) while the second can be easily calculated for any given l .For the critical dimension, associated with the operator(4.27), from the relation (2.47) we finally obtain:∆ l = l + 2∆ θ + γ ∗ l = l − y/ γ ∗ l , (4.47)where from (4.42) and (2.41) we find γ ∗ l = γ l ( g ∗ ) = − yd ( α − d − S l ( d ) , (4.48)with the higher-order corrections in y .For l = 0, expressions (4.42), (4.47) agree with theexact result (4.13) for the operator θ (we recall that S ( d ) = 0), while for l = 2 they agree with the results(4.23)–(4.25) with n = l = 2.Now let us turn to the density case. Then the factor p i p j in (4.31) should be replaced with q i q j (and, of course,moved into the integrand). It is convenient to write q i q j (cid:110) P ⊥ ij ( k ) + αP (cid:107) ij ( k ) (cid:111) = p i p j (cid:110) P ⊥ ij ( k ) + αP (cid:107) ij ( k ) (cid:111) + α ( q − p ) : (4.49)the first term gives the old expression (4.31), and the lastone is proportional to p and can be dropped. Thus weonly need to calculate the contribution of the term αq to the analog of expression (4.31). Then the analog of(4.32) takes on the form gµ y α i l (cid:90) k>m d k (2 π ) d ( λ q ) l k − d − y ( k + q ) . (4.50) Applying the expansions (4.33), (4.34) leads to the dou-ble sum l (cid:88) n =0 C nl ( λ p ) l − n ∞ (cid:88) m =0 ( − m ( pk ) m ( λ k ) n k m . We have to retain only the terms proportional to ( λ p ) l ,which leads to the restriction n = m and hence to the9finite sum: l (cid:88) m =0 ( − m C ml ( λ p ) l − m ( pk ) m ( λ k ) m k m . (4.51)Substituting it into (4.50) gives rise to the integrals (4.36)with all n ≥
0. In the sum (4.37) over all possible per-mutations we have to keep only n ! = m ! terms that giverise to the structure ( λ p ) n after the contraction with thevectors λ and p in (4.51). To avoid possible confusion,we will write down the terms with m = 0 and m = 1separately and for m ≥ l ( λ p ) m gµ y α (cid:26) − ld + M l ( d ) (cid:27) J ( m ) (4.52)with J ( m ) from (3.21) and the sum M l ( d ) from (4.45).Proceeding as before for the tracer case, we arrive atthe following expression for the renormalization constant Z l in the MS scheme: Z l = 1 + ( α −
1) ˆ g y S l ( d ) + α ˆ g y (cid:26) − ld + M l ( d ) (cid:27) . (4.53)Here the contribution with S l ( d ) comes from the firstterm in (4.49) and the last term with curly bracketscomes from (4.53). Then for the anomalous dimension,using the expressions (4.44)–(4.46), we obtain γ l ( g ) = − α ˆ g (cid:18) − ld (cid:19) + (1 − α ) ˆ g N l ( d ) − ˆ g M l ( d ) , (4.54)with higher-order corrections in g .In the expression (4.47) for the critical dimension onehas: γ ∗ l = − α y ( l − d )3( d −
1) + (1 − α ) 8 l ( l − y d − d + 2 l − − dy d − M l ( d ) , (4.55)with higher-order corrections in y . For l = 0 this result isin agreement with the expression (4.12) for the operator θ in the density case. V. OPERATOR PRODUCT EXPANSION ANDTHE ANOMALOUS SCALINGA. The case of a density field
Consider the equal-time pair correlation function oftwo UV finite quantities F , ( x ) with definite criticaldimensions, for example, those of the primary fields orrenormalized local composite operators. We restrict our-selves with equal-time correlators, because they are usu-ally Galilean invariant and do not bear strong depen-dence on the IR scale, caused by the so-called sweepingeffects. From the (canonical) dimensionality considera-tions it follows that (cid:104) F ( t, x ) F ( t, x ) (cid:105) = ν d ωF µ d F η ( µr, mr, c/ ( µν )) , (5.1)where d ωF and d F are the canonical dimensions of thecorrelation function, given by simple sums of the corre-sponding dimensions of the operators, r = | x − x | , and η ( . . . ) is a function of completely dimensionless variables.We have written the right hand side in terms of renor-malized parameters, when the reference mass substitutesthe typical UV momentum scale Λ. The behavior of thefunction η in the IR range, that is, for µr (cid:29)
