# One-dimensional scattering of two-dimensional fermions near quantum
criticality

Dimitri Pimenov<EMAIL_ADDRESS>Alex Kamenev  Andrey V. Chubukov  William I.
Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN
55455, USA

###### Abstract

Forward and backscattering play an exceptional role in the physics of two-
dimensional interacting fermions. In a Fermi liquid, both give rise to a non-
analytic $\omega^{2}\ln(\omega)$ form of the fermionic scattering rate at
second order in the interaction. Here we argue that higher powers of
$\ln(\omega)$ appear in the backscattering contribution at higher orders. We
show that these terms come from “planar” processes, which are effectively one-
dimensional. This is explicitly demonstrated by extending a Fermi liquid to
the limit of $N\gg 1$ fermionic flavors, when only planar processes survive.
We sum the leading logarithms for the case of a 2D Fermi liquid near a nematic
transition and obtain an expression for the scattering rate at $T=0$ to all
orders in the interaction. For a repulsive interaction, the resulting rate is
logarithmically suppressed, and the result is valid down to $\omega=0$. For an
attractive interaction, the ground state is an $s$-wave superconductor with a
gap $\Delta_{0}$. We show that in this case the scattering rate increases as
$\omega$ is reduced towards $\Delta_{0}$. At $\omega\geq\Delta_{0}$, the
behavior of the scattering rate is rather unconventional as many pairing
channels compete near a nematic critical point, and $s$-wave wins only by a
narrow margin. We take superconductivity into consideration and obtain the
scattering rate also at smaller $\omega\simeq\Delta_{0}$.

##  I Introduction

Quantum critical metals (QCM) constitute a rich and ever-evolving field of
modern condensed matter physics. The broad interest in their behavior is
fueled not only by the increasing number of experimentally probed systems,
such as cuprates and iron-based materials [1, 2, 3, 4], but also by the advent
of modern numerical techniques, in particular Quantum Monte-Carlo [5, 6, 7,
8]. One exciting aspect of QCMs is their prototypical non-Fermi liquid
behavior, and it is a theorist’s dream to determine the associated critical
behavior exactly. So far, this has only been achieved in the special cases
such as the SYK model [9, 10], a particular matrix large $N$ theory [11], a
scalar large $N$ model with a particular dispersion of a critical boson [12],
and models obtained by dimensional regularization [13]. Besides, a fully self-
consistent description of a spin density wave quantum critical point (QCP) in
(2+1)D at the lowest energies has also been proposed in Ref. [14] and Ref.
[15] presented the general description of critical theories with a chiral
Fermi surface.

A particular example of a QCM, which defies complete understanding despite
years of study [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 12, 13, 27, 28],
is the one at an Ising nematic QCP in (2+1)D. Here, the fermionic self-energy
both at one and two loop level scales as $\omega^{2/3}$, which is a hallmark
of a non-Fermi liquid. However, the loop expansion lacks a control parameter.
It was originally believed [16, 17] that one can justify the expansion by
extending the model to $N$ fermion flavors and taking the limit $N\to\infty$.
Indeed, at large $N$, the prefactor for $\omega^{2/3}$ at the two-loop order
is parametrically smaller than the one-loop result [17, 23]. However, it was
later recognized by S.-S. Lee [29] that this does not hold beyond the two-loop
order. He analyzed a simplified “one-patch” model without backscattering and
explicitly demonstrated that planar diagrams, which emerge at three-loop and
higher orders, are not small in $1/N$. Subsequent studies of the Ising nematic
model within the “two-patch theory”, which includes backscattering, found more
contributions from planar diagrams, not small in $1/N$, and discovered
additional logarithmic singularities (Refs. [25, 26]).

There are several equivalent characterizations of planar diagrams, which can
all be invoked to extract their special behaviour at large $N$. The original
name-giving criterion in the high-energy literature [30] is that they can be
drawn on a plane without holes. From the condensed matter perspective, the
planar diagrams describe a subset of scattering processes which have a
(1+1)-dimensional structure, despite the original problem being (2+1)D – more
specifically, in these diagrams the curvature term in the fermionic dispersion
cancels out [25],111We note that the two-dimensionality still shows up via
Landau damping, without it the fermionic self-energy would be more singular
than $\omega^{2/3}$ [22, 68] .

Because the leading one-loop self-energy $\omega^{2/3}$ also comes from (1+1)D
processes [32, 33], the $N\to\infty$ limit at a QCP should be described by
some effective (1+1)-dimensional theory. However, the detailed structure of
higher-loop planar diagrams is not yet known, and the story is further
complicated by apparent UV-divergences at higher-loop orders [25, 26]. As a
result, the form of fermionic and bosonic propagators at a QCP in the large
$N$ limit remains unclear.

In this paper we analyze the scattering rate in a (2+1)D Fermi liquid away
from a nematic QCP. Our first goal is to understand the role of planar
diagrams in a Fermi liquid regime. In a generic 2D Fermi liquid, the imaginary
part of the one-loop self-energy $\text{Im}\left[\Sigma(\omega)\right]$ at
$k=k_{F}$ scales as $\omega^{2}\ln(\omega)$ (as $(\omega^{2}+(\pi
T)^{2})\ln\left[{\text{max}}\left(\omega,T\right)\right]$ at a finite
temperature $T$). This non-analytic form comes from forward and backscattering
processes [34], which can be regarded as effectively one-dimensional. We
analyze $\text{Im}\left[\Sigma(\omega)\right]$ at $T=0$ beyond one-loop order,
and show that it contains a series of higher powers of $\ln(\omega)$. The
logarithms come from the Cooper part of backscattering, which is manifestly
(1+1)D, and we argue that these terms can be equally viewed as coming from
planar processes. We demonstrate this explicitly by analyzing the system
behavior in the limit of large $N$, when only planar processes survive. We
explicitly sum up series of logarithms and obtain the scattering rate to
logarithmic accuracy to all orders in the interaction. We show, however, that
in a Fermi liquid not all planar processes are (1+1)D – the ones that do not
contain the highest power of $\ln(\omega)$ at any loop order involve
scattering with deviations from a single direction by momenta $q\sim M$, where
$M^{2}$ scales with the distance from a QCP 222At a QCP, all planar processes
become (1+1)D, and, simultaneously, the argument of the logarithm becomes of
order one. More specifically, it becomes a function of the combination
$q^{3}/\omega$, where $q$ and $\omega$ are relevant internal momenta and
frequency in a planar diagram. Both are infinitesimally small at vanishing
external frequency, and typical $q^{3}$ are of order $\omega$.Then the
effective dimension of the full $N=\infty$ theory becomes (1+1)D, with no
separation into the leading and subleading terms..

For a toy model with repulsive interaction in the particle-particle channel,
the full $\text{Im}\left[\Sigma(\omega)\right]$ to logarithmic accuracy
remains finite and is reduced, roughly by a half, compared to the one-loop
expression. A similar result holds for the $T^{2}$ term in the specific heat
of a 2D Fermi gas [36, 37]. For the realistic case of an attractive pairing
interaction, the ground state is an $s$-wave superconductor [38, 39, 40, 41,
42, 43, 44, 45]. We show that, in this case, the prefactor of the $\omega^{2}$
term in $\text{Im}\left[\Sigma(\omega)\right]$ increases with $\ln(\omega)$,
and the summation of the logarithms breaks down at some energy scale
$\omega\geq\Delta_{0}$, where $\Delta_{0}$ is a pairing gap.

Our second goal is then to correctly incorporate the superconducting ground
state and obtain $\text{Im}\left[\Sigma(\omega)\right]$ for
$\omega\simeq\Delta_{0}$. This is of relevance both for experiments and for
Quantum Monte-Carlo studies, which found [6, 7] that superconductivity must be
included to understand the data for the self-energy at the lowest
frequencies/temperatures. We argue that in a 2D Fermi liquid, the frequency
range where logarithmic renormalizations from planar diagrams are relevant,
but superconductivity can be neglected, can be made wide even near a QCP, by
restricting to a weak coupling. We show that in this case one can incorporate
superconductivity in a controllable way at $\omega\simeq\Delta_{0}$ 333Exactly
at a QCP, the scale $\Delta_{0}$ is of the same order as the upper edge of the
non-Fermi-liquid [53], and there is no sizable frequency range for the
renormalization of the self-energy by planar diagrams without including
superconducting fluctuations, unless one makes specific assumptions about the
shape of the Fermi surface [29]..

The fermionic self-energy at $\omega\simeq\Delta_{0}$ has been analyzed in the
past for a conventional $s$-wave superconductor with momentum-independent
pairing interaction [47, 48]. We show that our case is qualitatively
different, because the interaction, mediated by soft nematic fluctuations,
gives rise to near-equal attraction in a large number of pairing channels, and
$s$-wave is only slightly preferable over other pairing symmetries [40, 49].
In this case, we show that $\text{Im}\left[\Sigma(\omega)\right]$ has the same
form as in a pure $s$-wave superconductor between $\Delta_{0}$ (below which
$\text{Im}\left[\Sigma(\omega)\right]=0$) and slightly over $3\Delta_{0}$,
where the system realizes that attraction in the $s$-wave channel is larger
than that in other channels. At larger $\omega$, all pairing channels
contribute equally, and the form of $\text{Im}\left[\Sigma(\omega)\right]$
changes qualitatively.

The paper is structured as follows: In Sec. II, we present the model,
recapitulate one-loop results for $\Sigma[\omega]$ and discuss the relevant
energy scales. In Sec. III, we analyze the structure of three-loop planar
diagrams and demonstrate that the theory is effectively one-dimensional at
logarithmic order, but not beyond. In the next Sec. IV, we sum up the series
of the leading $\omega^{2}(\ln{\omega})^{n}$ contributions to the self-energy
from backscattering at all loop orders. In Sec. IV.1, we consider a toy model
with repulsive pairing interaction and show that the summation of the leading
logarithms converges at any frequency. The realistic case of an attractive
interaction is analyzed in Sec. IV.2, where we show that the convergence only
holds up to $\omega\sim\Delta_{0}$. In Sec. V, we demonstrate how the results
get modified once we add superconductivity and obtain the expressions for
$\text{Im}\left[\Sigma(\omega)\right]$ at $\omega\simeq\Delta_{0}$.
Conclusions and an outlook are presented in Sec. VI. Various technical details
are relegated to the Appendices.

##  II Model and one-loop results

We consider a system of interacting $N$-flavor fermions in two dimensions at
$T=0$. Two spin projections are incorporated into $N$ such that the physical
case of spin-1/2 fermions corresponds to $N=2$. We assume that the system is
isotropic and close to a QCP towards a nematic order ($d$-wave Pomeranchuk
instability) and that near a QCP the dominant interaction between fermions is
mediated by soft fluctuations of a nematic order parameter. The corresponding
$T=0$ Euclidean action [20, 21] is given by

$\displaystyle\mathcal{S}$
$\displaystyle=\mathcal{S}_{0}+\mathcal{S}_{\text{int}}\ ,$ (1)
$\displaystyle\mathcal{S}_{0}$
$\displaystyle=\sum_{\sigma=1}^{N}\int_{k}\bar{\psi}_{\sigma}(k)(-i\omega_{m}+\xi_{\boldsymbol{k}})\psi_{\sigma}(k)\
,$ $\displaystyle\mathcal{S}_{\text{int}}$
$\displaystyle=\frac{g}{V}\sum_{\sigma,\sigma^{\prime}}\int_{k,p,q}D_{0}({\boldsymbol{q}})f_{\boldsymbol{q}}({\boldsymbol{k}})f_{\boldsymbol{q}}({\boldsymbol{p}})\times$
$\displaystyle\quad\bar{\psi}_{\sigma}(k+q/2)\bar{\psi}_{\sigma^{\prime}}(p-q/2)\psi_{\sigma^{\prime}}(p+q/2)\psi_{\sigma}(k-q/2)\
,$ $\displaystyle\int_{k}$
$\displaystyle\equiv\int\frac{d\omega_{m}}{2\pi}\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\
.$

Here $\xi_{\boldsymbol{k}}=v_{F}(|{\boldsymbol{k}}|-k_{F})$ is the flavor-
independent fermionic dispersion, linearized around the Fermi momentum,
$\omega_{m}$ is a Matsubara frequency, $g$ is the coupling, which we set to be
small compared to the Fermi energy, $g/E_{F}\ll 1$, and

$\displaystyle D_{0}({\boldsymbol{q}})=-\frac{1}{q^{2}+M^{2}}\ ,\quad
q\equiv|{\boldsymbol{q}}|\ ,$ (2)

is the bare bosonic propagator. The parameter $M$ (bosonic mass) measures the
deviation from the quantum critical point, and has units of momentum. Near a
QCP,

$\displaystyle\epsilon\equiv\frac{M}{k_{F}}\ll 1\ .$ (3)

Finally,
$f_{\boldsymbol{q}}({\boldsymbol{k}})=\cos(2\theta_{{\boldsymbol{k}}{\boldsymbol{q}}})$
(with
$\theta_{{\boldsymbol{k}}{\boldsymbol{q}}}=\measuredangle({\boldsymbol{k}},{\boldsymbol{q}}))$
is the $d$-wave nematic form factor (not related to pairing symmetry), which
specifies the symmetry of the ordered state. At a general boson-fermion vertex
with incoming and outgoing fermionic momenta
${\boldsymbol{k}}_{\text{in}},{\boldsymbol{k}}_{\text{out}}$, the half-sum
$({\boldsymbol{k}}_{\text{in}}+{\boldsymbol{k}}_{\text{out}})/2$ corresponds
to ${\boldsymbol{k}}$ in the above defintion of $f$, while
${\boldsymbol{k}}_{\text{out}}-{\boldsymbol{k}}_{\text{in}}$ corresponds to
${\boldsymbol{q}}$. In our analysis, all momenta will fulfill
$|{\boldsymbol{k}}_{\text{in}}|,|{\boldsymbol{k}}_{\text{out}}|\simeq k_{F}$,
resulting in ${\boldsymbol{k}}\perp{\boldsymbol{q}}$ and $f\simeq-1$. (see
Fig. 1). Since all relevant expressions contain $f^{2}$, we can set $f=1$ from
the start. The form factor is important at high energies of order
$\mathcal{O}(E_{F})$ or in the ordered state only, which are not considered in
this paper. For a short discussion of lattice effects, see Sec. VI.

Figure 1: Definition of form factor, see main text

The action of Eq. (1) is obtained from a microscopic Hamiltonian with
4-fermion interaction by integrating out fermions with energies above a
certain cutoff. To account for screening from fermions with energies smaller
than the cutoff, the bare propagator $D_{0}({\boldsymbol{q}})$ should be
dressed by a particle-hole bubble $\Pi_{\text{ph}}$, see Fig. 2(a) and App. A.
The dressed propagator $D(\omega_{m},{\boldsymbol{q}})$ is

$\displaystyle
D(\omega_{m},{\boldsymbol{q}})=-\frac{1}{q^{2}+M^{2}+g\Pi_{\text{ph}}(\omega_{m},{\boldsymbol{q}})},$
(4)
$\displaystyle{\text{where}}\quad\Pi_{\text{ph}}(\omega_{m},{\boldsymbol{q}})=N\rho\frac{|\omega_{m}|}{\sqrt{\omega^{2}_{m}+(v_{F}q)^{2}}}\
.$

$\rho=m/(2\pi)$ the density of states per flavor, and $m=k_{F}/v_{F}$ the
fermionic mass. We assume, following earlier works [25, 50], that the static
part of $\Pi_{\text{ph}}$ is already incorporated into
$D(\omega_{m},{\boldsymbol{q}})$. At $\omega_{m}\ll v_{F}q$, which will be
relevant to our analysis, $\Pi_{\text{ph}}(\omega_{m},{\boldsymbol{q}})\approx
N\rho|\omega_{m}|/(v_{F}q)$ describes (in real frequencies) the Landau damping
of a boson due to a decay into the particle-hole continuum.

Figure 2: One-loop diagrams. (a) Particle-hole bubble $\Pi_{\text{ph}}$, which
renormalizes the bare interaction $D_{0}$ (dashed wavy lines). (b) One-loop
self-energy $\Sigma^{(1)}$. Full wavy line represents the renormalized
interaction $D$. (c) Diagram which effectively determines
$\Sigma^{(1)}(\omega)$ for $\omega\ll{\omega_{\text{FL}}}$ (interaction lines
are static).

