so that we do not show ${\mathbb{I}_{S}}$ explicitly in the following. We
proceed with the derivation in detail for $L_{m}(p,R\mathbf{a})$, and for
$L_{d}(p,R\mathbf{a})$ we quote the final result because the procedure is
completely analogous.

$\displaystyle L_{m}(p,R\mathbf{a})$
$\displaystyle=-m\int\frac{d^{3}k}{(2\pi)^{3}}\left[\theta({k_{F_{1}}}-|R\mathbf{a}+\mathbf{k}|)+\theta({k_{F_{2}}}-|R\mathbf{a}-\mathbf{k}|)\right]\frac{|\mathbf{k}\rangle\langle\mathbf{k}|}{k^{2}-p^{2}-i\epsilon}$
(A.4)
$\displaystyle=-m\int\frac{d^{3}k}{(2\pi)^{3}}\left[\theta({k_{F_{1}}}-|\mathbf{a}+R^{-1}\mathbf{k}|)+\theta({k_{F_{2}}}-|\mathbf{a}-R^{-1}\mathbf{k}|)\right]\frac{|\mathbf{k}\rangle\langle\mathbf{k}|}{k^{2}-p^{2}-i\epsilon}~{}.$

We perform next the change of integration variable
$R^{-1}\mathbf{k}\to\mathbf{k}$ and take into account that
$|R\mathbf{k}\rangle=R|\mathbf{k}\rangle$. Thus, Eq. (A.1) results. Analogous
steps also lead to Eq. (A.2).

We use now Eq. (A.1) to study the transformation properties of
$t_{m}(\mathbf{a})$ under a rotation of $\mathbf{a}$. We also take into
account the rotational invariance of the vacuum $T$-matrix $t_{V}$,

$\displaystyle Rt_{V}R^{\dagger}$ $\displaystyle=t_{V}~{}.$ (A.5)

Multiplying both sides of Eq. (2.24) by $R$ and $R^{\dagger}$ to the left and
right, respectively, we have that

$\displaystyle Rt_{m}(\mathbf{a})R^{\dagger}$
$\displaystyle=t_{V}+Rt_{V}L_{m}(p,\mathbf{a})t_{m}(\mathbf{a})R^{\dagger}$
(A.6)
$\displaystyle=t_{V}+t_{V}L_{m}(p,R\mathbf{a})Rt_{m}(\mathbf{a})R^{\dagger}~{},$

so that $Rt_{m}(\mathbf{a})R^{\dagger}$ and $t_{m}(R\mathbf{a})$ satisfy the
same IE and then they must coincide,

$\displaystyle t_{m}(R\mathbf{a})$
$\displaystyle=Rt_{m}(\mathbf{a})R^{\dagger}~{}.$ (A.7)

##  Appendix B Some symmetry properties of the PWAs in the many-body
environment

We proceed with the demonstration of the relations in Eqs. (4.20), (4.21) and
(4.22). To demonstrate the equality

$\displaystyle\langle
J_{2}{-\mu_{1}}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}{-\mu_{1}}\ell_{1}S_{1}p\rangle$
$\displaystyle=(-1)^{J_{2}+J_{1}}\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle~{},$
(B.1)

let us write down the IE of Eq. (4.14) for the PWAs with $-\mu_{1}$,

$\displaystyle\langle
J_{2}-\mu_{1}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}-\mu_{1}\ell_{1}S_{1}p\rangle=\langle
J_{2}-\mu_{1}\ell_{2}S_{1}p^{\prime}|V|J_{1}-\mu_{1}\ell_{1}S_{1}p\rangle$
(B.2)
$\displaystyle+\sum_{J_{4}\ell_{3}\ell_{4}m_{3}\sigma_{3}}\chi(S_{1}\ell_{3})\chi(S_{1}\ell_{4})\frac{m}{(2\pi)^{2}}\int\frac{k^{2}dk}{k^{2}-p^{2}-i\epsilon}\langle
J_{2}-\mu_{1}\ell_{2}S_{1}p^{\prime}|V|J_{2}-\mu_{1}\ell_{3}S_{1}k\rangle$
$\displaystyle\times\langle
J_{4}-\mu_{1}\ell_{4}S_{1}k|t_{m}(a\hat{\mathbf{z}})|J_{1}-\mu_{1}\ell_{1}S_{1}p\rangle(m_{3}\sigma_{3}-\mu_{1}|\ell_{3}S_{1}J_{2})(m_{3}\sigma_{3}-\mu_{1}|\ell_{4}S_{1}J_{4})$
$\displaystyle\times\int
d\hat{\mathbf{k}}Y_{\ell_{3}}^{m_{3}}(\hat{\mathbf{k}})^{*}Y_{\ell_{4}}^{m_{3}}(\mathbf{k})\left[1-\theta({k_{F}}-|\mathbf{k}+a\hat{\mathbf{z}}|)-\theta({k_{F}}-|\mathbf{k}-a\hat{\mathbf{z}}|)\right]~{}.$

The next step is to take into account that the matrix elements of $V$ are
independent of $\mu_{1}$ and use the symmetry properties of the Clebsch-Gordan
coefficients

$\displaystyle(-m_{1}-m_{2}-m_{3}|j_{1}j_{2}j_{3})$
$\displaystyle=(-1)^{j_{1}+j_{2}-j_{3}}(m_{1}m_{2}m_{3}|j_{1}j_{2}j_{3})~{},$
(B.3)

so that

$\displaystyle(m_{3}\sigma_{3}-\mu_{1}|\ell_{3}S_{1}J_{2})(m_{3}\sigma_{3}-\mu_{1}|\ell_{4}S_{1}J_{4})$
$\displaystyle=(-1)^{J_{2}+J_{4}}(-m_{3}-\sigma_{3}\mu_{1}|\ell_{3}S_{1}J_{2})(-m_{3}-\sigma_{3}\mu_{1}|\ell_{4}S_{1}J_{4})~{},$
(B.4)

where we have used that $(-1)^{\ell_{3}+\ell_{4}}=+1$ because of Eq. (4.11).
The sum over the dummy indices $m_{3}$ and $\sigma_{3}$ is symmetric around 0,
so that by exchanging their signs simultaneously the only change is in the the
angular integration. But this does not introduce any change in the final
result because of the well-known property of the spherical harmonics

$\displaystyle Y_{\ell}^{m}(\mathbf{k})^{*}$
$\displaystyle=(-1)^{m}Y_{\ell}^{-m}(\mathbf{k})~{}.$ (B.5)

Therefore,

$\displaystyle\int
d\hat{\mathbf{k}}Y_{\ell_{3}}^{-m_{3}}(\hat{\mathbf{k}})^{*}Y_{\ell_{4}}^{-m_{3}}(\mathbf{k})\left[1-\theta({k_{F}}-|\mathbf{k}+a\hat{\mathbf{z}}|)-\theta({k_{F}}-|\mathbf{k}-a\hat{\mathbf{z}}|)\right]=\int
d\hat{\mathbf{k}}Y_{\ell_{3}}^{m_{3}}(\hat{\mathbf{k}})Y_{\ell_{4}}^{m_{3}}(\mathbf{k})^{*}$
(B.6)
$\displaystyle\times\left[1-\theta({k_{F}}-|\mathbf{k}+a\hat{\mathbf{z}}|)-\theta({k_{F}}-|\mathbf{k}-a\hat{\mathbf{z}}|)\right]~{}.$

However, the integrand in the previous equation is real, and by taking its
complex conjugate, we recover again the original expression in Eq. (B.2). We
can then rewrite it as