1, is deter-mined by the IR attractive fixed point of the RG equa- tion. Solving the RG equation in a standard way, onederives the following asymptotic expression: (cid:104) F ( t, x ) F ( t, x ) (cid:105) (cid:39) ν d ωF µ d F ( µr ) − ∆ F ζ ( mr, c ( r )) . (5.2)Here ∆ F is the critical dimension of the correlation func-tion, given by simple sum of the dimensions of the oper-ators. The RG equation does not determine the form ofthe scaling function ζ ; it only determines the form of itsarguments. They are canonically and critically dimen-sionless: in particular, c ( r ) = c ( µr ) ∆ c / ( µν ) (5.3)with ∆ c from (2.49) can be interpreted as effective speedof sound; more detailed discussion of this point can befound in [45].For the correlation functions of two operators of thetype θ n ( x ) the general expression (5.2) gives: (cid:104) θ p ( t, x ) θ k ( t, x ) (cid:105) (cid:39) µ − ( p + k ) ( µr ) − ∆ p − ∆ k ζ pk ( mr, c ( r ))(5.4)with the dimensions ∆ n from (4.12). In the following, wedo not display the dependence on the UV parameters µ and ν and omit the indices of the scaling functions.The inertial-convective range corresponds to the ad-ditional condition that mr (cid:28)
1. The behavior of thefunctions ζ at mr → F ( t, x ) F ( t, x ) (cid:39) (cid:88) F C F ( mr, c ( r )) F ( t, x ) , (5.5)0where x − x → x = ( x + x ) / C F being numerical coefficient functions analytical in mr and c ( r ). In our model, due to the linearity in the field θ ,the number of such fields in the operators F cannot ex-ceed their number on the left hand side. This restriction,which our model shares with the Kraichnan’s model andits relatives [10] will be very important in the following.The correlation function (5.2) is obtained by averag-ing (5.5) with the weight exp S R with the renormalizedaction functional from (3.4). The mean values (cid:104) F ( x ) (cid:105) appear on the right hand side. Without loss of gener-ality, it can be assumed that the expansion in (5.5) ismade in irreducible tensor operators. Then, if the modelis SO ( d ) covariant (the correlation function of the scalarnoise (3.3) depends only on r = | r | ), only scalar operatorssurvive the averaging. It can also be assumed that theexpansion is made in the operators with definite criticaldimensions. Then their mean values, in the asymptoticregion of small m , take on the forms (cid:104) F ( x ) (cid:105) (cid:39) m ∆ F ξ ( c (1 /m )) , (5.6)with another set of scaling functions ξ and the argument c ( . . . ) from (5.3). Since the diagrams of the perturbationtheory have finite limits both for c → ∞ and c →
0, wemay assume that the functions ξ are restricted for all val-ues of c and can be estimated by some constants. Whatis more, for y large enough, including the most realis-tic case y →
4, the dimension ∆ c becomes negative; seeexpression (2.49). Thus the argument c (1 /m ) ∼ cm − ∆ c becomes small for fixed c and m →
0, and the function ξ can be replaced by its (finite) limit value ξ (0). We finallyconclude that, in the IR range, (cid:104) F ( x ) (cid:105) ∼ m ∆ F . (5.7)Then combining expressions (5.2), (5.5) and (5.7) givesthe desired asymptotic expression for the scaling func-tions: ζ ( mr, c ( r )) (cid:39) (cid:88) F A F ( mr, c ( r )) ( mr ) ∆ F , (5.8)where the summation runs over Galilean invariant scalar operators, with the coefficient functions A F analytical intheir arguments.Divergences for mr → negative critical dimensions, termed “dangerous” in[13]. Clearly, the leading contribution is determined bythe operator with the lowest (minimal) dimension; theothers determine the corrections. All the operators θ n are dangerous, and the spectrum of their dimensions isnot restricted from below (there is no “most dangerous”operator); see expression (4.12). Fortunately, for a given correlation function, only a finite number of those oper-ators can contribute to the OPE. For (5.4), these are theoperators with n ≤ p + k . Thus, ζ ( mr, c ( r )) (cid:39) p + k (cid:88) n =0 A n ( mr, c ( r )) ( mr ) ∆ n + . . . (5.9)with ∆ n from (4.12); the ellipsis stands for the “moredistant” corrections, related to the operators with deriva-tives and other types of fields. The leading term of thesmall- mr behavior in (5.9) is given by the operator withthe maximum possible n = p + k , so that the final ex-pression has the form (cid:104) θ p ( t, x ) θ k ( t, x ) (cid:105) (cid:39) µ − ( p + k ) ( µr ) − ∆ p − ∆ k ( mr ) ∆ p + k . (5.10)It is worth noting that the set of operators θ n is “closedwith respect to the fusion” in the sense that the leadingterm in the OPE for the pair correlator of two such oper-ators is given by the operator from the same family withthe summed exponent. This fact along with the inequal-ity ∆ p + ∆ k > ∆ p + k , which follows from the explicit ex-pression (4.12), can be interpreted as the statement thatthe correlations of the scalar field in the density modelreveal multifractal behavior; see [52]. B. The case of the tracer field
