On the dynamical kernels of fermionic equations of motion in strongly-correlated media

We adopt a framework, where the many-body fermionic Hamiltonian serves as the only input, which determines uniquely all the properties of the system of interacting fermions. The starting point is the fermionic Hamiltonian in the field-theoretical representation:

$$H=H^{(1)}+V^{(2)}+W^{(3)}+...\ .$$

(1)

The operator $H^{(1)}$ describes the one-body contribution:

$$H^{(1)}=\sum_{12}t_{12}\psi^{{\dagger}}_{1}\psi_{2}+\sum_{12}v^{(MF)}_{12}\psi^{{\dagger}}_{1}\psi_{2}\equiv\sum_{12}h_{12}\psi^{{\dagger}}_{1}\psi_{2}$$

(2)

with matrix elements $h_{12}$ combining the kinetic energy $t$ and the mean-field $v^{(MF)}$ part of the interaction. The two-body sector is described by the two-fermion interaction operator $V^{(2)}$:

$$V^{(2)}=\frac{1}{4}\sum\limits_{1234}{\bar{v}}_{1234}{\psi^{\dagger}}_{1}{\psi^{\dagger}}_{2}\psi_{4}\psi_{3},$$

(3)

and $W^{(3)}$ represents the three-body forces, which will be neglected in this work, where we will eventually discuss implementations employing the meson-nucleon interaction in covariant form. In the many-body formalism of this work we will not manifestly use covariant notations, however, keep in mind that the relativistic nucleonic Hamiltonian with the meson-exchange interaction is defined by the terms of essentially the same form as $H^{(1)}$ and $V^{(2)}$ Brockmann (1978); Boyussy et al. (1987). Hence, the general structure of the EOMs remains the same, as it is shown explicitly, for instance, for the one-fermion EOM with the non-symmetric form of the dynamical kernel Poschenrieder and Weigel (1988a, b).

The formal covariant theory will be presented elsewhere. Here we utilize the fact that in a relativistic theory, the role of three-body forces is found to be considerably smaller than in a non-relativistic one. The necessity of the relativistic three-body forces in nuclear systems is still debatable, while the corresponding quantitative studies were reported only for few-body systems Danielewicz and Namyslowski (1979); Karmanov (2011). We conjecture that the many-body dynamics, non-perturbatively described by the in-medium fermionic propagators in the EOM framework, includes the three- and higher-body forces defined as in Ref. Karmanov (2011) and thus must adequately capture their effects, leaving the contributions associated with the subnucleon degrees of freedom for future studies.

The operators $\psi_{1}$ and $\psi^{\dagger}_{1}$ in Eqs. (2,3) stand for the fermionic fields in some basis of states completely characterized by the number indices. In Eq. (3) and in the following we use the antisymmetrized matrix elements ${\bar{v}}_{1234}={v}_{1234}-{v}_{1243}$. The fermionic fields obey the anticommutation relations

	$\displaystyle[\psi_{1},{\psi^{\dagger}}_{1^{\prime}}]_{+}\equiv\psi_{1}{\psi^{\dagger}}_{1^{\prime}}+{\psi^{\dagger}}_{1^{\prime}}\psi_{1}=\delta_{11^{\prime}},$
	$\displaystyle\left[\psi_{1},{\psi}_{1^{\prime}}\right]_{+}=\left[{\psi^{\dagger}}_{1},{\psi^{\dagger}}_{1^{\prime}}\right]_{+}=0,$		(4)

and the Heisenberg form defines their time evolution:

$$\psi(1)=e^{iHt_{1}}\psi_{1}e^{-iHt_{1}},\ \ \ \ \ \ {\psi^{\dagger}}(1)=e^{iHt_{1}}{\psi^{\dagger}}_{1}e^{-iHt_{1}}.$$

(5)

The fermionic in-medium propagator, or real-time Green function, is defined as a correlator of two fermionic field operators:

$$G(1,1^{\prime})\equiv G_{11^{\prime}}(t-t^{\prime})=-i\langle T\psi(1){\psi^{\dagger}}(1^{\prime})\rangle,$$

(6)

where $T$ is the chronological ordering operator, and the averaging $\langle...\rangle$ is performed over the formally exact ground state of the many-body system of $N$ particles. Eq. (6) describes the propagation of a single fermion in the medium of $N$ interacting fermions.

In the EOM method, it is convenient to use the basis of fermionic states $\{1\}$ which diagonalizes the one-body (single-particle) part of the Hamiltonian $H^{(1)}$: $h_{12}=\delta_{12}\varepsilon_{1}$. The propagator (6) depends explicitly on a single time difference $\tau=t-t^{\prime}$, so that its Fourier transform to the energy domain, after inserting the operator $\mathbb{1}=\sum_{n}|n\rangle\langle n|$ with the complete set of the many-body states, leads to the spectral (Lehmann) representation:

	$\displaystyle G_{11^{\prime}}(\varepsilon)=\sum\limits_{n}\frac{\eta^{n}_{1}\eta^{n\ast}_{1^{\prime}}}{\varepsilon-(E^{(N+1)}_{n}-E^{(N)}_{0})+i\delta}+$
	$\displaystyle+\sum\limits_{m}\frac{\chi^{m}_{1}\chi^{m\ast}_{1^{\prime}}}{\varepsilon+(E^{(N-1)}_{m}-E^{(N)}_{0})-i\delta}.$		(7)

$G_{11^{\prime}}(\varepsilon)$ thus consists of terms of the simple pole character with factorized residues, which is the common feature of the propagators. Its poles are located at the formally exact energies $E^{(N+1)}_{n}-E^{(N)}_{0}$ and $-(E^{(N-1)}_{m}-E^{(N)}_{0})$ of the neighboring $(N+1)$-particle and $(N-1)$-particle systems with respect to the ground state of the background $N$-particle system. The corresponding residues are the matrix elements of the field operators between the ground state $|0^{(N)}\rangle$ of the reference $N$-particle system and states $|n^{(N+1)}\rangle$ and $|m^{(N-1)}\rangle$:

$$\eta^{n}_{1}=\langle 0^{(N)}|\psi_{1}|n^{(N+1)}\rangle,\ \ \ \ \ \ \ \ \chi^{m}_{1}=\langle m^{(N-1)}|\psi_{1}|0^{(N)}\rangle.$$

(8)

As it follows from their definition, these matrix elements are the weights of the given single-particle (single-hole) configuration on top of the ground state $|0^{(N)}\rangle$ in the $n$-th ($m$-th) state of the $(N+1)$-particle ($(N-1)$-particle) systems. The residues are thus associated with the occupancies of the corresponding fermionic states.

