Bethe-Salpeter equations for the fully reducible vertex \(F\)
The Bethe-Salpeter equations for the fully reducible vertex \(F\) is defined for any given channel \(r\) as
\[F = \Gamma^{r} + \Gamma^{r} \chi_0^r F\]
The possible pairings of indices in the right-hand side product produces three non-equivalent equations labeled \(r \in \{ PH, \bar{PH}, PP\}\) standing for, the particle-hole (\(PH\)), vertical-particle-hole (\(\bar{PH}\)), and particle-particle (\(PP\)) channel, respectively.
Each equation and index pairing is associated with one “channel” within which the \(r\)-channel-irreducible vertex \(\Gamma^r\).
\[F(a\bar{b}c\bar{d}) = \Gamma^{PH}(a\bar{b}c\bar{d}) -
\Gamma^{PH}(a\bar{b}u\bar{v}) G(u\bar{u}) G(v\bar{v}) F(v\bar{u}c\bar{d})\]
\[F(a\bar{b}c\bar{d}) = \Gamma^{\bar{PH}}(a\bar{b}c\bar{d}) +
\Gamma^{\bar{PH}}(a\bar{u}v\bar{d}) G(u\bar{u}) G(v\bar{v}) F(u\bar{b}c\bar{v})\]
\[F(a\bar{b}c\bar{d}) = \Gamma^{PP}(a\bar{b}c\bar{d}) + \frac{1}{2}
\Gamma^{PP}(a\bar{u}c\bar{v}) G(u\bar{u}) G(v\bar{v}) F(v\bar{b}u\bar{d})\]
Collecting the two single-particle Green’s functions into a channel dependent “bare” susceptibility \(\chi^r_0\) the Bethe-Salpeter equations can be expressed as
\[F = \Gamma^r + \Gamma^r \stackrel{r}{*} \chi^r_0 \stackrel{r}{*} F\]
where the channel non-interacting vertex functions \(\chi_0^r\) are defined as
\[\chi_0^{PH}(\bar{a}b\bar{c}d) = G(b\bar{c}) G(d\bar{a})
\, , \quad
\big(
\chi_0^{PH}(\bar{v}u\bar{u}v) = G(u\bar{u}) G(v\bar{v})
\big)\]
\[\chi_0^{\bar{PH}}(\bar{a}b\bar{c}d) = G(b\bar{a}) G(d\bar{c})
\, , \quad
\big(
\chi_0^{\bar{PH}}(\bar{u}u\bar{v}v) = G(u\bar{u}) G(v\bar{v})
\big)\]
\[\chi_0^{PP}(\bar{a}b\bar{c}d) = G(d\bar{a}) G(b\bar{c})
\, , \quad
\big(
\chi_0^{PP}(\bar{u}v\bar{v}u) = G(u\bar{u}) G(v\bar{v})
\big)\]
yielding the Bethe-Salpeter equations expressed fully in two-particle quantities
(1)\[F(a\bar{b}c\bar{d}) = \Gamma^{PH}(a\bar{b}c\bar{d}) -
\Gamma^{PH}(a\bar{b}q\bar{p}) \, \chi^{PH}_0(\bar{p}q\bar{r}s) \, F(s\bar{r}c\bar{d})\]
(2)\[F(a\bar{b}c\bar{d}) = \Gamma^{\bar{PH}}(a\bar{b}c\bar{d}) +
\Gamma^{\bar{PH}}(a\bar{p}s\bar{d}) \, \chi^{\bar{PH}}_0(\bar{p}q\bar{r}s) \, F(q\bar{b}c\bar{r})\]
(3)\[F(a\bar{b}c\bar{d}) = \Gamma^{PP}(a\bar{b}c\bar{d}) + \frac{1}{2}
\Gamma^{PP}(a\bar{p}c\bar{r}) \, \chi^{PP}_0(\bar{p}q\bar{r}s) \, F(q\bar{b}s\bar{d})\]
Note that the notion of the product \(\stackrel{r}{*}\) is dependent on the channel \(r\) and consists of different choices of contracting two indices in Eqs. (1), (2), and (3).