For the tracer model, the critical dimensions of the op-erators θ n are linear in n : ∆[ θ n ] = n ∆ θ , see eq. (4.13).Then the dependence on the separation r in the asymp-totic expressions (5.10) disappears: the leading terms ofthe inertial-range behavior are constants. More “vivid”quantities are the equal-time structure functions definedas S n ( r ) = (cid:104) [ θ ( t, x ) − θ ( t, x (cid:48) ] n (cid:105) = ( νµ ) − n η ( µr, mr, c/ ( µν )) , r = | x (cid:48) = x | ; (5.11)the second equality with dimensionless functions η fol-lows from dimensionality considerations. Solving the RGequations gives the asymptotic expressions for µr (cid:29) S n ( r ) = ( νµ ) − n ( µr ) − n ∆ θ ζ ( mr, c ( r )) , (5.12) with c ( r ) from (5.3) and some scaling functions ζ . It isimportant here that the pair correlation functions (cid:104) θ p θ k (cid:105) with k + p = 2 n , appearing in the binomial decompositionof S n , have similar asymptotic representations (5.4) with1the same critical dimension ∆ k + ∆ p = 2 n ∆ θ , and to-gether they form the single asymptotic expression (5.12).The constant leading terms for those correlators, relatedto the contributions of the operator θ n in the correspond-ing OPE, cancel each other in the structure function, and the latter acquires nontrivial dependence on r in the in-ertial range.Indeed, both the functions (5.11) and the action (3.8)for the tracer (not for the density!) are invariant withrespect to the constant shift θ ( x ) → θ ( x ) + const. Thenthe operators entering the corresponding OPE,[ θ ( t, x ) − θ ( t, x (cid:48) ] n (cid:39) (cid:88) F C F ( mr, c ( r )) F ( t, x ) , r → , x = ( x + x (cid:48) ) / , (5.13)must also be all invariant, so that they can involve thefield θ only in the form of derivatives. Clearly, the leadingterm of the small- m behavior will be determined by thescalar operator with maximum possible number of thefields θ (namely, 2 n for the given S n ) and the minimumpossible number of spatial derivatives (namely, 2 n : onederivative for each θ ). This is nothing other than the op-erator F (2 n, = ( ∂ i θ∂ i θ ) n from (4.14). Thus the desiredleading-order expression for S n in the inertial range is S n ( r ) ∼ ( νµ ) − n ( µr ) − n ∆ θ ( mr ) ∆ (2 n, , (5.14)with the dimension ∆ (2 n, given in (4.25). The operators F (2 p, with p < n determine the main corrections to(5.14), the operators with extra derivatives and/or othertypes of fields correspond to more “distant” corrections(they all must be invariant with respect to the Galileantransformation and the shift of θ ).For the tracer, the “multifractal” behavior is demon-strated by the family of the operators F ( n, rather thanby the simple powers θ n ; see the end of the preceding sub-section. Indeed, it is easy to grasp that the inertial-rangebehavior of the pair correlation function (cid:104) F ( p, F ( k, (cid:105) oftwo such operators is determined by the contribution tothe OPE from their “elder brother” F ( n, with n = p + k and has the form (omitting the dependence on the UVparameters µ and ν ) (cid:104) F ( p, ( t, x ) F ( k, ( t, x (cid:48) ) (cid:105) ∼ r − ∆ ( p, − ∆ ( k, +∆ ( n, . (5.15)The required inequality ∆ ( n, < ∆ ( p, + ∆ ( k, [52] fol-lows from the explicit one-loop expression (4.25). It re-mains to note that the operator F (2 , can be interpretedas the local dissipation rate of fluctuations of our scalarfield. C. Effects of the large-scale anisotropy