In the next subsection, we will discuss the EOM for the propagator $G_{11^{\prime}}(t-t^{\prime})$ and find that it is connected to the higher-rank two-time (two-point) CFs. Of particular importance are the two two-fermion correlators: the particle-hole propagator, also known as response function, and the particle-particle, or fermionic pair, propagator. The response function characterizes the response of the many-body system to an external probe of the one-body character and it is defined as follows:

	$\displaystyle R(12,1^{\prime}2^{\prime})$	$\displaystyle\equiv$	$\displaystyle R_{12,1^{\prime}2^{\prime}}(t-t^{\prime})=-i\langle T\psi^{\dagger}(1)\psi(2)\psi^{\dagger}(2^{\prime})\psi(1^{\prime})\rangle$		(9)
		$\displaystyle=$	$\displaystyle-i\langle T(\psi^{\dagger}_{1}\psi_{2})(t)(\psi^{\dagger}_{2^{\prime}}\psi_{1^{\prime}})(t^{\prime})\rangle,$

while the pair propagator has the form:

	$\displaystyle G(12,1^{\prime}2^{\prime})$	$\displaystyle\equiv$	$\displaystyle G_{12,1^{\prime}2^{\prime}}(t-t^{\prime})=-i\langle T\psi(1)\psi(2){\psi^{\dagger}}(2^{\prime}){\psi^{\dagger}}(1^{\prime})\rangle$		(10)
		$\displaystyle=$	$\displaystyle-i\langle T(\psi_{1}\psi_{2})(t)(\psi^{\dagger}_{2^{\prime}}\psi^{\dagger}_{1^{\prime}})(t^{\prime})\rangle,$

where we imply that $t_{1}=t_{2}=t,t_{1^{\prime}}=t_{2^{\prime}}=t^{\prime}$ and adopt the same phase phase factor as in Eq. (9) for convenience. In analogy to the one-fermion case, inserting the completeness relation between the operator pairs and making the Fourier transformations of these CFs to the energy (frequency) domain lead to:

$$R_{12,1^{\prime}2^{\prime}}(\omega)=\sum\limits_{\nu>0}\Bigl{[}\frac{\rho^{\nu}_{21}\rho^{\nu\ast}_{2^{\prime}1^{\prime}}}{\omega-\omega_{\nu}+i\delta}-\frac{\rho^{\nu\ast}_{12}\rho^{\nu}_{1^{\prime}2^{\prime}}}{\omega+\omega_{\nu}-i\delta}\Bigr{]}$$

(11)

$$G_{12,1^{\prime}2^{\prime}}(\omega)=\sum\limits_{\mu}\frac{\alpha^{\mu}_{21}\alpha^{\mu\ast}_{2^{\prime}1^{\prime}}}{\omega-\omega_{\mu}^{(++)}+i\delta}-\sum\limits_{\varkappa}\frac{\beta^{\varkappa\ast}_{12}\beta^{\varkappa}_{1^{\prime}2^{\prime}}}{\omega+\omega_{\varkappa}^{(--)}-i\delta}.$$

(12)

Similarly to the one-fermion propagator of Eq. (7), Eqs. (11,12) satisfy the general requirements of locality and unitarity. The poles represent the energies $\omega_{\nu}=E_{\nu}-E_{0},\omega_{\mu}^{(++)}=E_{\mu}^{(N+2)}-E_{0}^{(N)}$, and $\omega_{\mu}^{(--)}=E_{\varkappa}^{(N-2)}-E_{0}^{(N)}$ of the states in the systems with $N$ and $N\pm 2$ particles, respectively. In Eqs. (7,11,12) the sums are formally complete, i.e., run over the discrete spectra and engage the corresponding integrals over the continuum states.

The matrix elements in the residues

	$\displaystyle\rho^{\nu}_{12}=\langle 0|\psi^{\dagger}_{2}\psi_{1}|\nu\rangle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$		(13)
	$\displaystyle\alpha_{12}^{\mu}=\langle 0^{(N)}|\psi_{2}\psi_{1}|\mu^{(N+2)}\rangle\ \ \ \ \ \ \beta_{12}^{\varkappa}=\langle 0^{(N)}|\psi^{\dagger}_{2}\psi^{\dagger}_{1}|\varkappa^{(N-2)}\rangle$
			(14)

are the normal $\rho^{\nu}_{12}$ and anomalous (pairing) $\alpha_{12}^{\mu},\beta_{12}^{\varkappa}$ transition densities. They give the weights of the pure particle-hole, two-particle and two-hole configurations on top of the ground state $|0^{(N)}\rangle$ in the excited states of the respective systems. These matrix elements play a central role in characterizing transition probabilities, underlying properties of the transitions, pair transfer, and superfluid pairing spectral gaps in nuclear structure applications.

Eqs. (7,11,12) are model-independent, i.e., remain valid for any physical approximations applied to determination of the many-body states $|n\rangle,|m\rangle$, $|\nu\rangle$, $|\mu\rangle$, and $|\varkappa\rangle$.

As we will see below, the two-fermion two-point functions of Eqs. (9,10) will appear in the cluster decomposition of the three-fermion (two-particle-one-hole, or $2p1h$) Green function, which defines the exact symmetric dynamical kernel of the EOM for the one-fermion Green function of Eq. (6).

The EOM for the fermionic propagator (6) can be generated by taking time derivatives with respect to the times $t$ and $t^{\prime}$. The detailed derivation procedure is described in Refs. Litvinova and Schuck (2019); Litvinova and Zhang (2021), which are in agreement with the major steps given in earlier works Schuck and Ethofer (1973); Schuck (1976); Adachi and Schuck (1989); Danielewicz and Schuck (1994); Dukelsky et al. (1998); Dickhoff and Barbieri (2004); Dickhoff and Neck (2005).