With these ingredients, we can evaluate the fermionic self-energy for the
external momentum ${\boldsymbol{k}}$ on the Fermi surface,
$\Sigma(\omega_{m},{\boldsymbol{k}}_{F})\equiv\Sigma(\omega_{m})$
$(\text{with}\ G^{-1}=G^{-1}_{0}-\Sigma$). The one-loop self-energy
$\Sigma^{(1)}(\omega_{m})$, represented by the diagram in Fig. 2(b), has been
obtained before, see, e.g., Refs. [23, 25], and we just present the result:

$\displaystyle\Sigma^{(1)}(\omega_{m})\simeq$ (5)
$\displaystyle\begin{dcases}-i\lambda\omega_{m}+i\text{sign}(\omega_{m})\frac{2\lambda}{\pi}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\ln\left(\frac{{\omega_{\text{FL}}}}{|\omega_{m}|}\right),\\!&\omega_{m}\ll{\omega_{\text{FL}}}\\\
\omega_{\text{IN}}^{1/3}\omega^{2/3}_{m},\\!&\omega_{m}\gg{\omega_{\text{FL}}}.\end{dcases}$

The first line in Eq. (5) has a familiar Fermi-liquid form in 2D. It contains
two parameters

$\displaystyle\lambda\equiv\frac{g}{4\pi
v_{F}M},\quad\quad{\omega_{\text{FL}}}\equiv\frac{v_{F}M^{3}}{Ng\rho}\ .$ (6)

Here, $\lambda$ is an effective coupling, which will be used to control
perturbation theory, and ${\omega_{\text{FL}}}$ serves as UV cutoff for the
logarithms and also marks the crossover from a Fermi-liquid to a non-Fermi-
liquid regime. The second line in (5) is the one-loop formula for the self-
energy in the non-Fermi liquid regime. The scale

$\omega_{\text{IN}}=\frac{8}{3\sqrt{3}}\lambda^{3}{\omega_{\text{FL}}}\ $ (7)

is the upper boundary of the true non-Fermi-liquid region, where
$\Sigma(\omega_{m})>\omega_{m}$. The relevant energy scales are summarized in
Fig. 3 for both $\lambda\ll 1$, $\lambda\gg 1$.

In deriving Eq. (5) we assumed that typical internal frequencies
$\omega^{\prime}_{m}$ and momenta $q$ satisfy $\omega^{\prime}_{m}\ll v_{F}q$.
This condition is naturally satisfied at a QCP, where typical
$\omega^{\prime}_{m}\sim q^{3}$, see e.g. [51], but in a weakly coupled Fermi
liquid at $\lambda\ll 1$ we must additionally require $\lambda>\epsilon/N$,
such that ${\omega_{\text{FL}}}\leq v_{F}M$.

For $\omega\ll{\omega_{\text{FL}}}$, the Landau damping is small compared to
the mass term $M^{2}$ in $D(\omega_{m},{\boldsymbol{q}})$, and the Fermi-
liquid form of $\Sigma^{(1)}(\omega_{m})$ is obtained by expanding in the
Landau damping. In effect, this reduces the evaluation of the diagram in Fig.
2(b) to the evaluation of the two-loop diagram with static interactions
$D_{0}({\boldsymbol{q}})$ of Fig. 2(c). Let us now zoom into the
$\omega^{2}$-part of the Fermi-liquid self-energy at these frequencies:

$\displaystyle\Sigma^{(1)}_{\omega^{2}}(\omega_{m})\equiv
i\text{sign}(\omega_{m})\frac{2\lambda}{\pi}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\ln\left(\frac{{\omega_{\text{FL}}}}{|\omega_{m}|}\right).$
(8)

Upon analytical continuation $i\omega_{m}\rightarrow\omega+i\delta$, it maps
to the imaginary part of the retarded self-energy

$\displaystyle\text{Im}\left[\Sigma^{R}(\omega)\right]=-\frac{2\lambda}{\pi}\frac{\omega^{2}}{{\omega_{\text{FL}}}}\ln\left(\frac{{\omega_{\text{FL}}}}{|\omega|}\right),$
(9)

which determines the fermionic scattering rate.

Figure 3: Energy scales. For $\lambda\ll 1$, the self-energy is always smaller
than the bare frequency, $\Sigma(\omega_{m})<\omega_{m}$, and there is no true
non-Fermi-liquid. The green sliver marks the superconducting region for
attractive interactions discussed in Sec. V, with
$\Delta_{0}\simeq{\omega_{\text{FL}}}\exp(-\lambda)$. For $\lambda\gg 1$, a
true non-Fermi-liquid region (red) develops where
$\Sigma(\omega_{m})>\omega_{m}$. Superconductivity in the strong coupling case
is outside the scope of this paper. The scale $v_{F}M>{\omega_{\text{FL}}}$ is
not explicitly shown.  Figure 4: One-dimensional scattering at two-loop level.
(a) Most singular momentum points (forward and backscattering) which
contribute to the logarithm in $\Sigma_{\omega^{2}}^{(1)}$. The grayed out
diagram is also part of backscattering, but its contribution can be neglected
since it involves a large momentum transfer $\sim 2k_{F}$. (b) Way to extract
the logarithm from the forward and backscattering processes (see main text).

For our future considerations it is important that the
$\omega_{m}^{2}\ln{(\omega_{m})}$ dependence in Eq. (8) can be traced to
effectively one-dimensional scattering, embedded in two dimensions (see
Appendix B for technical details). Namely, the processes that contribute to
Eq. (8) are forward scattering and backscattering, with momentum deviations
from ${\boldsymbol{k}}$ and ${-{\boldsymbol{k}}}$ on the order of external
$\omega_{m}$ (see Fig. 4(a)). As sketched in Fig. 4(b), instead of evaluating
the closed particle-hole bubble in the diagram one can explicitly obtain one
half of $\omega_{m}^{2}\ln{(\omega_{m})}$ from either of these two scattering
processes (the first diagram describes forward scattering and the second one
backscattering). In each case, the integration over the internal momentum
${\boldsymbol{q}}_{1}$ (assuming $|{\boldsymbol{q}}_{1}|\ll k_{F}$) and
corresponding frequency in the blue boxes in Fig. 4(b) produces a Landau
damping term $\sim\omega_{m}^{\prime}/q$ as the leading dynamical
contribution, where $q=|{\boldsymbol{q}}|$ and $\omega_{m}^{\prime}$ are the
momentum and the frequency of the remaining internal fermion in the diagrams
in Fig. 4. The integral over the angle between
${\boldsymbol{k}},{\boldsymbol{q}}$ (or, equivalently, the component of
${\boldsymbol{q}}$ parallel to ${\boldsymbol{k}}$) restricts internal
frequencies to $\omega_{m}^{\prime}\in(0,\omega_{m})$, and each diagram yields

$\displaystyle\Sigma_{\omega^{2}}^{(1)}(\omega_{m})\propto\int_{0}^{\omega_{m}}d\omega_{m}^{\prime}\int_{\omega^{\prime}_{m}/v_{F}}^{{\omega_{\text{FL}}}/v_{F}}dq\frac{\omega_{m}^{\prime}}{q}\propto\omega_{m}^{2}\ln(\omega_{m})\
,$ (10)

as in Eq. (8).

Relevant $q$ in the integral are of order $\omega^{\prime}_{m}/v_{F}$ (to
logarithmic accuracy), and relevant $\omega^{\prime}_{m}$ are of order of
external $\omega_{m}$, which we set to be the smallest scale in the problem.
Then relevant $q^{2}$ are much smaller than $M^{2}$, and hence the static part
of $D(\omega_{m},q)$ can be approximated by $1/M^{2}$, i.e., at this order
static interaction can be treated as momentum-independent. This explains why
the prefactor in Eq. (8) is $\lambda/{\omega_{\text{FL}}}\propto 1/M^{4}$ .

##  III Structure and effective dimensionality of higher order diagrams

There are multiple self-energy diagrams at higher loop orders, but we have two
tuning knobs to single out the most important ones: small external
$\omega_{m}$ and large $N$. Let us focus on large $N$ first. At the QCP, the
diagrams with the highest power of $N$ are the planar ones, as discussed in
Refs. [33, 25, 12]. We show below that the same holds away from a QCP. We
discuss the general structure of planar diagrams in Appendix D.

We compute the self-energy order-by-order in the loop expansion on the
Matsubara axis, and then convert the result onto the real axis. We associate
the full dynamical $D(\omega_{m},{\boldsymbol{q}})$ to each bosonic propagator
in the diagrams and absorb the $-i\lambda\omega_{m}$ term in the self-energy
into the bare fermionic propagator.

The first planar diagrams beyond one-loop appear at the three-loop order.
There are two such diagrams, we show them in Figs. 5(a),(b). Each diagram can
be computed explicitly (App C), and the result is (for $\omega_{m}>0$)

$\displaystyle\Sigma^{(3)}_{\omega^{2},\text{a}}(\omega_{m})$
$\displaystyle=i\frac{1}{\pi^{2}}\frac{\lambda^{2}}{1+\lambda}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\times(0.56329\ldots)\
,$ (11) $\displaystyle\Sigma^{(3)}_{\omega^{2},\text{b}}(\omega_{m})$
$\displaystyle=i\frac{1}{2\pi}\frac{\lambda^{2}}{1+\lambda}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\ln^{2}\left(\frac{{\omega_{\text{FL}}}}{\omega_{m}}\right)\
.$ (12)

Comparing these two terms with the one-loop result, Eq. (8), we find that the
power of $N$ is the same, and there is an additional factor
$\lambda/(1+\lambda)$. Besides,
$\Sigma^{(3)}_{\omega^{2},\text{a}}(\omega_{m})$ contains no logarithm, while
$\Sigma^{(3)}_{\omega^{2},\text{b}}(\omega_{m})$ contains
$\ln^{2}{(\omega_{m})}$.

Figure 5: Three-loop diagrams. Labels indicate four-momenta (a) Planar diagram
in the forward scattering channel (b) Planar diagram in the backscattering
channel (c) Non-planar diagram.

For comparison, consider one of the non-planar diagrams, like the one in Fig.
5(c). Evaluating the corresponding integrals, we obtain

$\displaystyle\Sigma^{(3)}_{\text{c}}(\omega_{m})=-i\frac{2}{N}\frac{\lambda}{(1+\lambda)\pi}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\ln\left(\frac{{\omega_{\text{FL}}}}{\omega_{m}}\right)\
.$ (13)

Comparing this result with Eq. (8), we see that
$\Sigma^{(3)}_{\text{c}}(\omega_{m})$ contains an additional factor $1/N$.

We now look more closely at the two planar diagrams and explore another knob:
small external $\omega_{m}$. We remind the reader that the
$\omega^{2}_{m}\ln{(\omega_{m})}$ term in the one-loop self-energy, Eq. (8),
comes from forward scattering and backscattering. The issue we want to address
is whether the self-energies $\Sigma^{(3)}_{\omega^{2},\text{a}}(\omega_{m})$
and $\Sigma^{(3)}_{\omega^{2},\text{b}}(\omega_{m})$ can be cast into the same
form as Eq. (8) with dressed static forward scattering and backscattering
amplitudes. Simple experimentation shows that this holds if two of the three
internal momenta and frequencies in Fig. 5 (a),(b), (either $q$ and $q_{1}$,
or $q$ and $q_{2}$, where $q=(\omega^{\prime}_{m},{\boldsymbol{q}})$), scale
with $\omega_{m}$ and hence are infinitesimally small at $\omega_{m}\to 0$. At
a QCP, $\omega_{m}$ is the only scale in the problem, and all internal
momenta/frequencies in the planar diagrams are necessary infinitesimally small
at $\omega_{m}\to 0$. However, in a Fermi liquid there exists an internal
momentum scale $M$ (energy scale $v_{F}M$), and some internal momenta may be
of order $M$. We illustrate this in Fig. 6, where a blue box labels a dressed
vertex.

Figure 6: Three-loop diagrams as in Fig. 5, with the dressed vertices marked
by blue boxes.

The computation of the dressed vertices at vanishing external $q$ and $q_{2}$
is straightforward and we skip the details. For the planar diagram of Fig.
6(b), we find that, to logarithmic accuracy, the result can be expressed via
the dressed static backscattering amplitude, i.e., the sum of the
backscattering part of $\Sigma^{(1)}_{\omega^{2}}(\omega_{m})$ and
$\Sigma^{(3)}_{\omega^{2},\text{b}}(\omega_{m})$ can be re-expressed as

$\displaystyle\Sigma_{\omega^{2},\text{bs}}(\omega_{m})\equiv$ (14)
$\displaystyle
i\text{sign}(\omega_{m})\frac{2\lambda}{\pi{\omega_{\text{FL}}}}\int_{0}^{|\omega_{m}|}d\omega_{m}^{\prime}\omega_{m}^{\prime}\int_{\omega_{m}^{\prime}}^{{\omega_{\text{FL}}}}\frac{dz}{z}\left[\frac{M^{2}}{g}\Gamma_{\text{bs}}(z)\right]^{2},$

where

$\Gamma_{\text{bs}}(z)=-\frac{g}{M^{2}}\left[1+\frac{1}{2}\lambda
Z\ln\left(\frac{{\omega_{\text{FL}}}}{z}\right)+...\right]$ (15)

is the dressed backscattering amplitude and $Z=1/(1+\lambda)$ is the
quasiparticle spectral weight. However, if we go beyond logarithmic accuracy,
we find that $\Sigma^{(3)}_{\omega^{2},\text{b}}(\omega_{m})$ contains
contributions that come from $|{\boldsymbol{q}}|\sim M$ and cannot be
expressed via the backscattering vertex. Only very close to a QCP, where $M$
is vanishingly small, these non-logarithmic terms can be absorbed into the
dressed backscattering amplitude.

For the diagram of Fig. 6(a), which is a candidate for the forward-scattering
contribution, the dressed vertex (the blue box) vanishes at external
$q,q_{2}=0$, and to reproduce $\Sigma^{(3)}_{\omega^{2},\text{a}}(\omega_{m})$
one has to keep $|{\boldsymbol{q}}_{2}|\sim M$. This implies that
$\Sigma^{(3)}_{\omega^{2},\text{a}}(\omega_{m})$ does not come from forward
scattering. This again holds at a finite $M$. The vanishing of the dressed
vertex at $q,q_{2}=0$ also explains why the three-loop self-energy
$\Sigma^{(3)}_{\omega^{2},\text{a}}(\omega_{m})$ does not contain a logarithm.

We therefore see that in a Fermi liquid, only planar diagrams survive in the
limit $N\to\infty$, but only a portion of the self-energy from these diagrams
comes from forward scattering and backscattering. The converse is also true:
one can easily verify that the self-energy
$\Sigma^{(3)}_{\text{a}}(\omega_{m})$ from the non-planar diagram in Fig. 6(c)
does come from backscattering. However, this vertex renormalization is small
in $1/N$.

Below we keep $M$ finite and compute the full self-energy to all orders in the
interaction to logarithmic accuracy. To this accuracy, we can neglect
renormalizations from “near-forward-scattering” and “near-back-scattering”
processes and compute the renormalization of the backscattering amplitude
keeping only the largest power of $\ln{(\omega_{m})}$ at each order in the
loop expansion. The fact that the self-energy can be obtained from the
renormalized backscattering amplitude (to the logarithmic accuracy) again
implies that the problem is effectively one-dimensional: the relevant internal
fermions have momenta $\pm{\boldsymbol{k}}$ up to small deviations of order
$\omega$, i.e., they only move along a single direction. We note that for this
calculation we do not need to set $N\to\infty$ as the $1/N$ terms are outside
the logarithmic accuracy.

##  IV The full result for the self-energy to all orders in the interaction

Let us now compute the fully renormalized backscattering contribution to the
self-energy by extracting the most logarithmically singular contributions to
$\Sigma_{\omega^{2},\text{bs}}$ at each loop order. One can straightforwardly
verify that these contributions arise from the diagrams shown in Fig. 7, which
particle-particle (Cooper) “bubbles”. We label them as “Cooper diagrams”.

Figure 7: Diagram contributing to $\Sigma(\omega)$ with 4 Cooper bubbles.
Labels indicate four-momenta, interaction lines are RPA dressed.

An example of a Cooper diagram is shown in Fig. 7. A generic diagram with $n$
Cooper bubbles contains $\ln({\omega_{\text{FL}}}/|\omega_{m}|)^{n}$. Our goal
is to sum up this series. We will see that the series is not geometrical,
because the interaction is momentum-dependent.