Now consider the effects of the anisotropy, introducedinto the system at large scales ∼ L through the corre-lation function of the random noise (3.3). As an illus-tration, consider first the case of uniaxial anisotropy: as-sume that the function C ( r /L ) in (3.3) depends also on aconstant unit vector n = { n i } that determines a certaindistinguished direction. Then the irreducible tensor composite operators ac-quire nonzero mean values, with the tensor factors builtof the vector n . For example, the mean value of theoperator (4.15) is proportional to the irreducible ten-sor n i n j − δ ij /d . In general, the mean value of any l thrank irreducible operator is proportional to the tensor n i . . . n i l + . . . , where the ellipsis stands for the contri-butions with the Kronecker δ symbols that make it ir-reducible. Upon substitution into the OPE (5.13), theirtensor indices are contracted with the corresponding in-dices of the coefficient functions C F ( r ). This gives riseto the ( d -dimensional generalizations of the) Legendrepolynomials P l (cos ϑ ), where ϑ is the angle between thevectors r and n .Thus, the OPE expansion in irreducible composite op-erators provides the expansion in the irreducible repre-sentations of the SO ( d ) group. The main contribution tothe “shell” with a given l is determined by the l th rankoperator with the lowest critical dimension (of course, itshould respect the symmetries of the model and of theleft hand side). Clearly, for the structure function S n and l ≤ n the needed operator is F (2 n,l ) i ...i l from (4.14). For l > n we need the operators that contain more deriva-tives than fields.The expansion that takes into account only the leadingterm in each shell has the form (again, we omit ν and µ ): S n = r − n ∆ θ n (cid:88) l =0 A l ( r ) P l (cos ϑ ) ( mr ) ∆ (2 n,l ) + . . . (5.16)with the dimension ∆ (2 n,l ) from (4.24); the ellipsis standsfor the contributions with l > n . For the general large-scale anisotropy, all the spherical harmonics Y ls will ap-pear in the expansion, with the exponents depending onlyon l .From the explicit leading-order expressions (4.25) itfollows that the dimensions (4.24), for a fixed n , mono-tonically increase with l :∆ n,l > ∆ n,p if l > p, (5.17)or, in the differential form, ∂ ∆ n,l /∂l >
0. Similar in-equalities were derived earlier in various models of pas-sively advected vector [33] and scalar [17] fields. Thisfact has a clear physical interpretation: in the presence2of the large-scale anisotropy, anisotropic contributions inthe inertial range exhibit an hierarchy, related to the “de-gree of anisotropy” l : the leading contribution is given bythe isotropic shell ( l = 0); the corresponding anomalousexponent is the same as for the purely isotropic case. Thecontributions with l > mr →
0, the faster the higherthe degree of anisotropy l is. This effect gives quanti-tative support for Kolmogorov’s hypothesis of the localisotropy restoration and appears rather robust, being ob-served for the real fluid turbulence [53].The hierarchy (5.17) becomes more strongly pro-nounced as the degree of compressibility α increases,which can be expressed by the inequality ∂ ∆ n,l /∂l∂α >
0. Thus the anisotropic corrections become further fromone another and from the isotropic term, in contrast tothe situation observed earlier for passive scalar [29, 30]and vector [34] fields, advected by Kraichnan’s ensem-ble. The same inequality holds for the “frozen” regimein the Gaussian model with finite correlation time, thefact overlooked in [29].For l > n , the leading contributions to the l th shellare determined by the operators that involve more deriva-tives than fields. The calculation of their dimensions isa difficult task because of the mixing of such operatorsin renormalization. The hierarchy relations remain validdue to the contributions of the canonical dimensions tothe general expression (2.47): clearly, their critical di-mensions have the forms l − n + O ( y ).Fortunately, for the pair correlation functions, the fullanalog of the expression (5.16) can be presented, with allthe shells included. Indeed, it is clear that the leadingterm of the l th shell now is determined by the singleoperator (4.27) with two fields and l tensor indices: itis unique up to derivatives, which have vanishing meanvalues and do not contribute to the quantities of interest.Thus the desired asymptotic expression has the form (cid:104) θ ( t, x ) θ ( t, x (cid:48) ) (cid:105) = r − θ ∞ (cid:88) l =0 A l ( r ) P l (cos ϑ ) ( mr ) ∆ l (5.18)with the dimensions ∆ l from (4.47), (4.48) for the tracerand (4.47), (4.55) for the density case. The hierarchyof anisotropic contributions, similar to (5.17), holds, atleast for small y , due to the contribution of the canoni-cal dimensions to (4.47): ∆ l = l − O ( y ). Thus theleading term in (5.18) is given by the scalar operator θ .When one passes to the structure function S for thetracer that term is subtracted, and the leading role is in-herited by the scalar operator F (2 , from (4.14) in agree-ment with (5.16). The hierarchy is getting weaker as thecompressibility parameter α grows: ∂ ∆ n,l /∂l∂α <