The differentiation with respect to $t$ leads to

	$\displaystyle\partial_{t}G_{11^{\prime}}(t-t^{\prime})=-i\delta(t-t^{\prime})\langle[\psi_{1}(t),{\psi^{\dagger}}_{1^{\prime}}(t^{\prime})]_{+}\rangle+$
	$\displaystyle+\langle T[H,\psi_{1}](t){\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\rangle,$
			(15)

where $[H,\psi_{1}](t)=e^{iHt}[H,\psi_{1}]e^{-iHt}.$ Evaluating explicitly the commutator with the one-body part of the Hamiltonian and collecting the terms with $G_{11^{\prime}}(t-t^{\prime})$, one obtains the equation:

$$(i\partial_{t}-\varepsilon_{1})G_{11^{\prime}}(t-t^{\prime})=\delta_{11^{\prime}}\delta(t-t^{\prime})+i\langle T[V,\psi_{1}](t){\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\rangle,$$

(16)

or, after evaluating the latter commutator,

	$\displaystyle(i\partial_{t}-\varepsilon_{1})G_{11^{\prime}}(t-t^{\prime})$	$\displaystyle=$	$\displaystyle\delta_{11^{\prime}}\delta(t-t^{\prime})$
		$\displaystyle+$	$\displaystyle\frac{i}{2}\sum\limits_{ikl}{\bar{v}}_{i1kl}\langle T(\psi^{\dagger}_{i}\psi_{l}\psi_{k})(t){\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\rangle$

which is commonly referred to as the first EOM, or EOM1. Here we have introduced the Latin dummy indices, which have the same meaning as the number indices, to mark the intermediate fermionic states in the same single-particle basis. The EOM1 of this form can be found in many articles and textbooks on the quantum many-body problem, for instance, in Refs. Schuck and Ethofer (1973); Dickhoff and Neck (2005); Poschenrieder and Weigel (1988a). The appearance of the two-fermion CF on the right-hand side of Eq. (LABEL:spEOM1) indicates that the one-fermion propagator and the associated single-particle in-medium trajectories and densities are fundamentally coupled to higher-rank propagators. In principle, an EOM for this two-fermion CF can be generated, but one sees immediately that this EOM further produces a higher-rank CF. The relevance of increasingly-complex CFs to the description of the motion of a single fermion is the characteristic feature of strongly-correlated systems. In weakly-coupled regimes, the perturbation theory is a viable solution discussed in many applications, so we will concentrate on non-perturbative solutions in this work.

Here one can note that the CF in the right-hand side of Eq. (LABEL:spEOM1)

$$G^{(2)}_{ilk,1^{\prime}}(t-t^{\prime})=-i\langle T(\psi^{\dagger}_{i}\psi_{l}\psi_{k})(t){\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\rangle$$

(18)

depends on the time difference $\tau=t-t^{\prime}$, so that the Fourier transform of Eq. (LABEL:spEOM1) reads

$$G_{11^{\prime}}(\omega)=G^{0}_{11^{\prime}}(\omega)+\frac{1}{2}\sum\limits_{2ikl}G^{0}_{12}(\omega){\bar{v}}_{2ikl}G^{(2)}_{ilk,1^{\prime}}(\omega),$$

(19)

where the free, or uncorrelated, fermionic propagator is introduced as $G^{0}_{11^{\prime}}(\omega)=\delta_{11^{\prime}}/(\omega-\varepsilon_{1})$. Eqs. (LABEL:spEOM1,19) can be further transformed to the Dyson form Dickhoff and Barbieri (2004); Dickhoff and Neck (2005). There are various possible treatments of the integral part of the Eq. (LABEL:spEOM1), such as the relativistic ”$\Lambda^{00},\Lambda^{10},\Lambda^{11}$” approximations Poschenrieder and Weigel (1988b), which factorize the CF (18) into two one-fermion CFs, correlated or uncorrelated. Another famous approach is the Gor’kov factorization Gorkov (1958); Kucharek and Ring (1991); Litvinova and Zhang (2021), which retains, in addition, one-fermion CFs with the same kind of field operators (anomalous Green functions).

More insights into the interacting part of the one-fermion EOM come with its symmetric form, which can be obtained via the second EOM, or EOM2. It is generated by the differentiation of the last term on the right-hand side of Eq. (16),

$$R_{11^{\prime}}(t-t^{\prime})=i\langle T[V,\psi_{1}](t){\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\rangle,$$

(20)

with respect to $t^{\prime}$:

	$\displaystyle R_{11^{\prime}}(t-t^{\prime})\overleftarrow{\partial_{t^{\prime}}}$	$\displaystyle=$	$\displaystyle-i\delta(t-t^{\prime})\langle\bigl{[}[V,\psi_{1}](t),{\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\bigr{]}_{+}\rangle-$		(21)
		$\displaystyle-$	$\displaystyle\langle T[V,\psi_{1}](t)[H,{\psi^{\dagger}}_{1^{\prime}}](t^{\prime})\rangle,$

which gives the EOM2:

	$\displaystyle R_{11^{\prime}}(t-t^{\prime})(-i\overleftarrow{\partial_{t^{\prime}}}-\varepsilon_{1^{\prime}})$	$\displaystyle=$	$\displaystyle-\delta(t-t^{\prime})\langle\bigl{[}[V,\psi_{1}](t),{\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\bigr{]}_{+}\rangle$		(22)
		$\displaystyle+$	$\displaystyle i\langle T[V,\psi_{1}](t)[V,{\psi^{\dagger}}_{1^{\prime}}](t^{\prime})\rangle.$

Finally, after acting on the EOM1 (16) by the operator $(-i\overleftarrow{\partial_{t^{\prime}}}-\varepsilon_{1^{\prime}})$ and Fourier transformation to the energy domain one obtains:

$$G_{11^{\prime}}(\omega)=G^{0}_{11^{\prime}}(\omega)+\sum\limits_{22^{\prime}}G^{0}_{12}(\omega)T_{22^{\prime}}(\omega)G^{0}_{2^{\prime}1^{\prime}}(\omega).$$

(23)

Thus, the complete in-medium one-fermion propagator is expressed via the free propagator $G^{0}$ and the $T$-matrix, which absorbs all possible interaction processes of the fermion with the correlated medium. The $T(\omega)$ is the Fourier image of the following time-dependent $T$-matrix:

	$\displaystyle T_{11^{\prime}}(t-t^{\prime})$	$\displaystyle=$	$\displaystyle T^{0}_{11^{\prime}}(t-t^{\prime})+T^{r}_{11^{\prime}}(t-t^{\prime}),$
	$\displaystyle T^{0}_{11^{\prime}}(t-t^{\prime})$	$\displaystyle=$	$\displaystyle-\delta(t-t^{\prime})\langle\bigl{[}[V,\psi_{1}](t),{\psi^{\dagger}}_{1^{\prime}}(t^{\prime})\bigr{]}_{+}\rangle,$
	$\displaystyle T^{r}_{11^{\prime}}(t-t^{\prime})$	$\displaystyle=$	$\displaystyle i\langle T[V,\psi_{1}](t)[V,{\psi^{\dagger}}_{1^{\prime}}](t^{\prime})\rangle.$		(24)