We proceed with the evaluation of the Cooper diagrams. An inspection of these
diagrams yields the following general strategy for the calculation of the
$O(\omega^{2}_{m})$ term in the self-energy to logarithmic accuracy (see App.
E for details): we select one cross-section, from which we take the Landau-
damping term (this gives the overall $\omega_{m}^{2}$ in
$\Sigma_{\omega^{2},\text{bs}}$) and sum up series of Cooper logarithms on
both sides of this cross-section, which produces a factor
$\Gamma^{2}_{\text{bs}}$. For the calculation of the Landau damping in the
selected cross-section with internal momentum ${\boldsymbol{p}}_{i}$ in the
Cooper bubble, we set ${{\boldsymbol{q}}}$ in Fig. 7 nearly transverse to the
external ${{\boldsymbol{k}}}$ and ${\boldsymbol{p}}_{i}$ nearly parallel to
${{\boldsymbol{k}}}$. For the other Cooper bubbles, we only use a static
interaction between fermions on the Fermi surface,

$\displaystyle D(p_{1}-p_{2})\simeq-
gk^{2}_{F}\left[\epsilon^{2}+4\sin^{2}\\!\left(\frac{\phi_{1}-\phi_{2}}{2}\right)\right]\
,$ (16)

where $\phi_{i}$ are the angles of ${\boldsymbol{p}}_{i}$ measured relative to
${\boldsymbol{k}}$. We evaluate each particle-particle bubble
$\Pi_{\text{pp}}$ to logarithmic accuracy and for
$v_{F}q\gg\omega_{m}^{\prime}$, where $\omega^{\prime}_{m}$ is the frequency
component of $q$ in Fig. 7. Under these conditions, the integration of the two
fermionic propagators in the particle-particle bubble with $q-p_{j}$ and
$p_{j}$ over internal frequency $\omega_{jm}$ and over
$|{\boldsymbol{p}}_{j}|$ yields (see App. A)

$\displaystyle\Pi_{\text{pp}}(|{\boldsymbol{q}}|,\phi_{j})=Z\rho\ln\left(\frac{{\omega_{\text{FL}}}}{v_{F}|{\boldsymbol{q}}||\sin{\phi_{j}}|}\right)\
.$ (17)

We assume and verify that typical values of all $\phi_{j}$ are of order
$\epsilon$. To logarithmic accuracy, we can then approximate
$\Pi_{\text{pp}}(|{\boldsymbol{q}}|,\phi_{j})\simeq\Pi_{\text{pp}}(z)=Z\rho\ln({\omega_{\text{FL}}}/z)$,
where $z=v_{F}|{\boldsymbol{q}}|\epsilon$. Then the remaining integrations
over $\phi_{j}$ in the Cooper ladder involve only the interactions. This leads
to the following expression for the backscattering amplitude, with
$L(z)=\ln({\omega_{\text{FL}}}/z)$:

$\displaystyle\Gamma_{\text{bs}}(z)\equiv-\frac{g}{k_{F}^{2}}\sum_{n=0}^{\infty}\left(\frac{g\rho
ZL(z)}{k^{2}_{F}}\right)^{n}\cdot\frac{1}{(2\pi)^{n}}\int_{0}^{2\pi}d\phi_{1}d\phi_{2}\ldots
d\phi_{n}\frac{1}{\epsilon^{2}+4\sin^{2}\left(\frac{\phi_{1}}{2}\right)}\times\frac{1}{\epsilon^{2}+4\sin^{2}\left(\frac{\phi_{2}-\phi_{1}}{2}\right)}\times\ldots$
(18)
$\displaystyle\times\frac{1}{\epsilon^{2}+4\sin^{2}\left(\frac{\phi_{n}-\phi_{n-1}}{2}\right)}\times\frac{1}{\epsilon^{2}+4\sin^{2}\left(\frac{\phi_{n}}{2}\right)}\
.$

To leading order in $\epsilon$, the integrand can be approximated as

$\displaystyle\frac{1}{(2\pi)^{n}\epsilon^{n+1}}\int_{-\infty}^{\infty}d\phi_{1}\ldots
d\phi_{n}\frac{\epsilon}{\epsilon^{2}+\phi_{1}^{2}}\times\frac{\epsilon}{\epsilon^{2}+(\phi_{1}-\phi_{2})^{2}}\times\frac{\epsilon}{\epsilon^{2}+(\phi_{2}-\phi_{3})^{2}}\times\ldots\times\frac{\epsilon}{\epsilon^{2}+(\phi_{n-1}-\phi_{n})^{2}}\times\frac{\epsilon}{\epsilon^{2}+\phi_{n}^{2}}$
$\displaystyle=\frac{1}{(2\pi)^{n}\epsilon^{n+2}}\times\frac{\pi^{n}}{(n+1)}=\frac{1}{n+1}\frac{1}{(2\epsilon)^{n}}\times\frac{k^{2}_{F}}{M^{2}}\
.$ (19)

The integrals in Eq. (IV) were solved by going to Fourier space, where the
convolutions of Lorentzians become products of exponentials. The expression
for the backscattering amplitude then reduces to

$\displaystyle\Gamma_{\text{bs}}(z)\simeq-\frac{g}{M^{2}}\sum_{n=0}^{\infty}\left(\tilde{\lambda}L(z)\right)^{n}\frac{1}{n+1},\quad\tilde{\lambda}\equiv\lambda
Z=\frac{\lambda}{1+\lambda}\ .$ (20)

We see that $\tilde{\lambda}$ becomes the effective coupling controlling the
perturbation theory. As long as $\tilde{\lambda}L(z)<1$, the summation in Eq.
(20) converges, and we obtain

$\displaystyle\Gamma_{\text{bs}}(z)=\frac{g}{M^{2}}\left(\frac{\ln\left(1-\tilde{\lambda}L(z)\right)}{\tilde{\lambda}L(z)}\right)\
.$ (21)

The full self-energy from backscattering then becomes

$\displaystyle\Sigma_{\omega^{2},\text{bs}}(\omega_{m})=i\text{sign}(\omega_{m})\frac{\lambda}{\pi}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\int_{\omega_{m}}^{{\omega_{\text{FL}}}}\frac{dz}{z}\left[\frac{M^{2}}{g}\Gamma_{\text{bs}}(z)\right]^{2}$
$\displaystyle=i\text{sign}(\omega_{m})\frac{\lambda}{\pi}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\int_{\omega_{m}}^{{\omega_{\text{FL}}}}\frac{dz}{z}\left(\frac{\ln\left(1-\tilde{\lambda}L(z)\right)}{\tilde{\lambda}L(z)}\right)^{2}\
.$ (22)

At $\tilde{\lambda}\rightarrow 0$, we get

$\displaystyle\Sigma_{\omega^{2},\text{bs}}(\omega_{m})=i\text{sign}(\omega_{m})\frac{\lambda}{\pi}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\ln\left(\frac{{\omega_{\text{FL}}}}{|\omega_{m}|}\right)\
.$ (23)

This is exactly a half of the one-loop result, which is expected, as Eq. (IV)
only contains the backscattering contribution. The weak-coupling result (23)
is only valid for $\lambda\ln({\omega_{\text{FL}}}/|\omega_{m}|)\ll 1$; for
small frequencies, $\lambda\ln({\omega_{\text{FL}}}/|\omega_{m}|)$ becomes
$O(1)$, and one has to solve the full integral (IV) instead.

###  IV.1 Repulsive pairing interaction

We first analyze Eq. (IV) for a toy model, in which we assume that
${\tilde{\lambda}}$ is negative, i.e., that the pairing interaction is
repulsive. In this case, the series in Eq. (20) converges for all $z$, even
the smallest ones. The integral over $z$ in Eq. (IV) can then be evaluated
exactly, and the result is

$\displaystyle\Sigma_{\omega^{2},\text{bs}}(\omega_{m})$
$\displaystyle=i\text{sign}(\omega_{m})\frac{\omega^{2}_{m}}{Z\pi{\omega_{\text{FL}}}}f_{\text{bs;r}}(\ell)\
,$ (24)

where

$\ell=|\tilde{\lambda}|\ln({\omega_{\text{FL}}}/|\omega_{m}|)$ (25)

and

$f_{\text{bs;r}}(\ell)=-\frac{(1+\ell)\ln^{2}(1+\ell)}{\ell}-2\text{Li}_{2}(-\ell)\
,$ (26)

with $\text{Li}_{2}$ the Polylogarithm. The forward scattering contribution,
in the same units, is

$\displaystyle\Sigma_{\text{fw}}(\omega_{m})=i\text{sign}(\omega_{m})\frac{\omega^{2}_{m}}{Z\pi{\omega_{\text{FL}}}}f_{\text{fs}}(\ell),\quad
f_{\text{fs}}(\ell)=\ell\ .$ (27)

A plot of $f_{\text{bs;r}},f_{\text{fs}}$ is shown in Fig. 8. For small
$\ell$, i.e., fairly large frequencies, $f_{\text{bs;r}}=f_{\text{fs}}$ as
expected. For smaller frequencies (increasing $\ell$), $f_{\text{bs;r}}(\ell)$
does not grow linearly, but is bounded, and can be approximated by the
limiting form

$\displaystyle
f_{\text{bs;r}}(\ell)=\frac{\pi^{2}}{3}-\frac{\ln^{2}(\ell)}{\ell}+\mathcal{O}\left(\frac{\ln(\ell)}{\ell}\right)\
.$ (28)

Therefore, for a repulsive pairing interaction, the backscattering rate
incures a logarithmic suppression compared to forward scattering; at large
$\ell$ the full rate is reduced to the forward scattering part, i.e., by a
half compared to the one-loop result.

Upon analytical continuation to the real axis, $i\omega_{m}\to\omega+i\delta$,
Eq. (24) becomes the expression for the scattering rate at a low frequency

$\displaystyle{\text{Im}}\left[\Sigma_{\text{bs;r}}^{R}(\omega)\right]$
$\displaystyle=-\frac{\omega^{2}}{Z\pi{\omega_{\text{FL}}}}f_{\text{
bs;r}}(\ell)\ ,$ (29)

where now
$\ell=|{\tilde{\lambda}}|\ln\left({\omega_{\text{FL}}}/|\omega|\right)$.

Figure 8: The functions $f_{\text{bs;r}}(\ell)$ and $f_{\text{fs}}(\ell)$ for
the toy model with repulsive pairing interaction.

###  IV.2 Attractive pairing interaction

Let us now consider the physical case of attractive pairing interaction,
$\tilde{\lambda}>0$. Now we have

$\displaystyle\Sigma_{\omega^{2},\text{bs}}$
$\displaystyle=i\text{sign}(\omega_{m})\frac{\omega^{2}_{m}}{Z\pi{\omega_{\text{FL}}}}f_{\text{bs;a}}(\ell)\
,$ $\displaystyle f_{\text{bs;a}}(\ell)$
$\displaystyle=\frac{(\ell-1)\ln^{2}(1-\ell)}{\ell}+2\text{Li}_{2}(\ell)\ ,$
(30)

and $f_{\text{bs;a}}(\ell)$ is well-defined only for $\ell\leq 1$. At larger
$\ell$ the results must be modified by including superconductivity, as we show
below. A plot of $f_{\text{bs;a}}(\ell)$ is presented in Fig. 9. Again, for
small $\ell$, $f_{\text{bs;a}}\simeq\ell$, but at $\ell\lesssim 1$ it grows
faster. In the limit $\ell\rightarrow 1$,

$\displaystyle
f_{\text{bs;a}}(\ell)=\frac{\pi^{2}}{3}-(1-\ell)\ln^{2}(1-\ell)+\mathcal{O}\left(\ln(1-\ell)(1-\ell)\right).$
(31)

$f_{\text{bs;a}}(1)=\pi^{2}/3$ is finite, but the slope
$df_{\text{bs;a}}(\ell)/d\ell$ is logarithmically divergent at
$\ell\rightarrow 1$. As a curiosity, we note that $f_{\text{bs;r}}$ and
$f_{\text{bs;a}}$ are related as

$f_{\text{bs;r}}(\ell)=f_{\text{bs;a}}\left(\frac{\ell}{1+\ell}\right)\ .$
(32)

This can be derived by a simple substitution in the integral defining
$f_{\text{bs;a}}$.

Figure 9:  The function $f_{\text{bs;a}}$ to leading order in $\epsilon$, from
Eq. (IV.2). $\ell=1$ corresponds to $\omega=\Delta_{0}$. The inset shows the
modification of $f_{\text{bs;a}}$ due to extra contributions beyond the
leading order in $\epsilon$, see Eq. (39).

Let us look at the behavior of $f_{\text{bs;a}}(\ell)$ as $\ell\rightarrow 1$
more carefully. We recall that $f_{\text{bs;a}}(\ell)$ has been obtained using
the expression for the backscattering amplitude $\Gamma_{\text{bs}}(z)$, Eq.
(21), valid to the leading order in $\epsilon=M/k_{F}$. We now check whether
the dependence of $f_{\text{bs;a}}(\ell)$ on $1-\ell$ gets modified if we
compute $\Gamma_{\text{bs}}(z)$ beyond $\mathcal{O}(\epsilon)$. The reasoning
for doing this is the following: to leading order in $\epsilon$, partial
amplitudes in pairing channels with different angular momentum are all equal,
and this gives rise to the appearance of the factor $1/(n+1)$ in the series
for $\Gamma_{\text{bs}}(z)$ in (21). Beyond the order $\mathcal{O}(\epsilon)$,
partial pairing amplitudes do differ, and the largest one is in the $s$-wave
channel. If we would only include this channel and ignore all other pairing
components, we would find that the logarithmical series becomes geometrical
and $f_{\text{bs;a}}(\ell)\propto 1/(1-\ell)$ (see Eq. (38) below). For a
small but finite $\epsilon$, we therefore expect a crossover from
$f_{\text{bs;a}}(\ell)=\pi^{2}/3-(1-\ell)\ln^{2}(1-\ell)+...$ at $1-\ell\leq
1$ to $f_{\text{bs;a}}(\ell)\propto 1/(1-\ell)$ at the smallest $1-\ell$. To
describe this crossover, we now evaluate $\Gamma_{\text{bs}}(z)$ in next-to-
leading order in $\epsilon$. This can be achieved by angular momentum
decomposition. Namely, the effective interaction on the Fermi surface,

$\displaystyle V(\phi)$
$\displaystyle=\tilde{\lambda}\frac{2\epsilon}{\epsilon^{2}+4\sin^{2}(\phi/2)}\
,$ (33)

can be decomposed into angular momenta as

$\displaystyle V_{m}=\int_{0}^{2\pi}\frac{d\phi}{2\pi}V(\phi)\exp(-i\phi m)\
.$ (34)

To a very good approximation (see App. F)

$\displaystyle V_{m}\approx\tilde{\lambda}\exp(-|m|\epsilon)\ .$ (35)

At $\epsilon\rightarrow 0$, all components $V_{m}$ become degenerate. However,
for a nonzero $\epsilon$, this is not quite the case yet, and the $s$-wave
component $V_{0}\simeq\tilde{\lambda}$ is largest. A straightforward analysis
then shows that the series for $\Gamma_{\text{bs}}(z)$ can be rewritten as

$\displaystyle\Gamma_{\text{bs}}(z)$
$\displaystyle=-\frac{1}{\rho}\sum_{m}V_{m}\sum_{n=0}^{\infty}\left[V_{m}L(z)\right]^{n}$
$\displaystyle=-\frac{1}{\rho}\frac{V_{0}}{1-V_{0}L(z)}-\frac{2}{\rho}\sum_{m=1}^{\infty}\frac{V_{m}}{1-V_{m}L(z)},$
(36)

where in the second line we singled out the $s$-wave part. The remaining sum
in Eq. (36) can be evaluated using the Euler-Maclaurin formula, and the result
is

$\displaystyle\Gamma_{\text{bs}}(z)\simeq$ (37)
$\displaystyle-\frac{g}{M^{2}}\left[-\frac{\ln\left(1-\tilde{\lambda}L(z)+\epsilon\right)}{\tilde{\lambda}L(z)}+\frac{\epsilon}{2}\frac{1}{1-\tilde{\lambda}L(z)}\right].$

Eq. (37) shows that the $\ln\left(1-\tilde{\lambda}L(z)\right)$ dependence of
the backscattering amplitude is the consequence of the near degeneracy of all
partial pairing amplitudes. Once this degeneracy is lifted at a finite
$\epsilon$, the backscattering amplitude develops a conventional $s$-wave
pole, albeit with a small weight $\epsilon/2$. Inserting this
$\Gamma_{\text{bs}}(z)$ into the integral for $\Sigma_{\omega^{2},\text{bs}}$,
Eq. (IV), we obtain,

$\displaystyle\Sigma_{\omega^{2},\text{bs}}=i\text{sign}(\omega_{m})\frac{\omega_{m}^{2}}{Z\pi{\omega_{\text{FL}}}}f^{*}_{\text{bs;a}}(\ell),$
(38)

where

$\displaystyle f^{*}_{\text{bs;a}}(\ell)\simeq
f_{\text{bs;a}}(\ell)+\frac{\epsilon^{2}}{4(1-\ell)}\ .$ (39)

At $1-\ell\lesssim\epsilon$,

$\displaystyle f^{*}_{\text{bs;a}}(\ell)\simeq$
$\displaystyle\frac{\epsilon^{2}}{4(1-\ell)}+\frac{\pi^{2}}{3}+{\epsilon}\ln{(1-\ell+\epsilon)}\ln{(1-\ell)}$
$\displaystyle-(1-\ell+\epsilon)\ln^{2}(1-\ell+\epsilon)\ .$ (40)

$f_{\text{bs;a}}(\ell)$ and $f^{*}_{\text{bs;a}}(\ell)$ are plotted in the
inset of Fig. 9. We see that the self-energy diverges as $1/(1-\ell)$ at
$\ell\to 1$, as is expected for an $s$-wave superconductor [47, 48], but in
our case the divergent term appears with the prefactor $\epsilon^{2}$ 444We
note that the prefactor of the $s$-wave term gets renormalized ($N\rightarrow
N-1$ in the expression for ${\omega_{\text{FL}}}$) by ladder diagrams without
a loop, in which a line, expressing an outgoing fermion, is attached to the
upper part of the Cooper ladder [57]. Such a diagram has the same number of
logarithms, but involves a large momentum transfer $2k_{F}$ and therefore
renormalizes the $s$-wave part only.

