Momentum-Space Probability Density of ${}^6$He in Halo Effective Field Theory

Abstract

We compute the momentum-space probability density of ${}^6$He at leading order in Halo EFT. In this framework, the ${}^6$He nucleus is treated as a three-body problem with a ${}^4$He core ($c$) and two valence neutrons ($n$). This requires the $nn$ and $n c$ t-matrices as well as a $cnn$ force as input in the Faddeev equations. Since the $n c$ t-matrix corresponds to an energy-dependent potential, we consider the consequent modifications to the standard normalization and orthogonality conditions. We find that these are small for momenta within the domain of validity of Halo EFT. In this regime, the ${}^6$He probability density is regulator independent, provided the cutoff is significantly above the EFT breakdown scale.

Appendices

Alternative Derivation of the Normalization Condition

In this appendix we give an alternative derivation of the normalization condition in the presence of an energy-dependent potential. This derivation focuses on the resolvent of H(E). It is equivalent to the normalization conditions obtained for two-particle vertex functions in relativistic bound-state equations in Refs. [40, 41].

The resolvent of H(E) is:

$$\begin{aligned} G(E) :=\frac{1}{E - H(E)} . \end{aligned}$$

(104)

This contrasts with the resolvent of the energy-independent Hamiltonian obtained by evaluating H(E) at $E=E_\alpha$, ${\bar{H}} :=H(E_\alpha )$. Denoting that resolvent by ${\bar{G}}(E)$ we have:

$$\begin{aligned} {\bar{G}}(E) :=\frac{1}{E-{\bar{H}}} . \end{aligned}$$

(105)

By construction, ${\bar{H}}$ and H(E) both have $|\psi _\alpha \rangle$ as a right eigenstate corresponding to energy $E_\alpha$. The rest of ${\bar{H}}$’s spectrum will, however, be different. And, while the eigenstate $|\psi _\alpha \rangle$ is common to the spectrum of both operators, it is not guaranteed that it should be normalized in the same way. Indeed, we will show here that it should not be.

Expanding H(E) around $E=E_\alpha$ yields:

$$\begin{aligned} G(E)=\frac{1}{E-{\bar{H}} - (E-E_\alpha ) V'(E_\alpha ) - \cdots } , \end{aligned}$$

(106)

where $'$ denotes differentiation with respect to E. This, in turn, allows us to write:

$$\begin{aligned} G(E)={\bar{G}}(E)\left\{ 1 + \sum _{n=1}^\infty [V'(E_\alpha ) (E-E_\alpha ) {\bar{G}}(E)]^n\right\} + \hbox { regular as}\ E \rightarrow E_\alpha . \end{aligned}$$

(107)

Since ${\bar{G}}(E)$ is the resolvent of an energy-independent Hermitian operator it has a standard spectral representation. If $E_\alpha$ is an isolated bound state then that is:

$$\begin{aligned} {\bar{G}}(E)=\frac{|{\bar{\psi }}_\alpha \rangle \langle {\bar{\psi }}_\alpha |}{E-E_\alpha } + \hbox { pieces in the space orthogonal to}\ | {\bar{\psi }}_\alpha \rangle . \end{aligned}$$

(108)

where we have used $|{\bar{\psi }}_\alpha \rangle$ to emphasize that this eigenstates of ${\bar{H}}$ obey the standard normalization condition $\langle {\bar{\psi }}_\beta |{\bar{\psi }}_\alpha \rangle =\delta _{\beta \alpha }$, since ${\bar{H}}$ is an energy-independent, Hermitian, operator. As discussed at length in Sect. 3 the states $|\psi _\alpha \rangle$ do not obey this condition, although for $\beta \ne \alpha$ this is not a surprise, since the spectrum of ${\bar{H}}$ differs from that of H apart from the one state at $E=E_\alpha$. But for that particular state we must write:

$$\begin{aligned} \mathcal{Z}^{1/2} |\psi _\alpha \rangle =|{\bar{\psi }}_\alpha \rangle , \end{aligned}$$

(109)

with $\mathcal{Z}$ an additional wave function renormalization associated with the energy dependence of V(E).