The superscript ”0” marks the static, or instantaneous, part of the $T$-matrix, and ”r” is associated with its dynamical, or time-dependent, part containing retardation effects. The EOM (23) in the operator form reads:

$$G(\omega)=G^{0}(\omega)+G^{0}(\omega)T(\omega)G^{0}(\omega).$$

(25)

It is often more instructive to have this EOM in the Dyson ansatz, for which one introduces the irreducible part of the $T$-matrix with respect to the uncorrelated one-fermion propagator $G^{0}$, the self-energy (called also interaction kernel): $\Sigma=T^{irr}$. This operation removes all the contributions containing parts connected by the one-fermion free propagator $G^{0}$ from the $T$-matrix by the following definition:

$$T(\omega)=\Sigma(\omega)+\Sigma(\omega)G^{0}(\omega)T(\omega).$$

(26)

Combining Eqs. (25) and (26), one arrives at the Dyson equation for the fermionic propagator:

$$G(\omega)=G^{0}(\omega)+G^{0}(\omega)\Sigma(\omega)G(\omega).$$

(27)

The self-energy holds the same decomposition as the $T$-matrix:

$$\Sigma_{11^{\prime}}(\omega)=\Sigma_{11^{\prime}}^{0}+\Sigma_{11^{\prime}}^{r}(\omega),$$

(28)

where

$$\Sigma^{0}_{11^{\prime}}=-\langle[[V,\psi_{1}],{\psi^{\dagger}}_{1^{\prime}}]_{+}\rangle=\sum\limits_{il}{\bar{v}}_{1i1^{\prime}l}\rho_{li},$$

(29)

with $\rho_{li}=\langle{\psi^{\dagger}}_{i}\psi_{l}\rangle$ being the ground-state one-body density, and $\Sigma_{11^{\prime}}^{r}(\omega)$ is the Fourier image of

	$\displaystyle\Sigma_{11^{\prime}}^{r}(t-t^{\prime})$	$\displaystyle=$	$\displaystyle-\frac{i}{4}\sum\limits_{npq}\sum\limits_{ikl}{\bar{v}}_{1ikl}\times$
		$\displaystyle\times$	$\displaystyle\langle T\bigl{(}\psi^{\dagger}_{i}\psi_{l}\psi_{k}\bigr{)}(t)\bigl{(}\psi^{\dagger}_{p}\psi^{\dagger}_{q}\psi_{n}\bigr{)}(t^{\prime})\rangle^{irr}{\bar{v}}_{qpn1^{\prime}}$
		$\displaystyle=$	$\displaystyle\frac{1}{4}\sum\limits_{npq}\sum\limits_{ikl}{\bar{v}}_{1ikl}G^{(pph)irr}_{ilk,nqp}(t-t^{\prime}){\bar{v}}_{qpn1^{\prime}}.$

Thus, the EOM (27) for the fermionic propagator $G(\omega)$ is formally a closed equation with respect to $G(\omega)$. However, its self-energy, which in this version has the symmetric form of a CF ”sandwiched” between two interaction matrix elements, contains the three-fermion Green function. The obtained form of the self-energy (28-LABEL:Tr) indicates a clear separation between the static (instantaneous) Hartree-Fock-like term (29) and the dynamical term (LABEL:Tr), which accumulates all the retardation effects. Accordingly, the former is responsible for the short-range and the latter generates long-range correlations. Here the short range is associated with the range of the input bare interaction $\bar{v}$, and the long range may extend to the size of the entire many-body system.

So far the theory is exact, but again, the EOM for the three-body propagator generates higher-rank propagators, which makes the exact solution of the many-body problem hardly tractable. There are various ways to approximate the three-fermion propagator in the dynamical kernel of the one-fermion EOM. Some approximations can be obtained by making use of the cluster decomposition of the CF in the dynamical kernel (LABEL:Tr). In a symbolic form, it reads:

$$G^{(pph)irr}=G^{(p)}G^{(p)}G^{(h)}+G^{(p)}R^{(ph)}+G^{(h)}G^{(pp)}+\sigma^{(pph)},$$

(31)

where the number of particles (holes) in the superscripts indicates the rank of the respective CF and the sum implicitly includes all the necessary antisymmetrizations, see Refs. Vinh Mau ; Mau and Bouyssy (1976); Litvinova and Schuck (2019) for details. Retaining the first term only is the approach, which truncates the many-body problem at the one-body level, which is sometimes called the self-consistent Green functions approach. Some implementations were presented in Refs. Poschenrieder and Weigel (1988a, b). The one-fermion EOM has then a closed form and can, in principle, be solved iteratively. The decomposition retaining all possible terms with one-fermion and two-fermion propagators was discussed in detail, for instance, in Refs. Martin and Schwinger (1959); Vinh Mau ; Mau and Bouyssy (1976); Ring and Schuck (1980); Rijsdijk et al. (1992); Litvinova and Schuck (2019). It was shown, in particular, that this approximation can be linked to the nuclear field theory Bohr and Mottelson (1969, 1975); Broglia and Bortignon (1976); Bortignon et al. (1977); Bertsch et al. (1983), which implies a coupling between particles and phonons of both particle-hole and particle-particle origins. In this approximation, the one-fermion dynamical kernel takes the form

	$\displaystyle\Sigma^{r}_{11^{\prime}}(\omega)=\Sigma^{r(ph)}_{11^{\prime}}(\omega)+\Sigma^{r(pp)}_{11^{\prime}}(\omega)+\Sigma^{r(0)}_{11^{\prime}}(\omega),$
			(32)

where

$$\Sigma^{r(ph)}_{11^{\prime}}(\omega)=-\sum\limits_{33^{\prime}}\int\limits_{-\infty}^{\infty}\frac{d\varepsilon}{2\pi i}\Gamma^{ph}_{13^{\prime},1^{\prime}3}(\omega-\varepsilon)G_{33^{\prime}}(\varepsilon),$$

(33)

$$\Sigma^{r(pp)}_{11^{\prime}}(\omega)=\sum\limits_{22^{\prime}}\int\limits_{-\infty}^{\infty}\frac{d\varepsilon}{2\pi i}\Gamma^{pp}_{12,1^{\prime}2^{\prime}}(\omega+\varepsilon)G_{2^{\prime}2}(\varepsilon),$$

(34)