Matsubara frequency parametrization
In a carefully choosen parametrization of Matsubara frequencies, the two-time integrals appearing in the products of the Bethe-Salpeter equations (1), (2), and (3) can be reduced to a single sum over one Matsubara frequency. This is achieved by using a channel dependent three frequency reparametrization that directly imposes total frequency conservation, the forms are
(4)\[\begin{split}\begin{array}{ll}
PH: \left\{
\begin{array}{rl}
\nu_1 &=& \nu \\
\nu_2 &=& \nu + \omega \\
\nu_3 &=& \nu' + \omega \\
\nu_4 &=& \nu'
\end{array}
\right.
\, , & \quad
\bar{PH}: \left\{
\begin{array}{rcl}
\nu_1 &=& \nu \\
\nu_2 &=& \nu'\\
\nu_3 &=& \nu' + \omega\\
\nu_4 &=& \nu + \omega
\end{array}\right.
\, , \quad
\\ \\
PP: \left\{
\begin{array}{rcl}
\nu_1 &=& \nu \\
\nu_2 &=& \nu' \\
\nu_3 &=& \omega - \nu \\
\nu_4 &=& \omega - \nu'
\end{array}\right.
\, , & \quad
PPx: \left\{
\begin{array}{rcl}
\nu_1 &=& \nu \\
\nu_2 &=& \omega - \nu' \\
\nu_3 &=& \omega - \nu \\
\nu_4 &=& \nu'
\end{array}\right.
\end{array}\end{split}\]
for the (horizontal) Particle-Hole (\(PH\)) channel, the (vertical) Particle-Hole (\(\bar{PH}\)) channel, the Particle-Particle (\(PP\)) channel, and the Crossed-Particle-Particle (\(PPx\)) channel, respectively.
In terms of imaginary time the channel dependent three frequency representation maps to the follwing pairing of the four imaginary times \(\tau_a\), \(\tau_\bar{b}\), \(\tau_c\), \(\tau_{\bar{d}}\) of a response function \(\chi_{a\bar{b}c\bar{d}}(\tau_a, \tau_{\bar{b}}, \tau_c, \tau_{\bar{d}})\)
\[PH : \,
+ i\omega (\tau_{\bar{b}} - \tau_c)
+ i\nu (-\tau_{a} + \tau_{\bar{b}})
+ i\nu' (-\tau_{c} + \tau_{\bar{d}})\]
\[\bar{PH} : \,
+ i\omega (-\tau_{c} + \tau_{\bar{d}})
+ i\nu (-\tau_{a} + \tau_{\bar{d}})
+ i\nu' (\tau_{\bar{b}} - \tau_{c})\]
\[PP : \,
+ i\omega (-\tau_{c} + \tau_{\bar{d}})
+ i\nu (-\tau_{a} + \tau_{c})
+ i\nu' (\tau_{\bar{b}} - \tau_{\bar{d}})\]
\[PPx : \,
+ i\omega (\tau_{\bar{b}} - \tau_{c})
+ i\nu (-\tau_{a} + \tau_{c})
+ i\nu' (-\tau_{\bar{b}} + \tau_{\bar{d}})\]
In a general product \(P = \Gamma \stackrel{r}{*} \chi_0\), the total frequency conservation of the components of the product \(\Gamma\) and \(\chi_0\) gives two constraints that when combined gives the total frequency conservation of the product \(P\) and a reduction of the frequency summation of the product from two frequencies to one. This is achieved by using the above global reparametrizations of the four fermionic Matsubara frequencies \(\nu_1 ,\, \nu_2 ,\, \nu_3 ,\, \nu_4\) of every response function \(Q(\nu_1\nu_2\nu_3\nu_4)\) for the particular channel \(r \in \{PH, \bar{PH}, PP\}\) in question.
