* For each $l$, let us denote by $\{s_{l,n}; n \in \mathbb{N}\}$ the allowed values of $s$, in increasing order ($s_{l,0} < s_{l,1} < ...$). The index $\nu$ can be identified with the set of quantum numbers $(l, m, n)$. All the eigenfunctions $\phi_{(l,m,n)}$ with $-l \le m \le l$ correspond to the same $s_{l,n}$, so that each $s_{l,n}$ is $2l+1$ times degenerate. For $l=1$, the $s_{l,n}$ are the real positive solutions different from 1 of the transcendental equation (4) [the solution $s=1$ should be discarded since it leads to an identically vanishing $\phi(\alpha)$]. For arbitrary $l$, the $s_{l,n}$ also solve transcendental equations (see, e.g., [38] and refs. therein); the smallest of all $s_{l,n}$ is $s_{l=1,n=0}$, which is denoted by $s$ for short throughout the article.

* The eigenfunctions are

$$ \phi(\Omega) = \mathcal{N}(1 - \hat{P}) \frac{\varphi(\alpha)}{\sin(2\alpha)} Y_l^m(\hat{\rho}) \quad (99) $$

where $l \in \mathbb{N}$ and $m \in \{-l; \dots; l\}$ are the total-angular-momentum quantum numbers,<fnref n="37" /> whereas $\mathcal{N} > 0$ is a normalization constant such that $(\phi|\phi) = 1$. Note that $\mathcal{N}$ does not depend on $m$, as follows from the relation $L_\pm \tilde{\phi}_m = \hbar\sqrt{l(l+1)-m(m+1)} \tilde{\phi}_{m\pm 1}$ where $L_\pm := L_x \pm iL_y$ and $\tilde{\phi}_m := (1-\hat{P}) \frac{\varphi(\alpha)}{\sin(2\alpha)} Y_l^m(\hat{\rho})$. For $l=1$,

$$ \varphi(\alpha) = -s \cos\left[s\left(\frac{\pi}{2} - \alpha\right)\right] + \sin\left[s\left(\frac{\pi}{2} - \alpha\right)\right] \tan\alpha \quad (100) $$

(for arbitrary $l$, see, e.g., [38] and refs. therein). We refer to the $\{\phi_v(\Omega)\}$ as unitary hyperangular wavefunctions.<fnref n="38" /> We use the shorthand notation $\phi_m := \phi_{(l=1,m,n=0)}$.

The value of the normalization constant, for $l=1$, is

$$ \mathcal{N} = 0.31149\dots \quad (101) $$

We computed this value as follows. We have $\mathcal{N} = \sqrt{3/J(s)}$ with $J(s) = \sum_{m=-1}^{1}(\tilde{\phi}_m|\tilde{\phi}_m)$.

We evaluate the sum over $m$ thanks to the identity: $\sum_{m=-1}^{1} Y_1^m(\hat{u}) * Y_1^m(\hat{u}') = \frac{3}{4\pi} P_1(\hat{u} \cdot \hat{u}')$ for any unit vectors $\hat{u}$ and $\hat{u}'$. This yields

$$ J(s) = \frac{3}{4\pi} \int d^5\Omega \left[ \left( \frac{\varphi(\alpha)}{\sin(2\alpha)} \right)^2 + \left( \frac{\varphi(\alpha')}{\sin(2\alpha')} \right)^2 - 2 P_1(\hat{\rho} \cdot \hat{\rho}') \frac{\varphi(\alpha)}{\sin(2\alpha)} \frac{\varphi(\alpha')}{\sin(2\alpha')} \right] \quad (102) $$

where $\alpha'$ and $\hat{\rho}'$ are obtained from $\alpha$ and $\hat{\rho}$ by permutation of particles 1 and 2. The integrand in (102) only depends on $\alpha$ and $\alpha'$, since $\hat{\rho} \cdot \hat{\rho}' = \frac{\sin^2(\alpha) + \sin^2(\alpha') - 5/4}{\cos(\alpha) \cos(\alpha')}$. Moreover $\alpha'$ is a function of $\alpha$ and of $u = \hat{r} \cdot \hat{\rho}$, since $u = \frac{2}{\sqrt{3}} \times \frac{3/4 - \sin^2(\alpha)/2 - \sin^2(\alpha')}{\cos(\alpha) \sin(\alpha)}$. Hence the integrand in (102) is a function of $\alpha$ and $u$. To evaluate the integral we use the formula $\int d^5\Omega f(\Omega) = \int_0^{\pi/2} d\alpha \sin^2(\alpha) \cos^2(\alpha) \int d\hat{r} d\hat{\rho} f(\alpha, \hat{r}, \hat{\rho})$. Since the integrand is independent of $\hat{r}$ and of the azimuthal angle of $\hat{\rho}$ w.r.t. $\hat{r}$, we can integrate over them, which just gives a factor $4\pi \times 2\pi$. We are left with the integral over $\alpha$ and $u$. The change of variable $u \to \alpha'$ then yields

$$ J(s) = 8\sqrt{3\pi} \int_{\mathcal{D}} d\alpha d\alpha' \cos(\alpha) \sin(\alpha) \cos(\alpha') \sin(\alpha') \\ \times \left[ \left( \frac{\varphi(\alpha)}{\sin(2\alpha)} \right)^2 + \left( \frac{\varphi(\alpha')}{\sin(2\alpha')} \right)^2 - 2 P_1 \left( \frac{\sin(\alpha)^2 + \sin(\alpha')^2 - \frac{5}{4}}{\cos(\alpha)\cos(\alpha')} \right) \frac{\varphi(\alpha)}{\sin(2\alpha)} \frac{\varphi(\alpha')}{\sin(2\alpha')} \right] \quad (103) $$

<footnote n="37">³⁷More explicitly, the considered wavefunction is an eigenstate of $\mathbf{L}^2$ with eigenvalue $l(l+1)\hbar^2$, and of $L_z$ with eigenvalue $m\hbar$, where $\mathbf{L}$ is the relative angular momentum of the 3 particles, $\mathbf{L} = -i\hbar(\mathbf{r} \times \nabla_{\mathbf{r}} + \mathbf{p} \times \nabla_{\mathbf{p}})$. Note that the total angular momentum of the three particles is the sum of their relative and center-of-mass angular momenta: $-i\hbar \sum_{j=1}^3 \mathbf{r}_j \times \nabla_{\mathbf{r}_j} = \mathbf{L} - i\hbar \mathbf{C} \times \nabla_{\mathbf{C}}$.</footnote>

<footnote n="38">³⁸Like the standard hyperspherical harmonics, the $\phi_v$ are eigenstates of the Laplacian on the hypersphere, Eq. (95); but they also satisfy the unitary-limit contact condition Eq. (96) (together with fermionic antisymmetry) which leads to non-integer eigenvalues $s^2$.</footnote>
* For each $l$, let us denote by $\{s_{l,n}; n \in \mathbb{N}\}$ the allowed values of $s$, in increasing order ($s_{l,0} < s_{l,1} < ...$). The index $\nu$ can be identified with the set of quantum numbers $(l, m, n)$. All the eigenfunctions $\phi_{(l,m,n)}$ with $-l \le m \le l$ correspond to the same $s_{l,n}$, so that each $s_{l,n}$ is $2l+1$ times degenerate. For $l=1$, the $s_{l,n}$ are the real positive solutions different from 1 of the transcendental equation (4) [the solution $s=1$ should be discarded since it leads to an identically vanishing $\phi(\alpha)$]. For arbitrary $l$, the $s_{l,n}$ also solve transcendental equations (see, e.g., [38] and refs. therein); the smallest of all $s_{l,n}$ is $s_{l=1,n=0}$, which is denoted by $s$ for short throughout the article.

* The eigenfunctions are

$$ \phi(\Omega) = \mathcal{N}(1 - \hat{P}) \frac{\varphi(\alpha)}{\sin(2\alpha)} Y_l^m(\hat{\rho}) \quad (99) $$

where $l \in \mathbb{N}$ and $m \in \{-l; \dots; l\}$ are the total-angular-momentum quantum numbers,<fnref n="37" /> whereas $\mathcal{N} > 0$ is a normalization constant such that $(\phi|\phi) = 1$. Note that $\mathcal{N}$ does not depend on $m$, as follows from the relation $L_\pm \tilde{\phi}_m = \hbar\sqrt{l(l+1)-m(m+1)} \tilde{\phi}_{m\pm 1}$ where $L_\pm := L_x \pm iL_y$ and $\tilde{\phi}_m := (1-\hat{P}) \frac{\varphi(\alpha)}{\sin(2\alpha)} Y_l^m(\hat{\rho})$. For $l=1$,

$$ \varphi(\alpha) = -s \cos\left[s\left(\frac{\pi}{2} - \alpha\right)\right] + \sin\left[s\left(\frac{\pi}{2} - \alpha\right)\right] \tan \alpha \quad (100) $$

(for arbitrary $l$, see, e.g., [38] and refs. therein). We refer to the $\{\phi_\nu(\Omega)\}$ as unitary hyperangular wavefunctions.<fnref n="38" /> We use the shorthand notation $\phi_m := \phi_{(l=1,m,n=0)}$.

The value of the normalization constant, for $l=1$, is

$$ \mathcal{N} = 0.31149\dots \quad (101) $$

We computed this value as follows. We have $\mathcal{N} = \sqrt{3/J(s)}$ with $J(s) = \sum_{m=-1}^{1}(\tilde{\phi}_m|\tilde{\phi}_m)$.

We evaluate the sum over $m$ thanks to the identity: $\sum_{m=-1}^{1} Y_1^m(\hat{u}) * Y_1^m(\hat{u}') = \frac{3}{4\pi} P_1(\hat{u} \cdot \hat{u}')$ for any unit vectors $\hat{u}$ and $\hat{u}'$. This yields

$$ J(s) = \frac{3}{4\pi} \int d^5\Omega \left[ \left( \frac{\varphi(\alpha)}{\sin(2\alpha)} \right)^2 + \left( \frac{\varphi(\alpha')}{\sin(2\alpha')} \right)^2 - 2 P_1(\hat{\rho} \cdot \hat{\rho}') \frac{\varphi(\alpha)}{\sin(2\alpha)} \frac{\varphi(\alpha')}{\sin(2\alpha')} \right] \quad (102) $$

where $\alpha'$ and $\hat{\rho}'$ are obtained from $\alpha$ and $\hat{\rho}$ by permutation of particles 1 and 2. The integrand in (102) only depends on $\alpha$ and $\alpha'$, since $\hat{\rho} \cdot \hat{\rho}' = \frac{\sin^2(\alpha) + \sin^2(\alpha') - 5/4}{\cos(\alpha) \cos(\alpha')}$. Moreover $\alpha'$ is a function of $\alpha$ and of $u = \hat{r} \cdot \hat{\rho}$, since $u = \frac{2}{\sqrt{3}} \times \frac{3/4 - \sin^2(\alpha)/2 - \sin^2(\alpha')}{\cos(\alpha) \sin(\alpha)}$. Hence the integrand in (102) is a function of $\alpha$ and $u$. To evaluate the integral we use the formula $\int d^5\Omega f(\Omega) = \int_0^{\pi/2} d\alpha \sin^2(\alpha) \cos^2(\alpha) \int d\hat{r} d\hat{\rho} f(\alpha, \hat{r}, \hat{\rho})$. Since the integrand is independent of $\hat{r}$ and of the azimuthal angle of $\hat{\rho}$ w.r.t. $\hat{r}$, we can integrate over them, which just gives a factor $4\pi \times 2\pi$. We are left with the integral over $\alpha$ and $u$. The change of variable $u \to \alpha'$ then yields

$$ J(s) = 8\sqrt{3\pi} \int_{\mathcal{D}} d\alpha d\alpha' \cos(\alpha) \sin(\alpha) \cos(\alpha') \sin(\alpha') \\ \times \left[ \left( \frac{\varphi(\alpha)}{\sin(2\alpha)} \right)^2 + \left( \frac{\varphi(\alpha')}{\sin(2\alpha')} \right)^2 - 2 P_1 \left( \frac{\sin(\alpha)^2 + \sin(\alpha')^2 - \frac{5}{4}}{\cos(\alpha)\cos(\alpha')} \right) \frac{\varphi(\alpha)}{\sin(2\alpha)} \frac{\varphi(\alpha')}{\sin(2\alpha')} \right] \quad (103) $$

<footnote n="37">³⁷More explicitly, the considered wavefunction is an eigenstate of $\mathbf{L}^2$ with eigenvalue $l(l+1)\hbar^2$, and of $L_z$ with eigenvalue $m\hbar$, where $\mathbf{L}$ is the relative angular momentum of the 3 particles, $\mathbf{L} = -i\hbar(\mathbf{r} \times \nabla_{\mathbf{r}} + \mathbf{p} \times \nabla_{\mathbf{p}})$. Note that the total angular momentum of the three particles is the sum of their relative and center-of-mass angular momenta: $-i\hbar \sum_{j=1}^3 \mathbf{r}_j \times \nabla_{\mathbf{r}_j} = \mathbf{L} - i\hbar \mathbf{C} \times \nabla_{\mathbf{C}}$.</footnote>

<footnote n="38">³⁸Like the standard hyperspherical harmonics, the $\phi_\nu$ are eigenstates of the Laplacian on the hypersphere, Eq. (95); but they also satisfy the unitary-limit contact condition Eq. (96) (together with fermionic antisymmetry) which leads to non-integer eigenvalues $s^2$.</footnote>
where $\mathcal{D}$ is the domain $\{(\alpha, \alpha') | \frac{\pi}{3} \le \alpha + \alpha' \le \frac{2\pi}{3}, |\alpha - \alpha'| \le \frac{\pi}{3}\}$. We evaluate analytically the integrals over $\alpha$ and $\alpha'$ for the first and second term of (103); for the third term, we evaluate analytically the integral over $\alpha'$, and perform numerically the remaining integration over $\alpha$.

## Appendix B. Homogeneous gas

Let us show that for the homogeneous gas at equilibrium, $C_{2,1}^{(m)}$ is independent of m. We will use the following lemma: Let $\hat{\mathcal{O}}$ be an operator acting on functions of $\Omega$, such that

$$ (\hat{\mathcal{O}} \phi_m)(\Omega) = \lambda_m \phi_m(\Omega). $$

Then,

$$ \frac{N_1 (N_1 - 1) N_1}{2} \langle \hat{\mathcal{O}} \theta(\epsilon - R) \rangle \underset{\epsilon \to 0}{\sim} \epsilon^{2s+2} \sum_{m=-1}^{1} \lambda_m C_{2,1}^{(m)} \quad (104) $$

with $\theta$ the Heaviside function. This follows from Eq. (15) and the fact that the $\phi_m(\Omega)$ are orthonormal.