Like before, upon analytical continuation to the real axis,
$i\omega_{m}\to\omega+i\delta$, Eq. (38) becomes the expression for the
backscattering contribution to Im[$\Sigma^{R}(\omega)$]:

$\text{Im}\left[\Sigma_{\text{bs;a}}^{R}(\omega)\right]=-\frac{\omega^{2}}{Z\pi{\omega_{\text{FL}}}}f^{*}_{\text{
bs;a}}(\ell)\ ,$ (41)

where on the real frequency axis
$\ell={\tilde{\lambda}}\ln({{\omega_{\text{FL}}}/|\omega|})$. Eq. (41) shows
that the scattering rate increases when $\omega$ decreases ($\ell$ increases)
and becomes singular at $\ell=1$.

The enhancement of $\text{Im}[\Sigma_{\text{bs;a}}^{R}(\omega)]$ near $\ell=1$
can be interpreted as follows: to the lowest order in the interaction, the
rate comes from the decay of a particle into two particles and one hole;
$\text{Im}[\Sigma_{\text{bs;a}}^{R}(\omega)]$ measures the phase space for
such a process. By energy conservation, the three outgoing states must have
energies smaller than $\omega$. When Cooper pair formation sets in, the two
electrons in the particle-particle ladder form a tightly bound Cooper pair,
and there are only two decay products left. Therefore, the phase space
restriction is less severe for small $\omega$, and the scattering rate
increases.

Before concluding this section, we discuss the range of validity of Eqs. (38)
and (41). The expression for the self-energy has been obtained with
logarithmic accuracy, i.e., at each loop order we neglected terms with the
same power of ${\tilde{\lambda}}$, but a smaller power of
$\ln({\omega_{\text{FL}}}/|\omega|)$. This is valid when ${\tilde{\lambda}}\ll
1$ and $|\omega|\ll{\omega_{\text{FL}}}$. The condition on $\omega$ does not
pose a restriction as throughout the paper we assume that
${\omega_{\text{FL}}}$ is finite and focus on the self-energy at the smallest
$\omega$. The condition ${\tilde{\lambda}}=\lambda/(1+\lambda)\ll 1$ implies
weak coupling, $\lambda\ll 1$. Because $\lambda=g/(4\pi v_{F}M)\sim
g/(E_{F}\epsilon)$, we need to keep $g/E_{F}$ small to satisfy both
$\lambda\ll 1$ and $\epsilon\ll 1$. Weak coupling is advantageous to us
because, for ${\tilde{\lambda}}\ll 1$, there is a wide range of frequencies
where on the one hand $|\omega|\ll{\omega_{\text{FL}}}$ and on the other
$\ell={\tilde{\lambda}}\ln({\omega_{\text{FL}}}/|\omega|)<1$, hence Eqs. (38)
and (41) are applicable. This is indeed the consequence of the fact that
superconductivity at weak coupling emerges at an exponentially small
$\omega\sim\Delta_{0}={\omega_{\text{FL}}}\exp(-1/{\tilde{\lambda}})$. We
emphasize that this only holds in a Fermi liquid regime, at small enough
$g/E_{F}$. Very close to a QCP, $\lambda$ necessarily becomes large, and the
whole Fermi liquid range $\omega<{\omega_{\text{FL}}}$ falls into
$\omega<\omega_{\text{IN}}\sim\lambda^{3}{\omega_{\text{FL}}}$. The pairing
instability in this case emerges at a larger $\omega\leq\omega_{\text{IN}}$
(Refs. [38, 39, 43, 53]), hence there is no range of applicability for our
analysis.

##  V Self-energy in the superconducting state

We now show that the singularity of the self-energy at $\ell=1$ gets
regularized once we take into consideration the fact that the ground state is
actually an $s$-wave superconductor. In practical terms, superconductivity
implies that for the calculation of the backscattering amplitude and the
fermionic self-energy one has to use normal and anomalous Green’s functions

$\displaystyle G(\omega_{m},{\boldsymbol{k}})$
$\displaystyle=-\frac{iZ^{-1}\omega_{m}+\xi_{\boldsymbol{k}}}{Z^{-2}\omega^{2}_{m}+\xi_{\boldsymbol{k}}^{2}+\Delta_{0}^{2}}\
,$ (42) $\displaystyle F(\omega_{m},{\boldsymbol{k}})$
$\displaystyle=\frac{\Delta_{0}}{Z^{-2}\omega^{2}_{m}+\xi_{\boldsymbol{k}}^{2}+\Delta_{0}^{2}}\
.$

The three key processes that determine the decay of a quasiparticle in a
superconductor are (i) scattering by amplitude and phase fluctuations of the
superconducting order parameter, (ii) scattering by a (potential) resonance
mode in the nematic $D(\omega,{{\boldsymbol{q}}})$ at (real)
$\omega=\omega_{\text{res}}<3\Delta_{0}$ and (iii) scattering due to a decay
into particles and holes (this process starts at $\omega=3\Delta_{0}$). For a
charged superconductor, scattering by phase fluctuations is affected by long-
range Coulomb interaction, which converts a Goldstone phase fluctuation mode
into a plasmon (still gapless in 2D). However, we neglected the effects of
long-range Coulomb interaction in the calculations assuming a normal state at
$T=0$, and for consistency we also have to neglect this interaction in a
superconductor.

In general, all three processes are important at
$\omega_{m}\gtrsim\Delta_{0}$, and the full-fledged evaluation of
$\Sigma(\omega_{m}\gtrsim\Delta_{0})$ is a complicated endeavor. Since our
primary goal is to understand how superconductivity affects singular behavior
of the self-energy that we found in the previous section, we limit ourselves
to the scattering processes which cut off the singularity.

We will directly focus on the imaginary part of the backscattering self-energy
in real frequencies, $\text{Im}[\Sigma_{\text{bs;a}}^{R}(\omega)]$. Consider
first the contribution to $\text{Im}[\Sigma_{\text{bs;a}}^{R}(\omega)]$
arising from the sum of all angular momentum channels, Eq. (IV.2). This
expression shows a slope singularity at $\ell=1$, however, since it was
obtained with the logarithmic accuracy, it is only valid for $\ell\lesssim
1-\tilde{\lambda}$. At the boundary of this regime, the self-energy (IV.2) is
of the same order as the one-loop result, Eq. (9). Therefore, to estimate this
part of the self-energy at $1-\ell<\tilde{\lambda}$, it is sufficient to re-
evaluate the one-loop diagram of Fig. 2(b) in the superconducting state. We
find (see App. G for details)

$\displaystyle-\text{Im}\left[\Sigma_{\text{bs;a}}^{R}(\omega)\right]\sim$
(43)
$\displaystyle\frac{\Delta_{0}^{2}}{{\omega_{\text{FL}}}}\tilde{\lambda}\ln\left(\frac{{\omega_{\text{FL}}}}{\Delta_{0}}\right)\times\sqrt{\frac{\omega-3\Delta_{0}}{\Delta_{0}}}\theta(\omega-3\Delta_{0})\
.$

We have suppressed the dependence on $Z\simeq 1$. The scattering rate of Eq.
(43) starts at $\omega=3\Delta_{0}$, which is transparent as the three
scattering products, two quasiparticles and a quasihole, all have an energy
gap $\Delta_{0}$. At $\omega-3\Delta_{0}\simeq\Delta_{0}$, i.e., at
$1-\ell\sim\tilde{\lambda}$, Eq. (43) becomes of the same order as the normal
state result, Eq. (IV.2). We see therefore that the portion of the self-energy
that comes from all scattering channels smoothly evolves from Eq. (IV.2) to
Eq. (43). We also computed the contribution to the self-energy from the
resonance in $D(\omega,{\boldsymbol{q}})$ in the superconducting state. The
resonance develops at
$\omega_{\text{res}}=\Delta_{0}\times\left[1+\mathcal{O}(\epsilon k_{F}/(N\rho
g)^{1/2})\right]$. Adding this contribution, we find that
$\text{Im}[\Sigma_{\text{bs;a}}^{R}(\omega)]$ extends down to
$\omega_{\text{res}}$, but near $\omega\sim 3\Delta_{0}$ it is given by Eq.
(43) up to small corrections. The form of
$\text{Im}[\Sigma_{\text{bs;a}}^{R}(\omega)]$ at
$\omega_{\text{res}}\leq\omega\leq 3\Delta_{0}$ is rather involved and we will
discuss it in a separate report. Ultimately, the complicated form of the self-
energy is tied to the abundance of low-energy collective excitations in a
superconductor with long-ranged interactions, as also recognized in Refs. [54,
55].

To find the correct cutoff of the $s$-wave pole, Eq. (38), evaluation of the
one-loop diagram is insufficient, and one needs to re-evaluate the full
backscattering amplitude $\Gamma_{\text{bs}}$ using $\Pi_{\text{pp}}=GG+FF$.
The key new effect coming from this calculation is the scattering of a
quasiparticle by massless (Goldstone) phase fluctuations of the
superconducting order parameter. Evaluating the $s$-wave component of
$\Gamma_{\text{bs}}$ and substituting the result into the self-energy, we
obtain

$\displaystyle-\text{Im}\left[\Sigma_{\text{bs;a}}^{R}(\omega)\right]\sim\frac{\epsilon^{2}}{\tilde{\lambda}}\frac{\Delta_{0}^{2}}{{\omega_{\text{FL}}}}\times\theta(\omega-\Delta_{0})\
.$ (44)

This contribution to the self-energy starts at $\omega=\Delta_{0}$, since the
phase mode is gapless. For $1-\ell\simeq\tilde{\lambda}$, i.e.,
$\omega\gtrsim\Delta_{0}$, this self-energy matches with the $s$-wave piece
$\omega^{2}/{\omega_{\text{FL}}}\times\epsilon^{2}/(1-\ell)$, obtained in the
normal state calculation, Eq. (38). A similar expression for the self-energy
at $T=0$ in a 3D superfluid has recently been computed in Ref. [56], building
on older work of Ref. [57]. The contribution from the self-energy to the
density of states of an $s$-wave superconductor has been obtained in Refs.
[47, 48].

Collecting the results, we find the full self-energy from backscattering in
the form

$\displaystyle 0<\ \ell\lesssim 1-\mathcal{O}(\tilde{\lambda}):\quad$
$\displaystyle\text{Im}\left[\Sigma_{\text{bs;a}}^{R}(\omega)\right]=-\frac{\omega^{2}}{\pi{\omega_{\text{FL}}}}f^{*}_{\text{bs;a}}(\ell),\quad
f^{*}_{\text{bs;a}}(\ell)\simeq\frac{(\ell-1)\ln^{2}(1-\ell)}{\ell}+2\text{Li}_{2}(\ell)+\frac{\epsilon^{2}}{4(1-\ell)},$
(45) $\displaystyle 1-\mathcal{O}(\tilde{\lambda})\lesssim\ \ell<1:\quad$
$\displaystyle\text{Im}\left[\Sigma_{\text{bs;a}}^{R}(\omega)\right]=-\frac{\Delta_{0}^{2}}{{\omega_{\text{FL}}}}\tilde{\lambda}\ln\left(\frac{{\omega_{\text{FL}}}}{\Delta_{0}}\right)\times\sqrt{\frac{\omega-3\Delta_{0}}{\Delta_{0}}}\theta(\omega-3\Delta_{0})-\frac{\epsilon^{2}}{\tilde{\lambda}}\frac{\Delta_{0}^{2}}{{\omega_{\text{FL}}}}\times\theta(\omega-\Delta_{0}).$
(46)

We sketch $[\Sigma_{\text{bs;a}}^{R}(\omega)]/\omega^{2}$ in Fig. 10. In panel
(a) we set $\epsilon<\tilde{\lambda}$. In this situation, the contribution
from all angular channels dominates over the $s$-wave part for all $\ell$. In
panel (b) we set $\epsilon>\sqrt{\tilde{\lambda}}$. Here the contribution from
the $s$-wave channel is the dominant one. We see that in this situation
Im$[\Sigma_{\text{bs;a}}^{R}(\omega)]/\omega^{2}$ has a sharp peak at
$\ell\leq 1$. This can be directly verified in ARPES experiments.

Note that while the imaginary part of the self-energy in real frequencies
vanishes at $|\omega|<\Delta_{0}$, the Matsubara self-energy (which can be
obtained from Quantum Monte Carlo) is non-zero. At $\omega_{m}\ll\Delta_{0}$,
$\Sigma_{\text{bs;a}}(\omega_{m})$ scales as $\omega_{m}/\Delta_{0}$. At
larger $\omega_{m}\sim\Delta_{0}$, $\Sigma_{\text{bs;a}}(\omega_{m})$ passes
through a maximum.

Figure 10: Backscattering self-energy divided by $\omega^{2}$ as a function of
the logarithmic variable $\ell$. Blue curves are the normal state result (Eq.
(45) and Fig. 9), green curves correspond to the superconducting result, Eq.
(46). The black line is a possible interpolation representing the correct
physical backscattering rate. (a) Sum of angular momentum channels dominates,
used parameters: $\epsilon=0.05,\tilde{\lambda}=0.1$. (b): $s$-wave channel
dominates, used parameters: $\epsilon=0.3,\tilde{\lambda}=0.05$.

##  VI Conclusions

In this work we computed the fermionic self-energy in a Fermi liquid near a
nematic QCP. We considered the model of fermions interacting with soft
fluctuations of a nematic order parameter, and extended it to $N$ fermionic
flavors. The leading contributions to the self-energy at $N\to\infty$ come
from planar diagrams, as was determined before at a QCP. In a Fermi liquid,
the contributions from planar diagrams contain series of logarithms. We
identified the leading logarithmic contributions at all orders as coming from
the fully dressed backscattering amplitude, i.e., they describe one-
dimensional scattering processes. We further argued that the subleading
contributions (the terms with a smaller power of $\ln(\omega)$ at each loop
order) come from scattering with a finite momentum transfer counted from
either backscattering or forward scattering.

We found that the logarithmic series are non-geometrical and summed them up
using several computational tricks. We first considered a toy model with
repulsive pairing interaction, and then a realistic model with attractive
pairing interaction. For the latter case we found two contributions to the
self-energy: one is the combined contribution from multiple pairing channels
with near-equal attraction, and the second is the special contribution from an
$s$-wave pairing channel, where the pairing interaction is a bit larger than
in other channels. We first computed the two terms assuming the normal state
and found that perturbation theory holds only above a certain frequency,
comparable to superconducting gap $\Delta_{0}$. At the edge, both
contributions become non-analytic and the one from $s$-wave channel diverges.
We then extended the analysis to include superconductivity at $T=0$ and showed
that the would-be divergencies get regularized. In particular, the divergence
in the $s$-wave channel is regularized by scattering of a fermion by phase
fluctuations of the superconducting order parameter. We obtained the full
expression for the imaginary part of the retarded self-energy in real
frequencies (the scattering rate) at $\omega\sim\Delta_{0}$ and argued that,
depending on parameters, $\text{Im}[\Sigma^{R}(\omega)]/\omega^{2}$ either
evolves smoothly at $\omega\geq\Delta_{0}$ or has a peak. This peak can be
detected in ARPES measurements.