To fix $\mathcal{Z}$ we insert Eq. (108) in Eq. (107) and sum the geometric series. This yields:

$$\begin{aligned} G(E)=\frac{|{\bar{\psi }}_\alpha \rangle \langle {\bar{\psi }}_\alpha |}{E-E_\alpha } \frac{1}{1 - \langle {\bar{\psi }}_\alpha | V'(E_\alpha ))|{\bar{\psi }}_\alpha \rangle } + \hbox { regular as}\ E \rightarrow E_\alpha . \end{aligned}$$

(110)

The factor in the denominator here ensures that the consistency condition:

$$\begin{aligned} G'(E)=-G(E) \left[ \frac{\partial }{\partial E} G^{-1}(E)\right] G(E) . \end{aligned}$$

(111)

is satisfied. The spectral decomposition (110) will then take the standard form:

$$\begin{aligned} G(E)=\frac{|\psi _\alpha \rangle \langle \psi _\alpha |}{E-E_\alpha } + \hbox { regular as}\ E \rightarrow E_\alpha , \end{aligned}$$

(112)

provided we identify:

$$\begin{aligned} \mathcal{Z}=1 - \langle {\bar{\psi }}_\alpha | V'(E_\alpha )|{\bar{\psi }}_\alpha \rangle =\langle {\bar{\psi }}_\alpha |\mathbb {1}- V'(E_\alpha )|{\bar{\psi }}_\alpha \rangle . \end{aligned}$$

(113)

In practice this means that we solve the Hamiltonian eigenvalue problem at $E=E_\alpha$, thereby obtaining an eigenvector that is, prior to normalization, proportional to both $|{\bar{\psi }}_\alpha \rangle$ and $|\psi _\alpha \rangle$. Then, if we wish to compute the latter, we must normalize such that:

$$\begin{aligned} \langle \psi |\mathbb {1}- V'(E_\alpha )|\psi \rangle =1 , \end{aligned}$$

(114)

instead of using the standard

$$\begin{aligned} \langle \psi |\psi \rangle =1 . \end{aligned}$$

(115)

Additional Formulas for the Probability Density

1.1 Transformation of Jacobi Coordinates

For the evaluation of overlaps of the type ${}^{}_{i}\! \left\langle {\varvec{p},\varvec{q}}\vert {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{j}^{}$ the following functions are useful, since they abbreviate many expressions:

$$\begin{aligned} \varvec{\pi }_{1}{\left( \varvec{p},\varvec{q}\right) }&:=\varvec{p} + \frac{A}{A+1} \varvec{q} ,\end{aligned}$$

(116)

$$\begin{aligned} \varvec{\pi }_{2}{\left( \varvec{p},\varvec{q}\right) }&:=\varvec{p} + \frac{1}{2} \varvec{q} ,\end{aligned}$$

(117)

$$\begin{aligned} \varvec{\pi }_{3}{\left( \varvec{p},\varvec{q}\right) }&:=\varvec{p} + \frac{1}{A+1} \varvec{q} . \end{aligned}$$

(118)

The transformations can be summarized by the expression

$$\begin{aligned} {}^{}_{i}\! \left\langle {\varvec{p},\varvec{q}}\vert {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{j}^{} =:\delta ^{(3)}{\left( \varvec{p^{\prime }}-\kappa _{ijp}{\left( \varvec{p},\varvec{q}\right) }\right) } \, \delta ^{(3)}{\left( \varvec{q^{\prime }}-\kappa _{ijq}{\left( \varvec{p},\varvec{q}\right) }\right) } , \end{aligned}$$

(119)

which defines $\varvec{\kappa }_{ijk}{\left( \varvec{p},\varvec{q}\right) }$ with $i,j \in \{n,c\}$, $i \ne j$ and $k \in \{p,q\}$. The functions $\varvec{\kappa }_{ijk}$ are given by

$$\begin{aligned} \varvec{\kappa }_{ncp}&:=\varvec{\pi }_{2}{\left( \varvec{q}, -\varvec{\pi }_{1}{\left( \varvec{p},\varvec{q}\right) }\right) } ,\end{aligned}$$

(120)

$$\begin{aligned} \varvec{\kappa }_{ncq}&:=-\varvec{\pi }_{1}{\left( \varvec{p},\varvec{q}\right) } ,\end{aligned}$$