We then consider $Q := \langle \hat{L}_z \theta(\epsilon - R) \rangle$. In the absence of time-reversal symmetry breaking, we have $Q = 0$. On the other hand, applying the above lemma to $\hat{\mathcal{O}} = \hat{L}_z$ gives $Q \propto C_{2,1}^{(1)} - C_{2,1}^{(-1)}$. Hence $C_{2,1}^{(1)} = C_{2,1}^{(-1)}$.

Since the system is isotropic, the quantity $\langle \hat{L}_i^2 \theta(R - \epsilon) \rangle$ is the same for $i = x, y$ and $z$; hence $\langle \hat{L}_z^2 \theta(R - \epsilon) \rangle = \langle \hat{L}^2 \theta(R - \epsilon) \rangle / 3$. Applying the lemma to both sides then yields $C_{2,1}^{(1)} + C_{2,1}^{(-1)} = 2 \sum_{m=-1}^{1} C_{2,1}^{(m)}/3$, which gives $C_{2,1}^{(1)} = C_{2,1}^{(0)}$.

Finally we note that for the unpolarized gas ($N_1 = N_1$) at equilibrium, the two states $\uparrow$ and $\downarrow$ play a symmetric role, so that $C_{2,1}^{(m)} = C_{1,2}^{(m)}$.

## Appendix C. Two-body contribution to the energy correction

In this Appendix we show that $\Delta_2$ is given by the r.h.s. of (82), to leading order in the zero-range regime. In the mass-balanced case, to leading order, we have $\delta E \approx \Delta_2$, so that we recover the result of [56]. We provide the present derivation because it is not identical (although similar) to the one in Ref. [56] (the quantity $\Delta_2$, as defined here, does not explicitly appear in [56]).

Let us introduce a length $d$ such that

$$ b \ll d \ll 1/k_{\text{typ}}. \quad (105) $$

$\Delta_2$ is given by the contribution to the integral (86) coming from the region of $(\mathbf{r}_1, \dots, \mathbf{r}_N)$ such that there is a pair of particles $(i, j)$ of spins $(\uparrow, \downarrow)$ separated by a distance $r_{ij} < d$ while all interparticle distances are $\gg d$ [the result will not depend on the value of $d$ within the range (105)].

Let us denote by $\mathcal{R}_{13}$ the region where these conditions are met for the pair (1,3),

$$ \mathcal{R}_{13} := \{ (\mathbf{r}_1, \dots, \mathbf{r}_N) \text{ such that } r \equiv r_{13} < d \text{ and all other } r_{ij} \gg d \}. $$

Since all $\uparrow\downarrow$ pairs of particles contribute equally,

$$ \Delta_2 = N_\uparrow N_\downarrow \int_{\mathcal{R}_{13}} d^3\mathbf{r}_1 \dots d^3\mathbf{r}_N (\psi^{(0)} * V \psi^{(1)})(\mathbf{r}_1, \dots, \mathbf{r}_N). \quad (106) $$

In $\mathcal{R}_{13}$, we expect a factorization of the many-body wavefunction of the finite-range model

$$ \psi^{(1)}(\mathbf{r}_1, \dots, \mathbf{r}_N) \approx \chi_{\mathcal{E}}(r) A(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (107) $$

where we can assume $\chi_{\mathcal{E}}$ to be rotationally invariant, as was checked in [56].

Injecting the ansatz (107) into the N-body Schrödinger equation (71) yields, in the region $\mathcal{R}_{13}$,

$$ -\frac{\hbar^2}{m} \Delta_{\mathbf{r}} \chi_{\mathcal{E}}(r) + V_2(r) \chi_{\mathcal{E}}(r) = \mathcal{E} \chi_{\mathcal{E}}(r) \quad (108) $$
where $\mathcal{D}$ is the domain $\{(\alpha, \alpha') | \frac{\pi}{3} \le \alpha + \alpha' \le \frac{2\pi}{3}, |\alpha - \alpha'| \le \frac{\pi}{3}\}$. We evaluate analytically the integrals over $\alpha$ and $\alpha'$ for the first and second term of (103); for the third term, we evaluate analytically the integral over $\alpha'$, and perform numerically the remaining integration over $\alpha$.

## Appendix B. Homogeneous gas

Let us show that for the homogeneous gas at equilibrium, $C_{2,1}^{(m)}$ is independent of m. We will use the following lemma: Let $\hat{\mathcal{O}}$ be an operator acting on functions of $\Omega$, such that

$$ (\hat{\mathcal{O}} \phi_m)(\Omega) = \lambda_m \phi_m(\Omega). $$

Then,

$$ \frac{N_1 (N_1 - 1) N_1}{2} \langle \hat{\mathcal{O}} \theta(\epsilon - R) \rangle \underset{\epsilon \to 0}{\sim} \epsilon^{2s+2} \sum_{m=-1}^{1} \lambda_m C_{2,1}^{(m)} \quad (104) $$

with $\theta$ the Heaviside function. This follows from Eq. (15) and the fact that the $\phi_m(\Omega)$ are orthonormal.

We then consider $Q := \langle \hat{L}_z \theta(\epsilon - R) \rangle$. In the absence of time-reversal symmetry breaking, we have $Q = 0$. On the other hand, applying the above lemma to $\hat{\mathcal{O}} = \hat{L}_z$ gives $Q \propto C_{2,1}^{(1)} - C_{2,1}^{(-1)}$. Hence $C_{2,1}^{(1)} = C_{2,1}^{(-1)}$.

Since the system is isotropic, the quantity $\langle \hat{L}_i^2 \theta(R - \epsilon) \rangle$ is the same for $i = x, y$ and $z$; hence $\langle \hat{L}_z^2 \theta(R - \epsilon) \rangle = \langle \hat{L}^2 \theta(R - \epsilon) \rangle / 3$. Applying the lemma to both sides then yields $C_{2,1}^{(1)} + C_{2,1}^{(-1)} = 2 \sum_{m=-1}^{1} C_{2,1}^{(m)}/3$, which gives $C_{2,1}^{(1)} = C_{2,1}^{(0)}$.

Finally we note that for the unpolarized gas ($N_1 = N_1$) at equilibrium, the two states $\uparrow$ and $\downarrow$ play a symmetric role, so that $C_{2,1}^{(m)} = C_{1,2}^{(m)}$.

## Appendix C. Two-body contribution to the energy correction

In this Appendix we show that $\Delta_2$ is given by the r.h.s. of (82), to leading order in the zero-range regime. In the mass-balanced case, to leading order, we have $\delta E \approx \Delta_2$, so that we recover the result of [56]. We provide the present derivation because it is not identical (although similar) to the one in Ref. [56] (the quantity $\Delta_2$, as defined here, does not explicitly appear in [56]).

Let us introduce a length $d$ such that

$$ b \ll d \ll 1/k_{\text{typ}}. \quad (105) $$

$\Delta_2$ is given by the contribution to the integral (86) coming from the region of $(\mathbf{r}_1, \dots, \mathbf{r}_N)$ such that there is a pair of particles $(i, j)$ of spins $(\uparrow, \downarrow)$ separated by a distance $r_{ij} < d$ while all interparticle distances are $\gg d$ [the result will not depend on the value of $d$ within the range (105)].

Let us denote by $\mathcal{R}_{13}$ the region where these conditions are met for the pair (1,3),

$$ \mathcal{R}_{13} := \{ (\mathbf{r}_1, \dots, \mathbf{r}_N) \text{ such that } r \equiv r_{13} < d \text{ and all other } r_{ij} \gg d \}. $$

Since all $\uparrow\downarrow$ pairs of particles contribute equally,

$$ \Delta_2 = N_1 N_1 \int_{\mathcal{R}_{13}} d^3 \mathbf{r}_1 \dots d^3 \mathbf{r}_N (\psi^{(0)} * V \psi^{(1)})(\mathbf{r}_1, \dots, \mathbf{r}_N). \quad (106) $$

In $\mathcal{R}_{13}$, we expect a factorization of the many-body wavefunction of the finite-range model

$$ \psi^{(1)}(\mathbf{r}_1, \dots, \mathbf{r}_N) \approx \chi_{\mathcal{E}}(r) A(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (107) $$

where we can assume $\chi_{\mathcal{E}}$ to be rotationally invariant, as was checked in [56].

Injecting the ansatz (107) into the N-body Schrödinger equation (71) yields, in the region $\mathcal{R}_{13}$,

$$ -\frac{\hbar^2}{m} \Delta_{\mathbf{r}} \chi_{\mathcal{E}}(r) + V_2(r) \chi_{\mathcal{E}}(r) = \mathcal{E} \chi_{\mathcal{E}}(r) \quad (108) $$
with

$$
\mathcal{E} := E - 2U(\mathbf{c}) - \sum_{k \in \{2;4;...;N\}} U(\mathbf{r}_k) \\
+ \frac{1}{A(\mathbf{c};\mathbf{r}_2, \mathbf{r}_4, ..., \mathbf{r}_N)} \left( \frac{\hbar^2}{4m} \Delta_{\mathbf{c}} + \frac{\hbar^2}{2m} \sum_{k \in \{2;4;...;N\}} \Delta_{\mathbf{r}_k} \right) A(\mathbf{c};\mathbf{r}_2, \mathbf{r}_4, ..., \mathbf{r}_N),
$$

up to corrections that are negligible in the zero-range regime, as was checked in [56]. We omitted the dependence of $\mathcal{E}$ on $(\mathbf{c};\mathbf{r}_2, \mathbf{r}_4, ..., \mathbf{r}_N)$ to alleviate notations. We neglected all interaction potentials other than $V_2(r)$, because we are far outside their ranges: In the considered region $\mathcal{R}_{13}$, all interparticle distances other than $r \equiv r_{13}$, and hence also all triplet hyperradii, are $\gg b$. For the same reason, we will replace $V$ by $V_2$ in (106).

For the zero-range model, we also expect a factorization of the many-body wavefunction
in $\mathcal{R}_{13}$,

$$
\psi^{(0)}(\mathbf{r}_1, \dots, \mathbf{r}_N) \approx \chi_{\mathcal{E}}^{(0)}(r) A(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (109)
$$

where $\chi_{\mathcal{E}}^{(0)}(r)$ is the s-wave two-body scattering state at energy $\mathcal{E}$ for the zero-range model, i.e. the
solution of

$$
-\frac{\hbar^2}{m} \Delta_{\mathbf{r}} \chi_{\mathcal{E}}^{(0)}(r) = \mathcal{E} \chi_{\mathcal{E}}^{(0)}(r) \quad (110)
$$

with the contact condition: $\chi_{\mathcal{E}}^{(0)}(r) = 1/r - 1/a_2 + O(r)$ for $r \to 0$. Here we normalized $\chi_{\mathcal{E}}^{(0)}$ in such a way that the function $A$ in (109) is the same one than in (14). The solution is

$$
r \chi_{\mathcal{E}}^{(0)}(r) = \frac{1}{f_k^{(0)}} \frac{\sin(kr)}{k} + e^{ikr} \quad (111)
$$

where $f_k^{(0)} = -(1/a_2 + ik)^{-1}$ is the scattering amplitude of the zero-range model, and $k := \sqrt{m\mathcal{E}/\hbar}$, with the determination $k = i\sqrt{-m\mathcal{E}/\hbar}$ if $\mathcal{E} < 0$.

We note that for $r \to 0$. from the Taylor expansion of (111), we get a subleading singular con-
tribution to the asymptotic expansion of $\psi^{(0)}$ given by $-k^2 r A/2$, in agreement with Eqs. (135,136)
of [56]. From this we can infer that $k$ is typically $\lesssim k_{\text{typ}}$.

Scattering theory gives the large-distance behavior of the finite-range scattering state $\chi_{\mathcal{E}}$:

$$
r \chi_{\mathcal{E}}(r) \approx_{r \gg b_2} \frac{1}{f_k} \frac{\sin(kr)}{r} + e^{ikr} \quad (112)
$$

where $f_k$ is the s-wave scattering amplitude associated to the interaction potential $V_2(r)$.

Here we normalized $\chi_{\mathcal{E}}$ in such a way that $\chi_{\mathcal{E}}(r) \sim 1/r$ for $\mathcal{E} \to 0$ and $r \to \infty$, in agreement with the assumption that the same function $A$ appears in (109) and (107).

Next, in (106), we can thus replace $V$ by $V_2$ and substitute ((109), (107)), which yields

$$
\Delta_2 \approx N_\uparrow N_\downarrow \int d^3 c \, d^3 r_2 \, d^3 r_4 \dots d^3 r_N |A(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N)|^2 W(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (113)
$$

where

$$
W := \int_{r<d} d^3 r (\chi_{\mathcal{E}}^{(0)} V_2 \chi_{\mathcal{E}})(r).
$$

It remains to evaluate $W$. Using (108), (110), we rewrite it as

$$
W = \frac{\hbar^2}{m} \int_{r<d} d^3 r \left( \chi_{\mathcal{E}}^{(0)} \Delta_{\mathbf{r}} \chi_{\mathcal{E}} - \chi_{\mathcal{E}} \Delta_{\mathbf{r}} \chi_{\mathcal{E}}^{(0)} \right) (r).
$$

This gives, by the divergence theorem, $W = \frac{4\pi\hbar^2}{m} \mathcal{W}$ with

$$
\mathcal{W} := \left( \chi_{\mathcal{E}}^{(0)} \frac{\partial \chi_{\mathcal{E}}}{\partial r} - \chi_{\mathcal{E}} \frac{\partial \chi_{\mathcal{E}}^{(0)}}{\partial r} \right)_{r=d} .
$$
with

$$
\mathcal{E} := E - 2U(\mathbf{c}) - \sum_{k \in \{2;4;\dots;N\}} U(\mathbf{r}_k) \\
\qquad + \frac{1}{A(\mathbf{c};\mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N)} \left( \frac{\hbar^2}{4m} \Delta_{\mathbf{c}} + \frac{\hbar^2}{2m} \sum_{k \in \{2;4;\dots;N\}} \Delta_{\mathbf{r}_k} \right) A(\mathbf{c};\mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N),
$$

up to corrections that are negligible in the zero-range regime, as was checked in [56]. We omitted the dependence of $\mathcal{E}$ on $(\mathbf{c};\mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N)$ to alleviate notations. We neglected all interaction potentials other than $V_2(r)$, because we are far outside their ranges: In the considered region $\mathcal{R}_{13}$, all interparticle distances other than $r \equiv r_{13}$, and hence also all triplet hyperradii, are $\gg b$. For the same reason, we will replace $V$ by $V_2$ in (106).

