$\displaystyle+...\Bigg{]}+c.c..$ (149)

The expression for $D(\omega_{m})$ in the main text is obtained by further
expressing $\sum_{j=k}^{\infty}Q_{2k+1}^{(2j+1)}(\epsilon^{*})^{2j+1}$ in each
line in (149) as

$\sum_{j=k}^{\infty}Q_{2k+1}^{(2j+1)}(\epsilon^{*})^{2j+1}=(\epsilon^{*})^{2k+1}Q_{2k+1}e^{i(\phi_{2k+1}+(2k+1)\psi)}$
(150)

where $Q_{2k+1}$, $\phi_{2k+1}$, and $\psi$ are given by series in
$\epsilon^{*}$. These series are presented in Eqs. (48)-(51) in the main text.

##  Appendix E Condensation energy near $\gamma=2$

The condensation energy for a stationary $\Delta_{n}(\omega_{m})$ is given by
Eq. (25). We reproduce this formula here for convenience of a reader. At
$T=0$, which we consider here, we have

$\displaystyle E_{c,n}$ $\displaystyle=$ $\displaystyle N_{F}\int
d\omega_{m}\left(|\omega_{m}|-\frac{\omega^{2}_{m}}{\sqrt{\omega^{2}_{m}+\Delta^{2}_{n}(\omega_{m})}}\right)-\frac{1}{4}N_{F}\int\int
d\omega_{m}d\omega^{\prime}_{m}V(\omega_{m}-\omega^{\prime}_{m})$ (151)
$\displaystyle\times\left(\frac{\omega_{m}\omega^{\prime}_{m}+\Delta_{n}(\omega_{m})\Delta_{n}(\omega^{\prime}_{m})}{\sqrt{\omega^{2}_{m}+\Delta^{2}_{n}(\omega_{m})}\sqrt{(\omega^{\prime}_{m})^{2}+\Delta^{2}_{n}(\omega^{\prime}_{m})}}-{\text{sign}}(\omega_{m}\omega^{\prime}_{m})\right).$

The first term in (151) is infra-red convergent and is determined by fermions
with $\omega_{m}\sim{\bar{g}}$. This term is not singular at $\gamma=2$ and
below we just skip it. We argue in this Appendix that the second term in (151)
becomes singular in the limit $\gamma\to 2$ To see this we first consider
$E_{c,0}$, which is the largest by magnitude.

###  E.1 Condensation energy for $\Delta_{0}(\omega_{m})$

The gap function $\Delta_{0}(\omega_{m})$ is sign-preserving and tends to
$\Delta_{0}(0)\sim{\bar{g}}$ at $\omega_{m}\ll{\bar{g}}$. Using this, we
express the condensation energy as

$E_{c,0}\approx-\frac{1}{4}\bar{g}^{\gamma}N_{F}\left[\int_{-O(\bar{g})}^{O(\bar{g})}\frac{d\omega_{m}d\omega_{m}^{\prime}}{\rvert\omega_{m}-\omega_{m}^{\prime}\rvert^{\gamma}}\left(1-{\text{sign}}(\omega_{m}){\text{sign}}(\omega_{m}^{\prime})\right)+\text{regular
terms}\right]$ (152)

This integral can be easily evaluated and for $\gamma\lesssim 2$ yields

$E_{c,0}\simeq-\bar{g}^{2}N_{F}[\frac{1}{2-\gamma}+\text{regular terms}].$
(153)

At a finite $\omega_{D}$, the demominator in (152) is replaced by
$((\omega_{m}-\omega^{\prime}_{m})^{2}+\omega^{2}_{D})^{\gamma/2}$. The
integral then remains finite even at $\gamma=2$, and $E_{c,0}$ becomes

$E_{c,0}\simeq-\bar{g}^{2}N_{F}[\ln{{\bar{g}}\over\omega_{D}}+\text{regular
terms}].$ (154)

The same happens when $\omega_{D}=0$, but temperture $T$ is finite. At
$\gamma=2$ we have

$E_{c,0}\simeq-\bar{g}^{2}N_{F}[\ln{{\bar{g}}\over T}+\text{regular terms}],$
(155)

###  E.2 Condensation energy for $\Delta_{n}(\omega_{m})$ with $n>0$

For any finite $n$, the gap function $\Delta_{n}(\omega_{m})$ still saturates
at a finite value $\Delta_{n}(0)$ at $\omega_{m}=0$. At low enough
frequencies, $\Delta_{n}(0)\gg\omega_{m}$, and the integrand of the
condensation energy $E_{c,n}$ has the same as for $n=0$. Then $E_{c,n}$
contains the same divergent piece as in (153).

We will also need the expression for the difference $E_{c,n}-E_{c,0}$. We show
below that it primarily comes from the regions near the nodal points in
$\Delta_{n}(\omega_{m})$, where
$\rvert\Delta_{n}(\omega_{m})\rvert\leq\rvert\omega_{m}\rvert$. To demonstrate
this, we divide the frequency range into two subranges – $\Omega_{N}$ around
the nodal points and $\Omega_{A}$ away from nodal points. We use the
convention

$\Omega_{N}:\;\cup_{p}\\{\omega^{(p)}-\delta\omega^{(p)}<\rvert\omega_{m}\rvert<\omega^{(p)}+\delta\omega^{(p)}\\},$
(156)

where $p=1,...,n$ and
$\rvert\Delta_{n}(\omega^{(p)}\pm\delta\omega^{(p)})\rvert=|\omega^{(p)}\pm\delta^{(p)}|$.
Then $\rvert\Delta_{n}(\omega_{m})\rvert<\rvert\omega_{m}\rvert$ for
$\omega_{m}\in\Omega_{N}$ and
$\rvert\Delta_{n}(\omega_{m})\rvert>\rvert\omega_{m}\rvert$ for
$\omega_{m}\in\Omega_{A}$. Because the positions of nodal points of
$\Delta_{n}(\omega_{m})$ depend on the logarithm of frequency, it is
convenient to introduce the logarithmic variable
$x=\log(\rvert\omega_{m}\rvert/{\bar{g}})^{\gamma}$. The neighborhood of the
nodal point $\omega_{p}$ transforms to $x_{p}-\delta_{p}<x<x_{p}+\delta_{p}$,
where $x_{p}\simeq-\pi p/\beta_{\epsilon}$. We argued in the main text that
for finite $n$ and $\gamma\to 2$, $\epsilon_{n}\approx 1/(2-\gamma)^{a}$
($a>0$), $\beta_{\epsilon}\sim\epsilon^{-1/3}\propto(2-\gamma)^{(a/3)}$, and
$\Delta_{n}(\omega^{(p)}\pm\delta\omega_{m})/|\omega_{p}|\simeq(-1)^{p}(x-x_{p})$.
The condition $\rvert\Delta_{n}(\omega_{m})\rvert\sim\rvert\omega_{m}\rvert$
corresponds to $|x-x_{p}|\sim 1$.

Below we consider contributions to $\delta E_{c,n}$ from different
combinations of $\omega_{m},\omega_{m}^{\prime}$, each within either
$\Omega_{N}$ or $\Omega_{A}$.

(1) $\omega_{m},\omega_{m}^{\prime}\in\Omega_{N}$

In this frequency domain,
$\rvert\Delta(\omega_{m})\rvert<\rvert\omega_{m}\rvert\ $ and
$\rvert\Delta(\omega_{m}^{\prime})\rvert<\rvert\omega_{m}^{\prime}\rvert$. The
the corresponding contribution to $\delta E_{c,n}$ is

$\delta
E_{c,n}^{(1)}\simeq-\frac{1}{4}\bar{g}^{\gamma}N_{F}\iint_{\omega_{m},\omega_{m}^{\prime}\in\Omega_{N}}\frac{d\omega_{m}d\omega_{m}^{\prime}}{\rvert\omega_{m}-\omega_{m}^{\prime}\rvert^{\gamma}}\left(\text{sign}(\omega_{m})\text{sign}(\omega_{m}^{\prime})-1\right),$
(157)

where the superscript $(1)$ refers to the integration domain. Converting the
integral in (157) to positive $\omega_{m}$ and negative $\omega_{m}^{\prime}$
or vice versa and changing the frequencies to $x=x_{p}+y$ and
$x^{\prime}=x_{p^{\prime}}+y^{\prime}$, we obtain to leading order in
$2-\gamma$

$\delta
E_{c,n}^{(1)}\simeq\frac{1}{4}\bar{g}^{\gamma}N_{F}\sum_{p,p^{\prime}=1}^{n}\iint_{0}^{O(1)}dydy^{\prime}\left[2\cosh\frac{x_{p}-x_{p^{\prime}}+y-y^{\prime}}{4}\right]^{-2}e^{\frac{2-\gamma}{4}(x_{p}+x_{p^{\prime}})}.$
(158)

The integrand decays exponentially as a function of $\rvert
p-p^{\prime}\rvert$, hence the dominant contribution to the integral comes
from $p=p^{\prime}$. Keeping only this term, we obtain

$\displaystyle\delta E_{c,n}^{(1)}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4}\bar{g}^{\gamma}N_{F}\iint_{0}^{O(1)}dydy^{\prime}\left[2\cosh\frac{y-y^{\prime}}{4}\right]^{-2}\sum_{p=1}^{n}e^{\frac{2-\gamma}{2}x_{p}}$
(159) $\displaystyle=$ $\displaystyle{{\bar{g}}^{2}N_{F}}{c_{1}\over
2-\gamma}J_{n},$

where $J_{n}={2\beta_{\epsilon_{n}}\over\pi}\left(1-e^{-{\pi n(2-\gamma)\over
2\beta_{\epsilon_{n}}}}\right)$ and $c_{1}=O(1)$. At $\gamma\to 2$ and finite
$n$, the exponential factor in $J_{n}$ can be expanded in $n$, yielding
$\delta E_{c,n}^{(1)}={\bar{g}}^{2}N_{F}c_{1}n$.

(2) $\omega_{m},\omega_{m}^{\prime}\in\Omega_{A}$

In this frequency range,
$\rvert\Delta(\omega_{m})\rvert>\rvert\omega_{m}\rvert$ and
$\rvert\Delta(\omega_{m}^{\prime})\rvert>\rvert\omega_{m}^{\prime}\rvert$. The
contribution to the condensation energy is

$\delta
E_{c,n}^{(2)}=-\frac{1}{4}\bar{g}^{\gamma}N_{F}\iint_{\omega_{m},\omega_{m}^{\prime}\in\Omega_{A}}\frac{d\omega_{m}d\omega_{m}^{\prime}}{\rvert\omega_{m}-\omega_{m}^{\prime}\rvert^{\gamma}}\left(\text{sign}(\Delta_{n}(\omega_{m}))\text{sign}(\Delta_{n}(\omega_{m}^{\prime}))-1\right).$
(160)

We further divide $\Omega_{A}$ into sub-intervals $\Omega_{A}^{p}$ between two
adjacent nodal points $\omega_{p}$ and $\omega_{p+1}$, where $p=1,2,...,n$ and
$\omega_{n+1}\equiv 0$. The sign structure of $\Delta_{n}(\omega_{m})$ i such
that $\omega_{m}$ and $\omega_{m}^{\prime}$ must come from intervals
$\Omega_{A}^{p}$ and $\Omega_{A}^{p^{\prime}}$ with odd $p+p^{\prime}$.
Introducing again logarithmic variables $x=x_{p}-y$ and
$x^{\prime}=x_{p^{\prime}+1}+y^{\prime}$, we obtain after simple algebra

$\displaystyle\delta E_{c,n}^{(2)}$ $\displaystyle\simeq$
$\displaystyle\frac{\bar{g}^{2}N_{F}}{4}\sum_{p+p^{\prime}\in\text{odd}}\int_{O(1)}^{O(\pi/\beta_{\epsilon})}dy\int_{O(1)}^{O(\pi/\beta_{\epsilon})}dy^{\prime}$
(161) $\displaystyle\left[\frac{1}{\rvert
2\sinh\frac{x_{p}-x_{p^{\prime}+1}-y-y^{\prime}}{2}\rvert^{2}}+\frac{1}{(2\cosh\frac{x_{p}-x_{p^{\prime}+1}-y+y^{\prime}}{2})^{2}}\right]e^{(x_{p}+x_{p^{\prime}+1})\left(\frac{2-\gamma}{4}\right)}$

Given that $x_{p}-x_{p^{\prime}+1}\propto(p^{\prime}-p+1)/\beta_{\epsilon}$
and $\beta_{\epsilon}\ll 1$, this integral is exponentially small unless
$p^{\prime}+1=p$. Keeping only such term, we obtain

$\displaystyle\delta E_{c,n}^{(2)}$
$\displaystyle\simeq\frac{\bar{g}^{\gamma}N_{F}}{4}\int_{O(1)}^{O(\pi/\beta_{\epsilon})}dy\int_{O(1)}^{O(\pi/\beta_{\epsilon})}dy^{\prime}\left[\frac{1}{\rvert
2\sinh\frac{y+y^{\prime}}{2}\rvert^{2}}+\frac{1}{(2\cosh\frac{y+y^{\prime}}{2})^{2}}\right]\sum_{p=1}^{n}e^{\frac{2-\gamma}{2}x_{p}}$
(162) $\displaystyle={\bar{g}}^{2}N_{F}{c_{2}\over 2-\gamma}J_{n},$

where $c_{2}=O(1)$. The integral over $y,y^{\prime}$ is confined to
$y,y^{\prime}\sim 1$, hence it is not surprising that $\delta E_{c,n}^{(2)}$
is of the sam order as $\delta E_{c,n}^{(1)}$.

