acquires a negative imaginary part related to the amplitudes of the outgoing waves describing the departure of the resultant bound pair and the remaining free fermion. For weak interaction potentials, we derive an approximate formula of the hypervolume by using the Born expansion. For the square-barrier and the square-well potentials and the Gaussian potential, we solve the three-body Schrödinger equation to compute

D_F

numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero

D_F

and the three-body recombination rate in one dimension.

DOI: 10.48550/arxiv.2302.13685

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Three-boson scattering hypervolume for a nonzero orbital angular momentum

Pui In Ip, Shina Tan

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D_F

appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states,

D_F

D_F

numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero

D_F

and the three-body recombination rate in one dimension.

{D}_{F}

appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states,

{D}_{F}

{D}_{F}

numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero

{D}_{F}

and the three-body recombination rate in one dimension.

We define the three-body scattering hypervolume

{D}_{F}

for identical spin-polarized fermions in two dimensions by considering the wave function of three such fermions colliding at zero energy and zero orbital angular momentum. We derive the asymptotic expansions of such a wave function when three fermions are far apart or one pair and the third fermion are far apart, and

{D}_{F}

appears in the coefficients of such expansions. For weak-interaction potentials, we derive an approximate formula of

{D}_{F}

by using the Born expansion. We then study the shift of energy of three such fermions in a large periodic area due to

{D}_{F}

. This shift is proportional to

{D}_{F}

times the square of the area of the triangle formed by the momenta of the fermions. We also calculate the shifts of energy and of pressure of spin-polarized two-dimensional Fermi gases due to a nonzero

{D}_{F}

and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases in two dimensions.

{D}_{F}

. The scattering hypervolume

D

was first defined for identical bosons in 2008. It is the three-body analog of the two-body scattering length. We solve the three-body Schr\"odinger equation asymptotically when the three fermions are far apart or one pair and the third fermion are far apart, deriving two asymptotic expansions of the wave function. Unlike the case of bosons for which

D

has the dimension of length to the fourth power, here the

{D}_{F}

we define has the dimension of length to the eighth power. We then analyze the interaction energy of three such fermions with momenta $\ensuremath{\hbar}{\mathbf{k}}_{1}$, $\ensuremath{\hbar}{\mathbf{k}}_{2}$, and $\ensuremath{\hbar}{\mathbf{k}}_{3}$ in a large periodic cubic box. The energy shift due to

{D}_{F}

is proportional to ${D}_{F}/{\mathrm{\ensuremath{\Omega}}}^{2}$, where $\mathrm{\ensuremath{\Omega}}$ is the volume of the box. We also calculate the shifts of energy and pressure of spin-polarized Fermi gases due to a nonzero

{D}_{F}

and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.

D

called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-$\ensuremath{\uparrow}$ (spin-$\ensuremath{\downarrow}$) fermion and one pair, formed by the other two fermions, are far apart and derive three asymptotic expansions of the wave function. The three-body scattering hypervolume

D

appears in the coefficients of such expansions at the order of ${B}^{\ensuremath{-}5}$, where

B={[({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2})/2]}^{1/2}

is the hyperradius of the triangle formed by the three fermions (we assume that the three fermions have the same mass), and

{s}_{1}, {s}_{2}, {s}_{3}

are the sides of the triangle. We compute the

T

-matrix element for three such fermions colliding at low energy in terms of

D

in the absence of two-body interactions. When the interactions are weak, we calculate

D

approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to

D

and generalize this result to the many-fermion system.

D

also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of

D

and the density and temperature of the gas.

We study the wave function

φ^{(3)}

of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length

a

. We derive asymptotic expansions of

φ^{(3)}

in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as

\ln^3(B/a)

at large hyperradius

B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}

. At order

B^{-2}\ln^{-3}(B/a)

, we identify a three-body parameter

D

with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states,

D

acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of

D

in terms of

φ^{(3)}

and the changes of the two-body and three-body potentials. We also study the effects of

D

and

φ^{(3)}

on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.

We analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum

L=2

, assuming short range interactions. By solving the Schrödinger equation asymptotically, we derive two expansions of the wave function when three bosons are far apart or a pair of bosons and the third boson are far apart. The scattering hypervolume

D

is defined for this collision. Unlike the scattering hypervolume defined by one of us in 2008, whose dimension is length to the fourth power, the dimension of

D

studied in the present paper is length to the eighth power. We then derive the expression of

D

when the interaction potentials are weak, using the Born's expansion. We also calculate the energy shift of such three bosons with three different momenta

\hbar \mathbf{k_{1}}

\hbar\mathbf{k_{2}}

and

\hbar\mathbf{k_{3}}

in a large periodic box. The obtained energy shift depends on

D^{(0)}/Ω^{2}

and

D/Ω^{2}

, where

D^{(0)}

is the three-body scattering hypervolume defined for the three-body

L=0

collision and

Ω

is the volume of the periodic box. We also calculate the contribution of

D

to the three-body T-matrix element for low-energy collisions. We then calculate the shift of the energy and the three-body recombination rate due to

D^{(0)}

and

D

in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from

D

is proportional to

T^2

if the temperature

T

is much larger than the quantum degeneracy temperature but still much lower than the temperature scale at which the thermal de Broglie wave length becomes comparable to the physical range of interaction.