For the zero-range model, we also expect a factorization of the many-body wavefunction in $\mathcal{R}_{13}$,

$$
\psi^{(0)}(\mathbf{r}_1, \dots, \mathbf{r}_N) \approx \chi_{\mathcal{E}}^{(0)}(r) A(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (109)
$$

where $\chi_{\mathcal{E}}^{(0)}(r)$ is the s-wave two-body scattering state at energy $\mathcal{E}$ for the zero-range model, i.e. the solution of

$$
-\frac{\hbar^2}{m} \Delta_{\mathbf{r}} \chi_{\mathcal{E}}^{(0)}(r) = \mathcal{E} \chi_{\mathcal{E}}^{(0)}(r) \quad (110)
$$

with the contact condition: $\chi_{\mathcal{E}}^{(0)}(r) = 1/r - 1/a_2 + O(r)$ for $r \to 0$. Here we normalized $\chi_{\mathcal{E}}^{(0)}$ in such a way that the function $A$ in (109) is the same one than in (14). The solution is

$$
r \chi_{\mathcal{E}}^{(0)}(r) = \frac{1}{f_k^{(0)}} \frac{\sin(kr)}{k} + e^{ikr} \quad (111)
$$

where $f_k^{(0)} = -(1/a_2 + ik)^{-1}$ is the scattering amplitude of the zero-range model, and $k := \sqrt{m\mathcal{E}/\hbar}$, with the determination $k = i\sqrt{-m\mathcal{E}/\hbar}$ if $\mathcal{E} < 0$.

We note that for $r \to 0$, from the Taylor expansion of (111), we get a subleading singular con-
tribution to the asymptotic expansion of $\psi^{(0)}$ given by $-k^2 r A/2$, in agreement with Eqs. (135,136)
of [56]. From this we can infer that $k$ is typically $\lesssim k_{\text{typ}}$.

Scattering theory gives the large-distance behavior of the finite-range scattering state $\chi_{\mathcal{E}}$:

$$
r \chi_{\mathcal{E}}(r) \approx_{r \gg b_2} \frac{1}{f_k} \frac{\sin(kr)}{r} + e^{ikr} \quad (112)
$$

where $f_k$ is the s-wave scattering amplitude associated to the interaction potential $V_2(r)$.

Here we normalized $\chi_{\mathcal{E}}$ in such a way that $\chi_{\mathcal{E}}(r) \sim 1/r$ for $\mathcal{E} \to 0$ and $r \to \infty$, in agreement with the assumption that the same function $A$ appears in (109) and (107).

Next, in (106), we can thus replace $V$ by $V_2$ and substitute ((109), (107)), which yields

$$
\Delta_2 \approx N_\uparrow N_\downarrow \int d^3 c \, d^3 r_2 \, d^3 r_4 \dots d^3 r_N |A(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N)|^2 W(\mathbf{c}; \mathbf{r}_2, \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (113)
$$

where

$$
W := \int_{r<d} d^3 r (\chi_{\mathcal{E}}^{(0)} V_2 \chi_{\mathcal{E}})(r).
$$

It remains to evaluate $W$. Using (108), (110), we rewrite it as

$$
W = \frac{\hbar^2}{m} \int_{r<d} d^3 r \left( \chi_{\mathcal{E}}^{(0)} \Delta_{\mathbf{r}} \chi_{\mathcal{E}} - \chi_{\mathcal{E}} \Delta_{\mathbf{r}} \chi_{\mathcal{E}}^{(0)} \right) (r).
$$

This gives, by the divergence theorem, $W = \frac{4\pi\hbar^2}{m} \mathcal{W}$ with

$$
\mathcal{W} := \left( \chi_{\mathcal{E}}^{(0)} \frac{\partial \chi_{\mathcal{E}}}{\partial r} - \chi_{\mathcal{E}} \frac{\partial \chi_{\mathcal{E}}^{(0)}}{\partial r} \right)_{r=d} .
$$
Equivalently,

$$W = \left( r\chi_{\varepsilon}^{(0)} \frac{\partial(r\chi_{\varepsilon})}{\partial r} - r\chi_{\varepsilon} \frac{\partial(r\chi_{\varepsilon}^{(0)})}{\partial r} \right)_{r=d};$$

since $d \gg b_2$, we can directly substitute (111), (112) and their derivatives, which yields $W \approx 1/f_k - 1/f_k^{(0)}$.

Since $kb_2 \lesssim k_{typ}b_2 \ll 1$, we can use the low-energy expansion of the scattering amplitude: $-1/f_k = 1/a_2 + ik - k^2r_e/2 + \dots$ in the limit $kb_2 \to 0$, where $r_e$ is by definition the effective range. This gives $W \approx 2\pi\varepsilon r_e$. Substituting this into (113), we conclude that $\Delta_2$ is indeed given by the r.h.s. of (82).

## Appendix D. Mass-imbalanced case

In this Appendix we consider the case where the ↑ and ↓ fermions have different masses, $m_↑ > m_↓$ for definiteness. Experimentally, this is realized in a mixture of two fermionic species, such as $^{40}$K-$^{6}$Li [181], $^{161}$Dy-$^{40}$K [182], or $^{53}$Cr-$^{6}$Li [183]. In Sec. D.1, we extend the relations obtained in the main text to the mass-imbalanced case. In Sec. D.2 we show that, when the mass ratio exceeds a critical value, a conceptual simplification takes place: A generalized zero-range model can be introduced, within which relations involving the three-body parameters can be formulated and derived more directly. The results of this Appendix are valid if $m_↑/m_↓$ is smaller than the threshold where the five-body Efimov effect appears [103], which implies that the four-body Efimov effect [102] and three-body Efimov effect [176] do not take place either; moreover $m_↑/m_↓$ should not be too close to the four-body and five-body Efimov thresholds, as discussed in Sec. D.3.

Obviously, in the N-body Schrödinger equation, the mass m is replaced by the mass $m_i$ of particle i ($m_i = m_↑$ or $m_↓$ depending on the spin of particle i):

$$\sum_{i=1}^{N} \left[ -\frac{\hbar^2}{2m_i} \Delta_{\mathbf{r}_i} + U(\mathbf{r}_i) \right] \psi = E \psi \quad (114)$$

for the zero-range model, and

$$\sum_{i=1}^{N} \left[ -\frac{\hbar^2}{2m_i} \Delta_{\mathbf{r}_i} + U(\mathbf{r}_i) \right] \psi + \sum_{i:\uparrow, j:\downarrow} V_2(r_{ij}) \psi = E \psi, \quad (115)$$

for the finite-range model.

It will prove convenient to introduce the angles $\varphi$ and $\tilde{\varphi}$ related to the mass ratio by

$$\sin \varphi = \frac{m_\uparrow}{m_\uparrow + m_\downarrow}, \qquad \sin \tilde{\varphi} = \frac{m_\downarrow}{m_\uparrow + m_\downarrow}.$$

The definitions of the center-of-mass and Jacobi coordinates should be generalized to the unequal-mass case. The center-of-mass of particles 1 and 3, which appears in the two-body contact condition (14), is now $\mathbf{c} = (m_↑\mathbf{r}_1 + m_↓\mathbf{r}_3)/(m_↑ + m_↓)$. The center-of-mass of particles 1,2,3 (assumed to have spins $↑, ↓, ↓$) is now $\mathbf{C} = [m_↑(\mathbf{r}_1 + \mathbf{r}_2) + m_↓\mathbf{r}_3]/(2m_↑ + m_↓)$, and the Jacobi coordinate $\rho$ is now $(\mathbf{r}_2 - \mathbf{c})/\cos\varphi$, while we still have $\mathbf{r} := \mathbf{r}_3 - \mathbf{r}_1$ and $\mathbf{R} := (\mathbf{r}, \rho)$. Similarly, the center-of-mass of particles 1,3,4 (of spins $↑, ↓, ↓$) is now $\tilde{\mathbf{C}} = [m_↑\mathbf{r}_1 + m_↓(\mathbf{r}_3 + \mathbf{r}_4)]/(m_↑ + 2m_↓)$, and the Jacobi coordinate $\tilde{\rho}$ is now $(\mathbf{r}_4 - \mathbf{c})/\cos\tilde{\varphi}$, while we still have $\tilde{\mathbf{r}} := \mathbf{r}_1 - \mathbf{r}_2$ and $\tilde{\mathbf{R}} := (\tilde{\mathbf{r}}, \tilde{\rho})$. The three-body Schrödinger equation is then still given by (92) provided we define m as twice the reduced mass,

$$m := 2 \frac{m_\uparrow m_\downarrow}{m_\uparrow + m_\downarrow}.$$
Equivalently,

$$W = \left( r\chi_{\varepsilon}^{(0)} \frac{\partial(r\chi_{\varepsilon})}{\partial r} - r\chi_{\varepsilon} \frac{\partial(r\chi_{\varepsilon}^{(0)})}{\partial r} \right)_{r=d} ;$$

since $d \gg b_2$, we can directly substitute (111),(112) and their derivatives, which yields $W \approx 1/f_k - 1/f_k^{(0)}$.

Since $kb_2 \lesssim k_{typ}b_2 \ll 1$, we can use the low-energy expansion of the scattering amplitude: $-1/f_k = 1/a_2 + ik - k^2r_e/2 + \dots$ in the limit $kb_2 \to 0$, where $r_e$ is by definition the effective range. This gives $W \approx 2\pi\varepsilon r_e$. Substituting this into (113), we conclude that $\Delta_2$ is indeed given by the r.h.s. of (82).

## Appendix D. Mass-imbalanced case

In this Appendix we consider the case where the ↑ and ↓ fermions have different masses, $m_↑ > m_↓$ for definiteness. Experimentally, this is realized in a mixture of two fermionic species, such as $^{40}$K-$^{6}$Li [181], $^{161}$Dy-$^{40}$K [182], or $^{53}$Cr-$^{6}$Li [183]. In Sec. D.1, we extend the relations obtained in the main text to the mass-imbalanced case. In Sec. D.2 we show that, when the mass ratio exceeds a critical value, a conceptual simplification takes place: A generalized zero-range model can be introduced, within which relations involving the three-body parameters can be formulated and derived more directly. The results of this Appendix are valid if $m_↑/m_↓$ is smaller than the threshold where the five-body Efimov effect appears [103], which implies that the four-body Efimov effect [102] and three-body Efimov effect [176] do not take place either; moreover $m_↑/m_↓$ should not be too close to the four-body and five-body Efimov thresholds, as discussed in Sec. D.3.

Obviously, in the N-body Schrödinger equation, the mass m is replaced by the mass $m_i$ of particle i ($m_i = m_↑$ or $m_↓$ depending on the spin of particle i):

$$\sum_{i=1}^{N} \left[ -\frac{\hbar^2}{2m_i} \Delta_{\mathbf{r}_i} + U(\mathbf{r}_i) \right] \psi = E \psi \quad (114)$$

for the zero-range model, and

$$\sum_{i=1}^{N} \left[ -\frac{\hbar^2}{2m_i} \Delta_{\mathbf{r}_i} + U(\mathbf{r}_i) \right] \psi + \sum_{i:\uparrow, j:\downarrow} V_2(r_{ij}) \psi = E \psi, \quad (115)$$

for the finite-range model.

It will prove convenient to introduce the angles $\varphi$ and $\tilde{\varphi}$ related to the mass ratio by

$$\sin \varphi = \frac{m_{\uparrow}}{m_{\uparrow} + m_{\downarrow}}, \qquad \sin \tilde{\varphi} = \frac{m_{\downarrow}}{m_{\uparrow} + m_{\downarrow}}.$$

The definitions of the center-of-mass and Jacobi coordinates should be generalized to the unequal-mass case. The center-of-mass of particles 1 and 3, which appears in the two-body contact condition (14), is now $\mathbf{c} = (m_↑\mathbf{r}_1 + m_↓\mathbf{r}_3)/(m_↑ + m_↓)$. The center-of-mass of particles 1,2,3 (assumed to have spins $↑, ↓, ↓$) is now $\mathbf{C} = [m_↑(\mathbf{r}_1 + \mathbf{r}_2) + m_↓\mathbf{r}_3]/(2m_↑ + m_↓)$, and the Jacobi coordinate $\rho$ is now $(\mathbf{r}_2 - \mathbf{c})/\cos\varphi$, while we still have $\mathbf{r} := \mathbf{r}_3 - \mathbf{r}_1$ and $\mathbf{R} := (\mathbf{r}, \rho)$. Similarly, the center-of-mass of particles 1,3,4 (of spins $↑, ↓, ↓$) is now $\tilde{\mathbf{C}} = [m_↑\mathbf{r}_1 + m_↓(\mathbf{r}_3 + \mathbf{r}_4)]/(m_↑ + 2m_↓)$, and the Jacobi coordinate $\tilde{\rho}$ is now $(\mathbf{r}_4 - \mathbf{c})/\cos\tilde{\varphi}$, while we still have $\tilde{\mathbf{r}} := \mathbf{r}_1 - \mathbf{r}_2$ and $\tilde{\mathbf{R}} := (\tilde{\mathbf{r}}, \tilde{\rho})$. The three-body Schrödinger equation is then still given by (92) provided we define m as twice the reduced mass,

$$m := 2 \frac{m_{\uparrow} m_{\downarrow}}{m_{\uparrow} + m_{\downarrow}}.$$
The continuity equation and the probability current are still given by (56), (57) provided we define

$$ \mathbf{X} := \left( \sqrt{\frac{m_1}{m}} \mathbf{r}_1, \dots, \sqrt{\frac{m_N}{m}} \mathbf{r}_N \right). \qquad (116) $$

While in the mass-balanced case, there was a single exponent $s$ associated to the unitary three-body problem, in the mass-imbalanced case we have two exponents $s$ and $\tilde{s}$, associated respectively to the $\uparrow\uparrow\downarrow$ and $\uparrow\downarrow\downarrow$ unitary three-body problems [97, 176]. As a function of $m_1/m_\downarrow$, $s$ is continuously decreasing and vanishes at the Efimov-effect threshold $m_1/m_\downarrow = 13.6069\dots$, while $\tilde{s}$ is continuously increasing and tends to 2 for $m_1/m_\downarrow \to \infty$.

### D.1. Extension of the relations from the mass-balanced case

The number of nearby $\uparrow\uparrow\downarrow$ triplets is still given by

$$ N_{2,1}(\epsilon) \sim_{\epsilon \to 0} C_{2,1} \epsilon^{2s+2} \qquad (117) $$

whereas the number of nearby $\uparrow\downarrow\downarrow$ triplets is now given by

$$ N_{1,2}(\epsilon) \sim_{\epsilon \to 0} C_{1,2} \epsilon^{2\tilde{s}+2}. \qquad (118) $$

For our convention $m_1 > m_\downarrow$, we have $s < \tilde{s}$. Hence the total number of nearby triplets $N_3(\epsilon)$ is dominated by the $\uparrow\uparrow\downarrow$ contribution, so that

$$ N_3(\epsilon) \sim_{\epsilon \to 0} C_{2,1} \epsilon^{2s+2}. \qquad (119) $$

The remarks at the end of Sec. 2.1 remain valid. In particular, the bunching effect due to the zero-range interactions still overcompensates the antibunching effect due to Pauli exclusion, both for $N_{2,1}(\epsilon)$ and $N_{1,2}(\epsilon)$, because $s < \tilde{s} < 2$.