(3) $\omega_{m}\in\Omega_{N},\omega_{m}^{\prime}\in\Omega_{A}$

On general grounds we expect that the contribution from (3) is the same as
from (1) and (2), but it is instructive to explicitly verify this. We compute
the contribution from $\rvert\Delta(\omega_{m})\rvert<\rvert\omega_{m}\rvert$
and $\rvert\Delta(\omega_{m}^{\prime})\rvert>\rvert\omega_{m}^{\prime}\rvert$
and multiply the result by 2. The contribution to the condensation energy is

$\delta
E_{c,n}^{(3)}\simeq\frac{1}{2}\bar{g}^{\gamma}N_{F}\iint_{\omega_{m}\in\Omega_{N},\omega_{m}^{\prime}\in\Omega_{A}}\frac{d\omega_{m}d\omega_{m}^{\prime}}{\rvert\omega_{m}-\omega_{m}^{\prime}\rvert^{\gamma}}.$
(163)

In terms of logarithmic variables, it becomes

$\displaystyle\delta E_{c,n}^{(3)}$
$\displaystyle\simeq\frac{1}{4}\bar{g}^{\gamma}N_{F}\sum_{p=1}^{n}\int_{x_{p}-\delta_{p}}^{x_{p}+\delta_{p}}dx\sum_{p^{\prime}=1}^{n}\int_{x_{p^{\prime}+1}+\delta_{p^{\prime}+1}}^{x_{p^{\prime}}-\delta_{p^{\prime}}}dx^{\prime}\left(\frac{1}{\rvert
2\sinh\frac{x-x^{\prime}}{2}\rvert^{2}}+\frac{1}{(2\cosh\frac{x-x^{\prime}}{2})^{2}}\right)$
$\displaystyle\times e^{(x+x^{\prime})\left(\frac{2-\gamma}{4}\right)}.$ (164)

The dominant contribution again comes from two neighboring patches. Keeping
only this contribution, we obtain

$\displaystyle\delta E_{c,n}^{(3)}$
$\displaystyle\simeq\frac{1}{4}\bar{g}^{\gamma}N_{F}\int_{-O(1)}^{O(1)}dy\int_{O(1)}^{O(\pi/\beta_{\epsilon})}dy^{\prime}\left(\frac{1}{\rvert
2\sinh\frac{y-y^{\prime}}{2}\rvert^{2}}+\frac{1}{(2\cosh\frac{y-y^{\prime}}{2})^{2}}\right)$
$\displaystyle\times\left(\sum_{p=2}^{n+1}+\sum_{p=1}^{n}\right)e^{\frac{2-\gamma}{2}x_{p}}={{\bar{g}}^{2}N_{F}}{c_{3}\over
2-\gamma}J_{n},$ (165)

The integral over $y,y^{\prime}$ is free from divergences as $y$ and $y$
belong to different ranges. It is again confined to $y^{\prime}\geq 1$, $y\leq
1$.

In total, $\delta E_{c,n}={{\bar{g}}^{2}N_{F}}{c\over 2-\gamma}J_{n}$, where
$c=c_{1}+c_{2}+c_{3}$. This is what we used in the main text.

###  E.3 Condensation energy for infinitesimally small gap function

The reasoning in the main text for the dispersion of $E_{c}(\epsilon)$ at
$\gamma=2-0$ implies that the overall factor $1/(2-\gamma)$ is present in
$E_{c}(\epsilon)$ for all non-zero $\epsilon$, including the smallest
$\epsilon\to 0$. At vanishing $\epsilon$,
$\Delta(\omega_{m})=\epsilon\Delta_{\infty}(\omega_{m})$ is the solution of
the linearized gap equation. This gap function does not saturate at
$\omega_{m}=0$ and instead keep oscillating down to the smallest frequencies.
In this sutuation, the argument about the overall factor $1/(2-\gamma)$, which
we presented in the previous two subsections, does not hold, and it is
a’priori unclear whether $E_{c}(\epsilon)$ at $\epsilon\to 0$ does contain
$1/(2-\gamma)$. We show that it does.

It is convenient to use the condensation energy in the form given by Eq. (26).
The first term in (26) is not singular for any $\gamma$, so we focus on the
second term. For small $D(\omega_{m})$ we have

$\displaystyle E_{c}\simeq-N_{F}\frac{{\bar{g}}^{\gamma}}{16}\iint
d\omega_{m}d\omega_{m}^{\prime}\left({1\over\rvert\omega_{m}-\omega_{m}^{\prime}\rvert^{\gamma}}-{1\over\rvert\omega_{m}+\omega_{m}^{\prime}\rvert^{\gamma}}\right)\left(D^{2}(\omega_{m})-D^{2}(\omega_{m}^{\prime})\right)^{2},$
(166)

where $D(\omega_{m})$

$|D(\omega_{m})|=2\epsilon^{*}\cos{\left[\left(\beta(\gamma)\log{({\bar{g}}/|\omega_{m}|)^{\gamma}}+\phi(\gamma)\right)\right]},$
(167)

and $\epsilon^{*}=\epsilon({\bar{g}}/|\omega_{m}|)^{1-\gamma/2}$ (see Eq.
(47). For simplicity, we absorbed the numerical factor $Q_{1}$ into
$\epsilon$). For $\gamma\leq 2$, $\epsilon^{*}$ increases for decreasing
$\omega_{m}$, and becomes of order one at
$\omega_{m}=\omega_{*}\sim{\bar{g}}\epsilon^{2/(2-\gamma)}$. At smaller
$\omega_{m}$, $|D(\omega_{m})|$ becomes larger than one. Eq. (166) is valid at
$\omega_{m}$ and $\omega_{m}^{\prime}$ larger than $\omega_{*}$, which then
sets the lower limit of integration in (166). At infinitesimally small
$\epsilon$, $\omega_{*}$ is also infinitesimally small and further decreases
at $\gamma\to 2$. Yet, for any finite $\epsilon$ and finite $2-\gamma$,
$\omega_{*}$ remains finite.

We now evaluate the 2D integral in (166). It is convenient to introduce
logarithmic variables $x=\log({\bar{g}}/\omega_{m})^{\gamma}$ and
$x^{\prime}=\log({\bar{g}}/\omega_{m}^{\prime})^{\gamma}$. The integration
over $x$ and over $x^{\prime}$ is between $O(1)$ and
$x_{*}=\gamma\log({\bar{g}}/\omega_{*})=2\gamma/(2-\gamma)\log(C/\epsilon)$,
where $C=O(1)$. We further introduce $a=x+x^{\prime}$ and $b=x-x^{\prime}$.
The condensation energy is re-expressed in terms of $a$ and $b$ as

$\displaystyle E_{c}=-N_{F}{\bar{g}}^{2}{\epsilon^{4}\over
16\gamma^{2}2^{\gamma}}\int_{O(1)}^{2x_{*}}dae^{{2-\gamma\over
2\gamma}a}\int_{0}^{2x_{*}-a}db\left({1\over(\sinh{b\over
2\gamma})^{\gamma}}-{1\over(\cosh{b\over 2\gamma})}\right)\times$
$\displaystyle\left[\sin(\beta(\gamma)a+2\phi(\gamma))\sin{\beta(\gamma)b}\cosh({\gamma-2\over
2\gamma}b)+(1+\cos(\beta(\gamma)a+2\phi(\gamma))\cos(\beta(\gamma)b))\sinh({\gamma-2\over
2\gamma}b)\right]^{2}.$ (168)

We see that the integral over $a$ is confined to the upper limit $a\sim
x_{*}$, but the one over $b$ is confined to $b=O(1)$. We can then safely
extend the upper limit of the integration over $b$ to infinity. Taking the
limit $\gamma\to 2$ in the last factor in (168) and averaging over rapidly
oscillating factor $\sin^{2}(\beta(\gamma)a+2\phi(\gamma))$, we obtain

$\displaystyle E_{c}-N_{F}{\bar{g}}^{2}{\epsilon^{4}\over
512}I\int_{O(1)}^{2x_{*}}dae^{{2-\gamma\over 4}a}$ (169)

where

$\displaystyle I=\int_{0}^{\infty}db\left({1\over\sinh^{2}{b\over
4}}-{1\over\cosh^{2}{b\over 4}}\right)\sin^{2}(\beta(2)b)\simeq 5.75683.$
(170)

Evaluating the remaining integral over $a$, we obtain

$\displaystyle E_{c}=-N_{F}{\bar{g}}^{2}{\epsilon^{4}\over
512}I\frac{4}{2-\gamma}e^{(2-\gamma)x_{*}/2}$ (171)

Substituting the expression for $x_{*}$, we obtain

$\displaystyle E_{c}=-N_{F}{\bar{g}}^{2}{C^{2}I\over
128}\frac{\epsilon^{2}}{2-\gamma}$ (172)

The computation of the prefactor $C$ requires more sophisticated analysis.

We see that $E_{c}$ scales as $1/(2-\gamma)$, even for the smallest
$\epsilon$. We cited this result in the main text.

The $1/(2-\gamma)$ divergence at small $\epsilon$ can be regularized by a
finite bosonic mass $\omega_{D}$, like we found for larger $\epsilon$. We
found that at a finite $\omega_{D}$ the factor $1/(2-\gamma)$ is replaced by
$\epsilon^{2}\log{{\bar{g}}/\omega_{D}}$. As a result, the condensation energy
scales as $E_{c}\propto\epsilon^{4}\log{{\bar{g}}/\omega_{D}}$.

##  Appendix F The solutions of the gap equation, found in Ref. [55] and their
causality

In the Ref. [55] the authors found the new class of solutions of Eliashberg
equations for $\gamma=2$ both in the normal and superconducting state. In the
spin-chain language, introduced in [55], these solutions correspond to “spin-
flip” configurations, in which spins on some lattice sites are antiparallel to
the local field, generated by other spins. In terms of the Green’s function
they correspond to sign-changing $G(k_{F},\omega_{m})$ along $\omega_{m}>0$.

In this Appendix we show that such solutions cannot be analytically continued
to the upper half plane (UHP) of complex frequency without singularities,
i.e., they violate causality principle. We also show that these solutions do
not exist at $T=0$, in the limit of vanishing bosonic mass.

The analysis of the analytic properties can be done most easily the normal
state. We focus on $k=k_{F}$ and define $G(\omega_{m})=G(k_{F},\omega_{m})$.
We recall that the solution on the Matsubara axis is casual if there exists a
function $G(z)$ of complex $z=\omega^{\prime}+i\omega^{{}^{\prime\prime}}$,
which is analytic and has no singularities in the UHP and for which
$G(z=i\omega_{m})=G(\omega_{m})$. On the real axis, the function $G(\omega)$
is a conventional retarded Green’s function; at large $|z|$ in the UHP,
$G(z\rightarrow\infty)\approx 1/z$.

Like in [55], we assume particle-hole symmetry, in which case
$G(z=i\omega^{{}^{\prime\prime}})$ must be real. Then if $G(\omega_{m})$
changes sign once between Matsubara points $\omega_{m}$ and $\omega_{m+1}$,
then the real $G(z=i\omega^{{}^{\prime\prime}})$ must either have a pole or
have a zero in the interval $z\in(i\omega_{m},i\omega_{m+1})$. The former
possibility immediately indicates that $G(z)$ has a pole in the UHP and thus
violate the causality principle. Now we discuss the latter possibility and
demonstrate that $G(z)$ still must have a pole somewhere in the UHP.

For the proof, consider the following integral

$\frac{1}{2\pi}\oint_{\Gamma}\partial_{z}\operatorname{\text{Im}}\log G(z)dz,$
(173)

where the closed contour $\Gamma$ is shown in Fig. 13. The contour $\Gamma$
consists of two parts: the large arc of radius $R\to\infty$ and the line along
the real axis. The integral along the large arc can be evaluated using the
asymptotic large $z$ form $G(z)\approx 1/z$. Substituting this form into (173)
and using $z=Re^{i\phi}$, we find
$\oint_{\text{arc}}\partial_{z}\operatorname{\text{Im}}\log G(z)dz=-\pi$.

