Validity limits of the hydrodynamic pole description of a zero-temperature Bose liquid: high-order correlation functions and nearly-collinear phonon scattering

Josephson (1972) proved that at zero temperature the diagrammatic expansion of the microscopic many-body theory of a Bose-condensed liquid and the expansion generated by quantum hydrodynamics are identical for cumulants of the mass density and current, for the on-shell phonon S-matrix, and for the nonlinear response, provided that (a) a gradient expansion applies and (b) the pole approximation holds: all singularities of correlation functions other than phonon poles are discarded from the compact (vertex) parts of the expansion. The paper itself left open two linked issues: whether non-pole singularities such as multiphonon thresholds and branch cuts remain negligible in correlation functions of arbitrarily high order, and whether the phonon S-matrix equivalence survives for almost collinear phonons, where the nearly linear dispersion makes multiphonon intermediate states degenerate with single phonons. The later literature treats both facets as one infrared problem of the low-energy effective theory of superfluids: the collinear regime is now known to dominate phonon damping and number-changing rates from superfluid 4He to neutron-star cores, and pragmatic regularizations (finite phonon width, next-to-leading-order dispersion resummation) exist, but no controlled, process-independent power counting or resummation scheme is available, and the all-orders validity of the pole approximation for high-order cumulants is unproven. Determining these validity limits remains open.

Background

A Bose liquid is a liquid made of identical bosons; below the lambda transition (the second-order transition, at about 2.2 K at saturated vapor pressure, separating the normal liquid from the superfluid phase) liquid $^4$He is the paradigm. At zero temperature such a liquid is a superfluid: it has a Bose condensate, and its only low-energy excitations are phonons, the Goldstone modes (gapless collective modes) associated with spontaneous breaking of the $U(1)$ particle-number symmetry, with (nearly) linear dispersion $\omega \simeq ck$ where $c$ is the sound speed. Two descriptions meet here. Microscopic many-body theory computes correlation functions of the mass density $\rho(\mathbf{r},t)$ and mass current $\mathbf{j}(\mathbf{r},t)$ by diagrammatic perturbation theory. Quantum hydrodynamics (QHD) instead quantizes the classical equations of an inviscid, irrotational fluid whose energy density is the equation-of-state function $E(\rho)$; its perturbation theory is built from a phonon propagator and hydrodynamic interaction vertices; the Landau–Khalatnikov Hamiltonian of phonon kinetic theory — the low-energy Hamiltonian of the superfluid written in terms of the density and the condensate phase — is of this type. Gavoret and Nozières (1964) had shown microscopically that the low-energy spectrum is phonon-like and that certain two-point functions reduce to the hydrodynamic forms. Josephson (1972) completed this program: introducing a generating functional $F(U)$ for the cumulants (connected correlation functions) of $\rho$ and $\mathbf{j}$, where $U$ denotes auxiliary external fields coupled to these operators, he proved via a diagrammatic variational principle that, after exact reorganizations, the microscopic diagram expansion equals the QHD expansion (with specific non-tree, 'renormalization' diagrams excluded), that propagators and vertices coincide, that the exact nonlinear response reduces to the classical hydrodynamic equations, and that the phonon S-matrix — the on-shell phonon scattering amplitude extracted from residues of multipoint correlation functions as external momenta approach the phonon mass shell — is the same in both theories. The proof rests on two approximations. First, the gradient expansion: all fields vary slowly on the scale set by the correlation length (the inverse gap to non-phonon excitations), so the microscopic functional becomes local. Second, the pole approximation: in the compact parts (the renormalized vertex building blocks of the diagram expansion), every singularity other than phonon poles — multiphonon thresholds, branch cuts, two-phonon continua — is neglected, so each compact part is analytic at small frequency and wavenumber. In modern language, the leading-order effective field theory (EFT) of a superfluid (Leutwyler 1997; Son and Wingate 2006; Nicolis, Penco and Rosen 2014), whose leading action is the pressure viewed as a functional of the phase gradient, is exactly the QHD Hamiltonian, and corrections are organized in a derivative expansion with coefficients matched to the microscopic liquid; Josephson's two approximations correspond to the locality of the EFT coefficients and to tree-level (in his language, Cayley-tree) dominance of low-energy correlators. The known obstruction is kinematic. For exactly linear dispersion, any set of phonons with mutually collinear momenta $k_1 \parallel k_2 \parallel \cdots$ and total momentum $k$ has total energy exactly $ck$: multiphonon states are degenerate with a single phonon. In the almost-collinear regime the multiphonon states that the pole approximation discards are therefore not energetically separated from the poles — internal propagators go nearly on shell, producing collinear singularities — and the small parameter controlling the expansion degenerates. In real $^4$He the weak upward curvature of the phonon dispersion (anomalous dispersion) lifts the degeneracy and is what makes Beliaev decay ($1 \to 2$ phonon splitting, kinematically allowed only when the dispersion is convex) finite. Josephson's discussion explicitly flags the two unresolved issues audited here: non-pole singularities could accumulate or gain weight in correlation functions of sufficiently high order, where the number of potentially collinear internal lines grows; and the S-matrix equivalence may fail for almost collinear phonons, where enhanced phase-space contributions could qualitatively modify collision amplitudes.

Problem Statement

For a zero-temperature Bose-condensed liquid whose only low-energy excitations are phonons (paradigmatically superfluid $^4$He; formulations within a controlled model class such as the weakly interacting Bose gas, or within the matched superfluid effective field theory, are admissible), determine the validity limits of the hydrodynamic description — the conjunction of the gradient expansion and the pole approximation — that underlies the microscopic-hydrodynamic equivalence for density/current correlation functions, the phonon S-matrix, and nonlinear response. Concretely: (i) Establish whether multiphonon thresholds and branch cuts in the microscopic compact parts remain uniformly suppressed as the order of the density/current cumulants grows, or whether they accumulate so that the pole-plus-gradient expansion ceases to be asymptotically accurate in the hydrodynamic regime at some order; equivalently, formulate the power counting of non-pole contributions to $n$-point functions at arbitrary order $n$. (ii) Determine whether the equality of the microscopic and hydrodynamic phonon S-matrices survives when the external phonons are almost collinear — the regime where the linear dispersion makes multiphonon states degenerate with single phonons — and construct the correct leading-order treatment there, including the power counting for multi-phonon amplitudes, or demonstrate by explicit computation that non-pole singularities qualitatively modify the collision amplitudes. These two facets concern the same approximation and share one objective: a controlled characterization, in order, in kinematic angle, and in momentum, of the domain in which the hydrodynamic pole description of a Bose superfluid is asymptotically valid.

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