When three particles of spins $\uparrow, \downarrow, \downarrow$ approach each other, the asymptotic behavior of the many-body wavefunction is still given by (15). This yields [using the Jacobian $|\partial(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3)/\partial(\mathbf{C}, \mathbf{R})| = \cos^3\varphi$]

$$ C_{2,1} = \sum_{m=-1}^{+1} C_{2,1}^{(m)} $$

with

$$ C_{2,1}^{(m)} = N_{\uparrow}(N_{\downarrow}-1)N_{\downarrow} \frac{\cos^3 \varphi}{4(s+1)} \int |B_m(\mathbf{C}; \mathbf{r}_4, \dots, \mathbf{r}_N)|^2 d^3\mathbf{C} d^3\mathbf{r}_4 \dots d^3\mathbf{r}_N. \qquad (120) $$

When three particles of spins $\uparrow, \downarrow, \downarrow$ approach each other, the asymptotic behavior of the many-body wavefunction is given by (18) with $s$ replaced by $\tilde{s}$, i.e.

$$ \psi(\mathbf{r}_1, \dots, \mathbf{r}_N) \sim_{\tilde{R} \to 0} \tilde{R}^{\tilde{s}-2} \sum_{m=-1}^{+1} \phi_m(\tilde{\Omega}) \tilde{B}_m(\tilde{C}; \mathbf{r}_2, \mathbf{r}_5, \dots, \mathbf{r}_N). \qquad (121) $$

This yields

$$ C_{1,2} = \sum_{m=-1}^{+1} C_{1,2}^{(m)} $$

with

$$ C_{1,2}^{(m)} = N_{\uparrow}(N_{\downarrow}-1)N_{\downarrow} \frac{\cos^3 \tilde{\varphi}}{4(\tilde{s}+1)} \int |\tilde{B}_m(\tilde{C}; \mathbf{r}_2, \mathbf{r}_5, \dots, \mathbf{r}_N)|^2 d^3\tilde{C} d^3\mathbf{r}_2 d^3\mathbf{r}_5 \dots d^3\mathbf{r}_N. $$
The continuity equation and the probability current are still given by (56), (57) provided we define

$$ \mathbf{X} := \left( \sqrt{\frac{m_1}{m}} \mathbf{r}_1, \dots, \sqrt{\frac{m_N}{m}} \mathbf{r}_N \right). \qquad (116) $$

While in the mass-balanced case, there was a single exponent $s$ associated to the unitary three-body problem, in the mass-imbalanced case we have two exponents $s$ and $\tilde{s}$, associated respectively to the $\uparrow\uparrow\downarrow$ and $\uparrow\downarrow\downarrow$ unitary three-body problems [97, 176]. As a function of $m_1/m_\downarrow$, $s$ is continuously decreasing and vanishes at the Efimov-effect threshold $m_1/m_\downarrow = 13.6069\dots$, while $\tilde{s}$ is continuously increasing and tends to 2 for $m_1/m_\downarrow \to \infty$.

### D.1. Extension of the relations from the mass-balanced case

The number of nearby $\uparrow\uparrow\downarrow$ triplets is still given by

$$ N_{2,1}(\epsilon) \sim_{\epsilon \to 0} C_{2,1} \epsilon^{2s+2} \qquad (117) $$

whereas the number of nearby $\uparrow\downarrow\downarrow$ triplets is now given by

$$ N_{1,2}(\epsilon) \sim_{\epsilon \to 0} C_{1,2} \epsilon^{2\tilde{s}+2}. \qquad (118) $$

For our convention $m_1 > m_\downarrow$, we have $s < \tilde{s}$. Hence the total number of nearby triplets $N_3(\epsilon)$ is dominated by the $\uparrow\uparrow\downarrow$ contribution, so that

$$ N_3(\epsilon) \sim_{\epsilon \to 0} C_{2,1} \epsilon^{2s+2}. \qquad (119) $$

The remarks at the end of Sec. 2.1 remain valid. In particular, the bunching effect due to the zero-range interactions still overcompensates the antibunching effect due to Pauli exclusion, both for $N_{2,1}(\epsilon)$ and $N_{1,2}(\epsilon)$, because $s < \tilde{s} < 2$.

When three particles of spins $\uparrow, \downarrow, \downarrow$ approach each other, the asymptotic behavior of the many-body wavefunction is still given by (15). This yields [using the Jacobian $|\partial(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3)/\partial(\mathbf{C}, \mathbf{R})| = \cos^3\varphi$]

$$ C_{2,1} = \sum_{m=-1}^{+1} C_{2,1}^{(m)} $$

with

$$ C_{2,1}^{(m)} = N_{\uparrow}(N_{\downarrow}-1)N_{\downarrow} \frac{\cos^3 \varphi}{4(s+1)} \int |B_m(\mathbf{C}; \mathbf{r}_4, \dots, \mathbf{r}_N)|^2 d^3\mathbf{C} d^3\mathbf{r}_4 \dots d^3\mathbf{r}_N. \qquad (120) $$

When three particles of spins $\uparrow, \downarrow, \downarrow$ approach each other, the asymptotic behavior of the many-body wavefunction is given by (18) with $s$ replaced by $\tilde{s}$, i.e.

$$ \psi(\mathbf{r}_1, \dots, \mathbf{r}_N) \sim_{\tilde{R} \to 0} \tilde{R}^{\tilde{s}-2} \sum_{m=-1}^{+1} \phi_m(\tilde{\Omega}) \tilde{B}_m(\tilde{\mathbf{C}}; \mathbf{r}_2, \mathbf{r}_5, \dots, \mathbf{r}_N). \qquad (121) $$

This yields

$$ C_{1,2} = \sum_{m=-1}^{+1} C_{1,2}^{(m)} $$

with

$$ C_{1,2}^{(m)} = N_{\uparrow}(N_{\downarrow}-1)N_{\downarrow} \frac{\cos^3 \tilde{\varphi}}{4(\tilde{s}+1)} \int |\tilde{B}_m(\tilde{\mathbf{C}}; \mathbf{r}_2, \mathbf{r}_5, \dots, \mathbf{r}_N)|^2 d^3\tilde{\mathbf{C}} d^3\mathbf{r}_2 d^3\mathbf{r}_5 \dots d^3\mathbf{r}_N. $$
The triplet correlation functions satisfy

$$ \int d^3C \, d^5\Omega \, g_{2,1}(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3) \underset{R \to 0}{\sim} C_{2,1} R^{2s-4} \frac{4(s+1)}{\cos^3 \varphi} \quad (122) $$

$$ \int d^3\tilde{C} \, d^5\tilde{\Omega} \, g_{1,2}(\mathbf{r}_1, \mathbf{r}_3, \mathbf{r}_4) \underset{\tilde{R} \to 0}{\sim} C_{1,2} \tilde{R}^{2\tilde{s}-4} \frac{4(\tilde{s}+1)}{\cos^3 \tilde{\varphi}} \quad (123) $$

The leading large-momentum tail of $\tilde{N}_P(K)$ is given by

$$ \tilde{N}_P(K) \underset{K \to \infty}{\sim} \mathcal{M} \frac{C_{2,1}}{K^{2s+4}} \quad (124) $$

where $\mathcal{M}$ is still given by the expression (24) in terms of $\mathcal{N}$, and $\mathcal{N}$ now stands for the normalization constant of the unitary hyperangular wavefunction of the $\uparrow\uparrow\downarrow\downarrow$ problem.<fnref n="39" />

The two-particle momentum distribution has the tail

$$ \lim_{K \to \infty} \lim_{k \to \infty} K^{2s+4} k^4 \frac{1}{4\pi} \int d\hat{\mathbf{K}} N \left( \frac{m_\uparrow}{m_\uparrow + m_\downarrow} \mathbf{K} - \mathbf{k}, \frac{m_\downarrow}{m_\uparrow + m_\downarrow} \mathbf{K} + \mathbf{k} \right) = \mathcal{M} C_{2,1}. \quad (125) $$

The expression (72) for the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate remains valid. In the expression (73) for the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate, $s$ is replaced by $\tilde{s}$, i.e. $\Gamma_{1,2} \approx -\frac{\hbar}{m} 8\tilde{s}(\tilde{s}+1) \sum_{m=-1}^{1} C_{1,2}^{(m)} \text{Im} a_{1,2}^{(m)}$. Since the $\uparrow\uparrow\downarrow\downarrow$ three-body parameters $a_{1,2}^{(m)}$ are now of order $b^{2\tilde{s}}$ whereas the $\uparrow\uparrow\downarrow\downarrow$ three-body parameters $a_{2,1}^{(m)}$ are of order $b^{2s}$, the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate $\Gamma_{1,2} \propto b^{2\tilde{s}}$ is negligible (in the zero-range regime) compared to the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate $\Gamma_{2,1} \propto b^{2s}$. Hence

$$ \Gamma_3 \approx \Gamma_{2,1} \approx -\frac{\hbar}{m} 8 s(s+1) \sum_{m=-1}^{1} C_{2,1}^{(m)} \text{Im} a_{2,1}^{(m)}. \quad (126) $$

The expression (82) for the two-body contribution to the energy correction remains valid. For the three-body contribution to the energy correction, we obtain

$$ \delta E_3 \approx \frac{\hbar^2}{m} 4s(s+1) \sum_{m=-1}^{1} C_{2,1}^{(m)} a_{2,1}^{(m)} \quad (127) $$

because the leading-order contribution again comes from $\uparrow\uparrow\downarrow\downarrow$ triplets.<fnref n="40" />

A peculiar situation takes place for $s < 1/2$ (i.e. for $m_\uparrow/m_\downarrow > 12.313...$): $\delta E_3 \propto b^{2s}$ dominates over $\delta E_2 \propto b$; i.e., the finite-range correction mainly comes from configurations with three nearby particles, rather than two nearby particles. This was already pointed out in [175] at the level of the third virial coefficient. Few-body and many-body numerical computations are often performed with finite-range interactions and extrapolated to the zero-range limit, see e.g. [146, 184] and [28, 62, 65, 67, 111, 112, 173, 174, 185–191] respectively; to accurately perform such $b \to 0$ extrapolations in the regime $s < 1/2$, it may be important to include the $b^{2s}$ scaling.<fnref n="41" />,<fnref n="42" />

<footnote n="39">³⁹The contribution from the $\uparrow\uparrow\downarrow\downarrow$ three-body problem gives rise to a higher-order subleading tail of $\tilde{N}_P(K)$, given by $C_{1,2} K^{-2\tilde{s}-4} \times 32\pi^3 4^{\tilde{s}} 3^{-\tilde{s}-1/2} (\tilde{s}+1) \Gamma(\tilde{s}+2) \sin^2(\tilde{s}\pi) \tilde{\mathcal{N}}^2$ with $\tilde{\mathcal{N}}$ the normalization constant of the unitary hyperangular wavefunction of the $\uparrow\uparrow\downarrow\downarrow$ problem.</footnote>

<footnote n="40">⁴⁰Indeed, the contribution to $\delta E_3$ from $\uparrow\uparrow\downarrow\downarrow$ triplets of nearby particles, given by the r.h.s. of (127), is $\propto b^{2s}$, whereas the $\uparrow\uparrow\downarrow\downarrow$ contribution is $\approx \frac{\hbar^2}{m} 4\tilde{s}(\tilde{s}+1) \sum_{m=-1}^{1} C_{1,2}^{(m)} a_{1,2}^{(m)} \propto b^{2\tilde{s}}$.</footnote>

<footnote n="41">⁴¹Naturally, the $\propto b^{2s}$ scaling may be hard to distinguish from the regular $\propto b$ scaling if $s$ is close to $1/2$ (see e.g. [146, Fig. 9(b)]).</footnote>

<footnote n="42">⁴²We take this opportunity to recall two other subtleties relevant to zero-range extrapolations (i.e. to continuum extrapolations in the case of lattice models, where the interaction range is set by the lattice spacing). Being due to breaking of Galilean invariance, they arise for any mass ratio. The first subtlety, reported in [38, 56, 186], is that for lattice models (where the interaction range $b$ is set by the lattice spacing), the $\propto r_e$ term given by the r.h.s. of (82) is not the only $\propto b$ contribution to the two-body finite-range energy-correction $\delta E_2$: There is a second contribution to $\delta E_2$, proportional to a parameter $R_e$ (that parameter $R_e$ quantifies the dependence of the two-body vacuum T-matrix on the center-of-mass</footnote>
The triplet correlation functions satisfy

$$ \int d^3C \, d^5\Omega \, g_{2,1}(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3) \underset{R \to 0}{\sim} C_{2,1} R^{2s-4} \frac{4(s+1)}{\cos^3 \varphi} \quad (122) $$

$$ \int d^3\tilde{C} \, d^5\tilde{\Omega} \, g_{1,2}(\mathbf{r}_1, \mathbf{r}_3, \mathbf{r}_4) \underset{\tilde{R} \to 0}{\sim} C_{1,2} \tilde{R}^{2\tilde{s}-4} \frac{4(\tilde{s}+1)}{\cos^3 \tilde{\varphi}} \quad (123) $$

The leading large-momentum tail of $\tilde{N}_P(K)$ is given by

$$ \tilde{N}_P(K) \underset{K \to \infty}{\sim} \mathcal{M} \frac{C_{2,1}}{K^{2s+4}} \quad (124) $$

where $\mathcal{M}$ is still given by the expression (24) in terms of $\mathcal{N}$, and $\mathcal{N}$ now stands for the normalization constant of the unitary hyperangular wavefunction of the $\uparrow\uparrow\downarrow\downarrow$ problem.<fnref n="39" />

The two-particle momentum distribution has the tail

$$ \lim_{K \to \infty} \lim_{k \to \infty} K^{2s+4} k^4 \frac{1}{4\pi} \int d\hat{\mathbf{K}} N \left( \frac{m_\uparrow}{m_\uparrow + m_\downarrow} \mathbf{K} - \mathbf{k}, \frac{m_\downarrow}{m_\uparrow + m_\downarrow} \mathbf{K} + \mathbf{k} \right) = \mathcal{M} C_{2,1}. \quad (125) $$