	$\displaystyle\Sigma^{r(0)}_{11^{\prime}}(\omega)$	$\displaystyle=$	$\displaystyle-\sum\limits_{2342^{\prime}3^{\prime}4^{\prime}}{\bar{v}}_{1234}$		(35)
		$\displaystyle\times~{}$	$\displaystyle\int\limits_{-\infty}^{\infty}\frac{d\varepsilon d\varepsilon^{\prime}}{(2\pi i)^{2}}G_{44^{\prime}}(\omega+\varepsilon^{\prime}-\varepsilon)G_{33^{\prime}}(\varepsilon)G_{2^{\prime}2}(\varepsilon^{\prime})$
		$\displaystyle\times$	$\displaystyle{\bar{v}}_{4^{\prime}3^{\prime}2^{\prime}1^{\prime}},$

and the amplitudes $\Gamma^{ph}$ and $\Gamma^{pp}$ are defined as

	$\displaystyle\Gamma^{ph}_{13^{\prime},1^{\prime}3}=\sum\limits_{242^{\prime}4^{\prime}}{\bar{v}}_{1234}R^{(ph)}_{24,2^{\prime}4^{\prime}}(\omega){\bar{v}}_{4^{\prime}3^{\prime}2^{\prime}1^{\prime}}=$
	$\displaystyle=\sum\limits_{\nu,\sigma=\pm 1}g^{{\nu}(\sigma)}_{13}D^{(\sigma)}_{\nu}(\omega)g^{\nu(\sigma)\ast}_{1^{\prime}3^{\prime}},$		(36)

where we introduced the phonon vertices $g^{\nu}$ and propagators $D_{\nu}(\omega)$:

	$\displaystyle g^{\nu(\sigma)}_{13}=\delta_{\sigma,+1}g^{\nu}_{13}+\delta_{\sigma,-1}g^{\nu\ast}_{31},\ \ \ \ g^{\nu}_{13}=\sum\limits_{24}{\bar{v}}_{1234}\rho^{\nu}_{42},$
			(37)
	$\displaystyle D_{\nu}^{(\sigma)}(\omega)=\frac{\sigma}{\omega-\sigma(\omega_{\nu}-i\delta)},\ \ \ \ \omega_{\nu}=E_{\nu}-E_{0},$
			(38)

and

	$\displaystyle\Gamma^{pp}_{12,1^{\prime}2^{\prime}}(\omega)=\sum\limits_{343^{\prime}4^{\prime}}{v}_{1234}G^{(pp)}_{43,3^{\prime}4^{\prime}}(\omega){v}_{4^{\prime}3^{\prime}2^{\prime}1^{\prime}}=$
	$\displaystyle=\sum\limits_{\mu,\sigma=\pm 1}\gamma^{\mu(\sigma)}_{12}\Delta^{(\sigma)}_{\mu}(\omega)\gamma^{\mu(\sigma)\ast}_{1^{\prime}2^{\prime}}$		(39)

with the pairing vertices $\gamma^{\mu(\pm)}$ and propagator $\Delta_{\mu}(\omega)$

$$\gamma^{\mu(+)}_{12}=\sum\limits_{34}v_{1234}\alpha_{34}^{\mu},\ \ \ \ \ \ \gamma_{12}^{\varkappa(-)}=\sum\limits_{34}\beta_{34}^{\varkappa}v_{3412}.$$

(40)

$$\Delta^{(\sigma)}_{\mu}(\omega)=\frac{\sigma}{\omega-\sigma(\omega_{\mu}^{(\sigma\sigma)}-i\delta)}.$$

(41)

In the definitions above, the particle-particle ($pp$) and particle-hole ($ph$) correlation functions defined by Eqs. (9,10) are employed, while the mappings of Eqs. (36,39) to the particle-vibration coupling (PVC) are displayed diagrammatically in Fig. 1.

Thus, Eq. (32) is the foundation for microscopic approaches to the single-particle self-energy, which refer to the phenomenon of PVC. The mappings (36,39) lead to the diagrammatic form of the self-energy shown in Fig. 2. The spectral representations of the respective terms read:

	$\displaystyle\Sigma^{r(ph)}_{11^{\prime}}(\omega)=\sum\limits_{33^{\prime}}\Bigl{[}\sum\limits_{\nu n}\frac{\eta_{3}^{n}{g}_{13}^{\nu}{g}_{1^{\prime}3^{\prime}}^{\nu\ast}\eta_{3^{\prime}}^{n\ast}}{\omega-\omega_{\nu}-\varepsilon_{n}^{(+)}+i\delta}+$
	$\displaystyle+\sum\limits_{\nu m}\frac{\chi_{3}^{m}g_{31}^{\nu\ast}g_{3^{\prime}1^{\prime}}^{\nu}\chi_{3^{\prime}}^{m\ast}}{\omega+\omega_{\nu}+\varepsilon_{m}^{(-)}-i\delta}\Bigr{]},$		(42)

	$\displaystyle\Sigma^{r(pp)}_{11^{\prime}}(\omega)=\sum\limits_{22^{\prime}}\Bigl{[}\sum\limits_{\mu m}\frac{\chi_{2}^{m\ast}\gamma_{12}^{\mu(+)}\gamma_{1^{\prime}2^{\prime}}^{\mu(+)\ast}\chi_{2^{\prime}}^{m}}{\omega-\omega_{\mu}^{(++)}-\varepsilon_{m}^{(-)}+i\delta}+$
	$\displaystyle+\sum\limits_{\varkappa n}\frac{\eta_{2}^{n\ast}{\gamma}_{21}^{\varkappa(-)\ast}{\gamma}_{2^{\prime}1^{\prime}}^{\varkappa(-)}\eta_{2^{\prime}}^{n}}{\omega+\omega_{\varkappa}^{(--)}+\varepsilon_{n}^{(+)}-i\delta}\Bigr{]},$		(43)

		$\displaystyle\Sigma$	${}^{r(0)}_{11^{\prime}}(\omega)=-\sum\limits_{2342^{\prime}3^{\prime}4^{\prime}}{\bar{v}}_{1234}\times$		(44)
		$\displaystyle\times$	$\displaystyle\Bigl{[}\sum\limits_{mn^{\prime}n^{\prime\prime}}\frac{\chi_{2^{\prime}}^{m}\chi_{2}^{m\ast}\eta_{3}^{n^{\prime}}\eta_{3^{\prime}}^{n^{\prime}\ast}\eta_{4}^{n^{\prime\prime}}\eta_{4^{\prime}}^{n^{\prime\prime}\ast}}{\omega-\varepsilon_{n^{\prime}}^{(+)}-\varepsilon_{n^{\prime\prime}}^{(+)}-\varepsilon_{m}^{(-)}+i\delta}$
		$\displaystyle+$	$\displaystyle\sum\limits_{nm^{\prime}m^{\prime\prime}}\frac{\eta_{2^{\prime}}^{n}\eta_{2}^{n\ast}\chi_{3}^{m^{\prime}}\chi_{3^{\prime}}^{m^{\prime}\ast}\chi_{4}^{m^{\prime\prime}}\chi_{4^{\prime}}^{m^{\prime\prime}\ast}}{\omega+\varepsilon_{n}^{(+)}+\varepsilon_{m^{\prime}}^{(-)}+\varepsilon_{m^{\prime\prime}}^{(-)}-i\delta}\Bigr{]}{\bar{v}}_{4^{\prime}3^{\prime}2^{\prime}1^{\prime}}.$

