Two-loop completion of the double-expansion renormalization-group analysis of passive-scalar anomalous scaling near two dimensions

The field-theoretic renormalization-group (RG) analysis of a passively advected scalar near spatial dimension $d=2$ requires the double expansion -- a simultaneous expansion in the forcing-exponent deviation $\varepsilon$ and the dimensional deviation $\delta=(d-2)/2$ -- together with a modified minimal-subtraction scheme, because additional ultraviolet divergences appear at $d=2$. Gladyshev, Jurčišinová, Jurčišin, Remecký and Zalom [Phys. Rev. E 86, 036302 (2012)] completed this program at one loop: fixed points, the one-loop turbulent Prandtl number (0.6404 at $d=2$, 0.7179 at $d=3$ in the modified scheme), and the critical dimensions of the composite operators that determine the anomalous exponents of the scalar structure functions. They stated explicitly that the two-loop level is open: the one-loop relation $g^*=g_{1*}+g_{2*}$ 'will surely' be destroyed by two-loop corrections, and the turbulent Prandtl number within the double-expansion scheme was posed as an open problem. The $d=2$ instance of the Prandtl leg was resolved by Jurčišinová, Jurčišin and Remecký [Theor. Math. Phys. 200, 1139 (2019)]: the two-loop double-expansion value is $\mathrm{Pr}_t=0.27472$ versus the one-loop $0.64039$, a correction of about 57%, showing that two-loop effects are large near $d=2$. The surviving open problem is the two-loop completion of the double expansion: the two-loop beta functions and anomalous dimensions of the full charge set and the coordinates of the kinetic fixed point, the two-loop critical dimensions $\Delta[N,p]$ of the composite operators governing structure-function anomalous scaling for $d\ge2$, and the two-loop Prandtl number $\mathrm{Pr}_t(d)$ for $d>2$, which together decide whether the one-loop predictions and the predicted 'nontrivial influence' of the $d=2$ divergences on $d>2$ results are quantitatively reliable.

Background

A passive scalar field $\theta(t,\mathbf{x})$ -- a temperature or concentration marker transported by a turbulent flow without reacting back on the velocity -- displays anomalous scaling (intermittency): its single-time structure functions $S_N(r)=\langle[\theta(t,\mathbf{x}+\mathbf{r})-\theta(t,\mathbf{x})]^N\rangle$ deviate in the inertial range from the dimensional (Kolmogorov-type) prediction. In the field-theoretic treatment the velocity obeys the stochastic Navier-Stokes equation driven by a Gaussian random force, and the model is rewritten as a path-integral field theory with response fields (auxiliary fields, denoted by primes, that encode the stochastic forcing in the action); an infrared-stable fixed point of the renormalization group (RG) determines the inertial-range exponents. The anomalous parts of the exponents are governed by the critical dimensions of composite operators -- local monomials $F[N,p]\propto[(\mathbf{n}\cdot\partial)\theta]^{p}(\partial_i\theta\,\partial_i\theta)^{(N-p)/2}$ built from $N$ derivative factors of the scalar field, where $p$ single derivatives are contracted with the anisotropy direction $\mathbf{n}$ -- via the operator-product expansion (OPE, the short-distance expansion whose coefficient functions carry the scale dependence).

For $d>2$ this program uses the standard $\varepsilon$ expansion: $\varepsilon$, the deviation of the velocity-forcing exponent from its Kolmogorov value, is the small parameter of an analytical regularization, and ultraviolet (UV) poles are removed by minimal subtraction (MS). The two-loop level of this standard scheme is known: the two-loop turbulent Prandtl number (the ratio of turbulent viscosity to turbulent scalar diffusivity, obtained as the inverse of the fixed-point value of the charge $u$: $\mathrm{Pr}_t^{(1)}=1/u_*^{(1)}$ at one loop and $\mathrm{Pr}_t=1/u_{\mathrm{eff}}$ at two loops, where $u_{\mathrm{eff}}$ is the scheme-independent two-loop fixed-point value of the inverse Prandtl charge) receives a correction below 2% at $d=3$, and the two-loop anomalous dimensions of the leading composite operators are available.