$\displaystyle\langle
J_{2}-\mu_{1}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}-\mu_{1}\ell_{1}S_{1}p\rangle=\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|V|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle$ (B.7)
$\displaystyle+\sum_{m_{3}\sigma_{3}J_{4}\ell_{3}\ell_{4}}\chi(S_{1}\ell_{3})\chi(S_{1}\ell_{4})\frac{m}{(2\pi)^{2}}\int\frac{k^{2}dk}{k^{2}-p^{2}-i\epsilon}\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|V|J_{2}\mu_{1}\ell_{3}S_{1}k\rangle$
$\displaystyle\times\langle
J_{4}-\mu_{1}\ell_{4}S_{1}k|t_{m}(a\hat{\mathbf{z}})|J_{1}-\mu_{1}\ell_{1}S_{1}p\rangle(-1)^{J_{2}+J_{4}}(m_{3}\sigma_{3}\mu_{1}|\ell_{3}S_{1}J_{2})(m_{3}\sigma_{3}\mu_{1}|\ell_{4}S_{1}J_{4})$
$\displaystyle\times\int
d\hat{\mathbf{k}}Y_{\ell_{3}}^{m_{3}}(\hat{\mathbf{k}})^{*}Y_{\ell_{4}}^{m_{3}}(\mathbf{k})\left[1-\theta({k_{F}}-|\mathbf{k}+a\hat{\mathbf{z}}|)-\theta({k_{F}}-|\mathbf{k}-a\hat{\mathbf{z}}|)\right]~{},$

where we have taken into account again that the matrix elements of $V$ between
PWAs are $\mu$ independent because it is an scalar operator. Let us multiply
both sides of Eq. (B.7) by $(-1)^{J_{2}+J_{1}}$ and then we conclude that
$(-1)^{J_{2}+J_{1}}\langle
J_{2}-\mu_{1}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}-\mu_{1}\ell_{1}S_{1}p\rangle$
satisfies the same IE as $\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle$,
which implies Eq. (B.1). Notice that when multiplying $\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|V|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle$ by
$(-1)^{J_{1}+J_{2}}$ we get the same result because the matrix element is
proportional to $\delta_{J_{1}J_{2}}$. Furthermore, for the matrix element of
$t_{m}(a\hat{\mathbf{z}})$ inside the integrand since the $J_{i}$ are integers
it results that $(-1)^{J_{1}+J_{2}}(-1)^{J_{2}+J_{4}}=(-1)^{J_{1}+J_{4}}$.

For the demonstration of the symmetric relation in Eq. (4.22),

$\displaystyle\langle
J_{1}\mu_{1}\ell_{1}S_{1}p|t_{m}(a\hat{\mathbf{z}})|J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}\rangle$
$\displaystyle=\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle~{},$
(B.8)

we start with the IE for $\langle
J_{1}\mu_{1}\ell_{1}S_{1}p|t_{m}(a\hat{\mathbf{z}})|J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}\rangle$,
cf. Eq. (4.14),

$\displaystyle\langle
J_{1}\mu_{1}\ell_{1}S_{1}p|t_{m}(a\hat{\mathbf{z}})|J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}\rangle=\langle
J_{1}\mu_{1}\ell_{1}S_{1}p|V|J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}\rangle$ (B.9)
$\displaystyle+\sum_{J_{4}\ell_{3}\ell_{4}m_{3}\sigma_{3}}\chi(S_{1}\ell_{3})\chi(S_{1}\ell_{4})\frac{m}{(2\pi)^{2}}\int_{0}^{\infty}\frac{k^{2}dk}{k^{2}-p^{2}-i\epsilon}\langle
J_{1}\mu_{1}\ell_{1}S_{1}p|V|J_{1}\mu_{1}\ell_{3}S_{1}k\rangle$
$\displaystyle\times\langle
J_{4}\mu_{1}\ell_{4}S_{1}k|t_{m}(a\hat{\mathbf{z}})|J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}\rangle(m_{3}\sigma_{3}\mu_{1}|\ell_{3}S_{1}J_{1})(m_{3}\sigma_{3}\mu_{1}|\ell_{4}S_{1}J_{4})$
$\displaystyle\times\int
d\hat{\mathbf{k}}Y_{\ell_{3}}^{m_{3}}(\hat{\mathbf{k}})^{*}Y_{\ell_{4}}^{m_{3}}(\hat{\mathbf{k}})[1-\theta({k_{F}}-|\mathbf{k}+a\hat{\mathbf{z}}|)-\theta({k_{F}}-|\mathbf{k}-a\hat{\mathbf{z}}|)]~{}.$

Because of time-reversal invariance the matrix elements of $V$ in PWAs are
symmetric. In addition, since the product of the two spherical harmonics is
real we take its complex conjugate. Then, we can rewrite Eq. (B.9) as

$\displaystyle\langle
J_{1}\mu_{1}\ell_{1}S_{1}p|t_{m}(a\hat{\mathbf{z}})|J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}\rangle=\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|V|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle$
(B.10)
$\displaystyle+\sum_{J_{4}\ell_{3}\ell_{4}m_{3}\sigma_{3}}\chi(S\ell_{3})\chi(S\ell_{4})\frac{m}{(2\pi)^{2}}\int_{0}^{\infty}\frac{k^{2}dk}{k^{2}-p^{2}-i\epsilon}\langle
J_{4}\mu_{1}\ell_{4}S_{1}k|t_{m}(a\hat{\mathbf{z}})|J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}\rangle$
$\displaystyle\times\langle
J_{1}\mu_{1}\ell_{3}S_{1}k|V|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle(m_{3}\sigma_{3}\mu_{1}|\ell_{3}S_{1}J_{1})(m_{3}\sigma_{3}\mu_{1}|\ell_{4}S_{1}J_{4})$
$\displaystyle\times\int
d\hat{\mathbf{k}}Y_{\ell_{4}}^{m_{3}}(\hat{\mathbf{k}})^{*}Y_{\ell_{3}}^{m_{3}}(\hat{\mathbf{k}})[1-\theta({k_{F}}-|\mathbf{k}+a\hat{\mathbf{z}}|)-\theta({k_{F}}-|\mathbf{k}-a\hat{\mathbf{z}}|)]~{},$

which is the same IE as the one satisfied by $\langle
J_{2}\mu_{1}\ell_{2}S_{1}p^{\prime}|t_{m}(a\hat{\mathbf{z}})|J_{1}\mu_{1}\ell_{1}S_{1}p\rangle$
as we wanted to show. In order to arrive to this conclusion we have used the
fact that the IE for $t_{m}(\mathbf{a})$ of Eq. (4.1) can also be written as

$\displaystyle t_{m}(\mathbf{a})$
$\displaystyle=V-t_{m}(\mathbf{a})[G-L_{m}(p,\mathbf{a})]V~{}.$ (B.11)

##  Appendix C Calculations employing dimensional regularization

Here we discuss the calculation with DR of the $S$-wave and $P$-wave
potentials up to ${\cal O}(p^{4})$ and ${\cal O}(p^{2})$, respectively.

###  C.1 $S$ waves

We take the potential in Eq. (6.16),

$\displaystyle v(k,p)$
$\displaystyle=c_{0}+\frac{1}{2}c_{2}(k^{2}+p^{2})+\frac{1}{2}c_{4}(k^{4}+p^{4})~{},$
(C.1)

and proceed with the calculation of ${\mathsf{t}_{m}}(k,p)$ by applying the
method of Sec. 5. The coefficients in the expansion of the potential in powers
of $k^{2}$ and $p^{2}$ are denoted in this case $v_{ij}$, which is enough
since it is uncoupled. Therefore, the only non-zero coefficients are

$\displaystyle
v_{11}=c_{0}~{},~{}v_{12}=v_{21}=\frac{1}{2}c_{2}~{},~{}v_{13}=v_{31}=\frac{1}{2}c_{4}~{}.$
(C.2)

Next one needs to implement the matrix $[\cal G]$, Eq. (5.7), given in this
case by

$\displaystyle[{\cal G}]$
$\displaystyle=-\frac{m}{(2\pi)^{3}}\int_{0}^{\infty}\frac{d^{3}k}{k^{2}-p^{2}-i\epsilon}\left(\begin{array}[]{lll}1&k^{2}&k^{4}\\\
k^{2}&k^{4}&k^{6}\\\
k^{4}&k^{6}&k^{8}\end{array}\right)\left(1-2\theta(k_{F}-|\mathbf{k}-a\hat{\mathbf{z}}|)\right)~{}.$
(C.6)

In DR the calculation of the free part of $[{\cal G}]$, called $[{\cal
G}_{f}]$, is straightforward because $\int_{0}^{\infty}d^{D}kk^{n}=0$ for
$n\geq 0$. The result is

$\displaystyle[{\cal G}_{f}]$
$\displaystyle=-i\frac{mp}{4\pi}\left(\begin{array}[]{lll}1&p^{2}&p^{4}\\\
p^{2}&p^{4}&p^{6}\\\ p^{4}&p^{6}&p^{8}\end{array}\right)~{}.$ (C.10)