To evaluate the integral along the interval $(-R,R)$ on the real axis, we note
that i) the Eliashberg self energy depends only on $\omega$, and ii) the
density of states, $\sim-\int dk\operatorname{\text{Im}}G(\omega,k)$, must be
positive at all real $\omega$. One can then make sure that
$\operatorname{\text{Im}}G$ is negative and does not change sign on the real
axis, while $\operatorname{\text{Re}}G$ is negative at
$\omega\rightarrow-\infty$ and positive at $\omega\rightarrow+\infty$. It then
immediately follows that
$\oint_{-R}^{R}\partial_{\omega}\operatorname{\text{Im}}\log
G(\omega)d\omega=\pi$. Hence

$\frac{1}{2\pi}\oint_{\Gamma}\partial_{z}\operatorname{\text{Im}}\log
G(z)dz=0$ (174)

On the other hand, the function $\partial_{z}\log G(z)$ is an analytic
function in the UHP with a simple pole of residue $+1$ at the point where
$G(z)=0$. Using the residue theorem we then immediately find that

$\frac{1}{2\pi}\oint_{\Gamma}\partial_{z}\operatorname{\text{Im}}\log
G(z)dz=1$ (175)

We see that (175) is incomparable with (174). This can only be avoided if
$G(z)$ has a pole in the UHP as at this point $\partial_{z}\log G(z)$ has a
pole with residue $-1$, and the r.h.s of (175) is $1-1=0$. In general, the
r.h.s of (175) is $n-p$, where $n$ is the number of zeros and $p$ is the
number of poles, counted with their multiplicity. Then if $G(\omega_{m})$
changes sign on the Matsubara axis more than once, there must be multiple
poles of $G(z)$ in the UHP. The requirement that $G(z)$ must have a poles in
the UHP contradicts the causality principle.

Figure 13:  The contour $\Gamma$ used in the definition (173). The radius of
the arc is very large. The blue dots represent the Matsubara points. The red
cross represents a zero of the function $G(z)$.

###  F.1 T=0

Now we return to Eliashberg equations and show that $Z_{\omega}$ cannot change
sign on the positive Matsubara axis at $T=0$ in the limit of vanishing bosonic
mass. At $T=0$ the system is in the superconducting state, and we analyze
Eliashberg equations for a non-zero $\Delta(\omega_{m})$.

First, we notice that at $T=0$, $Z_{\omega}$ is a real function of
$\omega_{m}$. It also must be a continuous function, as the Green’s function
in the superconducting state must be analytic in the UHP.

As the authors of [55] noticed, the r.h.s. of the Eliashberg equation (4)
$\text{sign}(Z_{\omega_{m^{\prime}}})$ in the numerator under the sum. In the
main text we set this term to 1 as we were only interested in the sign-
preserving solutions for $Z(\omega_{m})$. Now we keep this term. At $T=0$ the
equation for $Z(\omega_{m})$ becomes

$Z_{\omega_{m}}=1+\frac{g^{\gamma}}{2\omega}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}_{m}}{\sqrt{{\omega^{\prime}_{m}}^{2}+\Delta_{\omega^{\prime}_{m}}^{2}}}\text{sign}(Z_{\omega^{\prime}_{m}})\frac{1}{((\omega_{m}-\omega^{\prime}_{m})^{2}+\omega^{2}_{D})^{\gamma/2}}$
(176)

We will be interested in the limit $\omega_{D}\rightarrow 0$.

Both functions $Z_{\omega_{m}}$ and $\Delta_{\omega_{m}}$ are real and
$Z_{\omega_{m}\rightarrow\infty}\rightarrow 1$,
$\Delta_{\omega_{m}\rightarrow\infty}\rightarrow 0$.

We want to check, if it is possible that

  * •

both $Z_{\omega_{m}}$ and $\Delta_{\omega_{m}}$ are smooth functions.

  * •

$Z_{\omega_{m}}$ changes sign $N$ times at frequencies $\omega_{n}$, where
$n=0\dots N-1$.

We take the derivative of $Z_{\omega_{m}}$ over $\omega_{m}$ and evaluate the
integral by parts

$\displaystyle\partial_{\omega_{m}}Z_{\omega_{m}}=-\frac{1}{\omega_{m}}(Z_{\omega_{m}}-1)-\frac{g^{\gamma}}{2\omega_{m}}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}_{m}}{\sqrt{{\omega^{\prime}_{m}}^{2}+\Delta_{\omega^{\prime}_{m}}^{2}}}\text{sign}(Z_{\omega^{\prime}_{m}})\partial_{\omega^{\prime}_{m}}\frac{1}{((\omega_{m}-\omega^{\prime}_{m})^{2}+\omega^{2}_{D})^{\gamma/2}}=$
$\displaystyle-\frac{1}{\omega_{m}}(Z_{\omega_{m}}-1)-\frac{g^{\gamma}}{2\omega_{m}}\left[\left.\frac{\text{sign}(Z_{\omega^{\prime}_{m}})}{\sqrt{{\omega^{\prime}_{m}}^{2}+\Delta_{\omega^{\prime}_{m}}^{2}}}\frac{1}{((\omega_{m}-\omega^{\prime}_{m})^{2}+\omega^{2}_{D})^{\gamma/2}}\right|_{-\infty}^{\infty}-\int_{-\infty}^{\infty}d\omega^{\prime}_{m}\frac{1}{((\omega_{m}-\omega^{\prime}_{m})^{2}+\omega^{2}_{D})^{\gamma/2}}\partial_{\omega^{\prime}_{m}}\frac{\text{sign}(Z_{\omega^{\prime}_{m}})}{\sqrt{{\omega^{\prime}_{m}}^{2}+\Delta_{\omega^{\prime}_{m}}^{2}}}\right]=$
$\displaystyle-\frac{1}{\omega}(Z_{\omega}-1)+\frac{g^{\gamma}}{2\omega}\left[2(-1)^{N+1}\sum_{n=0}^{N-1}\frac{(-1)^{n}}{((\omega-\omega_{n})^{2}+\omega^{2}_{D})^{\gamma/2}}\frac{1}{\sqrt{{\omega_{n}}^{2}+\Delta_{\omega_{n}}^{2}}}+\int_{-\infty}^{\infty}d\omega^{\prime}_{m}\frac{\text{sign}(Z_{\omega^{\prime}_{m}})}{((\omega-\omega^{\prime}_{m})^{2}+\omega^{2}_{D})^{\gamma/2}}\partial_{\omega^{\prime}_{m}}\frac{1}{\sqrt{{\omega^{\prime}_{m}}^{2}+\Delta_{\omega^{\prime}_{m}}^{2}}}\right],$
(177)

where we used
$\partial_{\omega^{\prime}_{m}}\text{sign}(Z_{\omega^{\prime}_{m}})=2(-1)^{N+1}\sum_{n=0}^{N-1}(-1)^{n}\delta(\omega_{m}-\omega_{n})$.
By assumption, $Z_{\omega_{m}}$ is a smooth function, so $Z_{\omega_{n}}=0$,
for all $n$. Setting $\omega_{m}=\omega_{k}$ in (177), we obtain

$\displaystyle\left.\partial_{\omega_{m}}Z_{\omega_{m}}\right|_{\omega_{m}=\omega_{k}}=\frac{1}{\omega_{k}}+\frac{g^{\gamma}}{2\omega_{k}}\int_{-\infty}^{\infty}d\omega^{\prime}_{m}\frac{\text{sign}(Z_{\omega^{\prime}_{m}})}{((\omega_{k}-\omega^{\prime}_{m})^{2}+\omega^{2}_{D})^{\gamma/2}}\partial_{\omega^{\prime}_{m}}\frac{1}{\sqrt{{\omega^{\prime}_{m}}^{2}+\Delta_{\omega^{\prime}_{m}}^{2}}}$
$\displaystyle+\frac{g^{\gamma}}{\omega_{k}}(-1)^{N+1}\sum_{n=0}^{N-1}\frac{(-1)^{n}}{((\omega_{k}-\omega_{n})^{2}+\omega^{2}_{D})^{\gamma/2}}\frac{1}{\sqrt{{\omega_{n}}^{2}+\Delta_{\omega_{n}}^{2}}}.$
(178)

We see that in the limit $\omega_{D}\rightarrow 0$, the most singular term is
the term in the sum with $n=k$. Keeping only this term we get in this limit

$\lim\limits_{m\rightarrow
0}\left.\partial_{\omega_{m}}Z_{\omega_{m}}\right|_{\omega_{m}=\omega_{k}}\sim\frac{g^{\gamma}}{\omega_{k}}\frac{(-1)^{k+N+1}}{\omega^{\gamma/2}_{D}}\frac{1}{\sqrt{{\omega_{k}}^{2}+\Delta_{\omega_{k}}^{2}}}\rightarrow\pm\infty.$
(179)

This shows that in the limit $\omega_{D}\rightarrow 0$, the sign-changing
function $Z_{\omega_{m}}$ is not smooth function as its derivative over
$\omega_{m}$ diverges at any finite $\omega_{m}$.

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#  Four-loop critical properties of polymerized membranes

Andrey Pikelner<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical
Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

###### Abstract

We calculate four-loop order corrections to the critical exponent $\eta$ in
the two-field model of flat phase membranes. Obtained results show better
agreement with the other calculation methods and confirm the validity of the
perturbative approach to the considered problem.

##  I Introduction

Description of many real-life systems can be reduced to the models of
polymerized $D$-dimensional membranes in their low-temperature flat phase
embedded in the $d$-dimensional Euclidean space. Using renormalization group
methods, one can describe the flow of the theory between non-interacting UV-
stable Gaussian fixed point and IR-attractive flat phase fixed point. Many
different approaches describe the model near the IR fixed-point, non-
perturbative such as NPRG Kownacki and Mouhanna (2009), self-consistent
screening approximation (SCSA) Le Doussal and Radzihovsky (1992); Gazit
(2009); Le Doussal and Radzihovsky (2018), numerical SCSA Zakharchenko  _et
al._ (2010); Roldán  _et al._ (2011) and Monte Carlo simulations Zhang  _et
al._ (1993); Bowick  _et al._ (1996); Tröster (2013). Also, perturbative
methods based on small epsilon expansion have been successfully applied to the
problem starting from pioneering works Nelson, D.R. and Peliti, L. (1987);
Aronovitz and Lubensky (1988) and recently extended to three-loop order in the
series of papers Coquand  _et al._ (2020); Mauri and Katsnelson (2021);
Metayer  _et al._ (2021). The main object of interest is to estimate critical
exponents controlling power-law behavior of the phonon-phonon and flexural-
flexural correlation functions near the critical point.

The recent three-loop calculation Metayer  _et al._ (2021) provides us with a
new value of critical exponent very close to the results from other methods.
Moreover, it demonstrates an apparent convergence of the perturbative series
in the proper direction. The present paper aims to perform four-loop
calculations and check perturbative series behavior for the critical exponent
at the next loop order.

We carry out all our calculations in the two-field model described in detail
in Aronovitz and Lubensky (1988) with action given by:

$\displaystyle S=\frac{1}{2}\int
d^{D}x\left[(\partial^{2}\mathbf{h})^{2}+2\mu\,u_{ij}^{2}+\lambda\,u_{ii}^{2}\right],$
(1) $\displaystyle
u_{ij}=\frac{1}{2}\left(\partial_{i}u_{j}+\partial_{j}u_{i}+\partial_{i}\mathbf{h}\cdot\partial_{j}\mathbf{h}\right)$
(2)

Here we keep only relevant part of $u_{ij}$ and appropriately rescaled111To
simplify notations, we use the model in such a form that fields and Lame
coefficients are normalized by $\kappa$ and use simply $\lambda$ and $\mu$ for
quantities divided by $\kappa$. fields $u,\mathbf{h}$ and Lame coefficients
$\lambda,\mu$. Our goal is to calculate renormalization constants of fields
and couplings entering (1) as expansion about the upper critical dimension
$D_{\rm uc}=4$ in a small parameter $\varepsilon=2-D/2$. Zeroes of beta-
functions derived from the renormalization constant provide us with the
coordinates of the set of fixed points $\\{\mu^{*},\lambda^{*}\\}$ which after
substitution into the field anomalous dimension lead to the final answer for
the critical exponent of the interest.

Our main result is the a new value of $\eta$ for the $d_{c}=1$ case
corresponding to the $D=2$ membrane embedded into the $d=3$ dimensional space
at the four-loop order:

$\eta=0.8670$ (3)

In section II, we provide details of our four-loop calculation and in section
III we make a comparison with available results and present analytical
expressions for obtained results together with cross checks.

##  II Calculation details

The possibility of highly nontrivial four-loop calculations described below is
based on two essential features of the considered model:

  * •

In all our calculations, we are allowed to consider massless diagrams only.

  * •

Thanks to the Ward identities, all needed renormalization constants can be
derived from two-point functions only.

