Subleading instanton asymptotics of the $O(N)$ quartic (N-vector) model beyond two loops in one dimension: the $1/K^{2}$ (four-loop) correction, two-loop correlation functions in $D\ge2$, and the large-order behavior of renormalization constants and RG functions

Perturbation series of the $O(N)$-symmetric quartic field theory (the N-vector model) are factorially divergent, and their large-order growth is governed by instantons — classical tunneling solutions at negative coupling. Giorgini, Jentschura, Malatesta, Parisi, Rizzo, and Zinn-Justin [Phys. Rev. D 101, 125001 (2020)] computed the two-loop correction about the instanton, i.e. the relative-order $1/K$ correction to the factorial large-order growth of the perturbation coefficients of the correlation functions of the one-dimensional N-vector model (the partition function, the two-point function and its second momentum derivative, the four-point function, and the two-point function with a $q^{2}$ insertion), obtaining closed-form coefficients as functions of $N$. The paper explicitly identifies the continuation of this program as future work: the next-order correction of relative order $1/K^{2}$ (four-loop expansion about the instanton), the generalization of the two-loop correlation-function analysis to higher spatial dimensions $D\ge2$, and the derivation of the large-order behavior of the renormalization constants $Z_{\phi}$, $Z_{\phi^{2}}$, $Z_{g}$, $\delta m^{2}$ and hence of the renormalization-group functions $\beta(g)$, $\eta(g)$, $\eta_{2}(g)$, which is the gateway to large-order analysis of critical-exponent series. None of these continuations has been carried out.

Background

The N-vector model is the $O(N)$-symmetric quartic scalar field theory; in one spatial dimension it is the quantum mechanics of an $N$-component field $\vec q(t)=\{q_{1},\ldots,q_{N}\}$ with Euclidean action

$$S[q]=\int dt\,\left[\frac{1}{2}\left(\frac{\partial q}{\partial t}\right)^{2}+\frac{1}{2}q^{2}+\frac{g}{4}q^{4}\right],$$

in the normalization of Giorgini et al. [Phys. Rev. D 101, 125001 (2020)]. Its perturbation series in the coupling $g$ diverge factorially: since Dyson's observation for QED, and the quantitative analyses of Bender and Wu for the anharmonic oscillator and of Brézin, Le Guillou, and Zinn-Justin for $\phi^{2N}$ field theories, it is known that the coefficient $G_{K}$ of order $K$ grows like $\Gamma(K+b)$ for some offset $b$. The mechanism is that for negative coupling the classical potential is unstable: the field can tunnel through the barrier, and observables acquire an imaginary part generated by an instanton — a classical finite-action solution of the Euclidean equations of motion at $g<0$ (in one dimension, the familiar localized bounce of the double-well problem). A dispersion relation expresses each perturbative coefficient through this imaginary part,

$$G_{K}=\frac{(-1)^{K}}{\pi}\int_{0}^{\infty}\frac{dg}{g^{K+1}}\,\mathrm{Im}\,G(-g-i\epsilon),$$

so that large $K$ is dominated by the small-$|g|$ behavior of the imaginary part. Expanding the imaginary part about the instanton,

$$\mathrm{Im}\,G(g)=c(N,D)(-g)^{-(n+N+D-1)/2}\exp\!\left[\frac{A}{g}\right]\left[1+d(N,D)\,g+O(g^{2})\right],\quad g<0,$$

where $n$ is the number of fields in the correlator, $A$ is the instanton action ($A=4/3$ in the one-dimensional conventions above), and $D$ is the spatial dimension; since $A>0$ and $g<0$, the exponential $\exp[A/g]=\exp[-A/|g|]$ is exponentially small at weak coupling, the hallmark of a tunneling amplitude. Inserting this form into the dispersion relation yields

$$G_{K}=\frac{c(N,D)}{\pi}\,\Gamma(K+b)\left(\frac{1}{A}\right)^{(n+N+D-1)/2}\left(-\frac{1}{A}\right)^{K}\left[1-\frac{A\,d(N,D)}{K+b-1}+\cdots\right],\qquad b=\frac{n+N+D-1}{2}.$$

The one-loop term $c(N,D)$ (the plain instanton) gives the leading factorial growth; the two-loop correction $d(N,D)$ gives the first subleading correction of relative order $1/K$; the four-loop term would give the next one of relative order $1/K^{2}$. The loop expansion about the instanton requires the collective-coordinate formalism: the instanton breaks time-translation invariance and the internal $O(N)$ symmetry, producing one time zero mode and $N-1$ angular zero modes — fluctuation directions with vanishing eigenvalue, one for each broken symmetry, which cannot be integrated over by Gaussian fluctuation integrals and must instead be treated exactly, as collective coordinates (the instanton's start time and its orientation in internal space). The transformation of the path integral to these variables generates a Jacobian — a functional determinant mixing the longitudinal fluctuations (along the instanton's direction in internal space) with the transverse ones (orthogonal to it) — that must be kept to the required loop order.