In order to map the products to matrix products in index and frequency space the following index ordering has to be done
\[PH: \,
\chi^{PH}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{PH}_{\{\bar{\nu}, \bar{a}b \},\{ \nu, d\bar{c} \}}(\omega)\]
\[\bar{PH}: \,
\chi^{\bar{PH}}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{\bar{PH}}_{\{\bar{\nu}, \bar{a}d \}, \{\nu', b\bar{c} \} }(\omega)\]
\[PP: \,
\chi^{PP}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{PP}_{\{ \bar{\nu}, \bar{a}\bar{c} \}, \{ \nu', bd\}}
(\omega)\]
\[PPx: \,
\chi^{PPx}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{PPx}_{\{ \bar{\nu}, \bar{c}\bar{a} \}, \{ \nu', bd\}}
(\omega)\]
The resulting product formulas reads (see separate derivation chapter),
\[P^{PH}_{a\bar{b}\bar{c}d}(\omega, \nu,\nu') =
\frac{1}{\beta^2} \sum_{\bar{\nu} u\bar{v}}
\Gamma^{PH}_{ \{ \nu, a\bar{b} \},\{ \bar{\nu}, \bar{v}u \}}(\omega)
\,
\chi^{PH}_{0, \{\bar{\nu}, \bar{v}u \},\{ \nu, d\bar{c} \}}(\omega)\]
\[P^{\bar{PH}}_{ab\bar{c}\bar{d}}(\omega, \nu, \nu')
=
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u} v}
\Gamma^{\bar{PH}}_{\{ \nu, a\bar{d} \}, \{ \bar{\nu}, \bar{u}v \}}(\omega)
\,
\chi^{\bar{PH}}_{0, \{\bar{\nu}, \bar{u}v \}, \{\nu', b\bar{c} \} }(\omega)\]
\[P^{PP}_{abcd}(\omega, \nu, \nu')
=
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{ \{ \nu , ac \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PP}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega)\]
\[P^{PPx}_{abcd}(\omega, \nu, \nu')
=
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PPx}_{ \{ \nu , ca \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PPx}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega)\]
Derivation: Product relations
Particle-Hole channel (\(PH\))
Consider the PH product
\[\begin{split}P(a\bar{b}\bar{c}d)
& =
\Gamma^{PH}(a\bar{b}u\bar{v}) \, \chi^{PH}_0(\bar{v}u\bar{c}d)
\\
& =
\sum_{u\bar{v}}
\iint_0^\beta d\tau_{u} d\tau_{\bar{v}} \,
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\tau_{a} \tau_{\bar{b}} \tau_{u} \tau_{\bar{v}})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d}(\tau_{\bar{v}} \tau_{u} \tau_{\bar{c}} \tau_{d}).\end{split}\]
Fourier transforming \(\Gamma^{PH}\) and \(\chi^{PH}_0\) and explicitly inserting Kronecker delta functions for the total frequency conservation gives
\[\begin{split}P(a\bar{b}\bar{c}d)
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a + i \nu_{\bar{b}} \tau_{\bar{b}} + i \nu_{\bar{c}} \tau_{\bar{c}}- i \nu_{d} \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2}
\sum_{u \bar{v}}
\sum_{\nu_{u} \nu_{\bar{v}}}
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\nu_a \nu_{\bar{b}} \nu_{u} \nu_{\bar{v}})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d}(\nu_{\bar{v}} \nu_u \nu_{\bar{c}} \nu_d)