Recent two-loop Coquand  _et al._ (2020) and three-loop Metayer  _et al._
(2021) calculations successfully adopted both of these facts. Table 1 provides
us with the number of diagrams that need to be calculated up to the four-loop
level, which is an order of magnitude higher than in the three-loop problem
considered before Metayer  _et al._ (2021). Thus, our main improvement
consists of applying modern tools of multi-loop calculations and enhanced
renormalization strategy compared to Coquand  _et al._ (2020); Metayer  _et
al._ (2021).

| 1-loop | 2-loop | 3-loop | 4-loop
---|---|---|---|---
$\Pi_{hh}$ | 1 | 6 | 45 | 516
$\Pi_{uu}$ | 1 | 3 | 23 | 237
Table 1: Number of calculated diagrams

Feynman rules for the model (1) used in our calculation can be found in
Guitter, E.  _et al._ (1989). We generate all diagrams with DIANA Tentyukov
and Fleischer (2000) and calculate all massless integrals up to the four loops
with FORCER Ruijl  _et al._ (2020). Applying appropriate projectors we obtain
bare results for the sum of one-particle irreducible(1PI) diagrams. In case of
the $\mathbf{h}$-field the projector is trivial

$\Pi_{a,b}^{hh}=\delta_{a,b}\Pi_{hh},\quad a,b=1\dots d_{c}$ (4)

with $d_{c}=d-D$ but in the case of the $u$-field we split result of the
calculation into transverse and longitudial parts:

$\Pi_{i,j}^{uu}=\left(\delta_{i,j}-\frac{Q_{i}\cdot
Q_{j}}{Q^{2}}\right)\Pi_{uu}^{T}+\frac{Q_{i}\cdot Q_{j}}{Q^{2}}\Pi_{uu}^{L},$
(5)

where $i,j=1\dots D$. Bare quantities222We utilize subscript B for bare
quantities. are related to renormalized ones by $h_{B}=\sqrt{Z}h$, $u_{B}=Zu$,
$\lambda_{B}=Z_{\lambda}\lambda$, $\mu_{B}=Z_{\mu}\mu$.

Renormalizability of the theory allows us to greatly simplify renormalization
of the theory and extraction of constants $Z,Z_{\mu},Z_{\lambda}$. Instead of
explicit account of diagrams with counterterms insertions or any other ways of
subtraction of UV divergencies like BPHZ, it is possible to renormalize
couplings in the bare result for the calculated Green functions and multiply
it with an appropriate renormalization constant. Similar approach was
pioneered in the three-loop QCD renormalization Larin and Vermaseren (1993)
and proved to be especially useful for renormalization of the Standard Model
Bednyakov  _et al._ (2013).

Our starting point is the calculation of the sum of bare 1PI diagrams up to
the four-loop order denoted as $\Pi_{hh}$ for the $\mathbf{h}$-field and
$\Pi_{uu}^{T},\Pi_{uu}^{L}$ for transverse and longitudinal parts in the case
of the $u$-field respectively. Here it is important to keep terms in
$\varepsilon$-expansion up to the order $\varepsilon^{3-L}$ in the calculation
of the $L$-loop diagrams. The sum of 1PI diagrams with an appropriate tree
term after replacement of bare couplings with renormalized ones and
multiplication by the overall renormalization constant is finite and we have
three equations to fix three renormalization constants
$Z,Z_{\mu},Z_{\lambda}$:

$\displaystyle Z^{2}\left(\mu Z_{\mu}-\Pi_{uu}^{T}(\mu_{B}\to\mu
Z_{\mu},\lambda_{B}\to\lambda
Z_{\lambda})\right)=\mathcal{O}(\varepsilon^{0})$ $\displaystyle
Z^{2}\left(2\mu Z_{\mu}+\lambda Z_{\lambda}-\Pi_{uu}^{L}(\mu_{B}\to\mu
Z_{\mu},\lambda_{B}\to\lambda
Z_{\lambda})\right)=\mathcal{O}(\varepsilon^{0})$ $\displaystyle
Z\left(1-\Pi_{hh}(\mu_{B}\to\mu Z_{\mu},\lambda_{B}\to\lambda
Z_{\lambda})\right)=\mathcal{O}(\varepsilon^{0})$ (6)

Due to the complicated dependence of $Z_{i}$ on couplings $\mu,\lambda$, it is
usefull to introduce the loop counting parameter and solve equations (6)
perturbatively order by order to get four-loop renormalization constants
$Z,Z_{\mu},Z_{\lambda}$333All results are available in a computer-readable
form as ancillary files to the arxiv version of the paper..

Beta functions $\beta_{X}=\frac{\partial X}{\partial\log\mathfrak{m}}$ and
field anomalous dimension $\gamma=\frac{\partial\log
Z}{\partial\log\mathfrak{m}}$ are defined as logarithmic derivatives in
$\mathrm{\overline{MS}}$ scale parameter $\mathfrak{m}$. From calculated
$Z,Z_{\mu},Z_{\lambda}$ we can find beta functions:

$\displaystyle\beta_{\mu}$
$\displaystyle=\frac{2\varepsilon\partial_{\lambda}\log{\frac{\mu
Z_{\mu}}{\lambda
Z_{\lambda}}}}{\det\begin{pmatrix}\partial_{\lambda}\log{\lambda
Z_{\lambda}}&\partial_{\mu}\log{\lambda Z_{\lambda}}\\\
\partial_{\lambda}\log{\mu Z_{\mu}}&\partial_{\mu}\log{\mu
Z_{\mu}}\end{pmatrix}}$ (7) $\displaystyle\beta_{\lambda}$
$\displaystyle=\frac{2\varepsilon\partial_{\mu}\log{\frac{\lambda
Z_{\lambda}}{\mu Z_{\mu}}}}{\det\begin{pmatrix}\partial_{\lambda}\log{\lambda
Z_{\lambda}}&\partial_{\mu}\log{\lambda Z_{\lambda}}\\\
\partial_{\lambda}\log{\mu Z_{\mu}}&\partial_{\mu}\log{\mu
Z_{\mu}}\end{pmatrix}}$ (8)

and the field anomalous dimension

$\gamma=\beta_{\lambda}\partial_{\lambda}\log{Z}+\beta_{\mu}\partial_{\mu}\log{Z}$
(9)

The absense of $\varepsilon$ poles in (7),(8) and (9) is a strong indication
of the validity of the obtained results.

##  III Results

From the set of equations
$\beta_{\mu}(\mu^{*},\lambda^{*})=0,\beta_{\lambda}(\mu^{*},\lambda^{*})=0$ we
have found four different solutions $(\mu^{*},\lambda^{*})$ corresponding to
four different fixed points. We adopt the same notation as in Aronovitz and
Lubensky (1988), where the point $P_{1}$ is the Gaussian one, and $P_{4}$ is
IR attractive one we are interested in. In addition, we consider unstable
fixed point $P_{3}$, since it allows comparing the result of the calculations
with $1/d_{c}$ results available in the literature. To fix the notation and
simplify comparison with Coquand  _et al._ (2020); Metayer  _et al._ (2021),
we provide one-loop coordinates of the points $P_{3}$ and $P_{4}$:

$\displaystyle\mu_{3}^{*}=\frac{12}{20+d_{c}}\varepsilon+\mathcal{O}(\varepsilon^{2})\qquad$
$\displaystyle\lambda_{3}^{*}=-\frac{6}{20+d_{c}}\varepsilon+\mathcal{O}(\varepsilon^{2})$
(10)
$\displaystyle\mu_{4}^{*}=\frac{12}{24+d_{c}}\varepsilon+\mathcal{O}(\varepsilon^{2})\qquad$
$\displaystyle\lambda_{4}^{*}=-\frac{4}{24+d_{c}}\varepsilon+\mathcal{O}(\varepsilon^{2})$
(11)

Substituting four-loop results for fixed-point coordinates into four-loop
field anomalous dimension (9), we obtain critical exponents
$\eta=\gamma(\mu^{*},\lambda^{*})$ for two selected fixed points.

Full analytic four-loop results for the critical exponents $\eta_{3}$ (fixed
point $P_{3}$) and $\eta_{4}$ (fixed point $P_{4}$) can be found in the
Appendix in Eq.(17) and Eq.(18) respectively. For the case $d_{c}=1$ our
result reads:

$\displaystyle\eta_{3}=$ $\displaystyle
0.952\varepsilon-0.071\varepsilon^{2}-0.069\varepsilon^{3}-0.075\varepsilon^{4}+\mathcal{O}(\varepsilon^{5})$
(12) $\displaystyle\eta_{4}=$ $\displaystyle
0.96\varepsilon-0.0461\varepsilon^{2}-0.0267\varepsilon^{3}-0.02\varepsilon^{4}+\mathcal{O}(\varepsilon^{5})$
(13)

The three-loop parts of (12) and (13) coincide with result of three-loop
calculationMetayer  _et al._ (2021) and the four-loop term is new. For the
important case $D=2$, substituting $\varepsilon=1$ into (12) and (13) we
obtain our final result for the critical exponent $\eta$. Comparison with the
earlier two-loop Coquand  _et al._ (2020) and three-loopMetayer  _et al._
(2021) calculations and also with nonperturbative results obtained with NPRG
technique Kownacki and Mouhanna (2009), SCSA Le Doussal and Radzihovsky (1992)
and from MC simulation Tröster (2013) are summarized in table 2.

| 1-loop | 2-loop | 3-loop | 4-loop | $[2/2]$ | SCSA Le Doussal and Radzihovsky (1992) | NPRG Kownacki and Mouhanna (2009) | MC Tröster (2013)
---|---|---|---|---|---|---|---|---
$\eta_{3}$ | 0.9524 | 0.8813 | 0.8116 | 0.7368 | - | - | - | -
$\eta_{4}$ | 0.96 | 0.9139 | 0.8872 | 0.8670 | 0.806 | 0.821 | 0.849 | 0.795(10)
Table 2: Perturbative results for $\eta$ compared to other methods predictions
for the $D=2$ case

The obtained result provides additional support for apparent convergence of
the series for $\eta$ near $D=2$ without any additional resummation. As the
trivial exercise, we also construct a $[2/2]$ Pade aproximant for the 4-loop
result $\eta_{4}^{[2/2]}=0.806$, which provides us with a slightly different
result, and indicates on the possible need for more careful resummation of the
series. The growth of the last expansion term in (12) compared to the three-
loop one implies the possibility of the divergence of the series and motivates
for further more careful resummation of obtained series.

Another check of the validity of the obtained perturbative series stems from
the comparison with known $1/d_{c}$ expansions in the vicinity of chosen
critical points. Our main interest is the expression for $\eta_{4}$ but the
result for $\eta_{3}$ is also important, since it’s leading order expansion in
$1/d_{c}$ also can be verified with results available in the literature.

In Le Doussal and Radzihovsky (1992), the following result for the leading
term in $1/d_{c}$ expansion corresponding to the fixed point $P_{4}$ was
derived:

$\eta(D,d_{c})=\frac{8}{d_{c}}\frac{D-1}{D+2}\frac{\Gamma(D)}{\Gamma(D/2)^{3}\Gamma(2-D/2)}+\mathcal{O}\left(\frac{1}{d_{c}^{2}}\right)$
(14)

after expansion in $\varepsilon=2-D/2$ up to the $\varepsilon^{4}$, it
perfectly matches the leading term of (18) expansion in $1/d_{c}$ and reads:

$\displaystyle\eta_{4}=\frac{1}{d_{c}}$
$\displaystyle\left(24\varepsilon-24\varepsilon^{2}-\frac{64}{3}\varepsilon^{3}\right.$
$\displaystyle\left.-\frac{16}{9}(13-27\zeta_{3})\varepsilon^{4}+\mathcal{O}(\varepsilon^{5})\right)+\mathcal{O}\left(\frac{1}{d_{c}^{2}}\right)$
(15)

For the fixed point $P_{3}$ according to Le Doussal and Radzihovsky (2018)
leading term of the $1/d_{c}$ expansion is determined by
$\eta\left(D,\frac{D(D-1)}{(D-2)(D+1)}d_{c}\right)$, and after expansion in
$\varepsilon$ becomes equal to the $1/d_{c}$ expansion of (17):

$\displaystyle\eta_{3}=\frac{1}{d_{c}}$
$\displaystyle\left(20\varepsilon-\frac{74}{3}\varepsilon^{2}-\frac{155}{9}\varepsilon^{3}\right.$
$\displaystyle\left.-\frac{1}{54}(769-2160\zeta_{3})\varepsilon^{4}+\mathcal{O}(\varepsilon^{5})\right)+\mathcal{O}\left(\frac{1}{d_{c}^{2}}\right)$
(16)

##  IV Conclusion

We have calculated four-loop beta-functions and field anomalous dimension in
the two-field model of polymerized membranes. From the obtained perturbative
results, we have found a set of fixed points and derived the value of the
critical exponent $\eta$ for $D=2$ case. The obtained results are in good
agreement with other calculation methods and demonstrate apparent convergence
of the perturbative series. The validity of the result is confirmed by
comparison with known results in $1/d_{c}$ expansion.