(121)

$$\begin{aligned} \varvec{\kappa }_{cnp}&:=-\varvec{\pi }_{1}{\left( \varvec{q},\varvec{\pi }_{2}{\left( \varvec{p},-\varvec{q}\right) }\right) } ,\end{aligned}$$

(122)

$$\begin{aligned} \varvec{\kappa }_{cnq}&:=\varvec{\pi }_{2}{\left( \varvec{p},-\varvec{q}\right) } . \end{aligned}$$

(123)

Furthermore, we define the functions $\varvec{\kappa }^\prime _{ijk}{\left( \varvec{p},\varvec{q}\right) }$ with $i,j \in \{n,c\}$ and $k \in \{p,q\}$ by

$$\begin{aligned} {}^{}_{i}\! \left\langle {\varvec{p},\varvec{q}}\right| {\mathcal {P}_{nn}^\mathrm {(spatial)}}\left| {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{j}^{} =:\delta ^{(3)}{\left( \varvec{p^{\prime }}-\varvec{\kappa }^\prime _{ijp}{\left( \varvec{p},\varvec{q}\right) }\right) } \, \delta ^{(3)}{\left( \varvec{q^{\prime }}-\varvec{\kappa }^\prime _{ijq}{\left( \varvec{p},\varvec{q}\right) }\right) } , \end{aligned}$$

(124)

where the spatial part of the $nn$ permutation operator $\mathcal {P}_{nn}$ is given by $\mathcal {P}_{nn}^\mathrm {(spatial)}$. The used functions of the type $\varvec{\kappa }^\prime _{ijk}$ read

$$\begin{aligned} \varvec{\kappa }_{nnp}^\prime&:=\varvec{\pi }_{3}{\left( \varvec{q},\varvec{\pi }_{3}{\left( \varvec{p},-\varvec{q}\right) }\right) } , \end{aligned}$$

(125)

$$\begin{aligned} \varvec{\kappa }_{nnq}^\prime&:=\varvec{\pi }_{3}{\left( \varvec{p},-\varvec{q}\right) } , \end{aligned}$$

(126)

$$\begin{aligned} \varvec{\kappa }_{cnp}^\prime&:=-\varvec{\pi }_{1}{\left( \varvec{q},-\varvec{\pi }_{2}{\left( \varvec{p},\varvec{q}\right) }\right) } , \end{aligned}$$

(127)

$$\begin{aligned} \varvec{\kappa }_{cnq}^\prime&:=-\varvec{\pi }_{2}{\left( \varvec{p},\varvec{q}\right) } . \end{aligned}$$

(128)

1.2 Simplification of Certain Combinations of Coupled Spherical Harmonics

In this appendix identities simplifying the angular dependence of the probability density are given. The angle $\gamma$ between two vectors $\varvec{v}_1$ and $\varvec{v}_2$ is defined by

$$\begin{aligned}&\cos {\gamma \left( \theta _1, \varphi _1, \theta _2, \varphi _2 \right) } :=\varvec{\hat{v}_1} \cdot \varvec{\hat{v}_2}\, \end{aligned}$$

(129)

$$\begin{aligned}&\varvec{\hat{v}_i} = \begin{pmatrix} \sin {\theta _i} \cos {\varphi _i} \\ \sin {\theta _i} \sin {\varphi _i} \\ \cos {\theta _i} \end{pmatrix} . \end{aligned}$$

(130)

Expressed in terms of the angles of the two vectors it reads

$$\begin{aligned} \cos {\gamma \left( \theta _1,\varphi _1,\theta _2,\varphi _2\right) } :=\cos {\theta _1} \cos {\theta _2} + \sin {\theta _1} \sin {\theta _2} \cos {\left( \varphi _1 - \varphi _2\right) } . \end{aligned}$$

(131)

In fact, the angular dependence of the coupled spherical harmonic $\mathcal {Y}_{11}^{00}$ is captured entirely by this angle $\gamma$.

$$\begin{aligned} \mathcal {Y}_{11}^{00}{\left( \theta _1, \varphi _1, \theta _2, \varphi _2\right) } = \frac{-\sqrt{3}}{4\pi } \cos {\gamma \left( \theta _1, \varphi _1, \theta _2, \varphi _2\right) } . \end{aligned}$$