The expression (72) for the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate remains valid. In the expression (73) for the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate, $s$ is replaced by $\tilde{s}$, i.e. $\Gamma_{1,2} \approx -\frac{\hbar}{m} 8\tilde{s}(\tilde{s}+1) \sum_{m=-1}^{1} C_{1,2}^{(m)} \mathrm{Im} a_{1,2}^{(m)}$. Since the $\uparrow\uparrow\downarrow\downarrow$ three-body parameters $a_{1,2}^{(m)}$ are now of order $b^{2\tilde{s}}$ whereas the $\uparrow\uparrow\downarrow\downarrow$ three-body parameters $a_{2,1}^{(m)}$ are of order $b^{2s}$, the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate $\Gamma_{1,2} \propto b^{2\tilde{s}}$ is negligible (in the zero-range regime) compared to the $\uparrow\uparrow\downarrow\downarrow$ three-body loss rate $\Gamma_{2,1} \propto b^{2s}$. Hence

$$ \Gamma_3 \approx \Gamma_{2,1} \approx -\frac{\hbar}{m} 8 s(s+1) \sum_{m=-1}^{1} C_{2,1}^{(m)} \mathrm{Im} a_{2,1}^{(m)}. \quad (126) $$

The expression (82) for the two-body contribution to the energy correction remains valid. For the three-body contribution to the energy correction, we obtain

$$ \delta E_3 \approx \frac{\hbar^2}{m} 4s(s+1) \sum_{m=-1}^{1} C_{2,1}^{(m)} a_{2,1}^{(m)} \quad (127) $$

because the leading-order contribution again comes from $\uparrow\uparrow\downarrow\downarrow$ triplets.<fnref n="40" />

A peculiar situation takes place for $s < 1/2$ (i.e. for $m_\uparrow/m_\downarrow > 12.313...$): $\delta E_3 \propto b^{2s}$ dominates over $\delta E_2 \propto b$; i.e., the finite-range correction mainly comes from configurations with three nearby particles, rather than two nearby particles. This was already pointed out in [175] at the level of the third virial coefficient. Few-body and many-body numerical computations are often performed with finite-range interactions and extrapolated to the zero-range limit, see e.g. [146, 184] and [28, 62, 65, 67, 111, 112, 173, 174, 185–191] respectively; to accurately perform such $b \to 0$ extrapolations in the regime $s < 1/2$, it may be important to include the $b^{2s}$ scaling.<fnref n="41" />,<fnref n="42" />

<footnote n="39">³⁹The contribution from the $\uparrow\uparrow\downarrow\downarrow$ three-body problem gives rise to a higher-order subleading tail of $\tilde{N}_P(K)$, given by $C_{1,2} K^{-2\tilde{s}-4} \times 32\pi^3 4^{\tilde{s}} 3^{-\tilde{s}-1/2} (\tilde{s}+1) \Gamma(\tilde{s}+2) \sin^2(\tilde{s}\pi) \tilde{\mathcal{N}}^2$ with $\tilde{\mathcal{N}}$ the normalization constant of the unitary hyperangular wavefunction of the $\uparrow\uparrow\downarrow\downarrow$ problem.</footnote>

<footnote n="40">⁴⁰Indeed, the contribution to $\delta E_3$ from $\uparrow\uparrow\downarrow\downarrow$ triplets of nearby particles, given by the r.h.s. of (127), is $\propto b^{2s}$, whereas the $\uparrow\uparrow\downarrow\downarrow$ contribution is $\approx \frac{\hbar^2}{m} 4\tilde{s}(\tilde{s}+1) \sum_{m=-1}^{1} C_{1,2}^{(m)} a_{1,2}^{(m)} \propto b^{2\tilde{s}}$.</footnote>

<footnote n="41">⁴¹Naturally, the $\propto b^{2s}$ scaling may be hard to distinguish from the regular $\propto b$ scaling if $s$ is close to $1/2$ (see e.g. [146, Fig. 9(b)]).</footnote>

<footnote n="42">⁴²We take this opportunity to recall two other subtleties relevant to zero-range extrapolations (i.e. to continuum extrapolations in the case of lattice models, where the interaction range is set by the lattice spacing). Being due to breaking of Galilean invariance, they arise for any mass ratio. The first subtlety, reported in [38, 56, 186], is that for lattice models (where the interaction range $b$ is set by the lattice spacing), the $\propto r_e$ term given by the r.h.s. of (82) is not the only $\propto b$ contribution to the two-body finite-range energy-correction $\delta E_2$: There is a second contribution to $\delta E_2$, proportional to a parameter $R_e$ (that parameter $R_e$ quantifies the dependence of the two-body vacuum T-matrix on the center-of-mass</footnote>
## D.2. Relations within the generalized zero-range model

In this Section, we assume that $m_1/m_↓$ is larger than 8.619..., so that $s < 1$. This ensures that a three-body wavefunction diverging as $R^{-s-2}$ at small $R$ remains square integrable. This allows one to define a generalized zero-range model (GZRM) by supplementing the two-body contact condition (14) (involving the two-body scattering length) by the following three-body contact condition (involving the three-body parameters): There exists $B_m$ such that

$$ \psi(\mathbf{r}_1, \dots, \mathbf{r}_N) \underset{R \to 0}{\sim} \frac{1}{R^2} \sum_{m=-1}^{+1} \left( -\frac{a_3^{(m)}}{R^s} + R^s + o(R^s) \right) \phi_m(\Omega) B_m(\mathbf{C}; \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (128) $$

in the limit $R \to 0$ where particles 1,2,3 approach each other, while keeping fixed their hyperangles $\Omega$, their center-of-mass $\mathbf{C}$, and the positions of the other particles $\mathbf{r}_4, \dots, \mathbf{r}_N$. By antisymmetry, Eq. (128) imposes a similar condition when any triplet of particles with spins $↑↑↓$ approach each other. Apart from the two-body and three-body contact conditions (14), (128), the GZRM is defined (like the standard zero-range model) by the Schrödinger equation without any interaction potential (114). Note that for vanishing three-body parameters, the GZRM reduces to the standard zero-range model (ZRM).

In the regime where $k_{typ}^{2s}|a_3^{(m)}| \gtrsim 1$ (for at least one value of $m$), the ZRM becomes irrelevant, whereas the GZRM remains applicable (provided $k_{typ}b \ll 1$). This regime can be reached

* near a ($↑↑↓$) three-body resonance [38, 97, 117, 143–148]

* when the mass ratio $m_1/m_↓$ is only slightly smaller than the threshold 13.607... where the three-body Efimov effect appears, so that $s$ is small [175].

### Remarks.

* If the underlying microscopic interactions are rotationally-invariant, then $a_3^{(m)}$ does not depend on $m$.

* In the present GZRM, in the limit where three particles of spins $↑↓↓$ approach each other, the asymptotic behavior of the wavefunction has the same form (121) than for the ZRM. In other words, the behavior $\psi \propto R^{-2-\tilde{s}}$ is forbidden. Such a wavefunction would not be a normalizable at small $R$, since $\tilde{s}$ is always larger than one.43 This fact does not prevent one from rederiving the relations ((126),(127)), which come from configurations with nearby $↑↑↓$ triplets.

* Extensions of the GZRM to the regime $s \ge 1$, which were introduced recently [142], are beyond the scope of this work.

* For mathematically rigorous studies of the GZRM in the three-particle case, see [200, 201].

<footnote>momentum, which arises from lattice-induced breaking of Galilean invariance). Therefore, if one wishes to cancel the $\propto b$ term in the $b \to 0$ continuum extrapolation, one needs to tune to zero not only $r_e$ (as done in [173, 174, 192] and for one of the dispersion relations considered in [112]) but also $R_e$ (and if the dispersion relation has a cusp at the edge of the Brillouin zone, there is a third $\propto b$ contribution to $\delta E_2$, in a finite box with periodic boundary conditions [56]). The second subtlety, reported in [56] and further evidenced in [190], arises if one restricts single-particle momenta to a ball of radius $\Lambda$ (with $\Lambda = \pi/b$ for lattice models, with $b$ the lattice spacing), i.e. if one takes a dispersion relation equal to $+\infty$ for momenta larger than $\Lambda$, as was done in [187, 193–199]: The universal zero-range model is not obtained in the limit $\Lambda \to \infty$, because the hard cutoff induces a dependence of the two-body T-matrix on the center-of-mass momentum, and this dependence surprisingly survives for $\Lambda \to \infty$. We expect the same problem in [173, 174] where such a spherical cutoff was also used.</footnote>

<footnote n="43">43 Hence there are no $↑↑↓$ three-body parameters within the GZRM, which is why we denoted the $↑↑↓$ three-body parameters by $a_3^{(m)}$ instead of the more cumbersome notation $a_{2,1}^{(m)}$.</footnote>
## D.2. Relations within the generalized zero-range model

In this Section, we assume that $m_1/m_↓$ is larger than 8.619..., so that $s < 1$. This ensures that a three-body wavefunction diverging as $R^{-s-2}$ at small $R$ remains square integrable. This allows one to define a generalized zero-range model (GZRM) by supplementing the two-body contact condition (14) (involving the two-body scattering length) by the following three-body contact condition (involving the three-body parameters): There exists $B_m$ such that

$$ \psi(\mathbf{r}_1, \dots, \mathbf{r}_N) \underset{R \to 0}{\sim} \frac{1}{R^2} \sum_{m=-1}^{+1} \left( -\frac{a_3^{(m)}}{R^s} + R^s + o(R^s) \right) \phi_m(\Omega) B_m(\mathbf{C}; \mathbf{r}_4, \dots, \mathbf{r}_N) \quad (128) $$

in the limit $R \to 0$ where particles 1,2,3 approach each other, while keeping fixed their hyperangles $\Omega$, their center-of-mass $\mathbf{C}$, and the positions of the other particles $\mathbf{r}_4, \dots, \mathbf{r}_N$. By antisymmetry, Eq. (128) imposes a similar condition when any triplet of particles with spins $↑↑↓$ approach each other. Apart from the two-body and three-body contact conditions (14), (128), the GZRM is defined (like the standard zero-range model) by the Schrödinger equation without any interaction potential (114). Note that for vanishing three-body parameters, the GZRM reduces to the standard zero-range model (ZRM).

In the regime where $k_{typ}^{2s}|a_3^{(m)}| \gtrsim 1$ (for at least one value of $m$), the ZRM becomes irrelevant, whereas the GZRM remains applicable (provided $k_{typ}b \ll 1$). This regime can be reached

* near a ($↑↑↓$) three-body resonance [38, 97, 117, 143–148]

* when the mass ratio $m_1/m_↓$ is only slightly smaller than the threshold 13.607... where the three-body Efimov effect appears, so that $s$ is small [175].

## Remarks.

* If the underlying microscopic interactions are rotationally-invariant, then $a_3^{(m)}$ does not depend on $m$.

* In the present GZRM, in the limit where three particles of spins $↑↓↓$ approach each other, the asymptotic behavior of the wavefunction has the same form (121) than for the ZRM. In other words, the behavior $\psi \propto R^{-2-\tilde{s}}$ is forbidden. Such a wavefunction would not be a normalizable at small $R$, since $\tilde{s}$ is always larger than one.43 This fact does not prevent one from rederiving the relations ((126),(127)), which come from configurations with nearby $↑↑↓$ triplets.

* Extensions of the GZRM to the regime $s \ge 1$, which were introduced recently [142], are beyond the scope of this work.

* For mathematically rigorous studies of the GZRM in the three-particle case, see [200, 201].

<footnote>momentum, which arises from lattice-induced breaking of Galilean invariance). Therefore, if one wishes to cancel the $\propto b$ term in the $b \to 0$ continuum extrapolation, one needs to tune to zero not only $r_e$ (as done in [173, 174, 192] and for one of the dispersion relations considered in [112]) but also $R_e$ (and if the dispersion relation has a cusp at the edge of the Brillouin zone, there is a third $\propto b$ contribution to $\delta E_2$, in a finite box with periodic boundary conditions [56]). The second subtlety, reported in [56] and further evidenced in [190], arises if one restricts single-particle momenta to a ball of radius $\Lambda$ (with $\Lambda = \pi/b$ for lattice models, with $b$ the lattice spacing), i.e. if one takes a dispersion relation equal to $+\infty$ for momenta larger than $\Lambda$, as was done in [187, 193–199]: The universal zero-range model is not obtained in the limit $\Lambda \to \infty$, because the hard cutoff induces a dependence of the two-body T-matrix on the center-of-mass momentum, and this dependence surprisingly survives for $\Lambda \to \infty$. We expect the same problem in [173, 174] where such a spherical cutoff was also used.</footnote>

<footnote n="43">43 Hence there are no $↑↑↓$ three-body parameters within the GZRM, which is why we denoted the $↑↑↓$ three-body parameters by $a_3^{(m)}$ instead of the more cumbersome notation $a_{2,1}^{(m)}$.</footnote>
### D.2.1. Derivative of the energy with respect to the three-body parameters

Within the GZRM, the derivatives of the energy w.r.t. the three-body parameters are given by the three-body contacts,

$$ \frac{\partial E}{\partial a_3^{(m)}} = \frac{\hbar^2}{m} 4s(s+1) C_{2,1}^{(m)}. \qquad (129) $$

Here the derivative w.r.t. $a_3^{(m)}$ is taken at fixed value of $a_2$ and of the other $a_3^{(m')}$ with $m' \neq m$. The three-body contacts $C_{2,1}^{(m)}$ are still defined by (120).

**Remark 5.** Within the ZRM, the derivative of the energy w.r.t. $a_2$ is given by $C_2$, see (85). Within the GZRM, this relation also holds, with $dE/d(-1/a_2)$ replaced by the partial derivative $\partial E/\partial(-1/a_2)$ taken at fixed $a_3^{(m)}$. This can be justified by using the derivation presented in [56, Sec. IV.C] (at fixed three-body parameters, there is no additional contribution to the energy variation coming from nearby triplets of particles, as we will see below).