###### Acknowledgements.

We thank A.Bednyakov, N.Lebedev, G.Kalagov, and M.Kompaniets for fruitful
discussions and useful comments on the manuscript. Comments and discussions
regarding the work Metayer  _et al._ (2021) by S.Metayer and S.Teber are
kindly acknowledged. Furthermore, we are grateful to the Joint Institute for
Nuclear Research for using their supercomputer “Govorun.”

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##  Appendix A Analytical four-loop results for critical exponents

$\displaystyle\eta_{3}$
$\displaystyle=\frac{20\varepsilon}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}}+\left(\frac{2800}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{3}}+\frac{1060}{3{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{2}}-\frac{74}{3{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}}\right)\varepsilon^{2}$
$\displaystyle+\left(\frac{784000}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{5}}-\frac{40(615553-591624\zeta_{3})}{27{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{4}}+\frac{2(1024193-1006344\zeta_{3})}{27{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{3}}-\frac{2(17105-20736\zeta_{3})}{27{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{2}}-\frac{155}{9{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}}\right)\varepsilon^{3}$
$\displaystyle+\left(\frac{274400000}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{7}}-\frac{28000(648943-591624\zeta_{3})}{27{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{6}}-\frac{40(63897618439+174575927736\zeta_{3}-263951628480\zeta_{5})}{243{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{5}}\right.$
$\displaystyle+\frac{4(226859519881+611469803304\zeta_{3}+239607720\zeta_{4}-927893517120\zeta_{5})}{729{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{4}}$
$\displaystyle-\frac{2(15308397193+40857079644\zeta_{3}+40756932\zeta_{4}-62189551440\zeta_{5})}{729{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{3}}$
$\displaystyle\left.+\frac{24880019+65141136\zeta_{3}+186624\zeta_{4}-99921600\zeta_{5}}{81{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}^{2}}-\frac{769-2160\zeta_{3}}{54{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[20+d_{c}\right]}}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5})$
(17) $\displaystyle\eta_{4}$
$\displaystyle=\frac{24\varepsilon}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}}+\left(\frac{2880}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{3}}+\frac{456}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{2}}-\frac{24}{24+d_{c}}\right)\varepsilon^{2}$
$\displaystyle+\left(\frac{691200}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{5}}-\frac{576(234137-192096\zeta_{3})}{125{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{4}}+\frac{8(1031777-923616\zeta_{3})}{125{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{3}}-\frac{4(39029-86832\zeta_{3})}{375{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{2}}-\frac{64}{3{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}}\right)\varepsilon^{3}$
$\displaystyle+\left(\frac{207360000}{{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{7}}-\frac{165888(20501-16008\zeta_{3})}{5{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{6}}-\frac{32(1174399340197+3188610294336\zeta_{3}-4827670269120\zeta_{5})}{1875{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{5}}\right.$
$\displaystyle+\frac{4(2761899037843+7430870367648\zeta_{3}+1867173120\zeta_{4}-11277698973120\zeta_{5})}{5625{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{4}}$
$\displaystyle-\frac{2(52639017319+140359656168\zeta_{3}+83125440\zeta_{4}-213590260800\zeta_{5})}{1875{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{3}}$
$\displaystyle\left.+\frac{2(1074978101+2807145072\zeta_{3}+3907440\zeta_{4}-4302849600\zeta_{5})}{5625{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}^{2}}-\frac{16(13-27\zeta_{3})}{9{\color[rgb]{0.3828125,0.5859375,0.61328125}\left[24+d_{c}\right]}}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5})$
(18)


$\displaystyle\times\frac{1}{\pi\Delta\nu_{D}^{2}}\Biggl{[}\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{\mu_{u}f_{u},\mu_{u}^{\prime}f_{u}^{\prime}}}}-\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{\mu_{u}f_{u},\mu_{u}^{\prime}f_{u}^{\prime}}}}\Biggr{]}$
$\displaystyle\times\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{\mu_{u}^{\prime}f_{u}^{\prime},\mu_{\ell}f_{\ell}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{\mu_{u}f_{u},\mu_{\ell}f_{\ell}}\Bigr{)}^{\ast}\Biggr{]}\,\,\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u+u_{\mu_{u}^{\prime}f_{u}^{\prime},\mu_{\ell}^{\prime}f_{\ell}^{\prime}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u+u_{\mu_{u}f_{u},\mu_{\ell}^{\prime}f_{\ell}^{\prime}}\Bigr{)}^{\ast}\Biggr{]}\,.$
(57b)

Notable changes in notation with respect to the expressions given in Bommier
(2018) include the exchange of the indices $K$ and $K^{\prime}$ and that the
quantum number $M$ in that paper corresponds to the quantum number $f$ used
here. The reduced frequency splitting between states that belong to the same
term is given by

$u_{\mu_{u}f_{u},\mu_{u}^{\prime}f_{u}^{\prime}}=\frac{\nu_{\mu_{u}f_{u},\mu_{u}^{\prime}f_{u}^{\prime}}}{\Delta\nu_{D}}\,\quad\mathrm{and}\quad
u_{\mu_{\ell}f_{\ell},\mu_{\ell}^{\prime}f_{\ell}^{\prime}}=\frac{\nu_{\mu_{\ell}f_{\ell},\mu_{\ell}^{\prime}f_{\ell}^{\prime}}}{\Delta\nu_{D}}\,.$

The expressions for ${R}_{\mbox{\sc{ii}}}$ and $R_{\mbox{\sc{iii}}}$ contain
the same quantity $\mathcal{A}$, which depends on the quantum numbers of the
states involved in each of the terms of the redistribution matrix, and the
coupling between them. This quantity is given by

$\displaystyle{\mathcal{A}}^{KK^{\prime}}_{Q}(\mu_{u}f_{u},\mu^{\prime}_{u}f^{\prime}_{u},\mu_{\ell}f_{\ell},\mu^{\prime}_{\ell}f^{\prime}_{\ell})=\frac{3(2L_{u}+1)}{(2I+1)(2S+1)}\sqrt{(2K+1)(2K^{\prime}+1)}$
$\displaystyle\times(-1)^{f_{\ell}-f^{\prime}_{\ell}}\sum_{\begin{subarray}{c}J_{u}{\bar{J}}_{u}J^{\prime}_{u}{\bar{J}^{\prime}_{u}}\\\
F_{u}\bar{F}_{u}F^{\prime}_{u}{\bar{F}^{\prime}_{u}}\end{subarray}}C_{{\mu_{u}}}^{{J_{u}}{F_{u}}}({f_{u}})\,C_{{\mu_{u}}}^{{\bar{J}_{u}}{\bar{F}_{u}}}({f_{u}})\,C_{\mu^{\prime}_{u}}^{J^{\prime}_{u}F^{\prime}_{u}}(f^{\prime}_{u})\,C_{\mu^{\prime}_{u}}^{{\bar{J}^{\prime}_{u}}{\bar{F}^{\prime}_{u}}}(f^{\prime}_{u})\sum_{\begin{subarray}{c}J_{\ell}\bar{J}_{\ell}J^{\prime}_{\ell}{\bar{J}^{\prime}_{\ell}}\\\
F_{\ell}\bar{F}_{\ell}F^{\prime}_{\ell}{\bar{F}^{\prime}_{\ell}}\end{subarray}}\,{C_{{\mu_{\ell}}}^{{J_{\ell}}{F_{\ell}}}({f_{\ell}})\,C_{{\mu_{\ell}}}^{{\bar{J}_{\ell}}{\bar{F}_{\ell}}}({f_{\ell}})\,C_{\mu^{\prime}_{\ell}}^{J^{\prime}_{\ell}F^{\prime}_{\ell}}(f^{\prime}_{\ell})\,C_{\mu^{\prime}_{\ell}}^{\bar{J}^{\prime}_{\ell}{\bar{F}^{\prime}_{\ell}}}(f^{\prime}_{\ell})}$
$\displaystyle\times{(-1)^{J_{u}+J^{\prime}_{u}+\bar{J}_{u}+{\bar{J}^{\prime}_{u}}}(-1)^{J_{\ell}+J^{\prime}_{\ell}+\bar{J}_{\ell}+{\bar{J}^{\prime}_{\ell}}}}\sqrt{(2F_{u}+1)(2F^{\prime}_{u}+1)(2\bar{F}_{u}+1)(2{\bar{F}^{\prime}_{u}}+1)(2J_{u}+1)(2J^{\prime}_{u}+1)(2\bar{J}_{u}+1)(2{\bar{J}^{\prime}_{u}}+1)}$
$\displaystyle\times\sqrt{(2F_{\ell}+1)(2F^{\prime}_{\ell}+1)(2\bar{F}_{\ell}+1)(2{\bar{F}^{\prime}_{\ell}}+1)(2J_{\ell}+1)(2J^{\prime}_{\ell}+1)(2\bar{J}_{\ell}+1)(2{\bar{J}^{\prime}_{\ell}}+1)}$
$\displaystyle\times{\left\\{\begin{array}[]{c c c}F_{u}&F_{\ell}&1\\\
J_{\ell}&J_{u}&I\end{array}\right\\}\left\\{\begin{array}[]{c c
c}F^{\prime}_{u}&\bar{F}_{\ell}&1\\\
\bar{J}_{\ell}&J^{\prime}_{u}&I\end{array}\right\\}\left\\{\begin{array}[]{c c
c}{\bar{F}^{\prime}_{u}}&{\bar{F}^{\prime}_{\ell}}&1\\\
{\bar{J}^{\prime}_{\ell}}&{\bar{J}^{\prime}_{u}}&I\end{array}\right\\}\left\\{\begin{array}[]{c
c c}\bar{F}_{u}&F^{\prime}_{\ell}&1\\\
J^{\prime}_{\ell}&\bar{J}_{u}&I\end{array}\right\\}}\left\\{\begin{array}[]{c
c c}J_{u}&J_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J^{\prime}_{u}&\bar{J}_{\ell}&1\\\ L_{\ell}&L_{u}&S\end{array}\right\\}$
(70) $\displaystyle\times\left\\{\begin{array}[]{c c
c}{\bar{J}^{\prime}_{u}}&{\bar{J}^{\prime}_{\ell}}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}\bar{J}_{u}&J^{\prime}_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\sum_{pp^{\prime}p^{\prime\prime}p^{\prime\prime\prime}=-1}^{1}\left(\begin{array}[]{c
c c}1&1&K\\\ -p&p^{\prime}&Q\end{array}\right)\left(\begin{array}[]{c c
c}1&1&K^{\prime}\\\
-p^{\prime\prime\prime}&p^{\prime\prime}&Q\end{array}\right)$ (79)
$\displaystyle\times\left(\begin{array}[]{c c c}F_{u}&F_{\ell}&1\\\
-f_{u}&f_{\ell}&p\end{array}\right)\left(\begin{array}[]{c c
c}F_{u}^{\prime}&\bar{F}_{\ell}&1\\\
-f_{u}^{\prime}&f_{\ell}&p^{\prime}\end{array}\right)\left(\begin{array}[]{c c
c}{\bar{F}^{\prime}_{u}}&{\bar{F}^{\prime}_{\ell}}&1\\\
-f^{\prime}_{u}&f^{\prime}_{\ell}&p^{\prime\prime}\end{array}\right)\left(\begin{array}[]{c
c c}\bar{F}_{u}&{F^{\prime}_{\ell}}&1\\\
-f_{u}&f^{\prime}_{\ell}&p^{\prime\prime\prime}\end{array}\right)\,.$ (88)

We arrive at this expression by making use of the equivalence between the
coupling coefficients $C_{\mu}^{FJ}(f)$ and the product of the pair
$C^{J}_{J^{\ast}M(F^{\ast}f)}$ $C^{F}_{F^{\ast\ast}(JM)f}$ given in Bommier
(2018), in which the perturbed Hamiltonians for FS and HFS are diagonalized
separately. Here the quantity $M$ is the value of $\@vec{J}$ measured along
the selected quantization axis, and it is referred to as $M_{J}$ in Bommier
(2018). We also changed the set of labels $J^{\ast}F^{\ast}$ to the equivalent
$\mu$, as shown in Eqs. (57).

###  C.5 The limits of CRD and fully coherent scattering

A theoretical framework that has been extensively used to model spectral line
polarization is the one introduced by LL04. It was derived from first
principles of quantum electrodynamics and relies on a first-order perturbative
treatment of matter-radiation interaction. Thus, scattering processes are
treated as a succession of statistically independent absorption and emission
processes (Casini & Landi Degl’Innocenti 2008). This corresponds to the limit
of complete frequency redistribution (CRD), in which any frequency coherence
in scattering processes is effectively lost. As discussed in LL04, this
approach is valid under the flat-spectrum approximation, in which the incident
radiation field is assumed to be independent of frequency.