The unconventional behavior that we found stems from the competition between
different pairing channels. In our case this holds near a nematic QCP, but
similar behavior is expected when there is a competition between multiple
pairing channels for other reasons, like, e.g., a competition between $s$\-
and $d$-wave channels in iron-based superconductors [58, 59, 60]. Adapting our
computations to a model that only allows for finite number of pairing
channels, or, e.g., only selects pairing channels with even angular momentum
yielding singlet superconductivity could be an interesting future project. In
addition, a recent Monte Carlo study of a quantum critical metal [61] observed
a strong impact of pairing fluctuation on thermodynamic observables (e.g.,
specific heat), and it would be worthwhile to elucidate how the competition
between different pairing channels shows up in thermodynamics.

As a final remark, we emphasize that our analysis has been restricted to the
isotropic case. On a lattice, the nematic form factor $f$ (which we set to
one) becomes important, separating hot (where $f\simeq 1$) and cold ($f\ll 1$)
regions on the Fermi surface. In the normal state $(\omega\gg\Delta)$, our
results carry over to the hot regions. Deep in the superconducting state
($\omega<\Delta$), the gap function will be maximal in the hot regions, and
vanish in the cold ones, similar to Ref. [49]. Accordingly, in this limit our
results for the scattering rate have to be modified to account for the near-
gapless regions.

##  VII Acknowledgments

We thank E. Berg, M. Hecker, Y. B. Kim, A. Klein, S.-S. Lee and Y. Schattner
for useful discussions. D.P. is grateful to the Max-Planck-Institute for the
Physics of Complex Systems Dresden (MPIPKS) for hospitality during the initial
stage of this project. The work by A.V.C. was supported by the Office of Basic
Energy Sciences, U.S. Department of Energy, under award DE-SC0014402. A.K. was
supported by NSF grant DMR-2037654.

##  Appendix A Particle-hole and particle-particle bubbles in the normal state

The particle-hole bubble of Fig. 2(a) reads

Figure 11: Particle-particle bubble $\Pi_{\text{pp}}$ in the form required in
Sec. IV. Labels indicate four-momenta.

$\displaystyle\Pi_{\text{ph}}(\omega_{m},{\boldsymbol{q}})=N\int_{p}\frac{1}{i\omega_{m}^{\prime}-\xi_{\boldsymbol{p}}}\frac{1}{i(\omega_{m}^{\prime}+\omega_{m})-\xi_{{\boldsymbol{p}}+{\boldsymbol{q}}}}\
,$ (47)

where the factor of $N$ comes from summation over flavors. Evaluation of
$\Pi_{\text{ph}}$ is standard (see, e.g., [23]) and we omit the details here.
Performing the $\omega_{m}^{\prime}$-integral first, and subsequently the
integrals over the linearized dispersion and angular variable, we find

$\displaystyle\Pi_{\text{ph}}(\omega_{m},{\boldsymbol{q}})=-N\rho+N\rho\cdot\frac{|\omega_{m}|}{\sqrt{\omega_{m}^{2}+(v_{F}q)^{2}}}\
.$ (48)

Next, we discuss the particle-particle bubble $\Pi_{\text{pp}}$. For the
evaluation of Sec. IV it is required in the form shown in Fig. 11(b). The
internal bubble frequency $\omega_{m}^{\prime}$ and momentum modulus
$\delta_{p}=|{\boldsymbol{p}}|-k_{F}$ appears in both interaction lines.
However, as discussed in Sec. IV, this dependence only serves to cut off the
Cooper logarithm, and can otherwise be neglected. The frequency dependence of
the interaction (arising through the Landau damping) leads to the cutoff
${\omega_{\text{FL}}}$, while the $\delta_{p}$ dependence yields a cutoff
$v_{F}M$. Since ${\omega_{\text{FL}}}\leq v_{F}M$ (see discussion above Eq.
(8) in the main text), we take into account the frequency cutoff, but
integrate over $\delta_{p}$ in infinite limits.

The interaction lines also depend on the orientation of ${\boldsymbol{p}}$,
i.e. the angular variable $\phi_{p}$. This dependence needs to be considered
in detail. Therefore, as an important auxiliary object we define the particle-
particle bubble with a general angle $\phi_{p}$ as follows:

$\displaystyle\Pi_{\text{pp}}(\omega_{m},q,\phi_{q}-\phi_{p})=$ (49)
$\displaystyle
k_{F}\int\displaylimits_{-{\omega_{\text{FL}}}}^{{\omega_{\text{FL}}}}\\!\frac{d\omega_{m}^{\prime}}{2\pi}\int\frac{d\delta_{p}}{2\pi}\frac{1}{iZ^{-1}(\omega_{m}-\omega_{m}^{\prime})-\xi_{{\boldsymbol{q}}-{\boldsymbol{p}}}}\frac{1}{iZ^{-1}\omega_{m}^{\prime}-\xi_{\boldsymbol{p}}}\
,$

where we also incorporated the one-loop renormalized quasiparticle weight
$Z^{-1}=1+\lambda$. Linearizing the dispersion, the $\delta_{p}$ integral is
solved by contour integration. One obtains

$\displaystyle\Pi_{\text{pp}}(\omega_{m},q,\phi_{q}-\phi_{p})=$ (50)
$\displaystyle-i\rho\int_{-{\omega_{\text{FL}}}}^{\omega_{\text{FL}}}d\omega^{\prime}\frac{\theta(\omega_{m}^{\prime})-\theta(\omega_{m}-\omega_{m}^{\prime})}{iZ^{-1}(\omega_{m}-2\omega_{m}^{\prime})+qv_{F}\cos(\phi_{q}-\phi_{p})}\
.$

We focus on $\omega_{m}>0$; $\omega_{m}<0$ yields the same result. We have

$\displaystyle\Pi_{\text{pp}}(\omega_{m},q,\phi_{q}-\phi_{p})=$ (51)
$\displaystyle-i\rho\int_{\omega_{m}}^{\omega_{\text{FL}}}d\omega_{m}^{\prime}\frac{1}{iZ^{-1}(\omega_{m}-2\omega_{m}^{\prime})+qv_{F}\cos(\phi_{q}-\phi_{p})}$
$\displaystyle+i\rho\int_{-{\omega_{\text{FL}}}}^{0}d\omega_{m}^{\prime}\frac{1}{iZ^{-1}(\omega_{m}-2\omega_{m}^{\prime})+qv_{F}\cos(\phi_{q}-\phi_{p})}$
$\displaystyle=Z{\rho}\ln\left(\frac{2Z^{-1}{\omega_{\text{FL}}}}{\sqrt{(Z^{-1}\omega_{m})^{2}+(v_{F}q\cos(\phi_{q}-\phi_{p}))^{2}}}\right)\
.$

The factor $2Z^{-1}$ in the numerator can be neglected with logarithmic
accuracy. In addition, since $\phi_{q}\simeq\pm\pi/2$ plus small corrections
of order $\omega_{m}/v_{F}q$, we can approximate
$\cos^{2}(\phi_{q}-\phi_{p})\simeq\sin^{2}(\phi_{p})$ (see also App. E),
resulting in the static interaction Eq. (17) of the main text.

For a momentum-independent (BCS-like) interaction, one can integrate over the
internal angle and obtains

$\displaystyle\Pi_{\text{pp}}^{\text{BCS}}(\omega_{m},q)=$ (52) $\displaystyle
Z{\rho}\ln\left(\frac{{\omega_{\text{FL}}}}{|Z^{-1}\omega_{m}|+\sqrt{(v_{F}q)^{2}+|Z^{-1}\omega_{m}|^{2}}}\right)\simeq$
$\displaystyle
Z{\rho}\ln\left(\frac{{\omega_{\text{FL}}}}{v_{F}q}\right)-\rho\frac{|\omega_{m}|}{v_{F}q}\quad\text{for}\
|\omega_{m}|\ll v_{F}q\ .$

##  Appendix B One-loop fermion self-energy

Here we detail the evaluation of the one-loop self-energy. From Fig. 2(b), it
is derived as

$\displaystyle\Sigma^{(1)}(\omega)=-g\int_{q}G(\omega_{m}-\omega_{m}^{\prime},{\boldsymbol{k}}-{\boldsymbol{q}})D(\omega_{m}^{\prime},{\boldsymbol{q}})\simeq\frac{g}{(2\pi)^{3}}\int
d\omega_{m}^{\prime}dqd\phi_{q}\frac{q}{q^{2}+M^{2}+g\Pi_{\text{ph}}(\omega_{m}^{\prime},q)}\frac{1}{i(\omega_{m}-\omega_{m}^{\prime})+v_{F}q\cos(\phi_{q})}$
(53) $\displaystyle=-i\frac{g}{(2\pi)^{2}}\int d\omega_{m}^{\prime}dq\
\text{sign}(\omega_{m}-\omega_{m}^{\prime})\frac{q}{\sqrt{(\omega_{m}-\omega_{m}^{\prime})^{2}+(v_{F}q)^{2}}}\frac{1}{q^{2}+M^{2}+g\Pi_{\text{ph}}(\omega_{m}^{\prime},q)}\
.$

with $\phi_{q}=\measuredangle({\boldsymbol{k}},{\boldsymbol{q}})$. To further
evaluate Eq. (53), we need to assess the typical frequencies and momenta.
First, we note that the frequency integral receives contributions from both
$|\omega_{m}^{\prime}|\lesssim|\omega_{m}|$ and
$|\omega_{m}^{\prime}|\gg|\omega_{m}|$. However, the high-energy region only
contributes to the analytical terms in $\Sigma^{(1)}$, as also expected on
general grounds. Let us neglect these non-universal terms, and only consider
the low-energy contribution $|\omega_{m}^{\prime}|\lesssim|\omega|$, i.e., the
internal frequencies of interest are set by the external one. We will focus on
the limit of small frequencies such that

$\displaystyle|\omega_{m}|\ll\omega_{\text{FL}}\equiv\frac{v_{F}M^{3}}{N|g|\rho}\
.$ (54)

This scale can be extracted by equating the Landau damping term
$g\Pi_{\text{ph}}(\omega_{m},q)$ with $q^{2}$ for $q=M$. For $\omega_{m}$
fulfilling Eq. (54), $g\Pi_{\text{ph}}$ is typically small compared to the
bosonic mass $M^{2}$, and the typical momenta $q$ in Eq. (54) are given by
$q\simeq M$. In the opposite limit $|\omega_{m}|\gg{\omega_{\text{FL}}}$, the
Landau damping term dominates over the mass term, and we enter the non-Fermi-
liquid regime.

To simplify the evaluation, it is also convenient to require that for typical
$q$

$\displaystyle|\omega_{m}|\ll v_{F}q\ ,$ (55)

which is guaranteed if $\omega_{m}\ll{\omega_{\text{FL}}}\leq v_{F}M$. As a
result, we can neglect the frequency dependence under the square-root in Eq.
(53), and also simplify the Landau damping as
$\Pi_{\text{ph}}(\omega_{m}^{\prime},q)\simeq
N\rho|\omega_{m}^{\prime}|/v_{F}q$. This allows for the approximation

$\displaystyle\Sigma^{(1)}(\omega_{m})\simeq$ (56)
$\displaystyle-i\text{sign}(\omega_{m})\frac{g}{2\pi^{2}v_{F}}\int_{0}^{|\omega_{m}|}d\omega_{m}^{\prime}dq\frac{1}{q^{2}+M^{2}+Ng\rho\frac{\omega_{m}^{\prime}}{v_{F}q}}\
.$

Now $\Sigma^{(1)}(\omega_{m})$ can be systematically evaluated by expanding in
the Landau damping. By simply neglecting the Landau damping, we obtain the
term linear in frequency:

$\displaystyle\mathcal{O}(\omega_{m})\\!:\quad\Sigma^{(1)}(\omega_{m},{\boldsymbol{k}})=-i\lambda\omega_{m},\quad\lambda\equiv\frac{g}{4\pi
v_{F}M}\ .$ (57)

Here, $\lambda$ is an effective dimensionless coupling constant. We do not
place any restrictions on it a priori. The next order term is of order
$\omega_{m}^{2}\ln(\omega_{m})$, and requires more careful analysis: the
expansion in the Landau damping is strictly legitimate for

$\displaystyle M^{2}\gg Ng\rho\frac{\omega_{m}^{\prime}}{v_{F}q}\Rightarrow
q\gg
q_{\text{min}}(\omega_{m}^{\prime})\equiv\frac{Ng\rho\omega_{m}^{\prime}}{v_{F}M^{2}},$
(58)

while the momenta $q\lesssim q_{\text{min}}(\omega_{m}^{\prime})$ only
contribute to the $\omega_{m}^{2}$ term. Since
$q_{\text{min}}(\omega_{m}^{\prime})$ only enters a logarithm, we can replace
it by $q_{\text{min}}(|\omega_{m}|)$ with logarithmic accuracy (i.e.,
neglecting $\mathcal{O}(1)$ terms compared to large logarithms). As a result:

$\displaystyle\mathcal{O}\left(\omega_{m}^{2}\ln(\omega_{m})\right)\\!:\quad\Sigma^{(1)}(\omega_{m})=$
(59) $\displaystyle
i\text{sign}(\omega_{m})\frac{g^{2}\rho}{\pi^{2}v_{F}}\int_{0}^{|\omega_{m}|}d\omega_{m}^{\prime}\int_{q_{\text{min}}(|\omega_{m}|)}^{\infty}\\!dq\frac{1}{(q^{2}+M^{2})^{2}}\frac{\omega_{m}^{\prime}}{q}\simeq$
$\displaystyle
i\text{sign}(\omega_{m})\frac{g^{2}\rho}{2\pi^{2}M^{4}v_{F}}\omega_{m}^{2}\ln\left(\frac{M}{q_{\text{min}}(|\omega_{m}|)}\right)=$
$\displaystyle
i\text{sign}(\omega_{m})\frac{2|\lambda|}{\pi}\frac{\omega_{m}^{2}}{{\omega_{\text{FL}}}}\ln\left(\frac{{\omega_{\text{FL}}}}{|\omega_{m}|}\right)\
.$

Due to the condition (54), the logarithm is indeed large. Evaluation of
$\Sigma^{(1)}(\omega_{m})$ for $\omega_{m}\gg{\omega_{\text{FL}}}$ can be
found e.g. in Ref. [23].

In Eq. (59), internal momenta $q$ are small in a logarithmic sense: one can
think about $q$ as $q\propto\omega_{m}^{\alpha}$ with $0<\alpha\ll 1$, such
that $v_{F}q\gg\omega_{m}$, but at the same time
$v_{F}q\ll{\omega_{\text{FL}}},v_{F}M$ is much smaller than the UV scales in
the problem. Thus, given an external wave-vector ${\boldsymbol{k}}$, the
internal fermions are fixed to $\pm{\boldsymbol{k}}$, and momentum variations
around these points are small in the external $\omega_{m}$; the fermions
almost move along a line.