\\
& \times
\delta_{\nu_{a} - \nu_{\bar{b}} + \nu_{u} - \nu_{\bar{v}}, 0}
\delta_{\nu_{\bar{v}} - \nu_{u} + \nu_{\bar{c}} - \nu_{d}, 0}.\end{split}\]
Inserting the \(PH\) frequency pairing of (4) in this expression fulfills both Kronecker delta functions and reduce the summation by one frequency to
\[\begin{split}P(a\bar{b}\bar{c}d)
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu \tau_a + i (\nu + \omega) \tau_{\bar{b}} + i (\nu' + \omega) \tau_{\bar{c}} - i \nu' \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2} \sum_{u \bar{v}} \sum_{\bar{\nu}}
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\nu, \nu+\omega, \bar{\nu} + \omega, \bar{\nu})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d}(\bar{\nu}, \bar{\nu} + \omega, \nu' + \omega, \nu').\end{split}\]
Using the three frequency notation \(Q(\omega, \nu, \nu') \equiv Q(\nu, \nu+\omega, \nu'+\omega, \nu)\) we get the final product relation
\[\begin{split}P^{PH}_{a\bar{b}\bar{c}d}(\omega, \nu,\nu')
& =
\frac{1}{\beta^2} \sum_{\bar{\nu} u\bar{v}}
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\omega,\nu, \bar{\nu})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d }(\omega,\bar{\nu}, \nu)
\\
& =
\frac{1}{\beta^2} \sum_{\bar{\nu} u\bar{v}}
\Gamma^{PH}_{ \{ \nu, a\bar{b} \},\{ \bar{\nu}, \bar{v}u \}}(\omega)
\,
\chi^{PH}_{0, \{\bar{\nu}, \bar{v}u \},\{ \nu, d\bar{c} \}}(\omega).\end{split}\]
Note
The right hand side indices has to be permuted in order to make the product a direct matrix multiplication. I.e. the pairing reads
\[P^{PH}_{abcd}(\omega, \nu, \nu') = P^{PH}_{\{\nu, ab \}, \{\nu', dc\}}(\omega).\]
Writing the reversed product \(P = \chi^{PH}_0 * \Gamma^{PH}\) in slightly compressed notation we get
\[\mathcal{F} \big\{ P(\bar{a}bc\bar{d}) \big\} =
\frac{1}{\beta^2} \sum_{\bar{u}v} \sum_{\bar{\nu}}
\chi^{PH}_{0, \bar{a}b\bar{u}v}(\nu \nu+\omega, \bar{\nu} + \omega, \bar{\nu})
\,
\Gamma^{PH}_{v\bar{u}c\bar{d}}(\bar{\nu}, \bar{\nu} + \omega, \nu' + \omega, \nu').\]
where \(\mathcal{F}\{ \cdot \}\) denotes Fourier transformation to four fermionic Matsubara frequency space. Thus, the product with grouped indices becomes
\[\begin{split}P_{\bar{a}bc\bar{d}}(\omega, \nu, \nu')
& =
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u}v}
\chi^{PH}_{0, \bar{a}b\bar{u}v}(\omega, \nu, \bar{\nu})
\,
\Gamma^{PH}_{v\bar{u}c\bar{d}}(\omega, \bar{\nu}, \nu')
\\
& =
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u}v}
\chi^{PH}_{0, \{ \nu, \bar{a}b \}, \{\bar{\nu}, v\bar{u} \} }(\omega)
\,
\Gamma^{PH}_{\{ \bar{\nu} , v\bar{u} \}, \{ \nu', \bar{d}c\}}(\omega)\end{split}\]
which shows that the same index grouping relations hold for both products \(\chi_0^{PH} * \Gamma^{PH}\) and \(\Gamma^{PH} * \chi_0^{PH}\).