By contrast, the formalism considered in this work (see Bommier 2017) allows
one to take the spectral structure of the incident radiation fully into
account. It can account for scattering processes that are coherent in
frequency (i.e., that preserve the frequency correlation between incoming and
outgoing radiation) and, the same time, it accounts for loss of such coherence
due to elastic collisions. As can be seen from the redistribution matrices
shown in Appendix C.4, a higher elastic collisional rate $\Gamma_{E}$ implies
smaller branching ratios in $\bm{\mathcal{R}}_{\mbox{\sc{ii}}}$ (which
quantify coherent scattering processes) and larger ones in
$\bm{\mathcal{R}}_{\mbox{\sc{iii}}}$ (noncoherent processes). The CRD limit
can in principle be reached by artificially setting a very large value of
$\Gamma_{E}$ (i.e., $\Gamma_{E}\gg\Gamma_{R},\Gamma_{I}$) in the branching
ratios so that the contribution from $\bm{\mathcal{R}}_{\mbox{\sc{iii}}}$
dominates over that from $\bm{\mathcal{R}}_{\mbox{\sc{ii}}}$. However, one
must bear in mind that $\Gamma_{E}$ also enters the definition of the damping
constant $a$ (which could introduce inconsistencies) and that elastic
collisions may in principle relax atomic polarization (so that polarization
phenomena could become negligible in a highly collisional regime). We also
observe that the $\bm{\mathcal{R}}_{\mbox{\sc{iii}}}$ considered in this
framework was not obtained under the flat-spectrum assumption and could
potentially introduce a sensitivity to the spectral structure of the radiation
field. Conversely, the limit of fully coherent scattering (CS) can by reached
by setting $\Gamma_{E}=0$ so that the contribution from
$\bm{\mathcal{R}}_{\mbox{\sc{iii}}}$ vanishes. In this work, we have
considered calculations that correspond to this limit both accounting for the
contribution from $\Gamma_{E}$ to the damping constant and neglecting it (see
Sect C.8 for the approximation of additionally neglecting Doppler
redistribution). We note that accounting for this contribution in the damping
constant gives rise to an inconsistency, because the emission profiles
contained in the redistribution matrices no longer cancel perfectly with the
branching ratios in the far wings, which could introduce an artificial
magnetic sensitivity (see Sect. 10.4 of LL04; Alsina Ballester et al. 2018).

###  C.6 Micro-structured magnetic fields

The expressions presented above are suitable for the computation of the RT
coefficients when the external magnetic field has a fixed strength and
orientation at any given spatial point. Hereafter, such fields are referred to
as deterministic. However, in stellar atmospheres it is common for magnetic
fields to be tangled at small spatial scales, so that its orientation changes
multiple times across the mean free path of the line’s photons. Making the
assumption that the strength of the magnetic field remains constant over these
scales (i.e., the magnetic field is unimodal), the expressions for the RT
coefficients in the vertical reference system can be found through an
analytical integration over the considered distribution of magnetic field
orientations. The dependency on the magnetic field orientation is fully
contained in the rotation matrices, both in the elements of the propagation
matrix and in the line scattering emission coefficient (see Eqs. (15) and
(21)). Thus, these matrices are the only quantities entering the integration.
Below, we consider two distinct scenarios for the distribution of magnetic
field orientations and show the corresponding expressions.

#### Isotropic distribution of magnetic fields:

The analytical integration is carried out over all possible inclinations
$\theta_{B}$ and azimuths $\chi_{B}$.

$\displaystyle\frac{1}{4\pi}\int_{0}^{2\pi}\\!\mathrm{d}\chi_{B}\int_{0}^{\pi}\\!\mathrm{d}\theta_{B}\sin\theta_{B}\,{\mathcal{D}}^{K}_{0Q}(R_{B})^{\ast}=\delta_{K,0}\,\delta_{Q,0}\,,$
$\displaystyle\frac{1}{4\pi}\int_{0}^{2\pi}\\!\mathrm{d}\chi_{B}\int_{0}^{\pi}\\!\mathrm{d}\theta_{B}\sin\theta_{B}\,{\mathcal{D}}^{K}_{QQ^{\prime\prime}}(R_{B})\,{\mathcal{D}}^{K^{\prime}}_{QQ^{\prime}}(R_{B})^{\ast}=\frac{1}{2K+1}\delta_{K,K^{\prime}}\delta_{Q^{\prime},Q^{\prime\prime}}\,.$

For the latter integration we used the Weyl theorem (see Sect. 2.6 of LL04).
Making use of these integrals, the expression for the absorption coefficient
becomes

$\displaystyle\eta_{I}(\nu,\mathbf{\Omega})=\frac{k_{M}}{3\sqrt{\pi}\Delta\nu_{D}}\sum_{q}\sum_{\mu_{u}f_{u}\mu_{\ell}f_{\ell}}S^{\mu_{\ell}f_{\ell},\mu_{u}f_{u}}_{q}\,H\bigl{(}a,u+u_{\mu_{u}f_{u},\mu_{\ell}f_{\ell}}\bigr{)}\,.$
(89)

The propagation matrix elements $\rho_{i}$ and $\eta_{i}$ with subscripts
other than $I$ are zero and therefore their expressions are not shown. The
elements of the scattering phase matrix contained in the redistribution
matrices become

$\displaystyle\bm{\mathcal{P}}^{KK^{\prime}}_{Q}(\mathbf{\Omega}^{\prime},\mathbf{\Omega})_{ij}=\delta_{K,K^{\prime}}\frac{1}{2K+1}\sum_{Q^{\prime}=-K}^{K}(-1)^{Q^{\prime}}{\mathcal{T}}^{K}_{Q^{\prime}}(i,\mathbf{\Omega}){\mathcal{T}}^{K}_{-Q^{\prime}}(j,\mathbf{\Omega}^{\prime})\,.$
(90)

It is often useful to consider this distribution in order to investigate the
small-scale magnetic fields present in the quiescent solar photosphere.

#### Magnetic field with a fixed inclination and a random distribution of
azimuths:

In this case, the analytical integration is only carried out over $\chi_{B}$,
while $\theta_{B}$ is kept fixed, resulting in

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\\!\mathrm{d}\chi_{B}\,{\mathcal{D}}^{K}_{0Q}(R_{B})^{\ast}=\delta_{Q,0}\,d^{K}_{00}(\theta_{B}),$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\\!\mathrm{d}\chi_{B}\,{\mathcal{D}}^{K}_{QQ^{\prime\prime}}(R_{B})\,{\mathcal{D}}^{K^{\prime}}_{QQ^{\prime}}(R_{B})^{\ast}=(-1)^{Q-Q^{\prime\prime}}\sum_{\kappa=|K-K^{\prime}|}^{K+K^{\prime}}(2\kappa+1)\left(\begin{array}[]{c
c c}K&K^{\prime}&\kappa\\\ Q&-Q&0\end{array}\right)\left(\begin{array}[]{c c
c}K&K^{\prime}&\kappa\\\
Q^{\prime\prime}&-Q^{\prime}&0\end{array}\right)d^{\kappa}_{00}(\theta_{B})\,,$

where $d^{K}_{00}(\theta_{B})$ is the so-called reduced rotation matrix (see
Sect. 2.6 of LL04). Through these relations we reach

$\displaystyle\eta_{i}(\nu,\mathbf{\Omega})={\frac{k_{M}}{\sqrt{\pi}\Delta\nu_{D}}}$
$\displaystyle\sum_{K=0}^{2}\sqrt{\frac{2K+1}{3}}d^{K}_{00}(\theta_{B})\,\mathcal{T}^{K}_{0}(i,\mathbf{\Omega})\sum_{q=-1}^{1}(-1)^{1+q}\left(\begin{array}[]{c
c c}1&1&K\\\
q&-q&0\end{array}\right)\sum_{\mu_{u}f_{u}\mu_{\ell}f_{\ell}}S^{\mu_{\ell}f_{\ell},\mu_{u}f_{u}}_{q}\,H\bigl{(}a,u+u_{\mu_{u}f_{u},\mu_{\ell}f_{\ell}}\bigr{)}\,,$
(91c)
$\displaystyle\rho_{i}(\nu,\mathbf{\Omega})={\frac{k_{M}}{\sqrt{\pi}\Delta\nu_{D}}}$
$\displaystyle\sum_{K=0}^{2}\sqrt{\frac{2K+1}{3}}d^{K}_{00}(\theta_{B})\,\mathcal{T}^{K}_{0}(i,\mathbf{\Omega})\sum_{q=-1}^{1}(-1)^{1+q}\left(\begin{array}[]{c
c c}1&1&K\\\
q&-q&0\end{array}\right)\sum_{\mu_{u}f_{u}\mu_{\ell}f_{\ell}}S^{\mu_{\ell}f_{\ell},\mu_{u}f_{u}}_{q}\,L\bigl{(}a,u+u_{\mu_{u}f_{u},\mu_{\ell}f_{\ell}}\bigr{)}\,,$
(91f)

and

$\bm{\mathcal{P}}^{KK^{\prime}}_{Q}(\mathbf{\Omega}^{\prime},\mathbf{\Omega})_{ij}=(-1)^{Q}\\!\\!\sum_{Q^{\prime}=-K^{\prime}}^{K^{\prime}}\sum_{Q^{\prime\prime}=-K}^{K}\mathcal{T}^{K^{\prime}}_{Q^{\prime}}(i,\mathbf{\Omega})\,\mathcal{T}^{K}_{-Q^{\prime\prime}}(j,\mathbf{\Omega}^{\prime})\sum_{\kappa=|K-K^{\prime}|}^{K+K^{\prime}}(2\kappa+1)\left(\begin{array}[]{c
c c}K&K^{\prime}&\kappa\\\ Q&-Q&0\end{array}\right)\left(\begin{array}[]{c c
c}K&K^{\prime}&\kappa\\\
Q^{\prime\prime}&-Q^{\prime}&0\end{array}\right)d^{\kappa}_{00}(\theta_{B})\,,$
(92)

It is often useful to consider this distribution in order to investigate the
canopy-like magnetic fields in the outer layers of the solar atmosphere.

###  C.7 The redistribution matrices in the absence of a magnetic field,
including the depolarizing effect of elastic collisions

In the absence of magnetic fields, $J$ and $F$ remain good quantum numbers.
The energy eigenstates are of the form $\lvert\beta LSIJFf\rangle$ (or, with
shorthand notation, $\lvert JFf\rangle$) and states with different quantum
number $f$ but the same numbers $J$ and $F$ remain degenerate. As a result,
the number of transitions with different frequencies greatly decreases and the
line scattering emission coefficients can be computed at a considerably lower
computational cost. Simpler expressions for the redistribution matrix,
suitable for the nonmagnetic case, are found in Eq. (40) of Bommier (2017).
Following the approach described in Appendix C.4 and making the same changes
in notation we write them as

$\displaystyle\bigl{[}R_{\mbox{\sc{ii}}}$
$\displaystyle\bigr{]}^{KK^{\prime}}_{Q}(\nu^{\prime},\nu)=\delta_{KK^{\prime}}\sum_{\begin{subarray}{c}J_{u}J_{u}^{\prime}\\\
F_{u}F_{u}^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}J_{\ell}J_{\ell}^{\prime}\\\
F_{\ell}F_{\ell}^{\prime}\end{subarray}}\mathcal{A}^{KK}_{Q}\bigl{(}J_{u}F_{u},J_{u}^{\prime}F_{u}^{\prime},J_{\ell}F_{\ell},J_{\ell}^{\prime}F_{\ell}^{\prime}\bigr{)}$
$\displaystyle\;\times\frac{1}{{\pi}\Delta\nu_{D}^{2}}\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{J_{u}F_{u},J_{u}^{\prime}F_{u}^{\prime}}}}\frac{1}{2}\int_{0}^{\pi}\\!\\!\mathrm{d}\Theta\,\exp\Biggl{[}-\,\biggl{(}\frac{u-u^{\prime}+u_{J_{\ell}F_{\ell},J_{\ell}^{\prime}F_{\ell}^{\prime}}}{2\sin\Theta/2}\biggr{)}^{2}\Biggr{]}$
$\displaystyle\;\times\Biggl{[}\frac{1}{2}W\,\biggl{(}\frac{a}{\cos\Theta/2},\frac{u+u^{\prime}+u_{J_{u}^{\prime}F_{u}^{\prime},J_{\ell}F_{\ell}}+u_{J_{u}^{\prime}F_{u}^{\prime},J_{\ell}^{\prime}F_{\ell}^{\prime}}}{2\cos\Theta/2}\biggr{)}+\frac{1}{2}W\,\biggl{(}\frac{a}{\cos\Theta/2},\frac{u+u^{\prime}+u_{J_{u}F_{u},J_{\ell}F_{\ell}}+u_{J_{u}F_{u},J_{\ell}^{\prime}F_{\ell}^{\prime}}}{2\cos\Theta/2}\biggr{)}^{\ast}\Biggr{]}\,,$
(93a) $\displaystyle\bigl{[}R_{\mbox{\sc{iii}}}$
$\displaystyle\bigr{]}^{KK^{\prime}}_{Q}(\nu^{\prime},\nu)=\delta_{KK^{\prime}}\sum_{\begin{subarray}{c}J_{u}J_{u}^{\prime}\\\
F_{u}F_{u}^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}J_{\ell}J_{\ell}^{\prime}\\\
F_{\ell}F_{\ell}^{\prime}\end{subarray}}\mathcal{A}^{KK}_{Q}\bigl{(}J_{u}F_{u},J_{u}^{\prime}F_{u}^{\prime},J_{\ell}F_{\ell},J_{\ell}^{\prime}F_{\ell}^{\prime}\bigr{)}$
$\displaystyle\times\frac{1}{\pi\Delta\nu_{D}^{2}}\Biggl{[}\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\frac{1}{2}\bigl{[}D^{(K)}(J_{u}F_{u})+D^{(K)}(J_{u}^{\prime}F_{u}^{\prime})\bigr{]}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{J_{u}F_{u},J_{u}^{\prime}F_{u}^{\prime}}}}-\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{J_{u}F_{u},J_{u}^{\prime}F_{u}^{\prime}}}}\Biggr{]}$
$\displaystyle\times\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{J_{u}^{\prime}F_{u}^{\prime},J_{\ell}F_{\ell}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{J_{u}F_{u},J_{\ell}F_{\ell}}\Bigr{)}^{\ast}\Biggr{]}\,\,\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u+u_{J_{u}^{\prime}F_{u}^{\prime},J_{\ell}^{\prime}F_{\ell}^{\prime}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u+u_{J_{u}F_{u},J_{\ell}^{\prime}F_{\ell}^{\prime}}\Bigr{)}^{\ast}\Biggr{]}\,,$
(93b)