(132)

Also the following combination of coupled spherical harmonics can be purely expressed in terms of relative angles:

$$\begin{aligned} \sum _{M=-1}^{1} \mathcal {Y}_{11}^{1M}{\left( \theta _1, \varphi _1, \theta _2, \varphi _2\right) } \left( \mathcal {Y}_{11}^{1M}{\left( \theta _1^\prime , \varphi _1^\prime , \theta _2^\prime , \varphi _2^\prime \right) }\right) ^*&= \frac{1}{2} \left( \frac{3}{4\pi }\right) ^2 \bigg ( \cos {\gamma \left( \theta _1, \varphi _1, \theta _1^\prime , \varphi _1^\prime \right) } \cos {\gamma \left( \theta _2, \varphi _2, \theta _2^\prime , \varphi _2^\prime \right) } \nonumber \\&\quad -\, \cos {\gamma \left( \theta _1, \varphi _1, \theta _2^\prime , \varphi _2^\prime \right) } \cos {\gamma \left( \theta _1^\prime , \varphi _1^\prime , \theta _2, \varphi _2\right) } \bigg ) . \end{aligned}$$

(133)

Although these relative angles are angles between different vectors, their use simplifies the probability density a lot. If the identity is applied to our probability density all these relative angles will be only functions of $p$, $q$ and $\theta _{pq}$. Therefore angular integrations over the probability density simplify significantly. Instead of four angular integrals only the one over $\theta _{pq}$ has to be carried out numerically.

1.3 Numerical Calculation of $X_{ij}$

In the following expressions for the numerical calculation of the $X_{ij}$ are given for any type of form factor. In a first step the expression for $X_{nn}$ is derived using the identity

$$\begin{aligned} {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\right| {-\mathcal {P}_{nn}}\left| {p^{\prime },q^{\prime };\varOmega _n}\right\rangle _{n}^{}&= -\int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\right| { \left( \left| {\varvec{p},\varvec{q}}\right\rangle _{n}^{} {}^{}_{n}\! \left\langle {\varvec{p},\varvec{q}}\right| {\mathcal {P}_{nn}^\mathrm {(spatial)}}\left| {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{n}^{} {}^{}_{n}\! \langle {\varvec{p^{\prime }},\varvec{q^{\prime }}}|{\otimes }\mathcal {P}_{nn}^\mathrm {(spin)}\right) }\left| {p^{\prime },q^{\prime };\varOmega _n}\right\rangle _{n}^{} . \end{aligned}$$

(134)

The calculation yields

$$\begin{aligned} X_{nn}{\left( q,q^{\prime };E\right) }&= \int \mathrm{d}{p^{}} p^{2} \int \mathrm{d}{p^{\prime }} p^{\prime 2} g_{l_n}{\left( p\right) } G_0^{\left( n\right) }{\left( p,q;E\right) } g_{l_n}{\left( p^{\prime }\right) } {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\right| {-\mathcal {P}_{nn}}\left| {p^{\prime },q^{\prime };\varOmega _n}\right\rangle _{n}^{} \nonumber \\&=\, - \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} g_{l_n}{\left( \pi _{3}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }\right) } G_0^{\left( n\right) }{\left( \pi _{3}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }, q; E\right) } g_{l_n}{\left( \pi _{3}{\left( \varvec{q^{}},\varvec{q^{\prime }}\right) }\right) } \nonumber \\&\quad \times \, \sum _{L=0}^1 \sum _{M=-L}^L \frac{ \left( -2\right) ^{1-L} }{ 6L+3 } \left( \mathcal {Y}_{11}^{L M}{\left( \varvec{p},\varvec{q}\right) } \right) ^* \mathcal {Y}_{11}^{L M}{\left( \varvec{p^{\prime }},\varvec{q^{\prime }}\right) } \nonumber \\&\quad \times \, \delta ^{(\varOmega )}{\left( \varvec{p} - \varvec{\pi }_{3}{\left( \varvec{q^{\prime }}, \varvec{q}\right) }\right) } \delta ^{(\varOmega )}{\left( \varvec{p^{\prime }} - \varvec{\pi }_{3}{\left( \varvec{q}, \varvec{q^{\prime }}\right) }\right) } \nonumber \\&= - \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} g_{l_n}{\left( \pi _{3}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }\right) } G_0^{\left( n\right) }{\left( \pi _{3}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }, q; E\right) } g_{l_n}{\left( \pi _{3}{\left( \varvec{q^{}},\varvec{q^{\prime }}\right) }\right) } \nonumber \\&\quad \times \, \sum _{L=0}^1 \sum _{M_L=-L}^L \frac{ \left( -2\right) ^{1-L} }{ 6L+3 } \left( \mathcal {Y}_{11}^{L M_L}{\left( \varvec{\pi }_{3}{\left( \varvec{q^{\prime }}, \varvec{q}\right) }, \varvec{q}\right) } \right) ^* \mathcal {Y}_{11}^{L M_L}{\left( \varvec{\pi }_{3}{\left( \varvec{q}, \varvec{q^{\prime }}\right) }, \varvec{q^{\prime }}\right) } , \end{aligned}$$