##  Appendix C Three-loop calculations

To evaluate the three-loop diagrams, it is convenient to use patch coordinates
[62], expanding the fermionic dispersion around the points
${\boldsymbol{k}},-{\boldsymbol{k}}$ as

$\displaystyle\xi(\pm{\boldsymbol{k}}+{\boldsymbol{q}})=\pm
v_{F}q_{x}+q_{y}^{2}/(2m)\ .$ (60)

First, we compute the three-loop diagram in the forward scattering channel of
Fig. 5(a), with $k=(\omega_{m},{\boldsymbol{k}}_{F},0),\
q=(\omega_{m}^{\prime},q_{x},q_{y}),q_{i}=(\omega_{im},q_{ix},q_{iy}),i=1,2$.
It will be denoted $\Sigma^{(3)}_{\text{a}}(\omega_{m})$. This diagram reads

$\displaystyle\Sigma^{(3)}_{\text{a}}(\omega)=g^{3}N\int_{q_{1},q_{2},q}G(k-q)G(k+q_{1}-q)\times$
(61) $\displaystyle
G(k+q_{1})G(k+q_{2}-q)G(k+q_{2})D(q_{1})D(q_{2})D(q_{2}-q_{1})\ .$

We will use a simplified boson propagator:

$\displaystyle D(q_{1})\simeq-\frac{1}{M^{2}+q_{1y}^{2}+\frac{g\rho
N}{v_{F}}\frac{|\omega_{1m}|}{|q_{1y}|}}\ ,$ (62)

which holds for $M\ll k_{F}$ where typically $q_{1x}\ll q_{1y}$. The $q_{1x}$
integral in (61) can then be computed by contour integration, and leads to

$\displaystyle I_{1x}$
$\displaystyle\equiv\int\frac{dq_{1x}}{2\pi}\frac{1}{iZ^{-1}(\omega_{m}+\omega_{1m})-v_{F}q_{1x}-q_{1y}^{2}/{2m}}\times\frac{1}{iZ^{-1}(\omega_{m}+\omega_{1m}-\omega_{m}^{\prime})-v_{F}(q_{1x}-q_{x})-(q_{1y}-q_{y})^{2}/{2m}}$
$\displaystyle=-\frac{i}{v_{F}}\frac{\theta(\omega_{m}+\omega_{1m})-\theta(\omega_{m}+\omega_{1m}-\omega_{m}^{\prime})}{-Z^{-1}i\omega_{m}^{\prime}+v_{F}q_{x}+q_{1y}^{2}/2m-(q_{1y}-q_{y})^{2}/2m}\
.$ (63)

The $q_{2x}$ integral is performed alike. Now performing the $q_{x}$ integral

$\displaystyle I_{x}$
$\displaystyle\equiv\int\frac{dq_{x}}{2\pi}\frac{1}{-iZ^{-1}\omega_{m}^{\prime}+v_{F}q_{x}+q_{1y}^{2}/2m-(q_{1y}-q_{y})^{2}/2m}$
$\displaystyle\times\frac{1}{-iZ^{-1}\omega_{m}^{\prime}+v_{F}q_{x}+q_{2y}^{2}/2m-(q_{2y}-q_{y})^{2}/2m}\times\frac{1}{iZ^{-1}(\omega_{m}-\omega_{m}^{\prime})+v_{F}q_{x}-q_{y}^{2}/2m}$
$\displaystyle=-\frac{i}{v_{F}}\frac{\theta(\omega_{m}-\omega_{m}^{\prime})-\theta(-\omega_{m}^{\prime})}{-iZ^{-1}\omega_{m}+q_{y}^{2}/2m+q_{1y}^{2}/2m-(q_{1y}-q_{y})^{2}/2m}\times\frac{1}{-iZ^{-1}\omega_{m}+q_{y}^{2}/2m+q_{2y}^{2}/2m-(q_{2y}-q_{y})^{2}/2m}\
.$ (64)

Notice the partial cancellation of the curvature terms; only mixed terms
$q_{iy}q_{y}$ remain. Let $\omega_{m}>0$. The step-function in Eq. (64)
restricts $\omega_{m}^{\prime}\in(0,\omega_{m})$, and the two step-functions
in (63) then imply
$\omega_{im}\in(-\omega_{m},\omega_{m}^{\prime}-\omega_{m})$. Combining the
results obtained so far:

$\displaystyle\Sigma^{(3)}_{\text{a}}(\omega_{m})=$ (65) $\displaystyle
i\frac{g^{3}N}{(2\pi)^{6}v_{F}^{3}}\int_{0}^{\omega}d\omega_{m}^{\prime}\int\displaylimits_{-\omega_{m}}^{\omega_{m}^{\prime}-\omega_{m}}d\omega_{1m}d\omega_{2m}\int
dq_{y}dq_{1y}dq_{2y}\frac{1}{-iZ^{-1}\omega_{m}+q_{1y}q_{y}/m}\frac{1}{-iZ^{-1}\omega_{m}+q_{2y}q_{y}/m}D(q_{1})D(q_{2})D(q_{2}-q_{1})\
.$

The $q_{y}$-integral comes from poles $q_{y}=-iZ^{-1}\omega_{m}m/q_{iy}$, with
$q_{iy}=\mathcal{O}(M)$. I.e., it does not come from $|{\boldsymbol{q}}_{i}|$
which are small in $\omega$, as stated in the main text. It yields

$\displaystyle\Sigma^{(3)}_{\text{a}}(\omega_{m})$
$\displaystyle=i\frac{g^{3}N}{(2\pi)^{5}v_{F}^{3}}\int_{0}^{\omega_{m}}d\omega_{m}^{\prime}\int_{-\omega_{m}}^{\omega_{m}^{\prime}-\omega_{m}}d\omega_{1m}d\omega_{2m}\int
dq_{1y}dq_{2y}\frac{m(\theta(q_{1y})-\theta(q_{2y}))}{Z^{-1}\omega_{m}(q_{2y}-q_{1y})}D(q_{1})D(q_{2})D(q_{2}-q_{1})$
$\displaystyle=-i\frac{g^{3}ZNm}{(2\pi)^{5}v_{F}^{3}}\frac{2}{\omega}\int_{0}^{\omega_{m}}d\omega_{m}^{\prime}\int_{-\omega_{m}}^{\omega_{m}^{\prime}-\omega_{m}}d\omega_{1m}d\omega_{2m}\int_{0}^{\infty}dq_{1y}dq_{2y}\frac{1}{q_{1y}+q_{2y}}D(q_{1})D(q_{2})D(\omega_{1m}-\omega_{2m},q_{1y}+q_{2y})\
.$ (66)

To leading order in $\omega_{m}$ we can neglect the dynamics (Landau damping)
in the Boson propagator. Computing the trivial frequency integral, and
rescaling the remaining momenta $y_{i}\equiv q_{iy}/M$, we arrive at

$\displaystyle\Sigma^{(3)}_{\text{a}}(\omega_{m})={i\text{sign}(\lambda)\frac{1}{\pi^{2}}\frac{\lambda^{2}}{(1+\lambda)}\frac{\omega_{m}^{2}}{{\omega_{\text{FL}}}}\times\text{const.}}\
,$ (67)
$\displaystyle\text{const.}=\frac{2}{3}\\!\int_{0}^{\infty}\\!dy_{1}dy_{2}\frac{1}{y_{1}+y_{2}}\frac{1}{1+y_{1}^{2}}\frac{1}{1+y_{2}^{2}}\frac{1}{1+(y_{1}+y_{2})^{2}}$
$\displaystyle=0.56329\ldots.$

Since there is no linear contribution, we write
$\Sigma^{(3)}_{\text{a}}(\omega_{m})=\Sigma^{(3)}_{\text{a},\omega^{2}}(\omega_{m})$
as in Eq. (11) of the main text.

The $\omega_{m}^{2}$ part of the remaining three-loop diagrams is more
conveniently computed by first evaluating the dressed vertices (blue boxes in
Fig. 6). For the planar diagram in the backscattering channel (Fig. 5(b)), we
only need the vertex as function of $q_{y}$ and can set the other variables to
zero (see also Sec. IV). Thus, we have

$\displaystyle\Gamma_{\text{bs}}^{(1)}(q_{y})=-g^{2}\int_{q_{1}}D(q_{1})^{2}\
\frac{1}{iZ^{-1}\omega_{1m}-v_{F}q_{1x}-q_{1y}^{2}/2m}$
$\displaystyle\times\frac{1}{-iZ^{-1}\omega_{1m}-v_{F}q_{1x}-(q_{y}-q_{1y})^{2}/2m}\
.$ (68)

The $q_{1x}$-integral yields

$\displaystyle\Gamma_{\text{bs}}^{(1)}(q_{y})=g^{2}\frac{i}{v_{F}}\int\frac{dq_{1y}d\omega_{1m}}{(2\pi)^{2}}\text{sign}(\omega_{1m})\times$
(69)
$\displaystyle\frac{1}{-2iZ^{-1}\omega_{1m}+{(q_{1y})^{2}}/{2m}-(q_{y}-q_{1y})^{2}/2m}\times$
$\displaystyle\frac{1}{\left((q_{1y})^{2}+M^{2}+\frac{N\rho
g}{v_{F}}\frac{|\omega_{1m}|}{|q_{1y}|}\right)^{2}}\ .$

The $\omega_{1m}$ integral is logarithmic, and the boson propagators just set
the cutoff; since the typical value of $q_{1y}$ is $M$, the cutoff becomes
${\omega_{\text{FL}}}$. Then we find, with logarithmic accuracy:

$\displaystyle\Gamma_{\text{bs}}^{(1)}(q_{y})\simeq-\frac{Zg^{2}}{v_{F}(2\pi)^{2}}\int
dq_{1y}\frac{1}{(q_{1y}^{2}+M^{2})^{2}}$ (70)
$\displaystyle\times\ln\left(\frac{{\omega_{\text{FL}}}}{|q_{1y}q_{y}/m-q_{y}^{2}/2m|}\right)\
.$

In the logarithm, we can replace the argument by a typical value. We will need
$q_{y}\ll M$, therefore we can approximate it as $q_{1y}q_{y}/m\simeq
Mq_{y}/m=q_{y}v_{F}\epsilon$, with $\epsilon=M/k_{F}$. Then the integral
results in

$\displaystyle\Gamma_{\text{bs}}^{(1)}(q_{y})\simeq-\frac{g}{M^{2}}\times\frac{1}{2}\frac{\lambda}{1+\lambda}\ln\left(\frac{{\omega_{\text{FL}}}}{v_{F}q_{y}\epsilon}\right)\
.$ (71)

Inserting this vertex into Eq. (14) (with variable $z=v_{F}q\epsilon$, see
also discussion below Eq. (IV)) immediately yields
$\Sigma_{\text{b},\omega^{2}}^{(3)}$ from Eq. (12).

To evaluate the non-planar diagram of Fig. 6(c), we likewise evaluate the
vertex, to be denoted $\Gamma_{\text{np}}$. Here we can set external variables
to zero:

$\displaystyle\Gamma_{\text{np}}\equiv-g^{2}\int_{q_{1}}D(q_{1})^{2}\times$
(72)
$\displaystyle\frac{1}{iZ^{-1}\omega_{1m}+v_{F}q_{1x}-q_{1y}^{2}/2m}\frac{1}{iZ^{-1}\omega_{1m}-v_{F}q_{1x}-q_{1y}^{2}/2m}$
$\displaystyle=\frac{ig^{2}}{(2\pi)^{2}v_{F}}\int
d\omega_{1m}dq_{1y}\frac{\text{sign}(\omega_{1m})}{2iZ^{-1}\omega_{1m}-q_{y}^{2}/m}D(q_{1})^{2}=$
$\displaystyle\frac{g^{2}}{(2\pi)^{2}v_{F}}\int
d\omega_{1m}dq_{1y}\frac{2Z^{-1}|\omega_{1m}|}{4(Z^{-1}\omega_{1m})^{2}+\frac{q_{1y}^{4}}{m^{2}}}$
$\displaystyle\times\frac{1}{\left(M^{2}+q_{y}^{2}+\frac{g\rho
N}{v_{F}}\frac{|\omega_{1m}|}{|q_{1y}|}\right)^{2}}\ .$

To extract the limit $N\rightarrow\infty$, we introduce dimensionless
variables as

$\displaystyle y=\frac{q_{1y}}{M\alpha^{2}},\quad
x=\frac{Z^{-1}\omega_{1m}m}{M^{2}\alpha},\quad\alpha=\frac{m}{\rho N\lambda}\
.$ (73)

This yields

$\displaystyle\Gamma_{\text{np}}=\frac{1}{2\pi}\frac{g}{M^{2}}\tilde{\lambda}\alpha\times
2\int_{0}^{\infty}dxdy\frac{x}{4x^{2}+y^{4}}\frac{1}{1+y^{2}\alpha^{2}+\frac{x}{y}}\
.$ (74)

In the limit $N\rightarrow\infty$ where $\alpha\rightarrow 0$, the last factor
has a well-defined limit and one obtains

$\displaystyle\Gamma_{\text{np}}\rightarrow\frac{1}{2\pi}\frac{g}{M^{2}}\tilde{\lambda}\alpha=\frac{1}{2\pi}\frac{g}{M^{2}}\frac{1}{1+\lambda}\frac{m}{N\rho}=\frac{g}{M^{2}}\frac{1}{(1+\lambda)N}\
.$ (75)

Inserting this instead of the bare vertex at zero energy-momentum $(-g/M^{2})$
into the one-loop diagram, one obtains Eq. (13).

##  Appendix D Stucture of planar diagrams

Let us first recall the large-$N$ analysis of Refs. [33, 25, 12] at the QCP.
These authors rescale the bare coupling constant $g\rightarrow g/N$. Then the
Landau damping $\Pi$ is $\mathcal{O}(1)$, while the one-loop quantum critical
self-energy $\Sigma^{(1)}(\omega)=(\omega_{\text{IN}})^{1/3}\omega^{2/3}$ of
the fermions picks up a factor $1/N$. Naively, all diagrams in Fig. 5 then
have the power $1/N^{2}$ and are subleading compared to
$\Sigma^{(1)}(\omega)$. However, the two planar diagrams (Fig. 5 (a),(b))
contain a “singular manifold” of dimension two where all internal fermions are
on the Fermi surface – if the momentum ${\boldsymbol{q}}$ in the diagram is
set to zero, there are two free parameters for
${\boldsymbol{k}}+{\boldsymbol{q}}_{1},{\boldsymbol{k}}+{\boldsymbol{q}}_{2}$
to be on the Fermi surface [29]. By counting the strength of poles of
fermionic propagators, one sees that this zero-energy manifold would induce a
IR singularity were it not for the frequency dependence of the propagators.
Since the leading frequency-dependence comes from $\Sigma$, which scales a
$1/N$ as stated above, the diagrams acquire at least one additional power of
$N$ 555It turns out that the backscattering diagram is even of order
$\mathcal{O}(N^{0})$ due to a UV singularity arising in the quantum-critical
limit [25]. A graphical way to recognize planarity is by replacing the wavy
boson lines by two fermion lines, and inverting the fermion line direction at
the $(-{\boldsymbol{k}})$ patch [29, 25]. As visualized in Fig. 12, in this
double-line representation only the planar diagrams be drawn on a sphere
without any crossings; the non-planar diagrams can be “untwisted” on higher-
genus surfaces only. The additional powers of $N$ which the diagram contains
beyond the naive power counting are determined by the number of “single-line”
loops in this representation.

Figure 12: Double line representation of a planar (a) and non-planar diagram
(b). The planar diagram is free of crossings. Fig. adapted from Ref. [25].

From another perspective, the planar diagrams are distinguished by a
cancellation of Fermi surface curvature [25]. In the patch theory dispersion
$v_{F}k_{x}+k_{y}^{2}/2m$ we regard $v_{F}$ as fixed, and the Fermi surface
curvature is given by $1/m$. The cancellation of the curvature occurs in the
non-Fermi-liquid and Fermi-liquid cases alike, and is explicit in the three-
loop evaluations of App. C: in the forward-scattering planar diagram, after
the integrals over the $x$-momenta are taken, the terms
$q_{y}^{2},q_{1y}^{2},q_{2y}^{2}$ cancel out in Eq. (64); only mixed terms
remain. Such cancellation also occurs in the planar backscattering vertex
$\Gamma_{\text{bs}}^{(1)}$, Eq. (69). By contrast, in the planar
backscattering vertex $\Gamma_{\text{np}}$ there is no cancellation. see Eq.
(72). As a result, this vertex incures an additional factor $(m/N\rho)$
compared to the planar one.

For a general diagram, one can expect that the (partial) cancellation of
curvature is an equivalent criterion to the “singular manifold” introduced in
Ref. [29]: as shown in this reference for the one-patch model, by assigning
momenta such that all fermions are on the Fermi surface, the momenta of
propagators connected to the external legs in the double-line representation
are fixed. However, all fermions in the closed “single-line”-loops carry the
same unfixed momentum ${\boldsymbol{k}}+{\boldsymbol{p}}$. By integrating over
the $p_{x}$-component, the curvature in the single-line loop cancels as long
as one can neglect the $p_{x}$-dependence of the bosons. A similar argument
applies if the fermions are part of opposite patch and have momentum
$-{\boldsymbol{k}}-{\boldsymbol{p}}$, which results in the same pole structure
of the propagator. In conclusion, it is thus expected that the planar diagrams
are leading in the $\mathcal{O}(1/N)$ expansion not only at the QCP, but also
in a Fermi liquid.