To derive (129) we proceed as follows.<fnref n="44" /> We consider two wavefunctions $\psi$ and $\tilde{\psi}$, which are stationary states of the zero-range model for different sets of three-body parameters ($a_3^{(m)}$) and ($\tilde{a}_3^{(m)}$), but the same two-body scattering length $a_2$. Denoting the corresponding eigenenergies by $E$ and $\tilde{E}$, we have

$$ \delta := \langle \psi, H\tilde{\psi} \rangle - \langle H\psi, \tilde{\psi} \rangle = (\tilde{E} - E) \langle \psi|\tilde{\psi} \rangle. $$

On the other hand, from the Schrödinger equations for $\psi$ and $\tilde{\psi}$,

$$ \delta = - \sum_{i=1}^{N} \frac{\hbar^2}{2m_i} \int d^3 r_1 \dots d^3 r_N (\psi^* \Delta_{\mathbf{r}_i} \tilde{\psi} - \tilde{\psi} \Delta_{\mathbf{r}_i} \psi^*). \qquad (130) $$

As we will see below, there is a contribution $\delta_0$ to $\delta$ coming from the configurations where particles 1, 2 and 3 are close to each other. By symmetry, there is an identical contribution from the configurations where any set of three particles with spins $\uparrow\uparrow\downarrow$ are close to each other. Hence $\delta$ equals $\delta_0$ times the number of such three-particle sets, $N_1(N_1 - 1)N_1/2$. To evaluate $\delta_0$, we only need to keep the terms $i = 1, 2, 3$ in Eq. (130), which gives, after the change of coordinates $(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3) \to (C, R, \Omega)$,

$$ \delta_0 = -\frac{\hbar^2}{m} \cos^3 \varphi \int_0^\infty dR R^5 \int d^5\Omega d^3C d^3r_4 \dots d^3r_N \left\{ \mathcal{F}^* \left( \frac{\partial^2}{\partial R^2} + \frac{1}{R} \frac{\partial}{\partial R} + \frac{T_\Omega}{R^2} + \frac{\Delta C}{3} \right) \tilde{\mathcal{F}} - [\mathcal{F}^* \leftrightarrow \tilde{\mathcal{F}}] \right\} \qquad (131) $$

where $\mathcal{F} := R^2 \psi$, $\tilde{\mathcal{F}} := R^2 \tilde{\psi}$, and $T_\Omega$ is defined by (97). Since the two-body scattering length is the same for $\psi$ and $\tilde{\psi}$, we only need to keep the terms involving derivatives with respect to $R$ in Eq. (131).<fnref n="45" /> Transforming the integral over $R$ into a boundary term at $R \to 0$, we then get

$$ \delta_0 = \frac{\hbar^2}{m} \cos^3 \varphi \int d^3C d^3r_4 \dots d^3r_N \int d^5\Omega \lim_{R \to 0} R \left( \mathcal{F}^* \frac{\partial \tilde{\mathcal{F}}}{\partial R} - \tilde{\mathcal{F}} \frac{\partial \mathcal{F}^*}{\partial R} \right). $$

The result (129) follows by using the three-body contact conditions [given by Eq. (128) for $\psi$, and the same condition with $\tilde{a}_3^{(m)}$ for $\tilde{\psi}$], taking the limit $\tilde{a}_3^{(m)} \to a_3^{(m)}$, and using the expression (120) of the three-body contacts.

<footnote n="44">⁴⁴This derivation is similar to the case of three bosons treated in of [72, App. A].</footnote>

<footnote n="45">⁴⁵For a more detailed justification of this step, see the reasoning around [72, Eq. (A7)].</footnote>
### D.2.1. Derivative of the energy with respect to the three-body parameters

Within the GZRM, the derivatives of the energy w.r.t. the three-body parameters are given by the three-body contacts,

$$ \frac{\partial E}{\partial a_3^{(m)}} = \frac{\hbar^2}{m} 4s(s+1) C_{2,1}^{(m)}. \qquad (129) $$

Here the derivative w.r.t. $a_3^{(m)}$ is taken at fixed value of $a_2$ and of the other $a_3^{(m')}$ with $m' \neq m$. The three-body contacts $C_{2,1}^{(m)}$ are still defined by (120).

**Remark 5.** Within the ZRM, the derivative of the energy w.r.t. $a_2$ is given by $C_2$, see (85). Within the GZRM, this relation also holds, with $dE/d(-1/a_2)$ replaced by the partial derivative $\partial E/\partial(-1/a_2)$ taken at fixed $a_3^{(m)}$. This can be justified by using the derivation presented in [56, Sec. IV.C] (at fixed three-body parameters, there is no additional contribution to the energy variation coming from nearby triplets of particles, as we will see below).

To derive (129) we proceed as follows.<fnref n="44" /> We consider two wavefunctions $\psi$ and $\tilde{\psi}$, which are stationary states of the zero-range model for different sets of three-body parameters ($a_3^{(m)}$) and ($\tilde{a}_3^{(m)}$), but the same two-body scattering length $a_2$. Denoting the corresponding eigenenergies by $E$ and $\tilde{E}$, we have

$$ \delta := \langle \psi, H\tilde{\psi} \rangle - \langle H\psi, \tilde{\psi} \rangle = (\tilde{E} - E) \langle \psi|\tilde{\psi} \rangle. $$

On the other hand, from the Schrödinger equations for $\psi$ and $\tilde{\psi}$,

$$ \delta = - \sum_{i=1}^{N} \frac{\hbar^2}{2m_i} \int d^3 r_1 \dots d^3 r_N (\psi^* \Delta_{\mathbf{r}_i} \tilde{\psi} - \tilde{\psi} \Delta_{\mathbf{r}_i} \psi^*). \qquad (130) $$

As we will see below, there is a contribution $\delta_0$ to $\delta$ coming from the configurations where particles 1, 2 and 3 are close to each other. By symmetry, there is an identical contribution from the configurations where any set of three particles with spins $\uparrow\uparrow\downarrow$ are close to each other. Hence $\delta$ equals $\delta_0$ times the number of such three-particle sets, $N_1(N_1 - 1)N_1/2$. To evaluate $\delta_0$, we only need to keep the terms $i = 1, 2, 3$ in Eq. (130), which gives, after the change of coordinates $(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3) \to (\mathbf{C}, \mathbf{R}, \mathbf{\Omega})$,

$$ \delta_0 = -\frac{\hbar^2}{m} \cos^3 \varphi \int_0^\infty dR R^5 \int d^5\Omega d^3C d^3r_4 \dots d^3r_N \left\{ \mathcal{F}^* \left( \frac{\partial^2}{\partial R^2} + \frac{1}{R} \frac{\partial}{\partial R} + \frac{T_\Omega}{R^2} + \frac{\Delta C}{3} \right) \tilde{\mathcal{F}} - [\mathcal{F}^* \leftrightarrow \tilde{\mathcal{F}}] \right\} \qquad (131) $$

where $\mathcal{F} := R^2\psi$, $\tilde{\mathcal{F}} := R^2\tilde{\psi}$, and $T_\Omega$ is defined by (97). Since the two-body scattering length is the same for $\psi$ and $\tilde{\psi}$, we only need to keep the terms involving derivatives with respect to $R$ in Eq. (131).<fnref n="45" /> Transforming the integral over $R$ into a boundary term at $R \to 0$, we then get

$$ \delta_0 = \frac{\hbar^2}{m} \cos^3 \varphi \int d^3C d^3r_4 \dots d^3r_N \int d^5\Omega \lim_{R \to 0} R \left( \mathcal{F}^* \frac{\partial \tilde{\mathcal{F}}}{\partial R} - \tilde{\mathcal{F}} \frac{\partial \mathcal{F}^*}{\partial R} \right). $$

The result (129) follows by using the three-body contact conditions [given by Eq. (128) for $\psi$, and the same condition with $\tilde{a}_3^{(m)}$ for $\tilde{\psi}$], taking the limit $\tilde{a}_3^{(m)} \to a_3^{(m)}$, and using the expression (120) of the three-body contacts.

<footnote n="44">⁴⁴This derivation is similar to the case of three bosons treated in of [72, App. A].</footnote>

<footnote n="45">⁴⁵For a more detailed justification of this step, see the reasoning around [72, Eq. (A7)].</footnote>
### D.2.2. *Three-body loss rate*

Let us consider the GZRM with complex three-body parameters, $a_3^{(m)} \in \mathbb{C}$. The three-body loss rate is then given by

$$ \Gamma_3 = -\frac{\hbar}{m} 8s(s+1) \sum_{m=-1}^{1} C_{2,1}^{(m)} \operatorname{Im} a_3^{(m)}. \qquad (132) $$

Within the GZRM, there are only ↑↑↓ losses, and no ↑↓↓ losses, i.e. $\Gamma_3 = \Gamma_{2,1}$ and $\Gamma_{1,2} = 0$.

To derive this relation, we proceed very similarly to the bosonic case of [72]. We consider a stationary state of the GZRM, i.e. a solution $\psi(\mathbf{r}_1, \dots, \mathbf{r}_N)$ of the stationary Schrödinger equation (114) together with the two-body and three-body contact conditions ((14),(128)). The corresponding solution of the time-dependent Schrödinger equation is $\Psi(\mathbf{r}_1, \dots, \mathbf{r}_N; t) = \psi(\mathbf{r}_1, \dots, \mathbf{r}_N)e^{-iEt/\hbar}$, and $\Gamma_3 = -\frac{\partial}{\partial t}|_{t=0} \int d^3r_1 \dots d^3r_N |\Psi(\mathbf{r}_1, \dots, \mathbf{r}_N; t)|^2$, with $\psi$ normalized to unity. Excluding from the integration domain the regions where $R_{ijk} < \epsilon$ and taking the limit $\epsilon \to 0$, the continuity equation and the three-body contact conditions leads to the final expression, where all extra mass-ratio dependent factors, arising e.g. from (116), divide out.

### D.3. *Validity conditions*

Let us denote by $s_{j_1, j_\downarrow}$ the scaling exponent of the unitary $(j_\downarrow + j_\uparrow)$ body problem with $j_\uparrow$ particles of spin $\uparrow$ and $j_\downarrow$ particles of spin $\downarrow$ (so that $s_{2,1} \equiv s$ and $s_{1,2} \equiv \tilde{s}$). We expect the relations (117),(119),(124),(125) to be valid under the condition $\operatorname{Min}\{s_{j_1, j_\downarrow}; j_\uparrow \ge 2, j_\downarrow \ge 1, j_\uparrow + j_\downarrow > 3\} > s$. Indeed, we expect a contribution from the $(j_1 + j_\downarrow)$ body problem to $N_{2,1}(\epsilon)$ at small $\epsilon$ [resp. to $\tilde{N}_P(K)$ at large $K$] scaling as $\epsilon^{2s_{j_1, j_\downarrow}+2}$ (resp. $1/K^{2s_{j_1, j_\downarrow}+4}$), which dominates over the contribution from the three-body problem when $s_{j_1, j_\downarrow} < s$. Similarly, we expect relation (118) to be valid under the condition $\operatorname{Min}\{s_{j_1, j_\downarrow}; j_\uparrow \ge 1, j_\downarrow \ge 2, j_\uparrow + j_\downarrow > 3\} > \tilde{s}$. The former condition breaks down when $m_\uparrow / m_\downarrow$ is near the thresholds where the four-body [102] and five-body [103] Efimov effects appear, where $s_{3,1}$ and $s_{4,1}$ become small. Based on existing data, we expect that both of the above validity conditions are satisfied at least in the range $m_\uparrow / m_\downarrow \le 10$. Indeed, in this range, we have $s_{3,1} > s$ and $s_{4,1} > s$ [103], $s_{5,1} > s$ [104], $s_{2,2} > \tilde{s}$ [146], $s_{1,3} > \tilde{s}$ [103], and the trends of available data suggest that the conditions will also hold for larger values of $j_\uparrow$ or $j_\downarrow$. We conservatively restricted to $m_\uparrow / m_\downarrow \le 10$ because $s_{2,2}$ was not computed beyond this range, but the conditions $s_{3,1} > s$ and $s_{4,1} > s$ actually hold up to at least $m_\uparrow / m_\downarrow = 13.2$ [103].

## Appendix E. Generalization to statistical mixtures and non-stationary states

Many of the relations derived for stationary states in the main text and in App. D are directly generalizable to non-stationary states and statistical mixtures, similarly to the relations involving the two-body contact [35, 36, 56]. Indeed, Eqs. (15), (17), (18), (19) remain valid for any non-pathological linear combinations of stationary states (including solutions of time-dependent problems, where $a_2$ and the trapping potential $U$ can depend on time); and relations (3), (5), (6), (11), (12), (23) remain true for arbitrary non-pathological statistical mixtures of such pure states (including the case of thermal equilibrium). Here, non-pathological means that the occupation probabilities of stationary states should decay sufficiently quickly (which includes the simple case where only a finite number of states are populated); more specifically, for a pure state, it is necessary that Eqs. (14), (49) still hold, while for a statistical mixture $\hat{\rho} = \sum_i c_i |\psi_i\rangle\langle\psi_i|$, it is necessary that the three-body contact of the mixture [defined by Eq. (3)] equals $\sum_i c_i C_{3,i}$ with $C_{3,i}$ the three-body contact of state $\psi_i$.

Moreover, at thermal equilibrium, the thermally averaged loss rates are given by (43),(44), (45) for simple interactions; for more general interactions they are given by (72), (73), and by (80).
### D.2.2. *Three-body loss rate*

Let us consider the GZRM with complex three-body parameters, $a_3^{(m)} \in \mathbb{C}$. The three-body loss rate is then given by

$$ \Gamma_3 = -\frac{\hbar}{m} 8s(s+1) \sum_{m=-1}^{1} C_{2,1}^{(m)} \operatorname{Im} a_3^{(m)}. \qquad (132) $$

Within the GZRM, there are only ↑↑↓ losses, and no ↑↓↓ losses, i.e. $\Gamma_3 = \Gamma_{2,1}$ and $\Gamma_{1,2} = 0$.

To derive this relation, we proceed very similarly to the bosonic case of [72]. We consider a stationary state of the GZRM, i.e. a solution $\psi(\mathbf{r}_1, \dots, \mathbf{r}_N)$ of the stationary Schrödinger equation (114) together with the two-body and three-body contact conditions ((14),(128)). The corresponding solution of the time-dependent Schrödinger equation is $\Psi(\mathbf{r}_1, \dots, \mathbf{r}_N; t) = \psi(\mathbf{r}_1, \dots, \mathbf{r}_N)e^{-iEt/\hbar}$, and $\Gamma_3 = -\frac{\partial}{\partial t}|_{t=0} \int d^3r_1 \dots d^3r_N |\Psi(\mathbf{r}_1, \dots, \mathbf{r}_N; t)|^2$, with $\psi$ normalized to unity. Excluding from the integration domain the regions where $R_{ijk} < \epsilon$ and taking the limit $\epsilon \to 0$, the continuity equation and the three-body contact conditions leads to the final expression, where all extra mass-ratio dependent factors, arising e.g. from (116), divide out.