Vertical-Particle-Hole channel (\(\bar{PH}\))
The vertical-particle-hole product is defined in the channel’s Bethe-Salpeter equation as
\[\begin{split}P(ab\bar{c}\bar{d})
& =
\Gamma^{\bar{PH}}(a\bar{u}v\bar{d})
\,
\chi_0^{\bar{PH}}(\bar{u}b\bar{c}v)
\\
& =
\sum_{\bar{u}v} \iint_0^\beta d\tau_{\bar{u}} d\tau_v \,
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\tau_a, \tau_{\bar{u}}, \tau_v, \tau_{\bar{d}})
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\tau_{\bar{u}},\tau_b,\tau_{\bar{c}},\tau_v).\end{split}\]
Fourier expansion gives
\[\begin{split}P(ab\bar{c}\bar{d})
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a + i \nu_{\bar{b}} \tau_{\bar{b}} + i \nu_{\bar{c}} \tau_{\bar{c}} - i \nu_{d} \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2}
\sum_{\bar{u} v}
\sum_{\nu_{\bar{u}} \nu_{v}}
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\nu_a \nu_{\bar{u}} \nu_v \nu_{\bar{d}})
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\nu_{\bar{u}} \nu_b \nu_{\bar{c}} \nu_v)
\\
& \times
\delta_{\nu_a - \nu_{\bar{u}} + \nu_v - \nu_{\bar{d}}, 0}
\delta_{\nu_{\bar{u}} - \nu_b + \nu_{\bar{c}} - \nu_v, 0}.\end{split}\]
Inserting the \(\bar{PH}\) channel frequency parametrization of Eq. (4), gives
\[\begin{split}P(ab\bar{c}\bar{d})
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu \tau_a + i \nu' \tau_{\bar{b}} + i (\nu' + \omega) \tau_{\bar{c}} - i (\nu + \omega) \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2}
\sum_{\bar{u} v}
\sum_{\bar{\nu}}
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\nu, \bar{\nu}, \bar{\nu} + \omega, \nu + \omega)
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\bar{\nu}, \nu', \nu' + \omega, \bar{\nu} + \omega)\end{split}\]
using \(\bar{PH}\) frequency notation and grouping indices we get
\[\begin{split}P_{ab\bar{c}\bar{d}}(\omega, \nu, \nu')
& =
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u} v}
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\omega, \nu, \bar{\nu})
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\omega, \bar{\nu}, \nu')
\\
& =
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u} v}
\Gamma^{\bar{PH}}_{\{ \nu, a\bar{d} \}, \{ \bar{\nu}, \bar{u}v \}}(\omega)
\,
\chi^{\bar{PH}}_{0, \{\bar{\nu}, \bar{u}v \}, \{\nu', b\bar{c} \} }(\omega).\end{split}\]
The reversed product \(\chi^{\bar{PH}}_0 * \Gamma^{\bar{PH}}\) can be analysed in the same way and gives the same index pairing.
Particle-Particle channel (\(PP\))
\[\begin{split}P(abcd)
& =
\Gamma^{PP}(a\bar{u}c\bar{v})
\,
\chi^{PP}_0(\bar{u}b\bar{v}d)
\\
& =
\sum_{\bar{u}\bar{v}}
\iint_0^\beta d\tau_{\bar{u}} d\tau_{\bar{v}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}(\tau_a, \tau_{\bar{u}}, \tau_c, \tau_{\bar{v}})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}(\tau_{\bar{u}}, \tau_b, \tau_{\bar{v}}, \tau_d).\end{split}\]
Fourier transform
\[\begin{split}P(abcd)
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a - i \nu_{\bar{b}} \tau_{\bar{b}} - i \nu_{\bar{c}} \tau_{\bar{c}} - i \nu_{d} \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\nu_{\bar{u}} \nu_{\bar{v}}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}(\nu_a \nu_{\bar{u}} \nu_c \nu_{\bar{v}})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}(\nu_{\bar{u}} \nu_b \nu_{\bar{v}} \nu_d)
