# Impact of the X ray edge singularity on detection of relic neutrinos in the
PTOLEMY project

Zhiyang Tan (谭志阳)  Vadim Cheianov<EMAIL_ADDRESS>Lorentz
Institute, Leiden University

###### Abstract

Direct detection of relic neutrinos in a beta-decay experiment is an ambitious
goal, which has for a long time been beyond the reach of available technology.
One of the toughest practical difficulties that such an experiment has to
overcome is that it needs to deal with a large amount of radioactive material
in such a way as to not compromise the energy resolution required for the
separation of useful events from the massive beta-decay background. The
PTOLEMY project offers an innovative approach to this problem based on
deposition of radioactive material on graphene. While such an approach is
expected to resolve the main difficulty, new challenges arise from the
proximity of the beta decayers to a solid state system. In this work we focus
on the effect of the shakeup of the graphene electron system due to a beta-
decay event. We calculate the distortion of the relic neutrino peaks as
resulting from such a shakeup, analyse the impact of the distortion on the
visibility of neutrino capture events and discuss what technological solutions
could be used to improve the visibility of neutrino capture events.

††preprint: APS/123-QED

##  I Introduction

The discovery of the cosmic microwave background (CMB) by Penzias and Wilson
[1] was a pivotal event in Big Bang cosmology. The data from WMAP (Wilkinson
Microwave Anisotropy Probe), 2001 [2] and Planck in 2013 [3] gave us the
snapshot of the Universe dating back to about 13 billion years ago. However,
the CMB provides no direct access to the Universe within 300 000 years of the
Big Bang, because electromagnetic waves could not freely propagate in that
epoch. For this reason, astronomers look for an alternative messenger, a
particle that had decoupled from matter earlier than the CMB. In particular,
the decoupling of neutrinos is believed to have occurred just 1 second after
the Big Bang. Unfortunately, the extremely weak coupling of neutrinos with
matter makes the detection the Cosmic Neutrino Background
($\mathrm{C}\nu\mathrm{B}$) a challenging task. Various ideas have been put
forward as to how the $\mathrm{C}\nu\mathrm{B}$ could manifest itself in a
laboratory experiment [4, 5, 6]. The most practicable route to
$\mathrm{C}\nu\mathrm{B}$ detection today stems from Weinberg’s observation
that the processes of cosmic neutrino capture should leave an extremely weak
however potentially discernible feature in the beta spectra of radioactive
niclei [4]. Weinberg’s original idea was elaborated in several proposals [7,
8] centred around beta decay of Tritium, which has a number of advantages such
as the high neutrino capture cross-section, convenient half-life time,
sufficient abundance and relatively simple chemistry [9].

Beta-decay is a type of radioactive decay in which an atomic nucleus emits a
beta particle [10]. In this process, a neutron decays into a proton, and emits
an electron and an anti-neutrino as well. The induced beta decay process
occurs by absorbing a neutrino. The two processes are illustrated in the
following two reaction equations

${}^{3}\mathrm{H}\to\ ^{3}\mathrm{He}^{+}+e^{-}+\bar{\nu}_{e}$ (1)

and

$\nu_{e}+^{3}\mathrm{H}\to\ ^{3}\mathrm{He}^{+}+e^{-}.$ (2)

Fig. 1 depicts the theoretical beta-electron energy spectrum (called the beta-
spectrum) of free monoatomic ${}^{3}\rm H,$ showing both the beta decay
background and the $\mathrm{C}\nu\mathrm{B}$ peak[11]. Since in a spontaneous
beta decay process one particle decays in three, the energy spectrum of the
beta particle is continuous. In contrast, a low-energy neutrino is absorbed
and only two particles are produced in a relic-neutrino induced beta decay,
therefore the smoking gun signature of a relic neutrino capture is one or more
discrete peaks in the spectrum, shifted about $2m_{\nu}$ above the beta decay
endpoint. The broadening of the peaks seen in 1 reflects the finite energy
resolution of the experiment, which is assumed to be close to 40 meV [12, 13].
The two narrow peaks in the spectrum come from the contributions of different
neutrino mass eigenstates. The energy gap is only about 100 meV, so in order
for the C$\nu$B signal to be visible the energy resolution of the experiment
has to be the same order of magnitude or less than the neutrino mass,
otherwise the extremely weak cosmic neutrino feature will be submerged by the
massive beta decay background.

Although it has been demonstrated that a $40$ meV energy resolution of the
beta-electron detector is within the reach of today’s technology [14], the
task of the fabrication of the neutrino target remains a serious challenge.
The difficulty stems from the conflicting requirements of sufficiently high
sample size on the one hand and sufficiently low probability of information
loss to inelastic scattering on the other. In particular, detectors using gas-
phase Tritium as a neutrino target, such as KATRIN, fall short of the required
resolution [8]. Today, the most promising proposal addressing the issue is the
the solid state setup put forward by the PTOLEMY collaboration [15, 14, 16].
In this design the required density of radioactive material will achieved by
creating a stack of two-dimensional graphene sheets covered by ${}^{3}\rm H.$
Inelastic collisions will be avoided thanks to beta-electrons being emitted
into the empty space between the graphene sheets, whereupon they will guided
into the calorimeter by a cleverly designed electromagnetic guidance system.