##  Appendix E Detailed evaluation of $\text{Im}[\Sigma_{\text{bs}}]$

Let us fill in the details for the evaluation of
$\text{Im}[\Sigma_{\text{bs}}]$. Consider first an interaction line in Fig. 7,
say $D(p_{1}-p_{2})$:

$\displaystyle
D(p_{1}-p_{2})=-g\left[M^{2}+({\boldsymbol{p}}_{1}-{\boldsymbol{p}}_{2})^{2}+Ng\rho\frac{|\omega_{1m}-\omega_{2m}|}{v_{F}|{\boldsymbol{p}}_{1}-{\boldsymbol{p}}_{2}|}\right]^{-1}=-g\bigg{[}M^{2}+4k_{F}^{2}\sin^{2}\\!\left(\frac{\phi_{1}-\phi_{2}}{2}\right)$
(76)
$\displaystyle+4k_{F}(\delta_{p_{1}}+\delta_{p_{2}})\sin^{2}\\!\left(\frac{\phi_{1}-\phi_{2}}{2}\right)+\delta_{p_{1}}^{2}+\delta_{p_{2}}^{2}-2\delta_{p_{1}}\delta_{p_{2}}\cos(\phi_{2}-\phi_{1})+Ng\rho\frac{|\omega_{1m}-\omega_{2m}|}{v_{F}|{\boldsymbol{p}}_{1}-{\boldsymbol{p}}_{2}|}\bigg{]}^{-1},$

where $\delta_{p_{i}}=|{\boldsymbol{p}}_{i}|-k_{F}$, and angles are measured
relative to ${\boldsymbol{k}}$. Recall the parameter $\epsilon=M/k_{F}\ll 1$.
Eq. (76) and the remaining interaction lines in the diagram restrict the
typical angles to be $\phi_{i}\lesssim\epsilon$. Thus, in each Cooper bubble
the typical momentum along the Fermi surface at the point ${\boldsymbol{k}}$,
given by $|{\boldsymbol{p}}_{i}|\sin(\phi_{i})$, is of order $M$.

Fortunately, the detailed dependence on $\omega_{im},\delta_{p_{i}}$ in Eq.
(76) does not have to be taken into account, since it is slow compared to the
singular logarithmic dependence in the Cooper bubbles. However, the
interaction lines set the UV cutoff on the logarithmic Cooper integral:
$|\delta_{i}|<M$ and $|\omega_{im}|<{\omega_{\text{FL}}}$. For small $M$, the
latter cutoff is more important, and it is thus sufficient to evaluate the
particle-particle bubble $\Pi_{\text{pp}}$ restricting the internal frequency
integral only. This is done in App. A, Eq. (51), with the result

$\displaystyle\Pi_{\text{pp}}(\omega_{m}^{\prime},q,\phi_{q}-\phi_{i})=$ (77)
$\displaystyle
Z\rho\ln\left(\frac{{\omega_{\text{FL}}}}{\sqrt{(Z^{-1}\omega_{m}^{\prime})^{2}+(v_{F}q\cos(\phi_{q}-\phi_{i}))^{2}}}\right),$

where we did not perform the angular integral yet since the angular dependence
of the interaction lines is crucial per the above . Eq. (77) can be simplified
by inserting a typical value of $\phi_{q}$: the integral over $\phi_{q}$ is of
the form

$\displaystyle\int
d\phi_{q}\frac{1}{iZ^{-1}(\omega_{m}-\omega_{m}^{\prime})+v_{F}q\cos(\phi_{q})}\ldots$
(78)

In the most important limit $\omega_{m},\omega_{m}^{\prime}\ll v_{F}q$, it is
dominated by angles $\phi_{q}=\pm\pi/2$ up to corrections of order
$\omega_{m}/v_{F}q$, and we can therefore approximate

$\displaystyle\Pi_{\text{pp}}(\omega_{m}^{\prime},q,\phi_{q}-\phi_{i})\simeq$
(79) $\displaystyle
Z\rho\ln\left(\frac{{\omega_{\text{FL}}}}{\sqrt{(Z^{-1}\omega_{m}^{\prime})^{2}+(v_{F}q\sin(\phi_{i}))^{2}}}\right).$

Let us recall the form of $\Pi_{\text{pp}}$ for a momentum-independent
interaction (calling it BCS-like). Then the angular integral can simply be
taken, and one finds [64, *PhysRevB.74.079907]

$\displaystyle\Pi_{\text{pp}}^{\text{BCS}}(\omega_{m}^{\prime},q)=$
$\displaystyle
Z\rho\ln\left(\frac{{\omega_{\text{FL}}}}{|Z^{-1}\omega_{m}^{\prime}|+\sqrt{(v_{F}q)^{2}+|Z^{-1}\omega_{m}^{\prime}|^{2}}}\right)\simeq$
$\displaystyle
Z\rho\ln\left(\frac{{\omega_{\text{FL}}}}{v_{F}q}\right)-\rho\frac{|\omega_{m}^{\prime}|}{v_{F}q}\
.$ (80)

The dynamical term is nothing but the Landau damping up to a factor of $N$
(since our $\Pi_{\text{pp}}$ does not involve a flavor sum). As seen from Eq.
(79), this term arises from typical values
$\phi_{i}\simeq\omega_{m}^{\prime}/v_{F}q$ or
$\simeq\omega_{m}^{\prime}/v_{F}q+\pi$. With our momentum-dependent
interaction, $\phi_{i}\simeq\pi$ can be neglected as involves a large momentum
transfer. On the other hand, the static Cooper logarithm in (80) comes from
$\mathcal{O}(1)$ values of the angle. Put together: the Landau damping in the
Cooper bubble comes from internal momenta along the Fermi surface which are
small in $\omega_{m}$, while the Cooper logarithm comes from much larger
momenta of order $M$.

These considerations outline the following strategy for evaluation of
$\Sigma_{\omega^{2},\text{bs}}$: We need to select the Landau-damping part
from one particle-particle bubble, with a factor $1/2$ compared to Eq. (80);
since the angle $\phi_{i}$ corresponding to this bubble is small in
$\omega_{m}/v_{F}q$, we can set it to zero in the interactions (16). From the
remaining bubbles, we only take the static Cooper logarithm, setting
$\omega_{m}^{\prime}=0$ there. The angle dependence in the Cooper logarithm
can be replaced by a typical value, $\sin(\phi_{i})\simeq\epsilon$, and the
Cooper logarithms become

$\displaystyle L(z)\equiv\ln\left(\frac{{\omega_{\text{FL}}}}{z}\right)\ ,$
(81)

where $z=v_{F}q\epsilon$. All Cooper diagrams can then be summed by selecting
a cross-section in which we take the Landau damping piece from the particle-
particle bubble, and summing Cooper logarithms to the left and right of it,
corresponding to the evaluation of the dressed backscattering amplitude
$\Gamma_{\text{bs}}(z)$. Note that is crucial to retain the Landau-damping
piece once, without this we would miss the universal contribution with
$\omega_{m}^{2}$ in front of the logarithmic dependence.

Proceeding as in the main text, one arrives at Eq. (IV):

$\displaystyle\Sigma_{\omega^{2},\text{bs}}(\omega_{m})=i\text{sign}(\omega_{m})\frac{\lambda}{\pi}\frac{\omega^{2}_{m}}{{\omega_{\text{FL}}}}\int_{\omega_{m}}^{{\omega_{\text{FL}}}}\frac{dz}{z}\left[\frac{M^{2}}{g}\Gamma_{\text{bs}}(z)\right]^{2}$
(82)

In this equation, the integral boundaries have been chosen as follows: The
upper boundary is fixed in such a way that the form of the Cooper logarithms
remains valid; as per Eq. (79), this requires $z\ll{\omega_{\text{FL}}}$.
Likewise, we can only neglect $\omega_{m}^{\prime}\sim\omega_{m}$ in the
Cooper logarithms if $\omega_{m}\ll z$. This sets the lower boundary of
integration over $z$.

From Eq. (82), the final results for both repulsive and attractive
interactions can be obtained by substituting
$x=|\tilde{\lambda}|\ln({\omega_{\text{FL}}}/z)$, which results in

$\displaystyle\Sigma_{\omega^{2},\text{bs}}=$ $\displaystyle
i\text{sign}(\omega_{m})\frac{\omega^{2}_{m}}{\pi{\omega_{\text{FL}}}}\frac{|\lambda|}{|\tilde{\lambda}|}\int_{0}^{\ell}dx\left(\frac{\ln(1\pm
x)}{x}\right)^{2}$ (83)

where the $+(-)$ sign corresponds to repulsive (attractive) interactions, and
$\ell=|\tilde{\lambda}|\ln({\omega_{\text{FL}}}/|\omega_{m}|)$. Evaluating the
integrals we obtain Eqs. (24), (IV.2) of the main text. We note that the
integral for attractive interactions and integral boundary $\ell/(1+\ell)$ can
be mapped on the integral for repulsive interactions by substituting
$y=x/(1-x)$, which shows relation (32) of the main text.

##  Appendix F $\Gamma_{\text{bs}}$ beyond leading order

The backscattering amplitude $\Gamma_{\text{bs}}$ has been defined in Eq. (18)
as a function of the $z$-dependent logarithm $L(z)$, with $z=v_{F}q\epsilon$.
Suppressing the $z$-dependence, we can rewrite this definition as

$\displaystyle\Gamma_{\text{bs}}=-\frac{1}{\rho}\sum_{n=0}^{\infty}\left(L\right)^{n}\int_{0}^{2\pi}\frac{d\phi_{1}}{2\pi}\frac{d\phi_{2}}{2\pi}\ldots\frac{d\phi_{n}}{2\pi}V(\phi_{1})V(\phi_{2}-\phi_{1})\ldots
V(\phi_{n}-\phi_{n-1})V(\phi_{n})\ ,\quad
V(\phi)\equiv\frac{2\epsilon\tilde{\lambda}}{\epsilon^{2}+4\sin^{2}(\phi/2)}\
.$ (84)

Introducing the angular momentum decomposition, $\Gamma_{\text{bs}}$ can be
reformulated as in Eq. (36):

$\displaystyle\Gamma_{\text{bs}}=$ (85)
$\displaystyle-\frac{1}{\rho}\sum_{m}V_{m}\sum_{n=0}^{\infty}\left[V_{m}L\right]^{n}=-\frac{1}{\rho}\frac{V_{0}}{1-V_{0}L}-\frac{2}{\rho}\sum_{m=1}^{\infty}\frac{V_{m}}{1-V_{m}L},$

with $V_{m}$ the angular momentum components of $V$, which are defined as

$\displaystyle V_{m}=\int_{0}^{2\pi}\frac{d\phi}{2\pi}V(\phi)\exp(-i\phi
m)=V_{-m}\ .$ (86)

To compute $\Gamma_{\text{bs}}$, $V_{m}$ is needed as an input. Let $m>0$. For
fixed small $m$, $V_{m}$ can be evaluated explicitly:

$\displaystyle V_{0}=\tilde{\lambda}\frac{2}{\sqrt{4+\epsilon^{2}}}$ (87)
$\displaystyle
V_{1}=\tilde{\lambda}\left(\frac{\epsilon^{2}+2}{\sqrt{\epsilon^{2}+4}}-\epsilon\right)$
$\displaystyle
V_{2}=\tilde{\lambda}\left(\frac{2}{\sqrt{\epsilon^{2}+4}}-\epsilon\left(\epsilon^{2}-\sqrt{\epsilon^{2}+4}\epsilon+2\right)\right)\ldots$

To obtain a useful approximation to $V_{m}$ for general $m$ and small
$\epsilon$, we can evaluate (86) by extending boundaries to infinity and
replacing $4\sin^{2}(\phi/2)\simeq\phi^{2}$. This yields

$\displaystyle V_{m}^{(1)}\simeq\tilde{\lambda}\exp(-\epsilon m)\ .$ (88)

Comparison to Eq. (87) indicates that this expression is valid up to order
$\mathcal{O}(\epsilon^{2})$. From Eq. (87) one can also make an educated guess
for the next order terms in $V_{m}$:

$\displaystyle V_{m}^{(2)}=\tilde{\lambda}\exp(-\epsilon
m)\left(1-\epsilon^{2}/8+\frac{m}{24}\epsilon^{3}+\ldots\right).$ (89)

This improved approximation is numerically checked in Fig. 13, and has a
relative error of $\lesssim 10^{-6}$ even for $\epsilon=0.1$. As a takeaway
point, we conjecture that already $V_{m}^{(1)}$ from Eq. (88) reproduces the
leading (i.e., largest) power in $m$ at each order in $\epsilon$, which will
prove enough for our purposes.

Figure 13: Relative error $\delta V_{m}$ of $V_{m}^{(2)}$, Eq. (89), as
function of $m$, for three different values of $\epsilon$. $\delta V_{m}$ is
defined as $\delta V_{m}=V_{m}/V_{m}^{(2)}-1$, with $V_{m}$ obtained from
numerical integration.

We return to Eq. (85) and consider the sum over higher angular momenta:

$\displaystyle\frac{2}{\rho}\sum_{m=1}^{\infty}\frac{V_{m}}{1-V_{m}L}=\frac{2}{\rho}\sum_{m=1}^{\infty}\frac{1}{\tilde{\lambda}/V_{m}-1+(1-\tilde{\lambda}L)}\
.$ (90)

To evaluate the sum, we can apply the Euler-Maclaurin formula in the form [66]

$\displaystyle\sum_{m=1}^{\infty}f(m)=\int_{1}^{\infty}dmf(m)+\frac{1}{2}f(1)+R_{f},$
(91)

where $R_{f}$ is a rest term; we will numerically check that $R_{f}$ can be
neglected below. We approximate $V_{m}=\tilde{\lambda}\exp(-m\epsilon)$ and
therefore use

$\displaystyle f(m)=\frac{1}{\exp(m\epsilon)-1+(1-\tilde{\lambda}L)}\ .$ (92)

Then $f(1)\simeq 1/(\epsilon+(1-\tilde{\lambda}L))$, which will be subleading.
The integral in Eq. (91) be computed directly, but it is more transparent to
get the result by expanding $f$ in $\epsilon$, which is approximately valid as
long as $m\lesssim 1/\epsilon$:

$\displaystyle\int_{1}^{1/\epsilon}dm\frac{1}{\epsilon m+\frac{1}{2}(\epsilon
m)^{2}+(1-\tilde{\lambda}L)}\overset{x=m\epsilon}{=}$ (93)
$\displaystyle\frac{1}{\epsilon}\int_{\epsilon}^{1}dx\frac{1}{x}\frac{1}{1+x/2+(1-\tilde{\lambda}L)/x}\simeq$
$\displaystyle\frac{1}{\epsilon}\ln\left(\frac{1}{(1-\tilde{\lambda}L)+\epsilon}\right),$

where the upper cutoff of the logarithm is determined up to factors of order
$\mathcal{O}(1)$. The last form of the integral shows that knowledge of the
leading order coefficients in $m$, $\epsilon$ for $V_{m}$ is indeed
sufficient: for instance, in Eq. (89) it was claimed that the
$\epsilon^{2}$-term in $V_{m}$ actually reads $\epsilon^{2}(m^{2}/2-1/8)$.
However, the extra term $1/8\epsilon^{2}$ would only lead to an
$\mathcal{O}(\epsilon)$ correction to the upper cutoff in Eq. (93), which can
be neglected.

In Fig. 14 we compare the approximate result (93) to direct numerical
evaluation of the appropriately truncated sum in (91). We find very good
agreement as $\epsilon\rightarrow 0$, indicating that the remainder term
$R_{f}$ can be disregarded.

Figure 14: Numerical check of the Euler-MacLaurin approximation for $\delta
L=1-\tilde{\lambda}L=0.001$. The sum is truncated at
$m_{\text{max}}=10^{3}\left\lfloor{1/(\epsilon+\delta L)}\right\rfloor$.

We can combine Eq. (93) with the $s$-wave part and set
$V_{0}\simeq\tilde{\lambda}$ (which holds up to $\mathcal{O}(\epsilon^{2})$).
Dividing the logarithm by $\tilde{\lambda}L(z)$ to match the high-energy form
of $\Gamma_{\text{bs}}(z)$, we obtain Eq. (37) of the main text.