\\
& \times
\delta_{\nu_a - \nu_{\bar{u}} + \nu_c - \nu_{\bar{v}}, 0}
\delta_{\nu_{\bar{u}} - \nu_b + \nu_{\bar{v}} - \nu_d, 0}.\end{split}\]
Inserting the \(PP\) channel frequency parametrization of Eq. (4) gives
\[\begin{split}P(abcd)
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i(\nu) \tau_a - i (\nu') \tau_{\bar{b}}
- i (\omega - \nu') \tau_{\bar{c}} - i (\omega - \nu') \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}
(\nu, \bar{\nu}, \omega - \nu, \omega - \bar{\nu})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}
(\bar{\nu}, \nu', \omega - \bar{\nu}, \omega - \nu').\end{split}\]
Collecting indices
\[\begin{split}P_{abcd}(\omega, \nu, \nu')
& =
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}
(\omega, \nu, \bar{\nu})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}
(\omega, \bar{\nu}, \nu')
\\
& =
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{ \{ \nu , ac \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PP}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega).\end{split}\]
Crossed-Particle-Particle channel (\(PPx\))
\[\begin{split}P(abcd)
& =
\Gamma^{PPx}(a\bar{u}c\bar{v})
\
\chi^{PPx}_0(\bar{v}b\bar{u}d)
\\
& =
\sum_{\bar{u}\bar{v}}
\iint_0^\beta d\tau_{\bar{u}} d\tau_{\bar{v}}
\Gamma^{PPx}_{a\bar{u}c\bar{v}}(\tau_a, \tau_{\bar{u}}, \tau_c, \tau_{\bar{v}})
\,
\chi^{PPx}_{0, \bar{v}b\bar{u}d}(\tau_{\bar{v}}, \tau_b, \tau_{\bar{u}}, \tau_d).\end{split}\]
Fourier transform
\[\begin{split}P(abcd)
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a - i \nu_{\bar{b}} \tau_{\bar{b}} - i \nu_{\bar{c}} \tau_{\bar{c}} - i \nu_{d} \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\nu_{\bar{u}} \nu_{\bar{v}}}
\Gamma^{PPx}_{a\bar{u}c\bar{v}}(\nu_a \nu_{\bar{u}} \nu_c \nu_{\bar{v}})
\,
\chi^{PPx}_{0, \bar{v}b\bar{u}d}(\nu_{\bar{v}} \nu_b \nu_{\bar{u}} \nu_d)
\\
& \times
\delta_{\nu_a - \nu_{\bar{u}} + \nu_c - \nu_{\bar{v}}, 0}
\delta_{\nu_{\bar{v}} - \nu_b + \nu_{\bar{u}} - \nu_d, 0}.\end{split}\]
Inserting the \(PPx\) channel parametrization of Eq. (4) gives
\[\begin{split}\nu_a - \nu_{\bar{u}} + \nu_c - \nu_{\bar{v}}
& =
\nu - \omega + \bar{\nu} + \omega - \nu - \bar{\nu}
& = 0 \\
\nu_{\bar{v}} - \nu_b + \nu_{\bar{u}} - \nu_d
& =
\bar{\nu} - \omega + \nu' + \omega - \bar{\nu} - \nu'
& = 0,\end{split}\]
\[\begin{split}P(abcd)
& =
\frac{1}{\beta^4} \sum
\exp \Big[
-i(\nu) \tau_a - i (\omega - \nu') \tau_{\bar{b}}
- i (\omega - \nu) \tau_{\bar{c}} - i (\nu') \tau_d
\Big]
\\
& \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PPx}_{a\bar{u}c\bar{v}}
(\nu, \omega - \bar{\nu}, \omega - \nu, \bar{\nu})
\,
\chi^{PPx}_{0, \bar{u}b\bar{v}d}
(\bar{\nu}, \omega - \nu', \omega - \bar{\nu}, \nu').\end{split}\]
Collecting indices
\[\begin{split}P^{PPx}_{abcd}(\omega, \nu, \nu')
& =
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}
(\omega, \nu, \bar{\nu})
\,
\chi^{PP}_{0, \bar{v}b\bar{u}d}
(\omega, \bar{\nu}, \nu')
\\
& =
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PPx}_{ \{ \nu , ca \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PPx}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega).\end{split}\]
Note
The first index is permuted in the grouping, i.e.
\[P^{PPx}_{abcd}(\omega, \nu, \nu')
= P^{PPx}_{\{\nu, ca\}, \{ \nu', bd \}}(\omega).\]