(135)

where $\delta ^{(\varOmega )}{\left( \varvec{p}- \varvec{p^{\prime }}\right) } :=\delta {\left( \varphi -\varphi ^\prime \right) } \frac{\delta {\left( \theta -\theta ^\prime \right) }}{\sin {\theta }}$ holds. In order to evaluate this expression only one angular integral has to be computed numerically by using the identities given in “Appendix B.2”.

The calculation of $X_{nc}$ is based on an identity similar to Eq. (134):

$$\begin{aligned} {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\vert {p^{\prime },q^{\prime };\varOmega _c}\right\rangle _{c}^{}&= \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\right| { \left( \left| {\varvec{p},\varvec{q}}\right\rangle _{n}^{} {}^{}_{n}\! \left\langle {\varvec{p},\varvec{q}}\vert {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{c}^{} {}^{}_{c}\! \langle {\varvec{p^{\prime }},\varvec{q^{\prime }}}|{\otimes }\mathbb {1}^{\left( \mathrm {spin}\right) }\right) }\left| {p^{\prime },q^{\prime };\varOmega _c}\right\rangle _{c}^{} . \end{aligned}$$

(136)

With it one obtains

$$\begin{aligned} X_{nc}{\left( q,q^{\prime };E\right) }&= \int \mathrm{d}{p^{}} p^{2} \int \mathrm{d}{p^{\prime }} p^{\prime 2} g_{l_n}{\left( p\right) } G_0^{\left( n\right) }{\left( p,q;E\right) } g_{l_c}{\left( p^{\prime }\right) } {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\vert {p^{\prime },q^{\prime };\varOmega _c}\right\rangle _{c}^{} \nonumber \\&= - \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} g_{l_n}{\left( \pi _{1}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }\right) } G_0^{\left( n\right) }{\left( \pi _{1}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }, q; E\right) } g_{l_c}{\left( \pi _{2}{\left( \varvec{q^{}},\varvec{q^{\prime }}\right) }\right) } \nonumber \\&\quad \times \, \sqrt{\frac{2}{3}} \left( \mathcal {Y}_{11}^{00}{\left( \varvec{p},\varvec{q}\right) } \right) ^* \mathcal {Y}_{00}^{00}{\left( \varvec{p^{\prime }}, \varvec{q^{\prime }}\right) } \delta ^{(\varOmega )}{\left( \varvec{p} + \varvec{\pi }_{1}{\left( \varvec{q^{\prime }}, \varvec{q}\right) }\right) } \delta ^{(\varOmega )}{\left( \varvec{p^{\prime }} - \varvec{\pi }_{2}{\left( \varvec{q}, \varvec{q^{\prime }}\right) }\right) } \nonumber \\&= - \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} g_{l_n}{\left( \pi _{1}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }\right) } G_0^{\left( n\right) }{\left( \pi _{1}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }, q; E\right) } g_{l_c}{\left( \pi _{2}{\left( \varvec{q^{}},\varvec{q^{\prime }}\right) }\right) } \nonumber \\&\quad \times \, \sqrt{\frac{2}{3}} \left( \mathcal {Y}_{11}^{00}{\left( -\varvec{\pi }_{1}{\left( \varvec{q^{\prime }}, \varvec{q}\right) }, \varvec{q}\right) } \right) ^* \mathcal {Y}_{00}^{00}{\left( \varvec{\pi }_{2}{\left( \varvec{q}, \varvec{q^{\prime }}\right) }, \varvec{q^{\prime }}\right) }\nonumber \\&= \frac{1}{\sqrt{2}} \int _{-1}^{1} \mathrm{d}{\cos {\theta _{q,q^{\prime }}}} g_{l_n}{\left( \pi _{1}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }\right) } G_0^{\left( n\right) }{\left( \pi _{1}{\left( \varvec{q^{\prime }},\varvec{q^{}}\right) }, q; E\right) } g_{l_c}{\left( \pi _{2}{\left( \varvec{q^{}},\varvec{q^{\prime }}\right) }\right) } \cos {\theta _{-\varvec{\pi }_{1}{\left( \varvec{q^{\prime }}, \varvec{q}\right) },\varvec{q}}} , \end{aligned}$$