### D.3. *Validity conditions*

Let us denote by $s_{j_1, j_\downarrow}$ the scaling exponent of the unitary $(j_\downarrow + j_\uparrow)$ body problem with $j_\uparrow$ particles of spin $\uparrow$ and $j_\downarrow$ particles of spin $\downarrow$ (so that $s_{2,1} \equiv s$ and $s_{1,2} \equiv \tilde{s}$). We expect the relations (117),(119),(124),(125) to be valid under the condition $\mathrm{Min}\{s_{j_1, j_\downarrow}; j_\uparrow \ge 2, j_\downarrow \ge 1, j_\uparrow + j_\downarrow > 3\} > s$. Indeed, we expect a contribution from the $(j_1 + j_\downarrow)$ body problem to $N_{2,1}(\epsilon)$ at small $\epsilon$ [resp. to $\tilde{N}_P(K)$ at large $K$] scaling as $\epsilon^{2s_{j_1, j_\downarrow}+2}$ (resp. $1/K^{2s_{j_1, j_\downarrow}+4}$), which dominates over the contribution from the three-body problem when $s_{j_1, j_\downarrow} < s$. Similarly, we expect relation (118) to be valid under the condition $\mathrm{Min}\{s_{j_1, j_\downarrow}; j_\uparrow \ge 1, j_\downarrow \ge 2, j_\uparrow + j_\downarrow > 3\} > \tilde{s}$. The former condition breaks down when $m_\uparrow / m_\downarrow$ is near the thresholds where the four-body [102] and five-body [103] Efimov effects appear, where $s_{3,1}$ and $s_{4,1}$ become small. Based on existing data, we expect that both of the above validity conditions are satisfied at least in the range $m_\uparrow / m_\downarrow \le 10$. Indeed, in this range, we have $s_{3,1} > s$ and $s_{4,1} > s$ [103], $s_{5,1} > s$ [104], $s_{2,2} > \tilde{s}$ [146], $s_{1,3} > \tilde{s}$ [103], and the trends of available data suggest that the conditions will also hold for larger values of $j_\uparrow$ or $j_\downarrow$. We conservatively restricted to $m_\uparrow / m_\downarrow \le 10$ because $s_{2,2}$ was not computed beyond this range, but the conditions $s_{3,1} > s$ and $s_{4,1} > s$ actually hold up to at least $m_\uparrow / m_\downarrow = 13.2$ [103].

## Appendix E. Generalization to statistical mixtures and non-stationary states

Many of the relations derived for stationary states in the main text and in App. D are directly generalizable to non-stationary states and statistical mixtures, similarly to the relations involving the two-body contact [35, 36, 56]. Indeed, Eqs. (15), (17), (18), (19) remain valid for any non-pathological linear combinations of stationary states (including solutions of time-dependent problems, where $a_2$ and the trapping potential $U$ can depend on time); and relations (3), (5), (6), (11), (12), (23) remain true for arbitrary non-pathological statistical mixtures of such pure states (including the case of thermal equilibrium). Here, non-pathological means that the occupation probabilities of stationary states should decay sufficiently quickly (which includes the simple case where only a finite number of states are populated); more specifically, for a pure state, it is necessary that Eqs. (14), (49) still hold, while for a statistical mixture $\hat{\rho} = \sum_i c_i |\psi_i\rangle\langle\psi_i|$, it is necessary that the three-body contact of the mixture [defined by Eq. (3)] equals $\sum_i c_i C_{3,i}$ with $C_{3,i}$ the three-body contact of state $\psi_i$.

Moreover, at thermal equilibrium, the thermally averaged loss rates are given by (43),(44), (45) for simple interactions; for more general interactions they are given by (72), (73), and by (80)
for a homogeneous unpolarized gas. Furthermore, (83), (84) remain valid in the canonical ensemble, with $\delta E_3$ the energy difference (between finite-range and zero-range models) taken at fixed entropy (which equals the free-energy difference at fixed temperature).

As for the wavevector $k_{\text{typ}}$ that appears in the validity conditions, it should be defined by the same procedure as before and then taking the maximum over all significantly populated eigenstates; for example for the balanced unitary gas at thermal equilibrium, $k_{\text{typ}} = \max(k_F, k_T)$ with $k_T$ the thermal wavevector, defined by $\hbar^2 k_T^2/(2m) = k_B T$.

References

[1] A. J. Leggett, "Diatomic Molecules and Cooper Pairs", in *Modern Trends in the Theory of Condensed Matter* (A. Pekalski and J. A. Przystawa, eds.), Springer, 1980, pp. 13–27.

[2] R. Haussmann, "Crossover from BCS superconductivity to Bose-Einstein condensation: a self-consistent theory", *Z. Phys., B* **91** (1993), pp. 291–308.

[3] R. Haussmann, "Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose-Einstein condensation", *Phys. Rev. B* **49** (1994), no. 18, pp. 12975-12983.

[4] S. Giorgini, L. P. Pitaevskii and S. Stringari, "Theory of ultracold Fermi gases", *Rev. Mod. Phys.* **80** (2008), no. 4, pp. 1215-1274.

[5] *The BCS-BEC Crossover and the Unitary Fermi Gas*, (W. Zwerger, ed.), Springer, 1978.

[6] M. Randeria, W. Zwerger and M. W. Zwierlein, "The BCS-BEC Crossover and the Unitary Fermi Gas", in *The BCS-BEC Crossover and the Unitary Fermi Gas* (W. Zwerger, ed.), Springer, 2012, pp. 1–32.

[7] K. M. O'Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade and J. E. Thomas, "Observation of a Strongly Interacting Degenerate Fermi Gas of Atoms", *Science* **298** (2002), no. 5601, pp. 2179–2182.

[8] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag and R. Grimm, "Crossover from a Molecular Bose-Einstein Condensate to a Degenerate Fermi Gas", *Phys. Rev. Lett.* **92** (2004), no. 12, article no. 120401.

[9] C. A. Regal, M. Greiner and D. S. Jin, "Observation of Resonance Condensation of Fermionic Atom Pairs", *Phys. Rev. Lett.* **92** (2004), no. 4, article no. 040403.

[10] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman and W. Ketterle, "Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance", *Phys. Rev. Lett.* **92** (2004), no. 12, article no. 120403.

[11] T. Bourdel et al., "Experimental Study of the BEC-BCS Crossover Region in Lithium 6", *Phys. Rev. Lett.* **93** (2004), no. 5, article no. 050401.

[12] A. Altmeyer et al., "Precision Measurements of Collective Oscillations in the BEC-BCS Crossover", *Phys. Rev. Lett.* **98** (2007), no. 4, article no. 040401.

[13] Y. Shin, C. H. Schunck, A. Schirotzek and W. Ketterle, "Phase diagram of a two-component Fermi gas with resonant interactions", *Nature* **451** (2008), pp. 689–693.

[14] S. Nascimbène, N. Navon, K. J. Jiang, F. Chevy and C. Salomon, "Exploring the thermodynamics of a universal Fermi gas", *Nature* **463** (2010), pp. 1057–1060.

[15] N. Navon, S. Nascimbène, F. Chevy and C. Salomon, "The Equation of State of a Low-Temperature Fermi Gas with Tunable Interactions", *Science* **328** (2010), no. 5979, pp. 729–732.

[16] M. J. H. Ku, A. Sommer, L. W. Cheuk and M. W. Zwierlein, "Revealing the Superfluid Lambda Transition in the Universal Thermodynamics of a Unitary Fermi Gas", *Science* **335** (2012), pp. 563–567.

[17] F. Scazza et al., "Repulsive Fermi Polars in a Resonant Mixture of Ultracold $^6$Li Atoms", *Phys. Rev. Lett.* **118** (2017), article no. 083602.

[18] S. Hoinka, P. Dyke, M. G. Lingham, J. J. Kinnunen, G. M. Bruun and C. J. Vale, "Goldstone mode and pair-breaking excitations in atomic Fermi superfluids", *Nature Phys.* **13** (2017), pp. 943–946.

[19] D. Husmann, M. Lebrat, S. Häusler, J.-P. Brantut, L. Corman and T. Esslinger, "Breakdown of the Wiedemann-Franz law in a unitary Fermi gas", *Proc. Natl. Acad. Sci. USA* **115** (2018), pp. 8563–8568.

[20] Z. Yan, P. B. Patel, B. Mukherjee, R. J. Fletcher, J. Struck and M. W. Zwierlein, "Boiling a Unitary Fermi Liquid", *Phys. Rev. Lett.* **122** (2019), article no. 093401.

[21] P. B. Patel, Z. Yan, B. Mukherjee, R. J. Fletcher, J. Struck and M. W. Zwierlein, "Universal sound diffusion in a strongly interacting Fermi gas", *Science* **370** (2020), pp. 1222–1226.

[22] G. Ness, C. Shkedrov, Y. Florshaim, O. K. Diessel, J. von Milczewski, R. Schmidt and Y. Sagi, "Observation of a Smooth Polaron-Molecule Transition in a Degenerate Fermi Gas", *Phys. Rev. X* **10** (2020), no. 4, article no. 041019.

[23] H. Biss, L. Sobirey, N. Luick, M. Bohlen, J. J. Kinnunen, G. M. Bruun, T. Lompe and H. Moritz, "Excitation Spectrum and Superfluid Gap of an Ultracold Fermi Gas", *Phys. Rev. Lett.* **128** (2022), article no. 100401.

[24] Y. Ji, G. L. Schumacher, G. G. T. Assumpção, J. Chen, J. T. Mäkinen, F. J. Vivanco and N. Navon, "Stability of the Repulsive Fermi Gas with Contact Interactions", *Phys. Rev. Lett.* **129** (2022), article no. 203402.
for a homogeneous unpolarized gas. Furthermore, (83), (84) remain valid in the canonical ensemble, with $\delta E_3$ the energy difference (between finite-range and zero-range models) taken at fixed entropy (which equals the free-energy difference at fixed temperature).

As for the wavevector $k_{\text{typ}}$ that appears in the validity conditions, it should be defined by the same procedure as before and then taking the maximum over all significantly populated eigenstates; for example for the balanced unitary gas at thermal equilibrium, $k_{\text{typ}} = \max(k_F, k_T)$ with $k_T$ the thermal wavevector, defined by $\hbar^2 k_T^2/(2m) = k_B T$.

References

[1] A. J. Leggett, "Diatomic Molecules and Cooper Pairs", in *Modern Trends in the Theory of Condensed Matter* (A. Pekalski and J. A. Przystawa, eds.), Springer, 1980, pp. 13–27.

[2] R. Haussmann, "Crossover from BCS superconductivity to Bose-Einstein condensation: a self-consistent theory", *Z. Phys., B* **91** (1993), pp. 291–308.

[3] R. Haussmann, "Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose-Einstein condensation", *Phys. Rev. B* **49** (1994), no. 18, pp. 12975-12983.

[4] S. Giorgini, L. P. Pitaevskii and S. Stringari, "Theory of ultracold Fermi gases", *Rev. Mod. Phys.* **80** (2008), no. 4, pp. 1215-1274.

[5] *The BCS-BEC Crossover and the Unitary Fermi Gas*, (W. Zwerger, ed.), Springer, 1978.

[6] M. Randeria, W. Zwerger and M. W. Zwierlein, "The BCS-BEC Crossover and the Unitary Fermi Gas", in *The BCS-BEC Crossover and the Unitary Fermi Gas* (W. Zwerger, ed.), Springer, 2012, pp. 1–32.

[7] K. M. O'Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade and J. E. Thomas, "Observation of a Strongly Interacting Degenerate Fermi Gas of Atoms", *Science* **298** (2002), no. 5601, pp. 2179–2182.

[8] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag and R. Grimm, "Crossover from a Molecular Bose-Einstein Condensate to a Degenerate Fermi Gas", *Phys. Rev. Lett.* **92** (2004), no. 12, article no. 120401.

[9] C. A. Regal, M. Greiner and D. S. Jin, "Observation of Resonance Condensation of Fermionic Atom Pairs", *Phys. Rev. Lett.* **92** (2004), no. 4, article no. 040403.

[10] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman and W. Ketterle, "Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance", *Phys. Rev. Lett.* **92** (2004), no. 12, article no. 120403.

[11] T. Bourdel et al., "Experimental Study of the BEC-BCS Crossover Region in Lithium 6", *Phys. Rev. Lett.* **93** (2004), no. 5, article no. 050401.

[12] A. Altmeyer et al., "Precision Measurements of Collective Oscillations in the BEC-BCS Crossover", *Phys. Rev. Lett.* **98** (2007), no. 4, article no. 040401.

[13] Y. Shin, C. H. Schunck, A. Schirotzek and W. Ketterle, "Phase diagram of a two-component Fermi gas with resonant interactions", *Nature* **451** (2008), pp. 689–693.

[14] S. Nascimbène, N. Navon, K. J. Jiang, F. Chevy and C. Salomon, "Exploring the thermodynamics of a universal Fermi gas", *Nature* **463** (2010), pp. 1057–1060.

[15] N. Navon, S. Nascimbène, F. Chevy and C. Salomon, "The Equation of State of a Low-Temperature Fermi Gas with Tunable Interactions", *Science* **328** (2010), no. 5979, pp. 729–732.

[16] M. J. H. Ku, A. Sommer, L. W. Cheuk and M. W. Zwierlein, "Revealing the Superfluid Lambda Transition in the Universal Thermodynamics of a Unitary Fermi Gas", *Science* **335** (2012), pp. 563–567.

[17] F. Scazza et al., "Repulsive Fermi Polars in a Resonant Mixture of Ultracold $^6$Li Atoms", *Phys. Rev. Lett.* **118** (2017), article no. 083602.

[18] S. Hoinka, P. Dyke, M. G. Lingham, J. J. Kinnunen, G. M. Bruun and C. J. Vale, "Goldstone mode and pair-breaking excitations in atomic Fermi superfluids", *Nature Phys.* **13** (2017), pp. 943–946.

[19] D. Husmann, M. Lebrat, S. Häusler, J.-P. Brantut, L. Corman and T. Esslinger, "Breakdown of the Wiedemann-Franz law in a unitary Fermi gas", *Proc. Natl. Acad. Sci. USA* **115** (2018), pp. 8563–8568.

[20] Z. Yan, P. B. Patel, B. Mukherjee, R. J. Fletcher, J. Struck and M. W. Zwierlein, "Boiling a Unitary Fermi Liquid", *Phys. Rev. Lett.* **122** (2019), article no. 093401.

[21] P. B. Patel, Z. Yan, B. Mukherjee, R. J. Fletcher, J. Struck and M. W. Zwierlein, "Universal sound diffusion in a strongly interacting Fermi gas", *Science* **370** (2020), pp. 1222–1226.

[22] G. Ness, C. Shkedrov, Y. Florshaim, O. K. Diessel, J. von Milczewski, R. Schmidt and Y. Sagi, "Observation of a Smooth Polaron-Molecule Transition in a Degenerate Fermi Gas", *Phys. Rev. X* **10** (2020), no. 4, article no. 041019.

[23] H. Biss, L. Sobirey, N. Luick, M. Bohlen, J. J. Kinnunen, G. M. Bruun, T. Lompe and H. Moritz, "Excitation Spectrum and Superfluid Gap of an Ultracold Fermi Gas", *Phys. Rev. Lett.* **128** (2022), article no. 100401.