Applying Eq. (5.8) with $[{\cal G}_{f}]$ one obtains for
${\mathsf{t}_{V}}(k,p)$ the following expression

$\displaystyle{\mathsf{t}_{V}}(k,p)$
$\displaystyle=[k]^{T}\left([v]^{-1}+[{\cal G}_{f}]^{-1}\right)^{-1}[p]$
(C.11) $\displaystyle=v(k,p)\left(1-i\frac{mp}{4\pi}v(p,p)\right)^{-1}~{}.$

Indeed, the last formula is a general one for any uncoupled vacuum PWA
calculated with DR. Taking the explicit expression for $v(k,p)$ we have

$\displaystyle{\mathsf{t}_{V}}(k,p)$
$\displaystyle=(c_{0}+\frac{1}{2}c_{2}(k^{2}+p^{2})+\frac{1}{2}c_{4}(k^{4}+p^{4}))\left(1-i\frac{mp}{4\pi}(c_{0}+c_{2}p^{2}+c_{4}p^{4})\right)^{-1}~{}.$
(C.12)

The matching with the ERE expansion, Eq. (5.10), of the on-shell
${\mathsf{t}_{V}}(p,p)$ up to ${\cal O}(p^{4})$ is straightforward and then

$\displaystyle\frac{m}{4\pi}c_{0}$ $\displaystyle=-a_{0}~{},$ (C.13)
$\displaystyle\frac{m}{4\pi}c_{2}$
$\displaystyle=-\frac{a_{0}^{2}r_{0}}{2}~{},$
$\displaystyle\frac{m}{4\pi}c_{4}$
$\displaystyle=-\frac{a_{0}^{2}}{4}(a_{0}r_{0}^{2}+4v_{0}^{(2)})~{}.$

With these expressions for the couplings

$\displaystyle{\mathsf{t}_{V}}(p,p)$
$\displaystyle=-\frac{4\pi}{m}\left(\frac{1}{a_{0}+\frac{1}{2}a_{0}^{2}r_{0}p^{2}+a_{0}^{2}(v_{0}^{(2)}+\frac{1}{4}a_{0}r_{0}^{2})p^{4}}+ip\right)^{-1}~{}.$
(C.14)

The $k_{F}$ dependence of the matrix elements of $[\cal G]$, Eq. (C.6), can be
written in terms of the basic integrals

$\displaystyle J_{n}$ $\displaystyle=\frac{m}{4\pi^{3}}\int
d^{3}k\,k^{2n}\theta(k_{F}-|\mathbf{k}-a\hat{\mathbf{z}}|)=\frac{m}{4\pi^{3}}\int
d^{3}k\,(\mathbf{k}+a\hat{\mathbf{z}})^{2n}\theta(k_{F}-k)~{},$ (C.15)
$\displaystyle I_{n}$
$\displaystyle=\frac{m}{4\pi^{3}}\int\frac{d^{3}k}{k^{2}-p^{2}-i\epsilon}k^{2n}\theta(k_{F}-|\mathbf{k}-a\hat{\mathbf{z}}|)=\sum_{l=0}^{n-1}p^{2(n-1-l)}J_{l}+p^{2n}I_{0}~{},$

such that

$\displaystyle{[\cal G]}_{jl}$
$\displaystyle=-i\frac{mp^{2(j+l)-3}}{4\pi}+I_{j+l-2}~{}.$ (C.16)

For the basic $I_{0}$ integral we have the relation

$\displaystyle{[\cal G]}_{11}$
$\displaystyle=-i\frac{mp}{4\pi}+I_{0}=\frac{mk_{F}}{4\pi}(\frac{R}{\pi}+i\,I)~{},$
(C.17)

where we have related $I_{0}$ with the functions $R$ and $I$ of Ref. [43],
also used in Sec. 6.

To calculate ${\mathsf{t}_{m}}(p,p)$ we just have to express the couplings as
in Eq. (C.13) in terms of the ERE parameters and apply Eq. (5.8) with $[\cal
G]$ as in Eq. (C.16). Then,

$\displaystyle{\mathsf{t}_{m}}(p,p)=[p]^{T}[\hat{t}_{m}(p)][p]=\frac{4\pi}{m}\left(W_{0}^{-1}+\frac{4\pi}{m}I_{0}-ip\right)^{-1}~{},$
(C.18) $\displaystyle W_{0}^{-1}=\frac{A}{B}~{},$ (C.19) $\displaystyle
A=-28(-30\pi+a_{0}^{2}k_{F}^{3}(10r_{0}+a_{0}k_{F}^{2}r_{0}^{2}(3+5s^{2}+5\kappa^{2})+4k_{F}^{2}v_{0}^{(2)}(3+5s^{2}+5\kappa^{2})))^{2}~{},$
$\displaystyle
B=5a_{0}\pi(1260\pi(4+2a_{0}k_{F}^{2}r_{0}\kappa^{2}+a_{0}k_{F}^{4}(a_{0}r_{0}^{2}+4v_{0}^{(2)})\kappa^{4})+a_{0}^{3}k_{F}^{5}(60a_{0}k_{F}^{2}r_{0}^{3}(3+7s^{2}(2+s^{2})-7\kappa^{4})$
$\displaystyle+240k_{F}^{2}r_{0}v_{0}^{(2)}(3+7s^{2}(2+s^{2})-7\kappa^{4})+4r_{0}^{2}(21(3+5s^{2})+14a_{0}k_{F}^{4}(5+45s^{2}+63s^{4}+15s^{6})v_{0}^{(2)}$
$\displaystyle+15(-7+2a_{0}k_{F}^{4}(3+7s^{2}(2+s^{2}))v_{0}^{(2)})\kappa^{2}-42a_{0}k_{F}^{4}(3+5s^{2})v_{0}^{(2)}\kappa^{4}-210a_{0}k_{F}^{4}v_{0}^{(2)}\kappa^{6})+a_{0}^{2}k_{F}^{4}r_{0}^{4}(5(7+9\kappa^{2})$
$\displaystyle+21(5s^{6}-5s^{2}(-3+\kappa^{2})(1+\kappa^{2})-\kappa^{4}(3+5\kappa^{2})+s^{4}(21+5\kappa^{2})))+16k_{F}^{4}{v_{0}^{(2)2}}(5(7+9\kappa^{2})$
$\displaystyle+21(5s^{6}-5s^{2}(-3+\kappa^{2})(1+\kappa^{2})-\kappa^{4}(3+5\kappa^{2})+s^{4}(21+5\kappa^{2})))))~{}.$

Here we have used the dimensionless variables $s=a/k_{F}$ and
$\kappa=p/k_{F}$, already introduced in Sec. 6.1.

Applying Eq. (3.14), and recalling that $4\pi L_{d}/m=2iIk_{F}$, as discussed
in Sec. 6.1, we then have that

$\displaystyle\bar{\cal{E}}_{L}$
$\displaystyle=-\frac{4i}{m\pi^{3}}\int_{0}^{k_{F}}a^{2}da\int_{0}^{\sqrt{k_{F}^{2}-a^{2}}}pdp\log\left(1-\frac{4\pi/m}{W_{0}^{-1}+\frac{4\pi}{m}I_{0}-ip}L_{d}\right)$
(C.20)
$\displaystyle=-\frac{8k_{F}^{5}}{m\pi^{3}}\int_{0}^{1}s^{2}ds\int_{0}^{\sqrt{1-s^{2}}}\kappa
d\kappa\arctan\left(\frac{\pi I}{\frac{\pi}{k_{F}}W_{0}^{-1}+R}\right)~{}.$
(C.21)

If in Eq. (C.13) we take $v_{0}^{(2)}=-a_{0}^{2}r_{0}/4$ then $c_{4}=0$ and
the potential reduces to $v(k,p)=c_{0}+c_{2}(k^{2}+p^{2})/2$. The expression
for $W_{0}^{-1}$ simplifies to

$\displaystyle W_{0}^{-1}$
$\displaystyle=-\frac{20(-3\pi+a_{0}^{2}r_{0}k_{F}^{3})^{2}}{3a_{0}\pi(a_{0}^{3}r_{0}^{2}k_{F}^{5}(3+5s^{2}-5\kappa^{2})+30\pi(2+a_{0}r_{0}k_{F}^{2}\kappa^{2}))}~{}.$
(C.22)

The case up to including $r_{0}$ was studied by Kaiser in [44] to ascertain
the effects of the effective range on $\cal{E}$ and employing DR as here. The
expression for $W_{0}^{-1}$ times $\pi/k_{F}$, cf. Eq. (C.21), is the same as
his function $\Omega_{0}^{-1}$ and we reproduce his results, confirming the
correctness of his combinatorial conjecture on the proper resummation of the
potential to calculate $\bar{\cal{E}}$ and getting the $\arctan$ as in Eq.
(C.21).