Here the single-particle energies in the neighboring $(N+1)$-particle system are defined as $\varepsilon_{n}^{(+)}=E^{(N+1)}_{n}-E^{(N)}_{0}$ and those in the $(N-1)$-particle system as $\varepsilon_{m}^{(-)}=E^{(N-1)}_{m}-E^{(N)}_{0}$.

The complete dynamical part (32-35) of the fermionic self-energy truncated at the two-body level is shown in Fig. 2 in the diagrammatic form. Note that the signs in front of the diagrams are convention dependent, for instance, they may not respect Feynman’s convention. For instance, the last uncorrelated term is often shown with the ”-” sign in the literature, and the phonon vertices may include the multiplier ”$i$”. The first two diagrams on the right-hand side in Fig. 2 are the typical one-loop diagrams, which are analogous to the electron self-energy correction in quantum electrodynamics (QED). In electronic systems, an electron typically emits and reabsorbs a virtual photon or a phonon excitation of the lattice in a metal. In the nucleonic self-energy of quantum hadrodynamics (QHD) a single nucleon emits and reabsorbs mesons of various kinds, which in the lowest order can be described by the first diagram on the right-hand side if the wavy line is attributed to the meson propagator. In the applications discussed in this work, the meson exchange acts as a bare interaction (although slightly renormalized) described by the matrix elements of $\bar{v}$. The dynamical self-energy then expresses the medium-induced emerging contribution to the fermionic interaction. Quite remarkably, in the leading approximation dominated by the term $\Sigma^{r(ph)}$ (42) in a strong coupling regime, the dynamical fermionic self-energy takes an analogous form of the boson exchange, thereby illustrating how this type of interaction is driven across the energy scales. This process can be expressed by an effective Hamiltonian with the explicit phonon degrees of freedom, as it is done in the phenomenological NFT. The difference between the nuclear system and QED or QHD, besides the differences in the reference (”vacuum”) states, is that in the latter theories, the fermionic and bosonic degrees of freedom are independent. Interestingly, the PVC vertices in nuclear systems are not the effective parameters of the theory but are in principle calculable from the underlying fermionic bare interaction.

There are very few realizations of the PVC model based on the bare NN interaction Barbieri and Hjorth-Jensen (2009); Barbieri (2009). The implementations employing effective interactions Litvinova and Ring (2006); Litvinova (2012); Litvinova and Afanasjev (2011); Afanasjev and Litvinova (2015) are more accurate quantitatively and more feasible because the phonons can be reasonably described already on the level of (quasiparticle) random phase approximation ((Q)RPA) and successive iterations of the fermionic propagator in the Dyson equation may be omitted. However, such approaches inevitably imply an additional procedure to remove the double counting of PVC, implicitly contained in the effective interaction Tselyaev (2013). An elegant way of avoiding such double counting is the explicit subtraction of the dynamical PVC kernel taken in the static limit from the effective interaction. The subtraction method is widely applied in calculations of two-nucleon Green functions, in particular, the particle-hole response Tselyaev (2013); Litvinova and Tselyaev (2007); Litvinova et al. (2007a, 2008); Gambacurta et al. (2015); Litvinova and Schuck (2019), while a corresponding method has not been yet adopted to the case of the one-body propagator.

The information about the two-fermion propagators $R^{(ph)}$ and $G^{(pp)}$ in terms of the phonon vertices and propagators can be obtained from their direct computation by solving the EOMs for these correlation functions. The theory and implementations of the corresponding EOMs are presented in Refs. Dukelsky et al. (1998); Olevano et al. (2018); Litvinova and Schuck (2019, 2020).

The spectral forms (42-44) of the three terms with the explicit locality and unitarity are best suited for the accurate diagrammatic mapping and they reveal a different sign of the ”second-order” term $\Sigma^{r(0)}$ as compared to the ”radiative correction” terms $\Sigma^{r(ph)}$ and $\Sigma^{r(pp)}$ containing phonons. This means, in particular, that potentially the positivity can be violated in the optical theorem. As it was pointed out, for instance, in Refs. Schuck et al. (1973); Danielewicz and Schuck (1994), to prevent this violation it is important to keep the integrity of the spectral representation of the dynamical self-energy (LABEL:Tr)

	$\displaystyle\Sigma_{11^{\prime}}^{r}(\omega)$	$\displaystyle=$	$\displaystyle\frac{1}{4}\sum\limits_{rpq}\sum\limits_{ikl}{\bar{v}}_{1ikl}G^{(pph)irr}_{ilk,rqp}(\omega){\bar{v}}_{qpr1^{\prime}}$
	$\displaystyle G^{(pph)}_{ilk,rqp}(\omega)$	$\displaystyle=$	$\displaystyle\sum\limits_{n}\frac{\langle 0|\psi^{\dagger}_{i}\psi_{l}\psi_{k}|n\rangle\langle n|\psi^{\dagger}_{p}\psi^{\dagger}_{q}\psi_{r}|0\rangle}{\omega-(E^{(N+1)}_{n}-E^{(N)}_{0})+i\delta}$
		$\displaystyle+$	$\displaystyle\sum\limits_{m}\frac{\langle 0|\psi^{\dagger}_{p}\psi^{\dagger}_{q}\psi_{r}|m\rangle\langle m|\psi^{\dagger}_{i}\psi_{l}\psi_{k}|0\rangle}{\omega+(E^{(N-1)}_{m}-E^{(N)}_{0})-i\delta}.$

Comparing the exact dynamical self-energy of Eq. (LABEL:pphgfspec) to Eqs. (42-44), one can see clearly that the above-mentioned sign problem and potentially other inconsistencies may appear because of the different analytical structures of the exact and approximate solutions.