At $d=2$ the standard scheme fails: the correlation functions $\langle v'v'\rangle$ and $\langle\theta'\theta'\rangle$ acquire additional UV divergences, so the model is no longer multiplicatively renormalizable. The remedy, introduced by Honkonen and Nalimov, is the double expansion: a local term $v'_i k^2 v'_j$ (a second velocity charge $g_2$) is added to the forcing, so that the velocity-force correlator takes the form $D^v_{ij}(k)=\delta(t)\nu_0^3(g_{10}k^{2-2\delta-2\varepsilon}+g_{20}k^2)P_{ij}(k)$ with $2\delta=d-2$ and $2\varepsilon$ the departure of the forcing exponent from its reference $k^2$ form (the physical Kolmogorov value is $\varepsilon=2$, while $\varepsilon$ and $\delta$ act as formal small parameters); the scalar-force correlator is modified analogously as $D^{\theta}(k)=\delta(t)\nu_0^3 u_0^3(g_{30}k^{2-2\delta-2a\varepsilon}+g_{40}k^2)$, with its own nonlocal and local charges $g_{30},g_{40}$, an injection-spectrum parameter $a$, and $u_0$ the ratio of the molecular scalar diffusivity $\nu_0 u_0$ to the viscosity $\nu_0$. The parameter $a$ is free in the model definition; however, the exact all-order relations $\beta_{g_3}=g_3(-2a\varepsilon+3\gamma_2)$ and $\beta_u=u(\gamma_1-\gamma_2)$ give $\gamma_{2*}=2a\varepsilon/3$ at the fixed point, and a kinetic fixed point with $u_*\neq0$ requires $\gamma_{1*}=\gamma_{2*}$, hence $a=1$; the Prandtl-number benchmarks quoted here correspond to this $a=1$ kinetic fixed point. One then expands simultaneously in $\varepsilon$ and $\delta$, using a modified MS scheme that includes the $d=2$-specific counterterms while keeping the $d$-dependence explicit.

Gladyshev, Jurčišinová, Jurčišin, Remecký and Zalom completed the one-loop level of this program for the Navier-Stokes-advected passive scalar: they found the RG fixed points (the kinetic fixed point is the inertial-range one), computed the one-loop turbulent Prandtl number ($0.6404$ at $d=2$; $0.7179$ at $d=3$ in the modified scheme), and obtained the one-loop critical dimensions $\Delta[N,p]$ of the composite operators, which reproduce the standard $\varepsilon$-expansion results for $d>2$ and remain valid at $d=2$. The one-loop fixed-point structure is simplified by the linearity $g^*=g_{1*}+g_{2*}$, which the authors state holds only at one loop. They pose the two-loop level as open work and expect 'a nontrivial influence of the existence of the additional divergences in two dimensions on the physical results for $d>2$'. This expectation is sharpened by the only two-loop double-expansion result obtained so far: the turbulent Prandtl number at $d=2$ changes from $0.64039$ (one loop) to $0.27472$ (two loops).

Problem Statement

Within the double-expansion scheme -- the simultaneous expansion in the forcing-exponent parameter $\varepsilon$ and the dimensional parameter $\delta=(d-2)/2$, with the modified minimal-subtraction scheme that includes the additional $d=2$ ultraviolet counterterms -- compute the two-loop structure of the renormalization-group analysis of a passive scalar advected by the stochastic Navier-Stokes velocity field, to the same level of completeness as the existing one-loop analysis of Gladyshev et al. (2012). Concretely:

(1) The two-loop beta functions and anomalous dimensions of the full charge set of the model, the coordinates of the kinetic (inertial-range) fixed point to second order in the double expansion, and the fate of the one-loop relation $g^*=g_{1*}+g_{2*}$ and of the infrared stability of the fixed point, consistent with the exact fixed-point anomalous dimensions $\gamma_{1*}=\gamma_{2*}=2\varepsilon/3$ and $\gamma_{3*}=2\varepsilon+2\delta$ at the kinetic fixed point (the $a=1$ instance of the all-order relations; for general $a$ one has $\gamma_{2*}=2a\varepsilon/3$, and $u_*\neq0$ enforces $a=1$), which hold to all loop orders because the beta functions in their form in terms of the anomalous dimensions -- $\beta_{g_1}=g_1(-2\varepsilon+3\gamma_1)$, $\beta_{g_2}=g_2(2\delta+3\gamma_1-\gamma_3)$, $\beta_{g_3}=g_3(-2a\varepsilon+3\gamma_2)$, $\beta_{g_4}=g_4(2\delta+3\gamma_2-\gamma_4)$, $\beta_u=u(\gamma_1-\gamma_2)$ -- and the linearity of $\beta_{g_3}$ and $\beta_{g_4}$ in the scalar charges are loop-order independent.

(2) The two-loop critical dimensions $\Delta[N,p]$ of the composite operators $F[N,p]$ that govern the anomalous exponents of the single-time structure functions $S_N(r)$ of the passive scalar, as functions of $d\ge2$ (equivalently of $\varepsilon$ and $\delta$), including the required two-loop operator mixing.

(3) The two-loop turbulent Prandtl number $\mathrm{Pr}_t(d)$ within the double expansion for $d>2$ (the $d=2$ value, $\mathrm{Pr}_t=0.27472$, is already known), quantifying whether and how the additional $d=2$ divergences modify the physical $d>2$ results relative to the standard two-loop $\varepsilon$-expansion values.

Deliverables (1)-(3) are jointly the two-loop completion of one calculation: the anomalous exponents (2) are evaluated at the fixed point (1) through the same renormalization constants, and the Prandtl number (3) shares the same fixed-point and velocity-scalar cross-renormalization machinery; together they answer whether the one-loop double-expansion predictions survive two-loop corrections. The genuinely two-dimensional inverse-cascade regime, which the source itself flags as requiring a separate analysis, is not part of this problem.

References

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