We would like to stress that the proper results are those obtained with cutoff
regularization of any sort ($\theta_{n}\neq 0)$, as derived in Sec. 6.1, due
to the non-perturbative nature of the calculations. Of course, for
perturbative ones, DR is perfectly suited, see also Ref. [63] for related
discussions. In particular, our results in Sec. 6.1 clearly show that the
inclusion of $r_{0}$ is a perturbative effect, such that the corrections due
to the effective range vanishes as $r_{0}\to 0$, and one recovers the result
for $a_{0}\to\infty$ with $r_{0}=0$. The opposite conclusion reached in Ref.
[44], so that the two limits $a_{0}\to\infty$ and $r_{0}\to 0$ do not commute,
comes entirely from the use of DR in the calculation. This is why
$\lim_{a_{0}\to\infty}\left.\xi(k_{F})\right|_{r_{0}\neq 0}=0.876$ compared
with the value $\lim_{a_{0}\to\infty}\left.\xi(k_{F})\right|_{r_{0}=0}=0.507$,
as follows from Eqs. (C.20) and (C.22).

###  C.2 $P$ waves

We consider the $P$-wave spin-independent potential given in Eq. (6.20) with
$d_{2}=0$,

$\displaystyle V(\mathbf{k},\mathbf{p})$
$\displaystyle=\mathbf{k}\cdot\mathbf{p}d_{0}~{}.$ (C.23)

Its partial wave projection for ${}^{3}P_{0}$, ${}^{3}P_{1}$ and ${}^{3}P_{2}$
can be read from Eq. (6.21),

$\displaystyle v(k,p)$ $\displaystyle=\frac{d_{0}}{3}kp~{}.$ (C.24)

Instead of cutoff regularization as in Sec. 6.2 we proceed with DR and
reproduce the results already obtained in Ref. [44]. However, while this
reference develops a specific method for the resummation of the ladder
diagrams for the $P$-wave potential of Eq. (C.23), we obtain it as a
particular case of our general method, cf. Eq. (3.14) and Sec. 5.

For vacuum scattering in our present case the free part of the unitarity loop
function is just $-imp^{3}/4\pi$ and $[v]=d_{0}/3$. Then, we apply Eq. (5.8)
with $[{\cal G}(p)]\to-imp^{3}/4\pi$ in order to calculate
${\mathsf{t}_{V}}(k,p)$ with the result

$\displaystyle{\mathsf{t}_{V}}(k,p)$
$\displaystyle=\frac{kp}{\frac{3}{d_{0}}-i\frac{mp^{3}}{4\pi}}~{}.$ (C.25)

The reproduction of the ERE with $a_{1}$ the scattering volume implies

$\displaystyle d_{0}$ $\displaystyle=-\frac{12\pi}{m}a_{1}~{}.$ (C.26)

To simplify the notation we introduce the constant
$\alpha_{1}\equiv(-a_{1})^{1/3}$.

In order to work out ${\mathsf{t}_{m}}(p,p)$ in the medium by applying Eq.
(5.8) we need the full unitarity loop function $[{\cal G}_{J_{2}\mu 1,J_{1}\mu
1}(p)]$, Eq. (5.7), expressed in terms of ${\cal A}_{J_{2}\mu 1,J_{1}\mu 1}$,
Eq. (4.19),

$\displaystyle{\cal A}_{J_{2}\mu 1,J_{1}\mu 1}$
$\displaystyle=2\left(\delta_{J_{2}J_{1}}-2\sum_{m_{3}\sigma_{3}}(m_{3}\sigma_{3}\mu|11J_{2})(m_{3}\sigma_{3}\mu|11J_{1})\int
d\hat{\mathbf{k}}|Y_{1}^{m}(\hat{\mathbf{k}})|^{2}\theta(k_{F}-|\mathbf{k}-a\hat{\mathbf{z}}|)\right)~{}.$
(C.27)

Here we have taken into account that $S=\ell=1$ and $\chi(11)^{2}=2$.

Let us also recall that because of the symmetry relations in Eq. (4.20) we
only need to obtain explicitly the PWAs for $\mu=2$, 1 and 0, and for the
latter in addition $(-1)^{J_{2}}=(-1)^{J_{1}}$, cf. Eq. (4.21). The different
coupled values of the total angular momentum as a function of $\mu$ are:
$J_{2}=J_{1}=2$ for $\mu=2$; $J_{2},J_{1}=1$ or 2 for $\mu=1$; $J_{2},J_{1}=0$
or 2 or $J_{2}=J_{1}=1$ for $\mu=0$. Thus, either we have one or two-coupled
channel scattering.

We derive in detail the coupled channel case of $\mu=1$ and for the other
cases, since they can be worked out in complete analogy, just give their
contributions to ${\cal{E}_{L}}$. The two possible values of $m_{3}$ and
$\sigma_{3}$ for $\mu=1$ are 0 or 1. With respect to $\hat{\mathbf{z}}$ we
have for $m_{3}=0$ the longitudinal angular integration

$\displaystyle\int
d\hat{\mathbf{k}}|Y_{1}^{0}(\hat{\mathbf{k}})|^{2}\theta(k_{F}-|\hat{\mathbf{k}}-a\hat{\mathbf{z}}|)~{},$
(C.28)

and for $m_{3}=\pm 1$ the transversal one

$\displaystyle\int
d\hat{\mathbf{k}}|Y_{1}^{1}(\hat{\mathbf{k}})|^{2}\theta(k_{F}-|\hat{\mathbf{k}}-a\hat{\mathbf{z}}|)~{}.$
(C.29)

The two basic in-medium integrals that one needs here in connection with the
two previous equations are

$\displaystyle C_{\parallel}$
$\displaystyle=\frac{2m}{\pi}\int_{0}^{\infty}dk\frac{k^{4}}{k^{2}-p^{2}}\int_{-1}^{1}d\\!\cos\theta\,\cos^{2}\\!\theta\,\,\theta\\!\left(k_{F}-\sqrt{k^{2}+a^{2}-2ak\cos\theta}\right)~{},$
(C.30) $\displaystyle C_{\perp}$
$\displaystyle=\frac{2m}{\pi}\int_{0}^{\infty}dk\frac{k^{4}}{k^{2}-p^{2}}\int_{-1}^{1}d\\!\cos\theta\,\sin^{2}\\!\theta\,\,\theta\\!\left(k_{F}-\sqrt{k^{2}+a^{2}-2ak\cos\theta}\right)~{}.$

In terms of the real functions $R_{\perp}(s,\kappa)$, $I_{\perp}(s,\kappa)$,
$R_{\parallel}(s,\kappa)$ and $I_{\parallel}(s,\kappa)$ introduced by Kaiser
in Ref. [44], cf. Eqs. (27)–(30) in this reference, we have

$\displaystyle C_{\parallel}$
$\displaystyle=\frac{mk_{F}^{3}}{3\pi}\left\\{R_{\parallel}(s,\kappa)+i\pi(I_{\parallel}(s,\kappa)+\kappa^{3})\right\\}~{},$
(C.31) $\displaystyle C_{\perp}$
$\displaystyle=\frac{2mk_{F}^{3}}{3\pi}\left\\{R_{\perp}(s,\kappa)+i\pi(I_{\perp}(s,\kappa)+\kappa^{3})\right\\}~{}.$