One may notice that the poles of the $G^{(pph)}$ in Eq. (LABEL:pphgfspec) coincide with those of the one-fermion propagator $G$ in the form of Eq. (7), because both sums on the right-hand sides of these expressions run over the complete formally exact spectra of the same $N\pm 1$ systems. Combining these equations with Eq. (23) and considering the energy argument close to the exact pole $\varepsilon_{n}$, one obtains:

$$\langle 0|\psi_{1}|n\rangle=\frac{1}{2}\sum\limits_{ikl}{\bar{v}}_{1ikl}\frac{\langle 0|\psi^{\dagger}_{i}\psi_{l}\psi_{k}|n\rangle}{\varepsilon_{n}-\varepsilon_{1}},$$

(46)

which is consistent with the exact EOM for a single field operator Zelevinsky and Volya (2017).

Considerable effort on finding viable approximations for the irreducible part of $G^{(pph)}$ compatible with the spectral expansion (LABEL:pphgfspec) was undertaken in the past. The authors of Ref. Schuck et al. (1973) have formulated the two-particle-one-hole ($2p1h$) RPA for this correlation function, while a more accurate approximation to the $2p1h$ energies and matrix elements via Faddeev series has been developed in Ref. Danielewicz and Schuck (1994) to include emergent collectivity in the $ph$ and $pp$ channels. Nuclear structure implementations, however, are still quite limited and confined by the Tamm-Dancoff and random phase approximations to the $ph$ and $pp$ correlation functions Rijsdijk et al. (1992, 1996); Barbieri and Hjorth-Jensen (2009); Barbieri (2009).

The majority of strongly-correlated fermionic systems, including atomic nuclei, exhibit pronounced effects of superfluidity Ring and Schuck (1980), that need an extended treatment. The superfluid phase is generally characterized by the enhanced formation of Cooper pairs and pairing phonons, which appear to be a dynamical counterpart of the Cooper pairs. In calculations for normal systems within the PVC approach to the self-energy using effective interactions one usually neglects the pairing phonons because of their relatively low importance, but they should be kept for superfluid systems and also if a bare interaction is used.

Within the PVC approach discussed above the pairing interaction is fully dynamical and mediated by the pairing phonons emerging naturally in the one-fermion self-energy. In the traditional frameworks based on effective interactions, however, the pairing is included in the static approximations, such as the BCS or the Hartree-(Fock)-Bogoliubov (HFB) ones. On this level of description, the corresponding Green function technique is the Gor’kov Green functions, which extends the notion of the one-fermion propagator (6) by introducing anomalous propagators with the same kind of fermionic operators, see Eq. (67) below. These correlation functions do not vanish because correlated fermionic pairs are present in the ground state of the system.

The simplest approach to the Gor’kov Green functions can be obtained from the EOM1 if the two-body correlations are neglected, see, for instance, Gorkov (1958). In ref. Litvinova and Zhang (2021) it is shown how the Gor’kov theory is generalized beyond this approximation, in particular, to the inclusion of the PVC effects in the dynamical self-energy. It is convenient to introduce the HFB basis, or the basis of the Bogoliubov quasiparticles Bogolubov (1947). The states in this basis combine particle and hole states, i.e., the states above and below the Fermi energy:

	$\displaystyle\psi_{1}=U_{1\mu}\alpha_{\mu}+V^{\ast}_{1\mu}\alpha^{\dagger}_{\mu}$
	$\displaystyle\psi^{\dagger}_{1}=V_{1\mu}\alpha_{\mu}+U^{\ast}_{1\mu}\alpha^{\dagger}_{\mu}.$		(47)

Here and henceforth the Greek indices are used to denote fermionic states in the HFB basis, while the number indices and the Roman indices are reserved for the single-particle mean-field basis states. The repeated indices $\mu$ imply summation, so that Eq. (47) can be expressed in a matrix form:

$\displaystyle\left(\begin{array}[]{c}\psi\\ \psi^{\dagger}\end{array}\right)=\cal{W}\left(\begin{array}[]{c}\alpha\\ \alpha^{\dagger}\end{array}\right),$

(52)

where

$\displaystyle\cal{W}=\left(\begin{array}[]{cc}U&V^{\ast}\\ V&U^{\ast}\end{array}\right)\ \ \ \ \ \ \cal{W}^{\dagger}=\left(\begin{array}[]{cc}U^{\dagger}&V^{\dagger}\\ V^{T}&U^{T}\end{array}\right)$

(57)

are unitary matrices. The quasiparticle operators $\alpha$ and $\alpha^{\dagger}$ obey the same anticommutator algebra as the particle operators $\psi$ and $\psi^{\dagger}$, while the matrices $U$ and $V$ satisfy:

	$\displaystyle U^{\dagger}U+V^{\dagger}V=\mathbb{1}\ \ \ \ \ \ UU^{\dagger}+V^{\ast}V^{T}=\mathbb{1}$
	$\displaystyle U^{T}V+V^{T}U=0\ \ \ \ \ \ UV^{\dagger}+V^{\ast}U^{T}=0.$		(58)

The generalized fermionic propagator, therefore, takes the form

	$\displaystyle{\hat{G}}_{12}(t-t^{\prime})=-i\langle T\Psi_{1}(t)\Psi^{\dagger}_{2}(t^{\prime})\rangle=$
	$\displaystyle=-i\theta(t-t^{\prime})\left(\begin{array}[]{cc}\langle\psi_{1}(t)\psi^{\dagger}_{2}(t^{\prime})\rangle&\langle\psi_{1}(t)\psi_{2}(t^{\prime})\rangle\\ \langle\psi^{\dagger}_{1}(t)\psi^{\dagger}_{2}(t^{\prime})\rangle&\langle\psi^{\dagger}_{1}(t)\psi_{2}(t^{\prime})\rangle\end{array}\right)+$		(61)
	$\displaystyle+i\theta(t^{\prime}-t)\left(\begin{array}[]{cc}\langle\psi^{\dagger}_{2}(t^{\prime})\psi_{1}(t)\rangle&\langle\psi_{2}(t^{\prime})\psi_{1}(t)\rangle\\ \langle\psi^{\dagger}_{2}(t^{\prime})\psi^{\dagger}_{1}(t)\rangle&\langle\psi_{2}(t^{\prime})\psi^{\dagger}_{1}(t)\rangle\end{array}\right)$		(64)
	$\displaystyle=\left(\begin{array}[]{cc}G_{12}(t-t^{\prime})&F^{(1)}_{12}(t-t^{\prime})\\ F^{(2)}_{12}(t-t^{\prime})&G^{(h)}_{12}(t-t^{\prime})\end{array}\right).$		(67)

with

$$\Psi_{1}(t_{1})=\left(\begin{array}[]{c}\psi_{1}(t_{1})\\ \psi_{1}^{\dagger}(t_{1})\end{array}\right),\ \ \ \ \ \ \ \ \ \Psi^{\dagger}_{1}(t_{1})=\Bigl{(}\psi_{1}^{\dagger}(t_{1})\ \ \ \ \ \psi_{1}(t_{1})\Bigr{)}.$$