(137)

where the definition $\theta _{\varvec{a},\varvec{b}} :=\arccos { \left( \varvec{a} \cdot \varvec{b}/\left( ab\right) \right) }$ holds. Note that

$$\begin{aligned} X_{nc}{\left( q,q^{\prime };E\right) } = X_{cn}{\left( q^{\prime },q;E\right) } \end{aligned}$$

(138)

holds. This can be shown using that ${}^{}_{n}\! \left\langle {p,q;\varOmega _n}\vert {p^{\prime },q^{\prime };\varOmega _c}\right\rangle _{c}^{}$ is real.

The implementation of these results was checked by evaluating the expressions for Heaviside form factors, for which the analytic expressions have been calculated in [15]. As in the derivation of the analytic expressions the Heaviside form factors have been neglected, discrepancies from the numerical results can occur at momenta of the order of the regularization scale.

1.4 Overlaps in Partial Wave Basis

In the following expressions for the overlaps of partial wave states with different spectators are derived. Also matrix elements of the $nn$ permutation operator $\mathcal {P}_{nn}= \mathcal {P}_{nn}^\mathrm {(spatial)}\otimes \mathcal {P}_{nn}^\mathrm {(spin)}$ are given in this basis. We obtain

$$\begin{aligned} {}^{}_{c}\! \left\langle {p,q;\varOmega _c}\vert {p^{\prime },q^{\prime };\varOmega _n}\right\rangle _{n}^{}&= \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, {}^{}_{c}\! \left\langle {p,q;\varOmega _c}\right| { \left( \left| {\varvec{p},\varvec{q}}\right\rangle _{c}^{} {}^{}_{c}\! \left\langle {\varvec{p},\varvec{q}}\vert {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{n}^{} {}^{}_{n}\! \langle {\varvec{p^{\prime }},\varvec{q^{\prime }}}|{\otimes }\mathbb {1}^{\left( \mathrm {spin}\right) }\right) }\left| {p^{\prime },q^{\prime };\varOmega _n }\right\rangle _{n}^{} \nonumber \\&= \frac{ \sqrt{2} }{\left( 4\pi \right) ^2} \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, \cos {\theta _{\varvec{p^{\prime }},\varvec{q^{\prime }}}} \delta ^{(3)}{\left( \varvec{p^{\prime }} - \varvec{\kappa }_{cnp}{\left( \varvec{p},\varvec{q}\right) } \right) } \delta ^{(3)}{\left( \varvec{q^{\prime }} - \varvec{\kappa }_{cnq}{\left( \varvec{p},\varvec{q}\right) } \right) } ,\end{aligned}$$

(139)