When inserted in $[{\cal G}_{J_{2}11,J_{1}11}]$ this function becomes

$\displaystyle[{\cal G}_{211,211}]$ $\displaystyle=[{\cal
G}_{111,111}]=\frac{mk_{F}^{3}}{8\pi^{2}}\left\\{R_{\perp}+R_{\parallel}+i\pi(I_{\perp}+I_{\parallel})\right\\}~{},$
(C.32) $\displaystyle[{\cal G}_{211,111}]$ $\displaystyle=[{\cal
G}_{111,211}]=\frac{mk_{F}^{3}}{8\pi^{2}}\left\\{R_{\perp}-R_{\parallel}+i\pi(I_{\perp}-I_{\parallel})\right\\}~{}.$

We are interested in the eigenvalues $\lambda_{\parallel}$ and
$\lambda_{\perp}$ of $([v]^{-1}+[{\cal G}])^{-1}$ which read

$\displaystyle\lambda_{\parallel}$ $\displaystyle=\frac{4\pi
k_{F}^{-3}/m}{(\alpha_{1}k_{F})^{-3}+\pi^{-1}R_{\parallel}+iI_{\parallel}}~{},$
(C.33) $\displaystyle\lambda_{\perp}$ $\displaystyle=\frac{4\pi
k_{F}^{-3}/m}{(\alpha_{1}k_{F})^{-3}+\pi^{-1}R_{\perp}+iI_{\perp}}~{}.$

Notice that the argument of the $\log$ in the calculation of $\cal{E}_{L}$,
Eq. (3.14), is $([v]^{-1}+[{\cal G}])^{-1}([v]^{-1}+[{\cal
G}]-p^{2}L_{d})=([v]^{-1}+[{\cal G}])^{-1}([v]^{-1}+[{\cal G}])^{*}$, because
of Eq. (2.34), so that its eigenvalues are $e^{-2i\phi_{a}}$, where $\phi_{a}$
is the principal argument of the eigenvalues in Eq. (C.33) and $a=\parallel$
or $\perp$. Then, the contribution to $\cal{E}_{L}$ per particle from $\mu=\pm
1$, $\bar{\cal{E}}_{{\cal{L}}}^{\mu=\pm 1}$, is

$\displaystyle\bar{\cal{E}}_{\cal{L}}^{\mu=\pm 1}$
$\displaystyle=-\frac{48k_{F}^{2}}{m\pi}\int_{0}^{1}dss^{2}\int_{0}^{\sqrt{1-s^{2}}}d\kappa\kappa\left(\arctan\frac{I_{\parallel}}{(\alpha_{1}k_{F})^{-3}+\pi^{-1}R_{\parallel}}+\arctan\frac{I_{\perp}}{(\alpha_{1}k_{F})^{-3}+\pi^{-1}R_{\perp}}\right)~{}.$
(C.34)

For the other values of $\mu$ we have with similar notation,

$\displaystyle\bar{\cal{E}}_{{\cal{L}}}^{\mu=\pm 2}$
$\displaystyle=-\frac{48k_{F}^{2}}{m\pi}\int_{0}^{1}dss^{2}\int_{0}^{\sqrt{1-s^{2}}}d\kappa\kappa\arctan\frac{I_{\perp}}{(\alpha_{1}k_{F})^{-3}+\pi^{-1}R_{\perp}}~{},$
(C.35) $\displaystyle\bar{\cal{E}}_{{\cal{L}}}^{\mu=0}$
$\displaystyle=\frac{1}{2}\left(\bar{\cal{E}}_{{\cal{L}}}^{\mu=\pm
2}+\bar{\cal{E}}_{{\cal{L}}}^{\mu=\pm 1}\right)~{}.$

Summing over all the values of $\mu$ the total result is

$\displaystyle\bar{\cal{E}}_{{\cal{L}}}$
$\displaystyle=-\frac{72k_{F}^{2}}{m\pi}\int_{0}^{1}dss^{2}\int_{0}^{\sqrt{1-s^{2}}}d\kappa\kappa\left(\arctan\frac{I_{\parallel}}{(\alpha_{1}k_{F})^{-3}+\pi^{-1}R_{\parallel}}+2\arctan\frac{I_{\perp}}{(\alpha_{1}k_{F})^{-3}+\pi^{-1}R_{\perp}}\right)~{},$
(C.36)

which is the one obtained in Ref. [44]. Let us recall that our conclusions on
the impact of the $P$ waves in $\cal{E}_{\cal{L}}$ are those derived in Sec.
6.2 making use of cutoff regularization (with the cutoff sent to infinity) in
an unspecified scheme.

##  Appendix D Poles in the in-medium $S$ wave

The algebraic expressions for $\zeta$ and $\nu$ (and their numerical values
after performing the integrations) provided by Kaiser in Ref. [43] by
expanding the integrand in Eq. (6.4) are not right, as we have checked
numerically (e.g. when used in Eq. (6.9) they fail to reproduce
$\bar{\cal{E}}$ around the unitary limit). Mathematically this is due to fact
that the conditions for the application of the Leibniz rule (Theorem 10.39 of
Ref. [95]) for differentiation with respect to $(a_{0}k_{F})^{-1}$ under the
signs of integration in Eq. (6.4) are not met. Therefore, one has to use the
integral representation of $\bar{\cal{E}}$ in Eq. (6.4) and after the
integration evaluate the derivatives at the unitary limit. The singularity
stems from the fact that one finds a pole of $\tau_{m}(p)$, Eq. (6.2), at the
border of the integration region, where $\kappa=\sqrt{1-s^{2}}$ and
$s\in[0,1]$, when $R(s,\sqrt{1-s^{2}})=\pi(a_{0}k_{F})^{-1}$ (notice also that
$I(s,\sqrt{1-s^{2}})=0$). The solution can be obtained by solving the
transcendental equation

$\displaystyle\kappa=\tanh\left(\frac{1}{\kappa}\left[1-\frac{\pi}{2a_{0}k_{F}}\right]\right)~{},~{}\kappa\in[0,1]~{}.$
(D.1)

In the unitary limit the solution is $\kappa=\sqrt{1-s^{2}}=0.833557$ and then
$s=0.552434$. The fact that $s\neq 0$ implies that the fermionic pair has a
non-vanishing total momentum equal to $2sk_{F}$.

More generally Eq. (D.1) has solution as long as $(a_{0}k_{F})^{-1}<2/\pi$.
Notice that for $(a_{0}k_{F})^{-1}>2/\pi$ the $\tanh$ in the right-hand side
of Eq. (D.1) becomes negative, which cannot be because $\kappa\in[0,1]$. We
plot in Fig. 16 the resulting values for $\kappa$ and $s$ as a function of
$(a_{0}k_{F})^{-1}$. We notice an interesting continuous transition as
$(a_{0}k_{F})^{-1}$ grows from $-\infty$ up to $2/\pi$, from a Cooper-pair
like situation [96] with total momentum $2sk_{F}=0$ and relative momentum
$\kappa k_{F}=k_{F}$, passing through the unitary limit in which
$2sk_{F}\approx 1.11k_{F}$, $\kappa k_{F}\approx 0.88k_{F}$, up to
$(a_{0}k_{F})^{-1}=2/\pi$ where $2sk_{F}=2k_{F}$ and $\kappa k_{F}=0$. Since
this pole singularity in ${\mathsf{t}}_{m}$ happens exactly at the boundary of
the two Fermi seas the total energy of the pair is always equal to the Fermi
energy, $(s^{2}+\kappa^{2})k_{F}^{2}/m=E_{F}$. It is also worth noticing that
for $0^{+}<a_{0}k_{F}<\pi/2$ there is no pole that could spoil a perturbative
low-density calculation of ${\cal E}$ [33].

Figure 16: Pole of the in-medium $S$ wave including the scattering length, Eq.
(6.2), at the border of two Fermi surfaces, $\kappa=\sqrt{1-s^{2}}$,
$s\in[0,1]$, as as function of $(a_{0}k_{F})^{-1}<2/\pi$. The total momentum
is $2sk_{F}$ and the relative momentum is $\kappa k_{F}$.