In the 2020 paper, the six authors computed $d(N,1)$ in closed form (rational functions of $N$, powers of $\pi$, and Riemann zeta values, with the PSLQ integer-relation algorithm (a standard tool that reconstructs exact analytic expressions from high-precision numerical values) used in the final analytic steps) for the partition function, the two-point function at zero momentum and its second derivative with respect to momentum, the four-point function at zero momentum, and the two-point function with a 'wigglet' insertion — the insertion of the composite field $q^{2}=\vec q\cdot\vec q$, obtainable as a mass derivative of the two-point function; 'wigglet' is the term the paper itself uses for this insertion. All are computed at zero momentum transfer because these are the quantities that feed the Callan–Symanzik equation, the renormalization-group (RG) equation satisfied by the vertex functions: the renormalization constants — $Z_{\phi}$ (wave-function), $Z_{\phi^{2}}$ (the $q^{2}$-insertion), $Z_{g}$ (coupling), and the mass counterterm $\delta m^{2}$ — are fixed by renormalization conditions imposed on the vertex functions, which follow from the correlation functions by a Legendre transformation, and they determine the RG functions $\beta(g)$ (coupling flow), $\eta(g)$ (anomalous dimension of $\phi$), and $\eta_{2}(g)$ (anomalous dimension of $\phi^{2}$, controlling the correlation length).

In one dimension the calculation is an end in itself, because 1D fluctuations are too strong for a second-order phase transition and there is no critical behavior to analyze. In dimensions $D\ge2$ the model has nontrivial RG fixed points, and its critical exponents were historically extracted from resummed high-order perturbation series [Le Guillou and Zinn-Justin, Phys. Rev. B 21, 3976 (1980)]; the large-order growth of those series, and in particular of the RG functions themselves, is exactly what the instanton asymptotics would quantify. The free-energy (partition-function) sector was already generalized to $D=2,3$ at two-loop order by Malatesta, Parisi, and Rizzo [Nucl. Phys. B 922, 293 (2017)], where the instanton is no longer known in closed analytic form and the fluctuation eigenvalues must be summed numerically. The correlation-function analysis in $D\ge2$, the next order in $1/K$ in one dimension, and the renormalization-constant program remain to be done, and the 2020 paper closes by naming them: 'The generalization to relative order $1/K^{2}$ (four-loop order about the instanton) and the consideration of field theories in higher dimensions are the natural next steps in this program and currently under investigation.'

Problem Statement

Extend the instanton-based subleading large-order asymptotics of the $O(N)$ quartic (N-vector) model beyond the two-loop, one-dimensional, correlation-function level reached by Giorgini et al. [Phys. Rev. D 101, 125001 (2020)], along the two axes the paper itself names, with the renormalization-group application as the stated target. The objective of the program is to interpolate between finite-order perturbative calculations and the factorial large-order asymptotics of the perturbation coefficients, the latter being sharpened by subleading corrections about the instanton. The two axes are the $1/K^{2}$ coefficients of component (a) in one dimension, and the renormalization-constant and RG-function asymptotics of component (c) in $D=2$ and $D=3$, for which the correlation-function analysis of component (b) is the prerequisite step.

(a) Next order in $1/K$, one dimension: compute the correction of relative order $1/K^{2}$ to the large-order asymptotics of the perturbation coefficients of the correlation functions whose two-loop ($1/K$) coefficients were obtained in the 2020 paper — the partition function, the two-point function at zero momentum and its second derivative with respect to momentum, the four-point function at zero momentum, and the two-point function with a $q^{2}$ (wigglet) insertion — i.e. evaluate the four-loop expansion about the instanton for these correlators, expressing the new asymptotic coefficients as explicit functions of $N$ (the two-loop results closed into combinations of rationals in $N$, powers of $\pi$, and Riemann zeta values, in the normalization of the 2020 paper where the instanton action is $A=4/3$).

(b) Next dimension, two loops: generalize the two-loop instanton analysis of these correlation functions to the spatial dimensions $D=2$ and $D=3$, where the instanton profile admits no closed analytic form (only transseries representations — asymptotic expansions combining powers and exponentials of the expansion parameter — and high-precision numerical profiles are available), the collective coordinates include the $D$ translation zero modes, and the fluctuation determinants require renormalization. This component determines the two-loop ($1/K$) coefficient for the same five correlators as in component (a), evaluated numerically where the $D=2,3$ profile admits no closed form. The free-energy (partition-function) case in $D=2,3$ was completed at two loops by Malatesta–Parisi–Rizzo (2017); the correlation-function case, which is what the Callan–Symanzik input requires, is the open task.

(c) Renormalization constants and RG functions: combine the instanton-derived correlation-function asymptotics (at zero momentum transfer) with the vertex-function renormalization conditions — passing from correlation functions to vertex functions via the Legendre transform — to compute the large-order behavior of the renormalization constants $Z_{\phi}$, $Z_{\phi^{2}}$, $Z_{g}$, and the mass counterterm $\delta m^{2}$, and thereby of the renormalization-group functions $\beta(g)$, $\eta(g)$, and $\eta_{2}(g)$, in dimensions $D=2$ and $D=3$. This is the input needed to characterize the factorial growth of the perturbative series for critical exponents of the $O(N)$ model in two and three dimensions, in the tradition of the resummed series of Le Guillou and Zinn-Justin.

References

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