$$\begin{aligned} {}^{}_{c}\! \left\langle {p,q;\varOmega _c}\right| {\mathcal {P}_{nn}}\left| {p^{\prime },q^{\prime };\varOmega _n}\right\rangle _{n}^{}&= \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, {}^{}_{c}\! \left\langle {p,q;\varOmega _c}\right| { \left( \left| {\varvec{p},\varvec{q}}\right\rangle _{c}^{} {}^{}_{c}\! \left\langle {\varvec{p},\varvec{q}}\right| {\mathcal {P}_{nn}^\mathrm {(spatial)}}\left| {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{n}^{} {}^{}_{n}\! \langle {\varvec{p^{\prime }},\varvec{q^{\prime }}}|{\otimes }\mathcal {P}_{nn}^\mathrm {(spin)}\right) }\left| {p^{\prime },q^{\prime };\varOmega _n }\right\rangle _{n}^{} \nonumber \\&= -\frac{ \sqrt{2} }{\left( 4\pi \right) ^2} \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, \cos {\theta _{\varvec{p^{\prime }},\varvec{q^{\prime }}}} \delta ^{(3)}{\left( \varvec{p^{\prime }} - \varvec{\kappa }_{cnp}^{\prime }{\left( \varvec{p},\varvec{q}\right) } \right) } \delta ^{(3)}{\left( \varvec{q^{\prime }} - \varvec{\kappa }_{cnq}^{\prime }{\left( \varvec{p},\varvec{q}\right) } \right) } , \end{aligned}$$

(140)

$$\begin{aligned} {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\vert {p^{\prime },q^{\prime };\varOmega _c}\right\rangle _{c}^{}&= \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\right| { \left( \left| {\varvec{p},\varvec{q}}\right\rangle _{n}^{} {}^{}_{n}\! \left\langle {\varvec{p},\varvec{q}}\vert {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{c}^{} {}^{}_{c}\! \langle {\varvec{p^{\prime }},\varvec{q^{\prime }}}|{\otimes }\mathbb {1}^{\left( \mathrm {spin}\right) }\right) }\left| {p^{\prime },q^{\prime };\varOmega _c }\right\rangle _{c}^{} \nonumber \\&= \frac{\sqrt{2}}{\left( 4\pi \right) ^2} \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, \cos {\theta _{\varvec{p},\varvec{q}}} \delta ^{(3)}{\left( \varvec{p^{\prime }} - \varvec{\kappa }_{ncp}{\left( \varvec{p},\varvec{q}\right) } \right) } \delta ^{(3)}{\left( \varvec{q^{\prime }} - \varvec{\kappa }_{ncq}{\left( \varvec{p},\varvec{q}\right) } \right) } , \end{aligned}$$

(141)

$$\begin{aligned} {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\right| {\mathcal {P}_{nn}}\left| {p^{\prime },q^{\prime };\varOmega _n}\right\rangle _{n}^{}&= \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \nonumber \\&\quad \times \, {}^{}_{n}\! \left\langle {p,q;\varOmega _n}\right| { \left( \left| {\varvec{p},\varvec{q}}\right\rangle _{n}^{} {}^{}_{n}\! \left\langle {\varvec{p},\varvec{q}}\right| {\mathcal {P}_{nn}^\mathrm {(spatial)}}\left| {\varvec{p^{\prime }},\varvec{q^{\prime }}}\right\rangle _{n}^{} {}^{}_{n}\! \langle {\varvec{p^{\prime }},\varvec{q^{\prime }}}|{\otimes }\mathcal {P}_{nn}^\mathrm {(spin)}\right) }\left| {p^{\prime },q^{\prime };\varOmega _n }\right\rangle _{n}^{} \nonumber \\&= \int \mathrm{d}{\varOmega _{\varvec{p}}} \int \mathrm{d}{\varOmega _{\varvec{q}}} \int \mathrm{d}{\varOmega _{\varvec{p^{\prime }}}} \int \mathrm{d}{\varOmega _{\varvec{q^{\prime }}}} \sum _{L=0}^1 \sum _{M=-L}^{L} \left( -1\right) ^{1-L} \frac{2^{1-L}}{6L+3} \nonumber \\&\quad \times \, \left( \mathcal {Y}_{11}^{LM}{\left( \varvec{p},\varvec{q}\right) }\right) ^* \mathcal {Y}_{11}^{LM}{\left( \varvec{p^{\prime }},\varvec{q^{\prime }}\right) } \delta ^{(3)}{\left( \varvec{p^{\prime }} - \varvec{\kappa }_{nnp}^{\prime }{\left( \varvec{p},\varvec{q}\right) } \right) } \delta ^{(3)}{\left( \varvec{q^{\prime }} - \varvec{\kappa }_{nnq}^{\prime }{\left( \varvec{p},\varvec{q}\right) }\right) } . \end{aligned}$$

(142)