##  Appendix E Poles in the vacuum $P$ wave

Let us consider the ERE in $P$ wave up to including the effective range
$r_{1}$, Eq. (6.27), and look for its poles in the complex-$p$ plane by
determining the zeroes of $-1/a_{1}+r_{2}p^{2}/2-ip^{3}$. We include the
auxiliary variables

$\displaystyle\alpha_{1}$ $\displaystyle={\rm
sign}(a_{1})|a_{1}|^{1/3}=\frac{a_{1}}{|a_{1}|^{\frac{2}{3}}}~{},$ (E.1)
$\displaystyle t$ $\displaystyle=\frac{r_{1}\alpha_{1}}{3\cdot
2^{\frac{2}{3}}}~{},$ $\displaystyle u(t)$
$\displaystyle=\frac{t}{(1+t^{3}+\sqrt{1+2t^{3}})^{\frac{1}{3}}}~{}.$

The square and cubic roots in the function $u(t)$ are defined such that
$z^{1/2}=|z|^{1/2}\exp(i\,{\rm arg}z/2)$ and $z^{1/3}=|z|^{1/3}\exp(i\,{\rm
arg}z/3)$, respectively, with ${\rm arg}z\in(-\pi,\pi]$ and ${\arg(-1)}=\pi$.
Denoting the real and imaginary parts of $u$ as $u_{r}$ and $u_{i}$,
respectively, we can write the pole positions $p_{i}$, $i=1,\,2,\,3$, as

$\displaystyle p_{1}$
$\displaystyle=-\frac{ir_{1}}{6}(1+u+\frac{1}{u})=\frac{r_{1}u_{i}}{6(u_{r}^{2}+u_{i}^{2})}(u_{r}^{2}+u_{i}^{2}-1)-i\frac{r_{1}}{6(u_{r}^{2}+u_{i}^{2})}\left((1+u_{r})(u_{r}^{2}+u_{i}^{2})+u_{r}\right)~{},$
(E.2) $\displaystyle p_{2}$
$\displaystyle=\frac{r_{1}}{6}\left(e^{-i\frac{\pi}{6}}-e^{i\frac{\pi}{6}}+\frac{e^{i\frac{\pi}{6}}}{u}-ue^{-i\frac{\pi}{6}}\right)$
$\displaystyle=-\frac{r_{1}(u_{i}+\sqrt{3}u_{r})(u_{r}^{2}+u_{i}^{2}-1)}{12(u_{r}^{2}+u_{i}^{2})}+i\frac{r_{1}}{12(u_{r}^{2}+u_{i}^{2})}\left(u_{r}+(-2+u_{r})(u_{r}^{2}+u_{i}^{2})-\sqrt{3}u_{i}(1+u_{r}^{2}+u_{i}^{2})\right)~{},$
$\displaystyle p_{3}$
$\displaystyle=\frac{r_{1}}{6}\left(-e^{i\frac{\pi}{6}}+e^{-i\frac{\pi}{6}}-\frac{e^{-i\frac{\pi}{6}}}{u}+ue^{i\frac{\pi}{6}}\right)$
$\displaystyle=-\frac{r_{1}(u_{i}-\sqrt{3}u_{r})(u_{r}^{2}+u_{i}^{2}-1)}{12(u_{r}^{2}+u_{i}^{2})}+i\frac{r_{1}}{12(u_{r}^{2}+u_{i}^{2})}\left(u_{r}+(-2+u_{r})(u_{r}^{2}+u_{i}^{2})+\sqrt{3}u_{i}(1+u_{r}^{2}+u_{i}^{2})\right)~{}.$

It is clear that $u$ is real for $t>-1/2^{1/3}$ and then $p_{2}$ and $p_{3}$
are two resonant poles with the same imaginary part and opposite real parts.
For $t<-1/2^{1/3}$ it results that $|u(t)|=1$ since

$\displaystyle|u(t)^{3}|$
$\displaystyle=\sqrt{\frac{t^{6}}{(1+t^{3})^{2}-1-2t^{3}}}=1~{},$ (E.3)

Thus, it follows from Eq. (E.2) that all the $p_{i}$ are purely imaginary
since then $u_{r}^{2}+u_{i}^{2}=1$.

We are interested in the forbidden regions of the parameter space
$(a_{1},r_{1})$ that drives to resonant poles with positive imaginary part
that could happen for $t>-1/2^{1/3}$. For real $u$ the imaginary parts of
$p_{2,3}$ can also be written from Eq. (E.2) as

$\displaystyle\text{Im}\,p_{2,3}$
$\displaystyle=\frac{r_{1}}{12u}(1-u)^{2}~{}.$ (E.4)

Then, for $u>0$, which happens for $r_{1}\alpha$ or $r_{1}a_{1}>0$, the
forbidden regions are those with $r_{1}>0$. In the case of $u<0$ with
$-1/2^{1/3}<t<0$, i.e. for $0>r_{1}\alpha_{1}>-(54)^{1/3}$, it is not allowed
a negative $r_{1}$. In terms of the dimensionless variables
$-1/a_{1}k_{F}^{3}$ and $r_{1}/k_{F}$ used in Fig. 14 this implies that the
regions that are forbidden are the top left quadrant and
$0>r_{1}/k_{F}>-(54/|a_{1}|k_{F}^{3})^{1/3}$ in the bottom left quadrant. We
give the signs of $\text{Im}\,p_{1,2}$ for $t>-1/2^{1/3}$ in Table 2 as a
function of the signs of $r_{1}$ and $a_{1}$.

${\rm sign}(r_{1}a_{1})$ | $({\rm sign}(r_{1}),{\rm sign}(a_{1}))$ | ${\rm sign}(\text{Im}\,p_{2,3})$
---|---|---
$+$ | $(+,+)$ | $+$
| $(-,-)$ | $-$
$-$ | $(+,-)$ | $-$
| $(-,+)$ | $+$
Table 2: Sign of $\text{Im}\,p_{2,3}$ for $t>-1/2^{1/3}$ giving rise to a real
$u(t)$ and resonant poles $p_{2}$ and $p_{3}$. The signs of $a_{1}$ and
$\alpha_{1}$ are the same.

We also considered for generating the Fig. 15 the case with $a_{1}=\infty$ and
finite $r_{1}$ and $v_{2}^{(1)}$. In this case, the pole positions are given
by the zeroes of $r_{1}p^{2}/2+v_{1}^{(2)}p^{4}-ip^{3}$. Apart from the double
zero at $p=0$ there are two other zeros when $r_{1}/2+v_{1}^{(2)}p^{2}-ip=0$ ,
at the positions

$\displaystyle
p_{1,2}=\frac{i}{2v_{1}^{(2)}}\pm\frac{1}{2v_{1}^{(2)}}\sqrt{-1-2r_{1}v_{1}^{(2)}}~{}.$
(E.5)

The region to be excluded because of the appearance of resonant poles with
positive imaginary parts in their associated momenta stems from the conditions
i) $v_{1}^{(2)}>0$ and ii) $r_{1}<-1/2v_{1}^{(2)}$.