(68)

In Ref. Litvinova and Zhang (2021) it was shown in detail how the procedure of generating the one-fermion EOM is performed for all the components of the propagator (67). A unified EOM was obtained, in particular, by transforming the matrix equation for the ${\hat{G}}_{12}$ to the quasiparticle basis. The resulting Gor’kov-Dyson equation for the quasiparticle propagator in the energy domain reads:

$$G^{(\eta)}_{\nu\nu^{\prime}}(\varepsilon)={\tilde{G}}^{(\eta)}_{\nu\nu^{\prime}}(\varepsilon)+\sum\limits_{\mu\mu^{\prime}}{\tilde{G}}^{(\eta)}_{\nu\mu}(\varepsilon)\Sigma^{r(\eta)}_{\mu\mu^{\prime}}(\varepsilon)G^{(\eta)}_{\mu^{\prime}\nu^{\prime}}(\varepsilon).$$

(69)

The indices $\eta=+$ and $\eta=-$ are introduced for the quasiparticle forward and backward components, respectively. The mean-field ${\tilde{G}}^{(\eta)}_{\nu\nu^{\prime}}(\varepsilon)$ and exact $G^{(\eta)}_{\nu\nu^{\prime}}(\varepsilon)$ propagator components are defined as follows:

$${\tilde{G}}^{(\eta)}_{\nu\nu^{\prime}}(\varepsilon)=\frac{\delta_{\nu\nu^{\prime}}}{\varepsilon-\eta(E_{\nu}-E_{0}-i\delta)}$$

(70)

$${G}^{(\eta)}_{\nu\nu^{\prime}}(\varepsilon)=\sum\limits_{n}\frac{S^{\eta(n)}_{\nu\nu^{\prime}}}{\varepsilon-\eta(E_{n}-E_{0}-i\delta)},$$

(71)

with the mean-field energies of the Bogoliubov quasiparticles $E_{\nu}$ and formally exact quasiparticle energies $E_{n}$. The residues are the matrix element products $S^{+(n)}_{\nu\nu^{\prime}}=\langle 0|\alpha_{\nu}|n\rangle\langle n|\alpha^{\dagger}_{\nu^{\prime}}|0\rangle$ and $S^{-(m)}_{\nu\nu^{\prime}}=\langle 0|\alpha_{\nu}|m\rangle\langle m|\alpha^{\dagger}_{\nu^{\prime}}|0\rangle$ with the formally exact states $|n\rangle$ and $|m\rangle$. The components of the dynamical kernel in the quasiparticle basis are related to those in the single-particle basis as follows:

	$\displaystyle\Sigma^{r(+)}_{\mu\mu^{\prime}}(\varepsilon)$	$\displaystyle=$	$\displaystyle\sum\limits_{12}\Bigl{(}U^{\dagger}_{\mu 1}\ \ \ V^{\dagger}_{\mu 1}\Bigr{)}\left(\begin{array}[]{cc}\Sigma^{r}_{12}(\varepsilon)&\Sigma^{(1)r}_{12}(\varepsilon)\\ \Sigma^{(2)r}_{12}(\varepsilon)&\Sigma^{(h)r}_{12}(\varepsilon)\end{array}\right)\left(\begin{array}[]{c}U_{2\mu^{\prime}}\\ V_{2\mu^{\prime}}\end{array}\right)$
		$\displaystyle=$	$\displaystyle\sum\limits_{12}\Bigl{(}U^{\dagger}_{\mu 1}\Sigma^{r}_{12}(\varepsilon)U_{2\mu^{\prime}}+U^{\dagger}_{\mu 1}\Sigma^{(1)r}_{12}(\varepsilon)V_{2\mu^{\prime}}$
		$\displaystyle+$	$\displaystyle V^{\dagger}_{\mu 1}\Sigma^{(2)r}_{12}(\varepsilon)U_{2\mu^{\prime}}+V^{\dagger}_{\mu 1}\Sigma^{(h)r}_{12}(\varepsilon)V_{2\mu^{\prime}}\Bigr{)},$		(77)

	$\displaystyle\Sigma^{r(-)}_{\mu\mu^{\prime}}(\varepsilon)$	$\displaystyle=$	$\displaystyle\sum\limits_{12}\Bigl{(}V^{T}_{\mu 1}\ \ \ U^{T}_{\mu 1}\Bigr{)}\left(\begin{array}[]{cc}\Sigma^{r}_{12}(\varepsilon)&\Sigma^{(1)r}_{12}(\varepsilon)\\ \Sigma^{(2)r}_{12}(\varepsilon)&\Sigma^{(h)r}_{12}(\varepsilon)\end{array}\right)\left(\begin{array}[]{c}V^{\ast}_{2\mu^{\prime}}\\ U^{\ast}_{2\mu^{\prime}}\end{array}\right)$
		$\displaystyle=$	$\displaystyle\sum\limits_{12}\Bigl{(}V^{T}_{\mu 1}\Sigma^{r}_{12}(\varepsilon)V^{\ast}_{2\mu^{\prime}}+V^{T}_{\mu 1}\Sigma^{(1)r}_{12}(\varepsilon)U^{\ast}_{2\mu^{\prime}}$
		$\displaystyle+$	$\displaystyle U^{T}_{\mu 1}\Sigma^{(2)r}_{12}(\varepsilon)V^{\ast}_{2\mu^{\prime}}+U^{T}_{\mu 1}\Sigma^{(h)r}_{12}(\varepsilon)U^{\ast}_{2\mu^{\prime}}\Bigr{)},$		(83)

while the matrix structure of the dynamical self-energy ${\hat{\Sigma}}_{11^{\prime}}^{r}$ corresponds to the structure of the propagator matrix (67):

$${\hat{\Sigma}}_{12}^{r}(\varepsilon)=\left(\begin{array}[]{cc}\Sigma^{r}_{12}(\varepsilon)&\Sigma^{(1)r}_{12}(\varepsilon)\\ \Sigma^{(2)r}_{12}(\varepsilon)&\Sigma^{(h)r}_{12}(\varepsilon)\end{array}\right).$$

(84)