where

$\displaystyle\mathcal{A}^{KK}_{Q}\Bigl{(}J_{u}F_{u},$ $\displaystyle
J_{u}^{\prime}F_{u}^{\prime},J_{\ell}F_{\ell},J_{\ell}^{\prime}F_{\ell}^{\prime}\Bigr{)}=\frac{3(2L_{u}+1)}{(2S+1)(2I+1)}\,(-1)^{F_{\ell}-F_{\ell}^{\prime}}(2J_{u}+1)\,(2J_{u}^{\prime}+1)\,(2J_{\ell}+1)\,(2J_{\ell}^{\prime}+1)$
$\displaystyle\times(2F_{u}+1)\,(2F_{u}^{\prime}+1)\,(2F_{\ell}+1)\,(2F_{\ell}^{\prime}+1)\left\\{\begin{array}[]{c
c c}1&1&K\\\
F_{u}&F_{u}^{\prime}&F_{\ell}\end{array}\right\\}\left\\{\begin{array}[]{c c
c}1&1&K\\\ F_{u}&F_{u}^{\prime}&F_{\ell}^{\prime}\end{array}\right\\}$ (98)
$\displaystyle\times\left\\{\begin{array}[]{c c c}J_{u}&J_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J_{u}^{\prime}&J_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J_{u}^{\prime}&J_{\ell}^{\prime}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J_{u}&J_{\ell}^{\prime}&1\\\ L_{\ell}&L_{u}&S\end{array}\right\\}$ (107)
$\displaystyle\times\left\\{\begin{array}[]{c c c}F_{u}&F_{\ell}&1\\\
J_{\ell}&J_{u}&I\end{array}\right\\}\left\\{\begin{array}[]{c c
c}F_{u}^{\prime}&F_{\ell}&1\\\
J_{\ell}&J_{u}^{\prime}&I\end{array}\right\\}\left\\{\begin{array}[]{c c
c}F_{u}^{\prime}&F_{\ell}^{\prime}&1\\\
J_{\ell}^{\prime}&J_{u}^{\prime}&I\end{array}\right\\}\left\\{\begin{array}[]{c
c c}F_{u}&F_{\ell}^{\prime}&1\\\
J_{\ell}^{\prime}&J_{u}&I\end{array}\right\\}\,.$ (116)

Similar expressions can be obtained directly from those presented in Appendix
C.4, observing that in the unmagnetized case $C_{\mu}^{JF}=\delta_{\mu,JF}$.
The only notable difference is that the branching ratios in the expressions
obtained from Bommier (2017) contain $K$-multipole depolarization rates for
each HFS level $D^{(K)}(J_{u},F_{u})$ produced by elastic collisions. In that
work the $D^{(K)}$ rates could be included because the master equations for
the atomic states allow for a closed analytical solution even when accounting
for the collisional transitions between states belonging to the same HFS
level, provided there is no external magnetic field. We note that, even in the
absence of magnetic fields, no such solution can be found when accounting for
the collisional transitions between states that belong to the same term but
with different $J$ or $F$ quantum numbers, which are therefore not included in
the expressions introduced above.

#### Neglecting interference between states of the upper term

The expressions above can be straightforwardly modified to artificially
neglect the quantum interference between states belonging to the upper term
with different $J$ or $F$, which remain good quantum numbers of the system in
the absence of a magnetic field. In order to neglect the interference between
states with different $J$ (i.e., $J$-state interference), we simply introduce
a $\delta_{J_{u},J_{u}^{\prime}}$ in the quantity ${\mathcal{A}}$ shown above.
Similarly, the interference between states with different quantum number $F$
can be neglected by introducing $\delta_{F_{u},F_{u}^{\prime}}$ in
$\mathcal{A}$.

###  C.8 The two-term atomic model without HFS

The Mg ii doublet (see main text) and the H i Lyman-$\alpha$ line (see
Appendix B) can be suitably modeled considering atomic systems without HFS. In
this case, the expressions shown in the previous sections simplify
considerably. The matrix elements of the magnetic Hamiltonian also simplify,
taking the form shown in Sect. 3.4 of LL04, and the atomic Hamiltonian is
given by $H=H_{\mbox{\scriptsize{FS}}}+H_{B}$. For a given $(\beta,L,S)$ term,
its eigenstates are specified by quantum number $M$, which is the value of the
total electronic angular momentum $\@vec{J}$ measured along the specified
quantization axis, and by the label $j$. The energies of these states and
their couplings to the eigenstates of $\@vec{J}$ are given by

$\displaystyle H\lvert\beta LSjM\rangle$ $\displaystyle=E_{\beta
LS}\lvert\beta LSjM\rangle\,;$ $\displaystyle\lvert\beta LSjM\rangle$
$\displaystyle=\sum_{J}C^{J}_{j}(M)\lvert\beta LSJM\rangle\,.$

Hereafter, we use the shorthand labels $\lvert jM\rangle$ and $\lvert
JM\rangle$ for states $\lvert\beta LSjM\rangle$ and $\lvert\beta LSJM\rangle$,
respectively. In this case, the RT coefficients can be obtained from those
corresponding to an atomic system with HFS (see Appendices C.3 and C.4), but
setting the nuclear spin $I=0$ and making use of the relation shown in Eq.
(2.36a) of LL04 for the $\operatorname{6-\\!\mathnormal{j}}$ symbols. This
also requires the substitution of labels $\mu$ with $j$, quantum numbers $f$
with $M$, and coupling coefficients $C_{\mu}^{JF}(f)$ with $C_{j}^{J}(M)$. The
resulting expressions for the elements of the propagation matrix are found in
Eqs. (15d) and (15h), and those for the frequency-dependent terms of the
redistribution matrices are found in Eqs. (19) and (20). The corresponding
quantity $\mathcal{A}$ is given by

$\displaystyle\mathcal{A}^{KK^{\prime}}_{Q}$
$\displaystyle\Bigl{(}j_{u}M_{u},j_{u}^{\prime}M_{u}^{\prime},j_{\ell}M_{\ell},j_{\ell}^{\prime}M_{\ell}^{\prime}\Bigr{)}=\frac{3(2L_{u}+1)}{(2S+1)}\sqrt{(2K+1)(2K^{\prime}+1)}$
$\displaystyle\times\sum_{\begin{subarray}{c}J_{u}\bar{J}_{u}J_{u}^{\prime}\bar{J}_{u}^{\prime}\end{subarray}}C^{J_{u}}_{j_{u}}(M_{u})\,C^{\bar{J}_{u}}_{j_{u}}(M_{u})\,C^{J_{u}^{\prime}}_{j_{u}^{\prime}}(M_{u}^{\prime})\,C^{\bar{J}_{u}^{\prime}}_{j_{u}^{\prime}}(M_{u}^{\prime})\sqrt{(2J_{u}+1)(2J^{\prime}_{u}+1)(2\bar{J}_{u}+1)(2\bar{J}_{u}^{\prime}+1)}$
$\displaystyle\times\sum_{\begin{subarray}{c}J_{\ell}\bar{J}_{\ell}J_{\ell}^{\prime}\bar{J}_{\ell}^{\prime}\end{subarray}}(-1)^{M_{\ell}-M_{\ell}^{\prime}}C^{J_{\ell}}_{j_{\ell}}(M_{\ell})\,C^{\bar{J}_{\ell}}_{j_{\ell}}(M_{\ell})\,C^{J_{\ell}^{\prime}}_{j_{\ell}^{\prime}}(M_{\ell}^{\prime})\,C^{\bar{J}_{\ell}^{\prime}}_{j_{\ell}^{\prime}}(M_{\ell}^{\prime})\sqrt{(2J_{\ell}+1)(2J^{\prime}_{\ell}+1)(2\bar{J}_{\ell}+1)(2\bar{J}_{\ell}^{\prime}+1)}$
$\displaystyle\times\left\\{\begin{array}[]{c c c}J_{u}&J_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J_{u}^{\prime}&\bar{J}_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}\bar{J}_{u}^{\prime}&\bar{J}_{\ell}^{\prime}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}\bar{J}_{u}&J_{\ell}^{\prime}&1\\\ L_{\ell}&L_{u}&S\end{array}\right\\}$
(125)
$\displaystyle\times\sum_{pp^{\prime}p^{\prime\prime}p^{\prime\prime\prime}=-1}^{1}\left(\begin{array}[]{c
c c }1&1&K\\\ -p&p^{\prime}&Q\end{array}\right)\left(\begin{array}[]{c c c
}1&1&K^{\prime}\\\
-p^{\prime\prime\prime}&p^{\prime\prime}&Q\end{array}\right)$ (130)
$\displaystyle\times\left(\begin{array}[]{c c c}J_{u}&J_{\ell}&1\\\
-M_{u}&M_{\ell}&p\end{array}\right)\left(\begin{array}[]{c c
c}J_{u}^{\prime}&\bar{J}_{\ell}&1\\\
-M_{u}^{\prime}&M_{\ell}&p^{\prime}\end{array}\right)\left(\begin{array}[]{c c
c}\bar{J}_{u}^{\prime}&\bar{J}_{\ell}^{\prime}&1\\\
-M_{u}^{\prime}&M_{\ell}^{\prime}&p^{\prime\prime}\end{array}\right)\left(\begin{array}[]{c
c c}\bar{J}_{u}&J_{\ell}^{\prime}&1\\\
-M_{u}&M_{\ell}^{\prime}&p^{\prime\prime\prime}\end{array}\right)\,.$ (139)

We note that these expressions for the redistribution matrix coincide with
those that would be obtained from Eq. (A.1) of Bommier (2018), making the
abovementioned changes in notation and the approximations discussed in Sect.
2.4 for the reference PRD calculations. In the rest of this section, we
present the expressions for the redistribution matrices under the
approximations that are studied in the main text.