[24] Y. Ji, G. L. Schumacher, G. G. T. Assumpção, J. Chen, J. T. Mäkinen, F. J. Vivanco and N. Navon, "Stability of the Repulsive Fermi Gas with Contact Interactions", *Phys. Rev. Lett.* **129** (2022), article no. 203402.
[25] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack and R. G. Hulet, "Molecular Probe of Pairing in the BEC-BCS Crossover", Phys. Rev. Lett. 95 (2005), article no. 020404.

[26] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford and C. J. Vale, "Universal Behavior of Pair Correlations in a Strongly Interacting Fermi Gas", Phys. Rev. Lett. 105 (2010), no. 7, article no. 070402.

[27] E. D. Kuhnle, S. Hoinka, P. Dyke, H. Hu, P. Hannaford and C. J. Vale, "Temperature Dependence of the Universal Contact Parameter in a Unitary Fermi Gas", Phys. Rev. Lett. 106 (2011), article no. 170402.

[28] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale, J. E. Drut and S. Gandolfi, "Precise Determination of the Structure Factor and Contact in a Unitary Fermi Gas", Phys. Rev. Lett. 110 (2013), no. 5, article no. 055305.

[29] J. T. Stewart, J. P. Gaebler, T. E. Drake and D. S. Jin, "Verification of Universal Relations in a Strongly Interacting Fermi Gas", Phys. Rev. Lett. 104 (2010), no. 23, article no. 235301.

[30] Y. Sagi, T. E. Drake, R. Paudel and D. S. Jin, "Measurement of the Homogeneous Contact of a Unitary Fermi gas", Phys. Rev. Lett. 109 (2012), article no. 220402.

[31] C. Shkedrov, Y. Florshaim, G. Ness, A. Gandman and Y. Sagi, "High-Sensitivity rf Spectroscopy of a Strongly Interacting Fermi Gas", Phys. Rev. Lett. 121 (2018), article no. 093402.

[32] C. Carcy, S. Hoinka, M. G. Lingham, P. Dyke, C. C. N. Kuhn, H. Hu and C. J. Vale, "Contact and Sum Rules in a Near-Uniform Fermi Gas at Unitarity", Phys. Rev. Lett. 122 (2019), article no. 203401.

[33] B. Mukherjee, P. B. Patel, Z. Yan, R. J. Fletcher, J. Struck and M. W. Zwierlein, "Spectral Response and Contact of the Unitary Fermi Gas", Phys. Rev. Lett. 122 (2019), article no. 203402.

[34] S. Laurent, M. Pierce, M. Delehaye, T. Yefsah, F. Chevy and C. Salomon, "Connecting Few-Body Inelastic Decay to Quantum Correlations in a Many-Body System: A Weakly Coupled Impurity in a Resonant Fermi Gas", Phys. Rev. Lett. 118 (2017), article no. 103403.

[35] S. Tan, "Energetics of a strongly correlated Fermi gas", Ann. Phys. 323 (2008), no. 12, pp. 2952-2970.

[36] S. Tan, "Large momentum part of a strongly correlated Fermi gas", Ann. Phys. 323 (2008), no. 12, pp. 2971-2986.

[37] A. J. Leggett and S. Zhang, "The BEC-BCS Crossover: Some History and Some General Observations", in The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, 2012, pp. 33-47.

[38] Y. Castin and F. Werner, "The Unitary Gas and its Symmetry Properties", in The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, 2012, pp. 127-191.

[39] E. Braaten, "Universal Relations for Fermions with Large Scattering Length", in The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, 2011, pp. 193-231.

[40] W. Schneider, V. B. Shenoy and M. Randeria, "Theory of Radio Frequency Spectroscopy of Polarized Fermi Gases", 2009. preprint, arXiv:0903.3006.

[41] R. Haussmann, M. Punk and W. Zwerger, "Spectral functions and rf response of ultracold fermionic atoms", Phys. Rev. A 80 (2009), article no. 063612.

[42] W. Schneider and M. Randeria, "Universal short-distance structure of the single-particle spectral function of dilute Fermi gases", Phys. Rev. A 81 (2010), article no. 021601.

[43] G. Baym, C. J. Pethick, Z. Yu and M. W. Zwierlein, "Coherence and Clock Shifts in Ultracold Fermi Gases with Resonant Interactions", Phys. Rev. Lett. 99 (2007), no. 19, article no. 190407.

[44] E. Braaten and L. Platter, "Exact Relations for a Strongly Interacting Fermi Gas from the Operator Product Expansion", Phys. Rev. Lett. 100 (2008), no. 20, article no. 205301.

[45] S. Tan, "Generalized virial theorem and pressure relation for a strongly correlated Fermi gas", Ann. Phys. 323 (2008), no. 12, pp. 2987-2990.

[46] F. Werner, "Virial theorems for trapped cold atoms", Phys. Rev. A 78 (2008), article no. 025601.

[47] M. Punk and W. Zwerger, "Theory of rf-Spectroscopy of Strongly Interacting Fermions", Phys. Rev. Lett. 99 (2007), article no. 170404.

[48] E. Braaten, D. Kang and L. Platter, "Universal relations for a strongly interacting Fermi gas near a Feshbach resonance", Phys. Rev. A 78 (2008), article no. 053606.

[49] F. Werner, L. Tarruell and Y. Castin, "Number of closed-channel molecules in the BEC-BCS crossover", Eur. Phys. J. B, Condens. Matter Complex Syst. 68 (2009), pp. 401-415.

[50] S. Zhang and A. J. Leggett, "Universal properties of the ultracold Fermi gas", Phys. Rev. A 79 (2009), no. 2, article no. 023601.

[51] R. Combescot, F. Alzetto and X. Leyronas, "Particle distribution tail and related energy formula", Phys. Rev. A 79 (2009), no. 5, article no. 053640.

[52] F. Werner and Y. Castin, "Exact relations for quantum-mechanical few-body and many-body problems with short-range interactions in two and three dimensions", 2018. preprint, arXiv:1001.0774.

[53] J. Hofmann, "Quantum Anomaly, Universal Relations, and Breathing Mode of a Two-Dimensional Fermi Gas", Phys. Rev. Lett. 108 (2012), article no. 185303.

[54] S. Tan, "Universal Energy Functional for Trapped Fermi Gases with Short Range Interactions", Phys. Rev. Lett. 107 (2011), article no. 145302.
[25] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack and R. G. Hulet, "Molecular Probe of Pairing in the BEC-BCS Crossover", Phys. Rev. Lett. 95 (2005), article no. 020404.

[26] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford and C. J. Vale, "Universal Behavior of Pair Correlations in a Strongly Interacting Fermi Gas", Phys. Rev. Lett. 105 (2010), no. 7, article no. 070402.

[27] E. D. Kuhnle, S. Hoinka, P. Dyke, H. Hu, P. Hannaford and C. J. Vale, "Temperature Dependence of the Universal Contact Parameter in a Unitary Fermi Gas", Phys. Rev. Lett. 106 (2011), article no. 170402.

[28] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale, J. E. Drut and S. Gandolfi, "Precise Determination of the Structure Factor and Contact in a Unitary Fermi Gas", Phys. Rev. Lett. 110 (2013), no. 5, article no. 055305.

[29] J. T. Stewart, J. P. Gaebler, T. E. Drake and D. S. Jin, "Verification of Universal Relations in a Strongly Interacting Fermi Gas", Phys. Rev. Lett. 104 (2010), no. 23, article no. 235301.

[30] Y. Sagi, T. E. Drake, R. Paudel and D. S. Jin, "Measurement of the Homogeneous Contact of a Unitary Fermi gas", Phys. Rev. Lett. 109 (2012), article no. 220402.

[31] C. Shkedrov, Y. Florshaim, G. Ness, A. Gandman and Y. Sagi, "High-Sensitivity rf Spectroscopy of a Strongly Interacting Fermi Gas", Phys. Rev. Lett. 121 (2018), article no. 093402.

[32] C. Carcy, S. Hoinka, M. G. Lingham, P. Dyke, C. C. N. Kuhn, H. Hu and C. J. Vale, "Contact and Sum Rules in a Near-Uniform Fermi Gas at Unitarity", Phys. Rev. Lett. 122 (2019), article no. 203401.

[33] B. Mukherjee, P. B. Patel, Z. Yan, R. J. Fletcher, J. Struck and M. W. Zwierlein, "Spectral Response and Contact of the Unitary Fermi Gas", Phys. Rev. Lett. 122 (2019), article no. 203402.

[34] S. Laurent, M. Pierce, M. Delehaye, T. Yefsah, F. Chevy and C. Salomon, "Connecting Few-Body Inelastic Decay to Quantum Correlations in a Many-Body System: A Weakly Coupled Impurity in a Resonant Fermi Gas", Phys. Rev. Lett. 118 (2017), article no. 103403.

[35] S. Tan, "Energetics of a strongly correlated Fermi gas", Ann. Phys. 323 (2008), no. 12, pp. 2952-2970.

[36] S. Tan, "Large momentum part of a strongly correlated Fermi gas", Ann. Phys. 323 (2008), no. 12, pp. 2971-2986.

[37] A. J. Leggett and S. Zhang, "The BEC-BCS Crossover: Some History and Some General Observations", in The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, 2012, pp. 33-47.

[38] Y. Castin and F. Werner, "The Unitary Gas and its Symmetry Properties", in The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, 2012, pp. 127-191.

[39] E. Braaten, "Universal Relations for Fermions with Large Scattering Length", in The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, 2011, pp. 193-231.

[40] W. Schneider, V. B. Shenoy and M. Randeria, "Theory of Radio Frequency Spectroscopy of Polarized Fermi Gases", 2009. preprint, arXiv:0903.3006.

[41] R. Haussmann, M. Punk and W. Zwerger, "Spectral functions and rf response of ultracold fermionic atoms", Phys. Rev. A 80 (2009), article no. 063612.

[42] W. Schneider and M. Randeria, "Universal short-distance structure of the single-particle spectral function of dilute Fermi gases", Phys. Rev. A 81 (2010), article no. 021601.

[43] G. Baym, C. J. Pethick, Z. Yu and M. W. Zwierlein, "Coherence and Clock Shifts in Ultracold Fermi Gases with Resonant Interactions", Phys. Rev. Lett. 99 (2007), no. 19, article no. 190407.

[44] E. Braaten and L. Platter, "Exact Relations for a Strongly Interacting Fermi Gas from the Operator Product Expansion", Phys. Rev. Lett. 100 (2008), no. 20, article no. 205301.

[45] S. Tan, "Generalized virial theorem and pressure relation for a strongly correlated Fermi gas", Ann. Phys. 323 (2008), no. 12, pp. 2987-2990.

[46] F. Werner, "Virial theorems for trapped cold atoms", Phys. Rev. A 78 (2008), article no. 025601.

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[56] F. Werner and Y. Castin, "General relations for quantum gases in two and three dimensions. Two-component fermions", Phys. Rev. A 86 (2012), article no. 013626.

[57] Z. Yu, G. M. Bruun and G. Baym, "Short-range correlations and entropy in ultracold-atom Fermi gases", Phys. Rev. A 80 (2009), article no. 023615.

[58] H. Hu, X.-J. Liu and P. D. Drummond, "Universal contact of strongly interacting fermions at finite temperatures", New J. Phys. 13 (2011), article no. 035007.

[59] M. Sun and X. Leyronas, "High-temperature expansion for interacting fermions", Phys. Rev. A 92 (2015), article no. 053611.

[60] F. Palestini, A. Perali, P. Pieri and G. C. Strinati, "Temperature and coupling dependence of the universal contact intensity for an ultracold Fermi gas", Phys. Rev. A 82 (2010), no. 2, article no. 021605.

[61] T. Enss, R. Haussmann and W. Zwerger, "Viscosity and scale invariance in the unitary Fermi gas", Ann. Phys. 326 (2011), no. 3, pp. 770-796.

[62] S. Gandolfi, K. E. Schmidt and J. Carlson, "BEC-BCS crossover and universal relations in unitary Fermi gases", Phys. Rev. A 83 (2011), article no. 041601.

[63] J. E. Drut, T. A. Lähde and T. Ten, "Momentum Distribution and Contact of the Unitary Fermi Gas", Phys. Rev. Lett. 106 (2011), article no. 205302.

[64] I. Boettcher, S. Diehl, J. M. Pawlowski and C. Wetterich, "Tan contact and universal high momentum behavior of the fermion propagator in the BCS-BEC crossover", Phys. Rev. A 87 (2013), no. 2, article no. 023606.

[65] O. Goulko and M. Wingate, "Numerical study of the unitary Fermi gas across the superfluid transition", Phys. Rev. A 93 (2016), article no. 053604.

[66] R. Rossi, T. Ohgoe, E. Kozik, N. Prokof'ev, B. Svistunov, K. Van Houcke and F. Werner, "Contact and Momentum Distribution of the Unitary Fermi Gas", Phys. Rev. Lett. 121 (2018), no. 13, article no. 130406.

[67] S. Jensen, C. N. Gilbreth and Y. Alhassid, "Contact in the Unitary Fermi Gas across the Superfluid Phase Transition", Phys. Rev. Lett. 125 (2020), article no. 043402.

[68] C. Langmack, M. Barth, W. Zwerger and E. Braaten, "Shift in a Strongly Interacting Two-Dimensional Fermi Gas", Phys. Rev. Lett. 108 (2012), article no. 060402.

[69] M. Valiente, N. T. Zinner and K. Mølmer, "Universal properties of Fermi gases in arbitrary dimensions", Phys. Rev. A 86 (2012), article no. 043616.

[70] M. Barth and W. Zwerger, "Tan relations in one dimension", Ann. Phys. 326 (2011), no. 10, pp. 2544-2565.

[71] M. Olshanii and V. Dunjko, "Short-Distance Correlation Properties of the Lieb-Liniger System and Momentum Distributions of Trapped One-Dimensional Atomic Gases", Phys. Rev. Lett. 91 (2003), article no. 090401.

[72] F. Werner and Y. Castin, "General relations for quantum gases in two and three dimensions. II. Bosons and mixtures", Phys. Rev. A 86 (2012), article no. 053633.

[73] Y.-Q. Zou, B. Bakkali-Hassani, C. Maury, É. Le Cerf, S. Nascimbene, J. Dalibard and J. Beugnon, "Tan's two-body contact across the superfluid transition of a planar Bose gas", Nat. Commun. 12 (2021), article no. 760.

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[84] Y. Castin and F. Werner, "Single-Particle Momentum Distribution of an Efimov trimer", Phys. Rev. A 83 (2011), article no. 063614.
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