## References

  * [1] J. A. Oller, “Chiral Lagrangians at finite density,” Phys. Rev. C, vol. 65, p. 025204, 2002.
  * [2] W. Ketterle and M. Zwierlein, “Making, probing and understanding ultracold Fermi gases,” Rivista del Nuovo Cimento, vol. 31, pp. 247–422, 2002\.
  * [3] M. Zwierlein, J. Abo-Shaeer, A. Schirotzek, C. Schunck, and W. Ketterle, “Vortices and superfluidity in a strongly interacting Fermi gas,” Nature, vol. 435, pp. 1047–1051, 2005.
  * [4] C. Schunck, M. Zwierlein, A. Schirotzek, and W. Ketterle, “Superfluid expansion of a rotating Fermi gas,” Phys. Rev. Lett., vol. 98 5, p. 050404, 2007.
  * [5] M. Anderson, J. Ensher, M. Matthews, C. Wieman, and E. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science, vol. 269, pp. 198 – 201, 1995.
  * [6] K. Davis, M. Mewes, M. Andrews, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett., vol. 75, 1995.
  * [7] E. W. Zwerger, The BCS-BEC Crossover and the Unitary Fermi Gas.  Lecture Notes in Physics 836, Springer-Verlag Berlin Heidelberg, 2012\.
  * [8] S. Giorgini, L. Pitaevskiĭ, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys., vol. 80, pp. 1215–1274, 2008.
  * [9] M. Randeria and E. Taylor, “Crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein Condensation and the Unitary Fermi Gas,” Annu. Rev. Condens. Matter. Phys, vol. 5, pp. 209–32, 2014.
  * [10] H. Bethe, “Theory of the Effective Range in Nuclear Scattering,” Phys. Rev., vol. 76, p. 38, 1949.
  * [11] N. Navon, S. Nascimbène, F. Chevy, and C. Salomon, “The Equation of State of a Low-Temperature Fermi Gas with Tunable Interactions,” Science, vol. 328, pp. 729 – 732, 2010.
  * [12] M. Ku, A. T. Sommer, L. Cheuk, and M. Zwierlein, “Revealing the Superfluid Lambda Transition in the Universal Thermodynamics of a Unitary Fermi Gas,” Science, vol. 335, pp. 563 – 567, 2012.
  * [13] G. Zuern, T. Lompe, A. Wenz, S. Jochim, P. Julienne, and J. Hutson, “Precise characterization of 6Li Feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules.,” Phys. Rev. Lett., vol. 110 13, p. 135301, 2013.
  * [14] U. van Kolck, “Unitarity and Discrete Scale Invariance,” Few-Body Systems, vol. 58, pp. 1–12, 2017.
  * [15] T.-L. Ho, “Universal Thermodynamics of Degenerate Quantum Gases in the Unitarity Limit,” Phys. Rev. Lett., vol. 92, p. 090402, 2004.
  * [16] M. M. Forbes, S. Gandolfi, and A. Gezerlis, “Resonantly Interacting Fermions In a Box,” Phys. Rev. Lett., vol. 106, p. 235303, 2011.
  * [17] M. M. Forbes, S. Gandolfi, and A. Gezerlis, “Effective-Range Dependence of Resonantly Interacting Fermions,” Phys. Rev. A, vol. 86, p. 053603, 2012\.
  * [18] J. Carlson, S. Gandolfi, K. E. Schmidt, and S. Zhang, “Auxiliary Field Quantum Monte Carlo for Strongly Paired Fermions,” Phys. Rev. A, vol. 84, p. 061602, 2011.
  * [19] Y. Nishida and D. T. Son, “An Epsilon expansion for Fermi gas at infinite scattering length,” Phys. Rev. Lett., vol. 97, p. 050403, 2006.
  * [20] Y. Nishida and D. T. Son, “Fermi gas near unitarity around four and two spatial dimensions,” Phys. Rev. A, vol. 75, p. 063617, 2007.
  * [21] Y. Nishida, “Ground-state energy of the unitary Fermi gas from the epsilon expansion,” Phys. Rev. A, vol. 79, p. 013627, 2009.
  * [22] T. Papenbrock, “Density-functional theory for fermions in the unitary regime,” Phys. Rev. A, vol. 72, p. 041603, 2005.
  * [23] D. Lacroix, “Density-functional theory for resonantly interacting fermions with effective range and neutron matter,” Phys. Rev. A, vol. 94, no. 4, p. 043614, 2016.
  * [24] A. Boulet and D. Lacroix, “Approximate self-energy for Fermi systems with large s-wave scattering length: a step towards density functional theory,” J. Phys. G, vol. 46, no. 10, p. 105104, 2019.
  * [25] M. Grasso, “Effective density functionals beyond mean field,” Prog. Part. Nucl. Phys., vol. 106, pp. 256–311, 2019.
  * [26] A. Fetter and J. Walecka, Quantum Theory of Many-Particle Systems.  McGraw-Hill, New York, 1971.
  * [27] K. Huang and C. Yang, “Quantum-mechanical many-body problem with hard-sphere interaction,” Phys. Rev., vol. 105, pp. 767–775, 1957.
  * [28] T. Lee and C. Yang, “Many-Body Problem in Quantum Mechanics and Quantum Statistical Mechanics,” Phys. Rev., vol. 105, pp. 1119–1120, 1957.
  * [29] V. N. Efimov and M. Y. Amusya, “Ground state of a rarefied Fermi gas of rigid spheres,” Sov. Phys. JETP, vol. 20, p. 388, 1965.
  * [30] M. Y. Amusya and V. N. Efimov, “Pair collisions in a low-density fermi gas,” Ann. Phys. (NY), vol. 47, pp. 377–403, 1968.
  * [31] G. A. Baker, “Singularity structure of the perturbation series for the ground-state energy of a many-fermion system,” Rev. Mod. Phys., vol. 43, pp. 479–531, 1971.
  * [32] B. F. Bishop, “Ground-state energy of a dilute fermi gas,” Ann. Phys. (NY), vol. 77, pp. 106–138, 1973.
  * [33] H. Hammer and R. Furnstahl, “Effective field theory for dilute Fermi systems,” Nucl. Phys. A, vol. 678, pp. 277–294, 2000.
  * [34] Q. Chen et al., “Measurement of the neutron-neutron scattering length using the pi-d capture reaction,” Phys. Rev. C, vol. 77, p. 054002, 2008\.
  * [35] A. Lacour, J. A. Oller, and U.-G. Meißner, “Non-perturbative methods for a chiral effective field theory of finite density nuclear systems,” Annals Phys., vol. 326, pp. 241–306, 2011.
  * [36] A. Dobado, F. J. Llanes-Estrada, and J. A. Oller, “The existence of a two-solar mass neutron star constrains the gravitational constant $G_{N}$ at strong field,” Phys. Rev. C, vol. 85, p. 012801, 2012.
  * [37] H. Bethe, “Introduction to the Brueckner theory,” Physics, vol. 22, pp. 987 – 993, 1956.
  * [38] K. Brueckner, C. Levinson, and H. Mahmoud, “Two-Body Forces and Nuclear Saturation. I. Central Forces,” Phys. Rev., vol. 95, pp. 217–228, 1954\.
  * [39] K. Brueckner, “Nuclear Saturation and Two-Body Forces. II. Tensor Forces,” Phys. Rev., vol. 96, pp. 508–516, 1954.
  * [40] K. Brueckner, “Two-Body Forces and Nuclear Saturation. III. Details of the Structure of the Nucleus,” Phys. Rev., vol. 97, pp. 1353–1366, 1955.
  * [41] D. Thouless, “Perturbation Theory in Statistical Mechanics and the Theory of Superconductivity,” Ann. Phys., vol. 10, pp. 553–588, 1960.
  * [42] J. V. Steele, “Effective field theory power counting at finite density,” nucl-th/0010066, 2000.
  * [43] N. Kaiser, “Resummation of fermionic in-medium ladder diagrams to all orders,” Nucl. Phys. A, vol. 860, pp. 41–55, 2011.
  * [44] N. Kaiser, “Resummation of in-medium ladder diagrams: s-wave effective range and p-wave interaction,” Eur. Phys. J. A, vol. 48, p. 148, 2012.
  * [45] D. Pines and P. Nozieres, Theory of Quantum Liquids: Normal Fermi Liquids.  W.A. Benjaim, INC., New York, 1966.
  * [46] A. B. Migdal, Nuclear Theory: The Quasiparticle Method.  W.A. Benjaim, INC., New York, 1968.
  * [47] J. Carlson, J. Morales, J., V. Pandharipande, and D. Ravenhall, “Quantum Monte Carlo calculations of neutron matter,” Phys. Rev. C, vol. 68, p. 025802, 2003.
  * [48] A. Gezerlis and J. Carlson, “Low-density neutron matter,” Phys. Rev. C, vol. 81, p. 025803, 2010.
  * [49] A. Schwenk and C. Pethick, “Resonant Fermi gases with a large effective range,” Phys. Rev. Lett., vol. 95, p. 160401, 2005.