#### Coherent scattering in the reference frame of the observer:

The first approximation considered in the main text is that of treating the
scattering processes quantified by $\bm{\mathcal{R}}_{\mbox{\sc{ii}}}$ as
coherent in the reference frame of the observer, or equivalently, neglecting
Doppler redistribution in this matrix. Under this approximation we still
account for the broadening effect of small-scale atomic motions in the
absorption profiles. This can be achieved by neglecting the Doppler shifts in
the Dirac delta function in Eq. (A.1) of Bommier (2018). Thus, when
integrating over velocities with a Maxwellian distribution, the Lorentzian and
associated dispersion profiles in the $R_{\mbox{\sc{ii}}}$ terms are replaced
with Voigt and associated dispersion profiles, respectively, multiplied by a
$1\\!/\\!\sqrt{\pi}\Delta\nu_{D}$ normalization factor. Making the
abovementioned changes in notation, the frequency-dependent components become

$\displaystyle\bigl{[}\hat{R}_{\mbox{\sc{ii}}}$
$\displaystyle\bigr{]}^{KK^{\prime}}_{Q}(\nu^{\prime},\nu)=\sum_{\begin{subarray}{c}j_{u}j_{u}^{\prime}\\\
M_{u}M_{u}^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}j_{\ell}j_{\ell}^{\prime}\\\
M_{\ell}M_{\ell}^{\prime}\end{subarray}}\mathcal{A}^{KK^{\prime}}_{Q}(j_{u}M_{u},j_{u}^{\prime}M_{u}^{\prime},j_{\ell}M_{\ell},j_{\ell}^{\prime}M_{\ell}^{\prime})\,\,\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{j_{u}M_{u},j_{u}^{\prime}M_{u}^{\prime}}}}$
$\displaystyle\times\frac{1}{\sqrt{\pi}\Delta\nu_{D}^{2}}\,\delta\,\Bigl{(}u-u^{\prime}+u_{j_{\ell}M_{\ell},j_{\ell}^{\prime}M_{\ell}^{\prime}}\Bigr{)}\,\,\,\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{j_{u}^{\prime}M_{u}^{\prime},j_{\ell}M_{\ell}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{j_{u}M_{u},j_{\ell}M_{\ell}}\Bigr{)}^{\ast}\Biggr{]}\,.$
(140)

Like in the main text (see Sect. 3.1), we write the components of this
approximate redistribution matrix as $\hat{R}_{\mbox{\sc{ii}}}$. The
additional $\Delta\nu_{D}$ in the denominator is obtained when changing the
argument of the Dirac delta function from frequency to reduced frequency. It
can easily be recognized that, under this approximation, the coupling between
the incoming and scattered radiation is contained only in the Dirac delta.
Thus, the radiation scattered at reduced frequency $u$ receives contributions
only from the incident radiation at the few reduced frequencies
$u^{\prime}=u+u_{j_{\ell}M_{\ell},j_{\ell}^{\prime}M_{\ell}^{\prime}}$
determined by the energy difference between the initial and final state. This
represents a significant decrease in numerical complexity. Indeed, if all
states of the lower term are degenerate, then these scattering processes
introduce no coupling between the radiation field at different frequencies. We
recall that this approximation is only applied to
$\bm{\mathcal{R}}_{\mbox{\sc{ii}}}$, but it can be combined with the
approximation of treating all scattering processes as coherent, discussed in
Appendix. C.5. Indeed, this is done in Sect. 3.2 of the main text.

We finally observe that in the line wings, where this approximation is valid,

$H(a,u)\sim{\sqrt{\pi}\Delta\nu_{D}}\,\phi(\nu_{0}-\nu)\,\quad\mathrm{and}\quad
L(a,u)\sim{\sqrt{\pi}\Delta\nu_{D}}\,\psi(\nu_{0}-\nu)\,,$

where $\phi$ and $\psi$ are the Lorentzian and associated dispersion profiles
(see Sect. 5.3 of LL04). Therefore, at these spectral ranges, one should
expect the results to be the same as if
$\bm{\hat{{\mathcal{R}}}}_{\mbox{\sc{ii}}}$ were considered directly in the
atomic rest frame without accounting for Doppler broadening.

#### Expressions in the absence of a magnetic field:

The expressions for the redistribution matrix in the unmagnetized case,
accounting for the depolarizing effect of elastic collisions (see Appendix
C.7) can be further simplified for atomic systems without HFS. In analogy to
what is discussed in Appendix C.7, they can be written as

$\displaystyle\bigl{[}R_{\mbox{\sc{ii}}}$
$\displaystyle\bigr{]}^{KK^{\prime}}_{Q}(\nu^{\prime},\nu)=\delta_{KK^{\prime}}\sum_{J_{u}J_{u}^{\prime}}\sum_{J_{\ell}J_{\ell}^{\prime}}\mathcal{A}^{KK}_{Q}\bigl{(}J_{u},J_{u}^{\prime},J_{\ell},J_{\ell}^{\prime}\bigr{)}$
$\displaystyle\;\times\frac{1}{\pi\Delta\nu_{D}^{2}}\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{J_{u},J_{u}^{\prime}}}}\frac{1}{2}\int_{0}^{\pi}\\!\\!\mathrm{d}\Theta\,\exp\Biggl{[}-\,\biggl{(}\frac{u-u^{\prime}+u_{J_{\ell},J_{\ell}^{\prime}}}{2\sin\Theta/2}\biggr{)}^{2}\Biggr{]}$
$\displaystyle\;\times\Biggl{[}\frac{1}{2}W\,\biggl{(}\frac{a}{\cos\Theta/2},\frac{u+u^{\prime}+u_{J_{u}^{\prime},J_{\ell}}+u_{J_{u}^{\prime},J_{\ell}^{\prime}}}{2\cos\Theta/2}\biggr{)}+\frac{1}{2}W\,\biggl{(}\frac{a}{\cos\Theta/2},\frac{u+u^{\prime}+u_{J_{u},J_{\ell}}+u_{J_{u},J_{\ell}^{\prime}}}{2\cos\Theta/2}\biggr{)}^{\ast}\Biggr{]}\,,$
(141a) $\displaystyle\bigl{[}R_{\mbox{\sc{iii}}}$
$\displaystyle\bigr{]}^{KK^{\prime}}_{Q}(\nu^{\prime},\nu)=\delta_{KK^{\prime}}\sum_{J_{u}J_{u}^{\prime}}\sum_{J_{\ell}J_{\ell}^{\prime}}\mathcal{A}^{KK}_{Q}\bigl{(}J_{u},J_{u}^{\prime},J_{\ell},J_{\ell}^{\prime}\bigr{)}$
$\displaystyle\times\frac{1}{\pi\Delta\nu_{D}^{2}}\Biggl{[}\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\frac{1}{2}\bigl{[}D^{(K)}(J_{u})+D^{(K)}(J_{u}^{\prime})\bigr{]}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{J_{u},J_{u}^{\prime}}}}-\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{J_{u},J_{u}^{\prime}}}}\Biggr{]}$
$\displaystyle\times\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{J_{u}^{\prime},J_{\ell}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{J_{u},J_{\ell}}\Bigr{)}^{\ast}\Biggr{]}\,\,\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u+u_{J_{u}^{\prime},J_{\ell}^{\prime}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u+u_{J_{u},J_{\ell}^{\prime}}\Bigr{)}^{\ast}\Biggr{]}\,,$
(141b)

and the ${\mathcal{A}}$ quantity contained within them is given by

$\displaystyle\mathcal{A}^{KK}_{Q}\Bigl{(}J_{u},J_{u}^{\prime},J_{\ell},$
$\displaystyle
J_{\ell}^{\prime}\Bigr{)}=\frac{3(2L_{u}+1)}{(2S+1)}\,(-1)^{J_{\ell}-J_{\ell}^{\prime}}(2J_{u}+1)\,(2J_{u}^{\prime}+1)\,(2J_{\ell}+1)\,(2J_{\ell}^{\prime}+1)$
$\displaystyle\times\left\\{\begin{array}[]{c c c}1&1&K\\\
J_{u}&J_{u}^{\prime}&J_{\ell}\end{array}\right\\}\left\\{\begin{array}[]{c c
c}1&1&K\\\
J_{u}&J_{u}^{\prime}&J_{\ell}^{\prime}\end{array}\right\\}\left\\{\begin{array}[]{c
c c}J_{u}&J_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J_{u}^{\prime}&J_{\ell}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J_{u}^{\prime}&J_{\ell}^{\prime}&1\\\
L_{\ell}&L_{u}&S\end{array}\right\\}\left\\{\begin{array}[]{c c
c}J_{u}&J_{\ell}^{\prime}&1\\\ L_{\ell}&L_{u}&S\end{array}\right\\}\,.$ (154)

These expressions can also be obtained from the redistribution matrix
presented in Eq. (34) of Bommier (2017), making the abovementioned changes in
notation and approximations for the reference PRD case (see Sect. 2.4). These
redistribution matrices were used for the calculations presented in Sect. 3.3,
in which the depolarizing effect of elastic collisions was fully neglected by
setting all $D^{(K)}(J_{u})$ rates to zero.

#### Reproducing the redistribution matrix of Belluzzi & Trujillo Bueno
(2014):

These authors proposed a redistribution matrix for the case of a two-term
atomic model with an unpolarized and infinitely sharp lower term in the
absence of a magnetic field. The branching ratios and
$\bm{\mathcal{R}}_{\mbox{\sc{iii}}}$ were obtained from heuristic arguments.
In order to recover this redistribution matrix using the present formalism, we
consider the following frequency-dependent factors for
$\bm{\mathcal{R}}_{\mbox{\sc{ii}}}$ and $\bm{\mathcal{R}}_{\mbox{\sc{iii}}}$

$\displaystyle\bigl{[}R_{\mbox{\sc{ii}}}$
$\displaystyle\bigr{]}^{KK^{\prime}}_{Q}(\nu^{\prime},\nu)=\delta_{KK^{\prime}}\sum_{J_{u}J_{u}^{\prime}}\sum_{J_{\ell}J_{\ell}^{\prime}}\mathcal{A}^{KK}_{Q}\bigl{(}J_{u},J_{u}^{\prime},J_{\ell},J_{\ell}^{\prime}\bigl{)}$
$\displaystyle\times\frac{1}{\pi\Delta\nu_{D}^{2}}\frac{\Gamma_{R}+\Gamma_{I}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}}\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+2\pi\,\mathrm{i}\,{\Delta\nu_{D}\,u_{J_{u},J_{u}^{\prime}}}}\frac{1}{2}\int_{0}^{\pi}\\!\\!\mathrm{d}\Theta\,\mathrm{exp}\Biggl{[}-\,\Biggl{(}\frac{u-u^{\prime}+u_{J_{\ell},J_{\ell}^{\prime}}}{2\sin\Theta/2}\Biggr{)}^{2}\Biggr{]}$
$\displaystyle\;\;\times\left[\frac{1}{2}W\,\Biggl{(}\frac{a}{\cos\Theta/2},\frac{u+u^{\prime}+u_{J_{u}^{\prime},J_{\ell}}+u_{J_{u}^{\prime},J_{\ell}^{\prime}}}{2\cos\Theta/2}\Biggr{)}+\frac{1}{2}W\,\Biggl{(}\frac{a}{\cos\Theta/2},\frac{u+u^{\prime}+u_{J_{u},J_{\ell}}+u_{J_{u},J_{\ell}^{\prime}}}{2\cos\Theta/2}\Biggr{)}^{\ast}\right]\,,$
(155a) $\displaystyle\bigl{[}R_{\mbox{\sc{iii}}}$
$\displaystyle\bigr{]}^{KK^{\prime}}_{Q}(\nu^{\prime},\nu)=\delta_{KK^{\prime}}\sum_{J_{u}J_{u}^{\prime}}\sum_{J_{\ell}J_{\ell}^{\prime}}\mathcal{A}^{KK}_{Q}(J_{u},J_{u}^{\prime},J_{\ell},J_{\ell}^{\prime})$
$\displaystyle\times\frac{1}{\pi\Delta\nu_{D}^{2}}\frac{\Gamma_{E}}{\Gamma_{R}+\Gamma_{I}+\Gamma_{E}}\frac{\Gamma_{R}}{\Gamma_{R}+\Gamma_{I}+2\pi\,\mathrm{i}\,\Delta\nu_{D}\,u_{J_{u},J_{u}^{\prime}}}$
$\displaystyle\times\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{J_{u}^{\prime},J_{\ell}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u^{\prime}+u_{J_{u},J_{\ell}}\Bigr{)}^{\ast}\Biggr{]}\,\,\Biggl{[}\frac{1}{2}W\,\Bigl{(}a,u+u_{J_{u}^{\prime},J_{\ell}^{\prime}}\Bigr{)}+\frac{1}{2}W\,\Bigl{(}a,u+u_{J_{u},J_{\ell}^{\prime}}\Bigr{)}^{\ast}\Biggr{]}\,,$
(155b)

where the quantity ${\mathcal{A}}$ is exactly as given in Eq. (154). We
observe that the only difference between these expressions and those given in
Eq. (141) is found in their branching ratios. Neglecting depolarizing
collisions in the latter, we note that the branching ratios of the two
formulations coincide if (i) the CS limit is taken ($\Gamma_{E}=0$), (ii) the
CRD limit is taken ($\Gamma_{E}\gg\Gamma_{R},\Gamma_{I}$), or (iii) the fine
structure energy splitting of both the upper and lower terms (contained in
$u_{J_{u},J_{\ell}}$ and $u_{J_{\ell},J_{\ell}^{\prime}}$) is negligible.

The $\bm{\mathcal{R}}_{\mbox{\sc{ii}}}$ that results from Eq. (155a) perfectly
coincides with the analogous quantity in Belluzzi & Trujillo Bueno (2014),
which is the first term on the right-hand side of Eq. (36) in that paper. On
the other hand, discrepancies remain between the
$\bm{\mathcal{R}}_{\mbox{\sc{iii}}}$ resulting from Eq. (155b) and the second
term of Eq. (36) in Belluzzi & Trujillo Bueno (2014). These discrepancies are
contained in the absorption profiles of the two expressions (i.e., the
profiles whose arguments depend on the incident radiation); in the latter work
this profile was introduced heuristically. Nevertheless, the two expressions
fully coincide if we make the assumption that the incident radiation field is
spectrally flat, in which case the frequency shifts in their absorption
profiles can be neglected.