0, asfollows from the analysis of the explicit one-loop expres-sions (4.47), (4.48), (4.55). Here our results agree withthose for the Kraichnan model: anisotropic correctionsbecome closer to each other and to the isotropic term;cf. [28].
VI. DISCUSSION AND CONCLUSION
We have studied two models of passive scalar advec-tion: the case of the density of a conserved quantity andthe case of a tracer, described by the advection-diffusionequations (3.1) and (3.2), respectively, and subject to arandom large-scale forcing (3.3). The advecting velocityfield is described by the Navier–Stokes equations for acompressible fluid (2.7), (2.8) with an external stirringforce with the correlation function ∝ k − d − y ; see (2.10),(2.11).The full stochastic problems can be formulated as fieldtheoretic models with the action functionals specifiedin (2.12), (3.5) and (3.8). Those models appear multi-plicatively renormalizable, so that the corresponding RGequations can be derived in a standard fashion. Theyhave the only IR attractive fixed point in the physicalrange of the model parameters, and the correlation func-tions reveal scaling behavior in the IR region (inertialand energy-containing ranges).Their inertial-range behavior was studied by means ofthe OPE; existence of anomalous scaling (singular power-like dependence on the integral scale L ) was established.The corresponding anomalous exponents were identifiedwith the scaling (critical) dimensions of certain compositefields (composite operators): powers of the scalar fieldfor the density and powers of its spatial gradients for thetracer, so that they can be systematically calculated asseries in the exponent y . The practical calculations wereperformed in the leading order (one-loop approximation)and are presented in (4.12), (4.25). The results (2.48),(3.27) for primary fields and (4.13) for the operators θ n for the tracer are given by this approximation exactly.Thus we removed two important restrictions of the pre-vious treatments of the passive compressible problem:absence of time correlations and Gaussianity of the ad-vecting velocity field. We stress that in contrast to pre-vious studies that combined compressibility with finitecorrelation time [29, 30], the present model is manifestlyGalilean covariant, and this fact holds in all orders of theperturbation theory.In a few respects, however, the results obtained hereare very similar to those obtained earlier for the com-pressible version of Kraichnan’s rapid-change model [26–28] and the Gaussian model with finite correlation time[29, 30]. First of all, the mechanism of the origin ofanomalous scaling is essentially the same: the anomalousexponents are identified with the dimensions of individual composite operators.Second, those dimensions are insensitive to the specificchoice of the random force (3.3), because the propagator (cid:104) θθ (cid:105) does not enter into the relevant Green functions.(In particular, this means that the anomalous exponentsremain intact if the artificial noise is replaced by an im-posed linear gradient, a more realistic formulation of theproblem). The force maintains the steady state and thusprovides nonvanishing mean values for the composite op-erators, but it does not affect their dimensions.3For the rapid-change case, this fact is naturally in-terpreted within the zero-mode approach, where theequal-time correlation functions satisfy certain differen-tial equations, and the anomalous exponents are relatedto the solutions of their homogeneous analogs, where theforcing terms are discarded; see [7, 9]. On the contrary,the amplitudes are found by matching of these inertial-range zero-mode solutions with the large-scale solutionsof the full inhomogeneous equations, which are nontrivialonly in the presence of the forcing terms.The close resemblance in the RG+OPE pictures of theorigin of anomalous scaling for the present model and itsrapid-change predecessors suggests that for the former,the concept of zero modes (and thus that of statisticalconservation laws) is also applicable, although