  * [50] S. König, H. W. Grießhammer, H. Hammer, and U. van Kolck, “Nuclear Physics Around the Unitarity Limit,” Phys. Rev. Lett., vol. 118, no. 20, p. 202501, 2017.
  * [51] U. van Kolck, “Effective field theory of short range forces,” Nucl. Phys. A, vol. 645, pp. 273–302, 1999.
  * [52] T. Schäfer, C.-W. Kao, and S. R. Cotanch, “Many body methods and effective field theory,” Nucl. Phys. A, vol. 762, pp. 82–101, 2005.
  * [53] D. B. Kaplan, M. J. Savage, and M. B. Wise, “Two nucleon systems from effective field theory,” Nucl. Phys. B, vol. 534, pp. 329–355, 1998.
  * [54] R. Stratonovich, “On a Method of Calculating Quantum Distribution Functions,” Soviet Physics Doklady, vol. 2, p. 416, 1957.
  * [55] J. Hubbard, “Calculation of partition functions,” Phys. Rev. Lett., vol. 3, pp. 77–80, 1959.
  * [56] V. Bernard, N. Kaiser, and U.-G. Meissner, “Chiral dynamics in nucleons and nuclei,” Int. J. Mod. Phys. E, vol. 4, pp. 193–346, 1995.
  * [57] R. Machleidt and D. R. Entem, “Chiral effective field theory and nuclear forces,” Phys. Rept., vol. 503, pp. 1–75, 2011.
  * [58] U.-G. Meißner, J. A. Oller, and A. Wirzba, “In-medium chiral perturbation theory beyond the mean field approximation,” Annals Phys., vol. 297, pp. 27–66, 2002.
  * [59] J. M. Alarcón and J. A. Oller, “Properties of nuclear matter,” in preparation, 2021.
  * [60] J. A. Oller, A Brief Introduction to Dispersion Relations.  SpringerBriefs in Physics, Springer, 2019.
  * [61] J. A. Oller, “New results from a number operator interpretation of the compositeness of bound and resonant states,” Annals Phys., vol. 396, pp. 429–458, 2018.
  * [62] J. A. Oller, “Coupled-channel approach in hadron–hadron scattering,” Prog. Part. Nucl. Phys., vol. 110, p. 103728, 2020.
  * [63] D. R. Phillips, S. R. Beane, and T. D. Cohen, “Nonperturbative regularization and renormalization: Simple examples from nonrelativistic quantum mechanics,” Annals Phys., vol. 263, pp. 255–275, 1998.
  * [64] D. R. Entem, E. Ruiz Arriola, M. Pavon Valderrama, and R. Machleidt, “Renormalization of chiral two-pion exchange NN interactions. momentum versus coordinate space,” Phys. Rev. C, vol. 77, p. 044006, 2008.
  * [65] J. B. Habashi, S. Sen, S. Fleming, and U. van Kolck, “Effective Field Theory for Two-Body Systems with Shallow S-Wave Resonances,” Annals Phys., vol. 422, p. 168283, 2020.
  * [66] J. A. Oller and D. Entem, “The exact discontinuity of a partial wave along the left-hand cut and the exact $N/D$ method in non-relativistic scattering,” Annals Phys., vol. 411, p. 167965, 2019.
  * [67] G. A. Baker Jr., “Neutron matter model,” Phys. Rev. C, vol. 60, p. 054311, 1999.
  * [68] K. O’Hara and et al, “Observation of a strongly interacting degenerate Fermi gas of atoms,” Science, vol. 298, p. 2179, 2002.
  * [69] J. Maki and S. Zhang, “The Role of the Effective Range in Resonantly Interacting Fermi Gases: How Breaking Scale Symmetry Affects the Bulk Viscosity,” Phys. Rev. Lett., vol. 125, p. 240402, 2020.
  * [70] J. A. Oller, “Nucleon-Nucleon scattering from dispersion relations: next-to-next-to-leading order study,” Phys. Rev. C, vol. 93, p. 024002, 2016.
  * [71] C. A. Bertulani, H. W. Hammer, and U. Van Kolck, “Effective field theory for halo nuclei,” Nucl. Phys. A, vol. 712, pp. 37–58, 2002.
  * [72] J. Hu, F. Wu, L. He, X.-J. Liu, and H. Hu, “Theory of strongly paired fermions with arbitrary short-range interactions,” Phys. Rev. A, vol. 101, no. 1, p. 013615, 2020.
  * [73] W. Haibin and J. Thomas, “Optical Control of Feshbach Resonances in Fermi Gases Using Molecular Dark States,” Phys. Rev. Lett., vol. 108, p. 010401, 2012.
  * [74] K. Gottfried, Quantum Mechanics. Vol. I. Fundamentals.  Advanced Book Classics. Addison-Wesley Publishing Company, Reading, Massachusetts, 1989.
  * [75] S. T. Ma, “Redundant Zeros in the Discrete Energy Spectra in Heisenberg’s Theory of Characteristic Matrix,” Phys. Rev., vol. 69, p. 668, 1946.
  * [76] S. T. Ma, “On a General Condition of Heisenberg for the S Matrix,” Phys. Rev., vol. 71, pp. 195–200, 1947.
  * [77] S. Chang, V. Pandharipande, J. Carlson, and K. Schmidt, “Quantum Monte Carlo Studies of Superfluid Fermi Gases,” Phys. Rev. A, vol. 70, p. 043602, 2004.
  * [78] G. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, “Equation of State of a Fermi Gas in the BEC-BCS Crossover: A Quantum Monte Carlo Study,” Phys. Rev. Lett., vol. 93, p. 200404, 2004.
  * [79] A. Bulgac and G. F. Bertsch, “Collective Oscillations of a Trapped Fermi Gas near the Unitary Limit,” Phys. Rev. Lett., vol. 94, p. 070401, 2004\.
  * [80] S. Tan, “Energetics of a strongly correlated Fermi gas,” Ann. Phys. (N.Y.), vol. 323, pp. 2925–2970, 2008.
  * [81] S. Gandolfi, K. E. Schmidt, and J. Carlson, “BEC-BCS crossover and universal relations in unitary Fermi gases,” Phys. Rev. A, vol. 83, p. 041601(R), 2011.
  * [82] A. Gezerlis and J. Carlson, “Strongly paired fermions: Cold atoms and neutron matter,” Phys. Rev. C, vol. 77, p. 032801, 2008.
  * [83] G. Miller, B. Nefkens, and I. Slaus, “Charge symmetry, quarks and mesons,” Phys. Rept., vol. 194, pp. 1–116, 1990.
  * [84] R. Navarro Pérez, J. E. Amaro, and E. Ruiz Arriola, “The low-energy structure of the nucleon–nucleon interaction: statistical versus systematic uncertainties,” J. Phys. G, vol. 43, no. 11, p. 114001, 2016\.
  * [85] H. Rothe, Lattice Gauge Theories: An Introduction.  World Scientific, Singapure, 1992.
  * [86] D. Lacroix, A. Boulet, M. Grasso, and C.-J. Yang, “From bare interactions, low-energy constants, and unitary gas to nuclear density functionals without free parameters: Application to neutron matter,” Phys. Rev. C, vol. 95, no. 5, p. 054306, 2017.
  * [87] F. Werner and Y. Castin, “General relations for quantum gases in two and three dimensions: Two-component fermions,” Phys. Rev. A, vol. 86, p. 013626, 2012.
  * [88] S. R. Beane and M. J. Savage, “Rearranging pionless effective field theory,” Nucl. Phys. A, vol. 694, pp. 511–524, 2001.
  * [89] A. Akmal, V. Pandharipande, and D. Ravenhall, “The Equation of state of nucleon matter and neutron star structure,” Phys. Rev. C, vol. 58, pp. 1804–1828, 1998.
  * [90] F. Werner, L. Tarruell, and Y. Castin, “Number of closed-channel molecules in the BEC-BCS crossover,” Eur. Phys. J. B, vol. 68, p. 401, 2009.
  * [91] R. Haussmann, M. Punk, and W. Zwerger, “Spectral Functions and rf Response of Ultracold Fermionic Atoms,” Phys. Rev. A, vol. 80, p. 063612, 2009.
  * [92] E. et al.. Kuhnle, “Universal behavior of pair correlations in a strongly interacting Fermi gas,” Phys. Rev. Lett., vol. 105, p. 070402, 2010.
  * [93] E. Braaten, In The BCS-BEC Crossover and the Unitary Fermi Gas, edited by W. Zwerger, Lectures Notes in Physics, vol. 86, Chap. 6.  Springer-Verlag, Berlin, 2012.
  * [94] D. B. Kaplan, “More effective field theory for nonrelativistic scattering,” Nucl. Phys. B, vol. 494, pp. 471–484, 1997.
  * [95] T. M. Apostol, Mathematical analysis, 2nd edition.  Pearson, 1974.
  * [96] H. Heiselberg, “Fermi systems with long scattering lengths,” Phys. Rev. A, vol. 63, p. 043606, 2001.