##  Appendix G Self-energy in the superconducting state: contribution of
higher angular momentum channels

To obtain Eq. (43) in the main text, we need to re-evaluate the one-loop
diagram of Fig. 2(b) in the superconducting state. This is easiest done in the
spectral representation, where, for $\omega>0$:

$\displaystyle\text{Im}\left[\Sigma_{\text{bs}}^{R}(\omega)\right]=(-g)\times$
(94)
$\displaystyle\int\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}\int_{0}^{\omega}\frac{d\omega^{\prime}}{\pi}\text{Im}\left[G^{R}(\omega-\omega^{\prime},{\boldsymbol{k}}+{\boldsymbol{q}})\right]\text{Im}\left[D^{R}(\omega^{\prime},{\boldsymbol{q}})\right]\
.$

To evaluate this formula, we need to find
$\text{Im}\left[D^{R}(\omega^{\prime},{\boldsymbol{q}})\right]$ (the RPA
dressed interaction), which requires knowledge of the particle-hole bubble
$\Pi_{\text{ph}}$. In the superconducting state, it becomes $GG-FF$ (see,
e.g., Sec. 10.2.4 of Ref. [67]), with propagators as in Eq. (42). In the
Matsubara domain, after evaluating lengthy but straightforward momentum
integrals, one arrives at

$\displaystyle\Pi_{\text{ph}}(\omega_{m}^{\prime},{\boldsymbol{q}})=N\int_{p}\left[G(p+q)G(p)-F(p+q)F(p)\right]=$
$\displaystyle
N\rho\int_{0}^{\infty}d\tilde{\omega}_{m}\frac{\Omega_{+}\Omega_{-}+(\omega_{m}^{\prime})^{2}/4-(\tilde{\omega}_{m})^{2}-\Delta_{0}^{2}}{\Omega_{+}\Omega_{-}\left(\left(\Omega_{+}+\Omega_{-}\right)^{2}+(v_{F}q)^{2}\right)^{1/2}}\
,$
$\displaystyle\Omega_{\pm}\equiv\sqrt{(\tilde{\omega}_{m}\pm\omega_{m}^{\prime}/2)^{2}+\Delta_{0}^{2}}\
,$ (95)

which reduces to the normal state expression, Eq. (4), for $\Delta_{0}=0$. We
have suppressed the dependence on the quasiparticle weight
$Z=1/(1+\lambda)\simeq 1$. To match
$\text{Im}\left[\Sigma_{\text{bs}}^{R}(\omega)\right]$ with the self-energy
obtained using normal state propagators, we need $\Pi_{\text{ph}}$ for
$v_{F}q\gg\Delta_{0}$. To extract the singular part of $\Pi_{\text{ph}}$ for
real frequencies, we can take a suitable $\tilde{\omega}_{m}\rightarrow 0$
limit in the integrand, and also perform the analytical continuation
$i\omega^{\prime}_{m}\rightarrow\omega^{\prime}_{m}$ before integration; this
results in the correct real part for $\Pi_{\text{ph}}(\omega^{\prime})$, and
the imaginary part can then be restored from Kramers-Kronig relations. We have

$\displaystyle\Pi_{\text{ph}}(\omega^{\prime},v_{F}q)\simeq-\frac{\rho
N}{v_{F}q}\frac{(\omega^{\prime})^{2}}{2}\times$ (96)
$\displaystyle\int_{0}^{\Lambda}d\tilde{\omega}_{m}\frac{1}{\sqrt{(\Delta_{0}^{2}-(\omega^{\prime})^{2}/4)^{2}+2(\tilde{\omega}_{m})^{2}(\Delta_{0}^{2}+(\omega^{\prime})^{2}/4)}}\
,$

where $\Lambda$ is some arbitrary cutoff. This integral is singular for
$\omega^{\prime}\rightarrow 2\Delta_{0}$, and we find

$\displaystyle\Pi_{\text{ph}}(\omega^{\prime},q)\simeq\frac{\rho
N}{v_{F}q}\Delta_{0}\ln\left(\frac{|\omega^{\prime}-2\Delta_{0}|}{\Lambda}\right).$
(97)

Restoring the imaginary part to get a retarded function, we obtain

$\displaystyle\text{Im}[\Pi_{\text{ph}}^{R}(\omega^{\prime},q)]\simeq-\pi\frac{\rho
N}{v_{F}q}\Delta_{0}\times\theta(\omega^{\prime}-2\Delta_{0})\ .$ (98)

This is the expected behavior: since quasiparticles and holes are gapped with
gap $\Delta_{0}$, the imaginary part of the polarization bubble should start
at $2\Delta_{0}$. With this expression at hand, we can approximate

$\displaystyle\text{Im}\left[D^{R}(\omega^{\prime},{\boldsymbol{q}})\right]\simeq\frac{g\text{Im}\left[\Pi_{\text{ph}}^{R}(\omega^{\prime},q)\right]}{\left(q^{2}+M^{2}\right)^{2}},$
(99)

which holds for $q>q_{\text{min}}\equiv(g\rho N\Delta)/v_{F}M^{2}$, similar to
the normal state case (compare Eq. (58)). Further, we have, for
$\omega-\omega^{\prime}>0$:

$\displaystyle\text{Im}\left[G^{R}(\omega-\omega^{\prime},{\boldsymbol{k}}+{\boldsymbol{q}})\right]=-\pi
u_{{\boldsymbol{k}}+{\boldsymbol{q}}}^{2}\delta\left(\omega-\omega^{\prime}-E_{{\boldsymbol{k}}+{\boldsymbol{q}}}\right),$
$\displaystyle\
E_{{\boldsymbol{k}}+{\boldsymbol{q}}}=\sqrt{\Delta_{0}^{2}+\xi_{{\boldsymbol{k}}+{\boldsymbol{q}}}^{2}},\
u_{{\boldsymbol{k}}+{\boldsymbol{q}}}^{2}=\frac{1}{2}\left(1+\frac{\xi_{{\boldsymbol{k}}+{\boldsymbol{q}}}}{E_{{\boldsymbol{k}}+{\boldsymbol{q}}}}\right).$
(100)

Inserting Eqs. (99), (100) into Eq. (94), we find, taking the
$\omega^{\prime}$ integral:

$\displaystyle\text{Im}\left[\Sigma_{\text{bs}}^{R}(\omega)\right]=-\pi\frac{g^{2}\rho
N\Delta_{0}}{v_{F}}\int_{q>q_{\text{min}}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}$
(101) $\displaystyle\theta\left(\omega-
E_{{\boldsymbol{k}}+{\boldsymbol{q}}}-2\Delta_{0}\right)\frac{u_{{\boldsymbol{k}}+{\boldsymbol{q}}}^{2}}{q(q^{2}+M^{2})^{2}}\
.$

It is readily seen that the self-energy starts at $\omega=3\Delta_{0}$. For
$0<\omega-3\Delta_{0}\ll\Delta_{0}$, we can expand
$E_{{\boldsymbol{k}}+{\boldsymbol{q}}}\simeq\Delta_{0}+(v_{F}q\cos(\phi))^{2}/2\Delta_{0}$,
and the integral is dominated by angles
$\phi=\measuredangle({\boldsymbol{k}},{\boldsymbol{q}})\simeq\pm\pi/2$. We can
then write:

$\displaystyle\text{Im}\left[\Sigma_{\text{bs}}^{R}(\omega)\right]=-\frac{g^{2}\rho
N\Delta_{0}}{v_{F}}\frac{1}{4\pi}\int_{q_{\text{min}}}dq\frac{1}{(q^{2}+M^{2})}$
$\displaystyle\int
dx\theta\left(\omega-3\Delta_{0}-(v_{F}q)^{2}/2\Delta_{0}\times x^{2}\right)=$
$\displaystyle-\frac{g^{2}\rho
N\Delta_{0}^{3/2}}{v_{F}^{2}}\frac{\sqrt{(\omega-3\Delta_{0})}}{\sqrt{2}\pi}\theta(3-\Delta_{0})\int_{q_{\text{min}}}dq\frac{1}{(q^{2}+M^{2})}\frac{1}{q}$
$\displaystyle=-2\sqrt{2}\frac{\Delta_{0}^{2}}{{\omega_{\text{FL}}}}\lambda\ln\left(\frac{{\omega_{\text{FL}}}}{\Delta_{0}}\right)\times\theta(\omega-3\Delta_{0})\sqrt{\frac{\omega-3\Delta_{0}}{\Delta_{0}}}\
,$ (102)

as in Eq. (43)

##  Appendix H Self-energy in the superconducting state: $s$-wave contribution

The proper cutoff to the $s$-wave contribution in the normal state can be
obtained by evaluating the scattering of a particle with a phase fluctuation
of the order parameter (Goldstone mode), described by the diagram of Fig. 15.
It corresponds to

$\displaystyle\Sigma_{\text{bs,s}}(\omega_{m})=N\int_{q}G(\omega_{m}^{\prime}-\omega_{m},{\boldsymbol{q}}-{\boldsymbol{k}})B(\omega^{\prime}_{m},{\boldsymbol{q}})\
\ ,$ (103)

Figure 15: Self-energy diagram which determines the $s$-wave part of the self-
energy in the superconducting state. The double wavy line represents a
Goldstone mode propagator, defined such that the vertex between it and the
fermionic propagators is unity.

Here, the $B$ is the Goldstone mode propagator, which can be obtained from the
$s$-wave part of the backscattering amplitude, Eq. (36),

$\displaystyle\Gamma_{\text{bs}}=-\frac{1}{\rho}\frac{\tilde{\lambda}}{1-\tilde{\lambda}\ln\left(\frac{{\omega_{\text{FL}}}}{v_{F}q\epsilon}\right)}\
,$ (104)

by replacing the logarithm with the $s$-wave piece of the particle-particle
bubble $\Pi_{\text{pp}}(\omega_{m}^{\prime},{\boldsymbol{q}})/\rho$. In the
superconducting state, $\Pi_{\text{pp}}(\omega_{m}^{\prime},{\boldsymbol{q}})$
corresponds to $GG+FF$:

$\displaystyle\Pi_{\text{pp}}(\omega_{m}^{\prime},{\boldsymbol{q}})$
$\displaystyle=\int_{p}\left[G(q-p)G(p)+F(q-p)F(p)\right]\ .$ (105)

Let us first consider $\Pi_{\text{pp}}(0)$. It fulfills

$\displaystyle\Pi_{\text{pp}}(0)\simeq\rho\ln\left({\omega_{\text{FL}}}/\Delta_{0}\right)=\rho/\tilde{\lambda}\
,$ (106)

where the last equation holds by definition of the $s$-wave gap. The energy-
momentum dependent piece is only needed for
$\omega^{\prime}_{m},v_{F}q\ll\Delta_{0}$. The integrands can then safely be
expanded in $\omega^{\prime}_{m},v_{F}q$, and the frequency integration can be
extended to infinity. Neglecting all parts which are odd in the integration
variables $\xi_{\boldsymbol{p}},\tilde{\omega}_{m},\cos(\phi)$ since they
vanish upon integration, one obtains

$\displaystyle\Pi_{\text{pp}}(\omega_{m}^{\prime},q)-\Pi_{\text{pp}}(0)=\rho\int\frac{d\phi}{2\pi}\frac{d\tilde{\omega}_{m}}{2\pi}d\xi_{\boldsymbol{p}}\left\\{-\frac{\left[(\omega_{m}^{\prime})^{2}+(v_{F}q\cos(\phi))^{2}\right]}{\left((\tilde{\omega}_{m})^{2}+\xi_{\boldsymbol{p}}^{2}+\Delta_{0}^{2}\right)^{2}}+\frac{\left[2(\omega_{m}^{\prime})^{2}(\tilde{\omega}_{m})^{2}+2\xi_{\boldsymbol{p}}^{2}(v_{F}q\cos(\phi))^{2}\right]}{\left((\tilde{\omega}_{m})^{2}+\xi_{\boldsymbol{p}}^{2}+\Delta_{0}^{2})\right)^{3}}\right\\}$
(107)
$\displaystyle=-\frac{\rho}{4\Delta_{0}^{2}}\left[(\omega_{m}^{\prime})^{2}+\frac{1}{2}(v_{F}q)^{2}\right]\
.$

Diving this result by $\rho$ and inserting into Eq. (104) yields a Goldstone
mode propagator

$\displaystyle
B(\omega_{m}^{\prime},{\boldsymbol{q}})=-\frac{1}{\rho}\frac{4\Delta_{0}^{2}}{(\omega_{m}^{\prime})^{2}+\frac{1}{2}\left(v_{F}q\right)^{2}}\
.$ (108)

Inserting this into Eq. (103), we have

$\displaystyle\Sigma_{\text{bs,s}}(\omega_{m})=$ (109)
$\displaystyle-N\int_{q}\left(\frac{u_{{\boldsymbol{q}}-{\boldsymbol{k}}}^{2}}{i(\omega_{m}^{\prime}-\omega_{m})-E_{{\boldsymbol{q}}-{\boldsymbol{k}}}}+\frac{v_{{\boldsymbol{q}}-{\boldsymbol{k}}}^{2}}{i(\omega_{m}^{\prime}-\omega_{m})+E_{{\boldsymbol{q}}-{\boldsymbol{k}}}}\right)$
$\displaystyle\quad\times\frac{1}{\rho}\frac{4\Delta_{0}^{2}}{(\omega_{m}^{\prime})^{2}+\frac{1}{2}\left(v_{F}q\right)^{2}}\
,\quad
v_{{\boldsymbol{q}}-{\boldsymbol{k}}}^{2}=\frac{1}{2}\left(1-\frac{\xi_{{\boldsymbol{q}}-{\boldsymbol{k}}}}{E_{{\boldsymbol{q}}-{\boldsymbol{k}}}}\right)\
.$

We perform the frequency integral, closing the contour such that we only
encircle one bosonic pole for both the
$u_{{\boldsymbol{q}}-{\boldsymbol{k}}},v_{{\boldsymbol{q}}-{\boldsymbol{k}}}$
terms. This yields

$\displaystyle\Sigma_{\text{bs,s}}(\omega_{m})=-\frac{N\Delta_{0}^{2}}{\sqrt{2}\pi^{2}v_{F}\rho}\times$
(110) $\displaystyle\int
d{\boldsymbol{q}}\frac{1}{q}\left(\frac{u_{{\boldsymbol{q}}-{\boldsymbol{k}}}^{2}}{-i\omega_{m}-E_{{\boldsymbol{q}}-{\boldsymbol{k}}}-\frac{v_{F}q}{\sqrt{2}}}+\frac{v_{{\boldsymbol{q}}-{\boldsymbol{k}}}^{2}}{-i\omega_{m}+E_{{\boldsymbol{q}}-{\boldsymbol{k}}}+\frac{v_{F}q}{\sqrt{2}}}\right).$

We perform the analytical continuation and take the imaginary part:

$\displaystyle\text{Im}\left[\Sigma^{R}_{\text{bs,s}}(\omega)\right]=$ (111)
$\displaystyle-\frac{N\Delta_{0}^{2}}{\sqrt{2}\pi v_{F}\rho}\int
d{\boldsymbol{q}}\frac{1}{q}\bigg{[}$ $\displaystyle
u_{{\boldsymbol{q}}-{\boldsymbol{k}}}^{2}\delta\\!\left(\omega+E_{{\boldsymbol{q}}-{\boldsymbol{k}}}+\frac{v_{F}q}{\sqrt{2}}\right)$
$\displaystyle+v_{{\boldsymbol{q}}-{\boldsymbol{k}}}^{2}\delta\\!\left(\omega-
E_{{\boldsymbol{q}}-{\boldsymbol{k}}}-\frac{v_{F}q}{\sqrt{2}}\right)\\!\bigg{]}\
.$

We focus on $\omega\gtrsim\Delta_{0}$, which comes from the part $\propto
v_{{\boldsymbol{q}}-{\boldsymbol{k}}}$ and $v_{F}q\ll\Delta_{0}$. To leading
order in $v_{F}q/\Delta_{0}$, one can approximate
$E_{{\boldsymbol{q}}-{\boldsymbol{k}}}\simeq\Delta_{0}$ and
$v_{{\boldsymbol{q}}-{\boldsymbol{k}}}\simeq 1/2$, which gives

$\displaystyle\text{Im}\left[\Sigma_{\text{bs,s}}^{R}(\omega)\right]$
$\displaystyle\simeq-\frac{N\Delta_{0}^{2}}{\sqrt{2}v_{F}\rho}\int_{0}^{\infty}dq\
\delta\\!\left(\omega-\Delta_{0}-\frac{v_{F}q}{\sqrt{2}}\right)$
$\displaystyle=-\pi\frac{\epsilon^{2}}{\lambda}\theta(\omega-\Delta_{0})\frac{\Delta_{0}^{2}}{{\omega_{\text{FL}}}},$

as stated in the main text.

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