no closeddifferential equations can be derived for the equal-timecorrelation functions.Although the anomalous exponents are independent ofthe specific choice of the noise, they do depend on the ex-ponent y , the dimension of space d , and on the parameter α that measures the degree of compressibility. In this re-spect, our results are also similar to those obtained forsimpler models. An important difference with Gaussianmodels appears when possible dependence on the timescales is studied. It was argued that the exponents candepend on more details of the velocity ensemble thanonly the exponents, namely, on the dimensionless ratioof the correlation times of the scalar and velocity fields;see e.g. the discussion in [6]. Indeed, analytic results ob-tained for Gaussian models with a finite correlation timewithin the zero-mode technique [54] and the RG+OPEapproach [17, 29, 30] show that such a dependence in-deed takes place, at least for some of the possible scalingregimes.In the present case, the exponents could depend, inprinciple, on the dimensionless parameters u , v , w ,the ratios of various viscosity and diffusion coefficients.After the RG treatment, these parameters are replacedwith the corresponding invariant variables, which exactlyhave the meaning of the ratios of the correlation times ofthe transverse and longitudinal components of the veloc-ity field, the pressure and the scalar field; for a detaileddiscussion of this issue see [17]. Existence of the uniqueIR attractive fixed point shows that in the IR range theseratios tend to their fixed-point values u ∗ , v ∗ , w ∗ irre-spective of the initial values u etc. We conclude thatthe anomalous exponents are independent on the timescale; the dependence observed in previous treatments isan artifact of simplified Gaussian statistics.Another essential difference between our results and those obtained for Kraichnan’s rapid-change model isthat in the latter the anomalous exponents have a finitelimit when the parameter that measures degree of com-pressibility (analog of α from (2.11) in our model) goesto infinity, that is, for the purely potential velocity field.In our case all the nontrivial anomalous dimensions growwith α without bound. Formally, the difference is due tothe fact that the coordinate of the fixed point (2.41) inour model is independent on α and the dependence onit appears only in the numerators of the expressions like(4.12), (4.25). This fact also means that the one-loopcontributions in the critical dimensions become large as α grows, and the one-loop approximation can hardly betrusted even for small y . One may think that the realRG expansion parameter then becomes yα rather than y . In this connection we also recall that the IR attractivefixed point in the one-loop approximation exists for all α but ceases to exist for α = ∞ (purely potential forcing).These facts suggest that, beyond the one-loop approxi-mation, the fixed point (2.41) in fact disappears or losesits stability, and the corresponding scaling regime un-dergoes some qualitative changeover, the possibility sup-ported by the phase transition to a purely chaotic stateobserved in [25] for a simplified model.To investigate this issue, it is necessary to go beyondthe leading one-loop approximation (of course, startingwith the compressible Navier–Stokes equation itself) andto discuss the existence, stability and the dependence on α of the fixed point at least at the two-loop level, whichseems to be a difficult technical task. Another interestinggeneralization of our present investigation is to derive amore realistic expression for the random force correlator(2.11) in order to determine realistic values of α and toexpress it in terms of measurable quantities. Here thecombination of the RG techniques and the energy balanceequation seems promising; see [55] for the incompressiblecase. This work remains for the future and is partly inprogress. ACKNOWLEDGMENTS
The authors are indebted to L.Ts. Adzhemyan, MichalHnatich, Juha Honkonen and M.Yu. Nalimov for discus-sion. The authors acknowledge Saint Petersburg StateUniversity for the research grant 11.38.185.2014. Thework was also supported by the Russian Foundation forBasic Research within the project 12-02-00874-a. [1] U. Frisch,
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