While the PTOELMY design offers an appealing solution to one issue, it also
encounters a range of challenges of its own. Recently, it was pointed out that
trapping Tritium on a solid state surface leads to a substantial loss of
energy resolution due to the zero-point motion of the nucleus [17]. Different
strategies of resolving this problem have been discussed such as ingeneering
of the trapping geometry [18], or using heavier emitters may work [19, 20],
however none of them is free of difficulty.

The main conclusion one can draw from these recent developments is that the
energy resolution requirements imposed by relic neutrino detection are so
stringent that it is impossible to build a viable solid state design without
full understanding and control of the energy balance of beta-decay processes
on a solid state surface down to the energy scale of about $10$ meV. Clearly,
the first step towards this goal should be identification and preliminary
theoretical analysis of all solid state effects affecting the beta-electron on
this energy scale. This, presumably, constitutes a considerable research
program consisting of a range of tasks each addressing a separate solid state
effect or interplay of several effects, if necessary. Some early and by no
means comprehensive discussion of potentially harmful effects can be found in
Refs. [17, 18, 12].

In this work we aim to contribute to the theoretical analysis of solid state
effects affecting the resolution of the PTOLEMY experiment by looking into a
particular set of phenomena associated with the shakeup of the electron Fermi
sea in graphene due to the beta-decay process. Ref. [12] identifies the Fermi
sea shakeup as potentially very harmful due to the large bandwidth of
graphene’s conduction band. Our preliminary analysis here shows that the
shakeup effect may not be as dangerous as it may seem from dimensional
considerations. The main reason for this is that the shakeup of the Fermi sea
results in a peculiar distortion of the line shape of a beta-electron, which
is highly asymmetric and contains a power-law divergence at the energy of the
process where no energy is transferred to the Fermi sea. Such a line shape is
reminiscent of the famous X-ray edge singularity in the X-ray emission spectra
(XES) of metals [21, 22, 23, 24, 25, 26] and it is explained by the same
physical mechanism. We demonstrate that the X-ray edge type broadening,
although unpleasant, is not fatal for the PTOLEMY experiment, moreover there
are reasonably straightforward in situ ways to mitigate it. What we find a lot
more dangerous is the effect of the core hole recombination. Our main
conclusion here is that the requirements on the core hole life time are so
stringent that one may need to consider ways to make the ion formed after the
beta-decay chemically stable.

We would like to stress that here we do not address any effects associated
with zero-point motion of the nucleus. The reader may assume that we are
considering beta-decay of a heavy isotope, such as ${}^{171}\text{Tm}$ or
${}^{151}\text{Sm}.$ We also set aside the issues of practicability of the use
of heavy isotopes on graphene.

Figure 1: Beta decay spectrum and the sketch of the beta decay process. The
blue curve depicts the C$\nu$B spectrum, and the red curve represents the beta
decay spectrum, and they are separated by an energy gap, which has an order of
magnitude twice the neutrino mass ($2m_{\nu}$). The annotation of the
horizontal axis is the kinetic energy of the emitted electron measured from
the endpoint of the beta decay spectrum, and the annotation of the vertical
axis is the events observed at a certain energy per yer. The area behind each
cure represents the events per year, and the area behind the C$\nu$B curve is
only 4.

##  II Formulation of the problem

In this section, we formulate the mathematical model of beta-emission on
graphene. This section is structured as follows, firstly, we give the physical
description of the problem; then we list our main assumptions; then, we
explain the approximation scheme used for the calculation of the observed
emission line.

We consider a beta-emitter atom attached to a graphene sheet either by the Van
der Waals force or through the formation of a chemical bond. After capturing
an incoming neutrino, the beta-emitter converts into a daughter isotope,
releasing a fast outgoing electron as depicted in Fig(2). The sudden emergence
of a positively charged daughter isotope ion next to the graphene sheet
triggers local rearrangement the material’s electronic structure, in the exact
same manner as a core hole in an X-ray emission experiment does. In the latter
case, such a rearrangement is known to result in the X-ray edge singularity in
the emission spectrum, which is seen as broadening of the emission line into
an asymmetric shape having a power-law singularity at the emission edge. A
similar phenomenon will occur in the $\beta$-spectrum of a radioactive atom on
graphene.

One remark is in order here. Due to the work function difference, an atom
attached to graphene would typically donate or accept a number of electrons.
This effect is characterised by the atom’s equilibrium charge state, that is
the atomic number less the average total number of electrons occupying its
atomic shells in equilibrium. For an mother isotope $\prescript{A}{Z}{X}$ in
the charge state $Q$ the dominant decay channel will be into a daughter
isotope $\prescript{A}{Z+1}{X}$ in the charge state $Q+1,$

$\nu_{e}+\prescript{A}{Z}{X}^{Q}\to\ \prescript{A}{Z+1}{X}^{Q+1}+e^{-}.$ (3)

Depending on the chemistry of the graphene-atom interaction
$\prescript{A}{Z+1}{X}^{Q+1}$ may or may not be an equilibrium charge state.
If this state is not equilibrium, it will decay into an equilibrium charge
state after some time $\tau.$ By virtue of Heisenberg’s principle, this will
introduce an additional $\hbar/\tau$ uncertainty into the energy conservation
law for the products of the decay and consequently into the energy of the
beta-electron. As will be discussed later in this work, in order for the
neutrino capture peak to be observable, the life time of
$\prescript{A}{Z+1}{X}^{Q+1}$ has to be quite long, for example it would have
to to be longer than $10^{-2}\,\rm s$ for the ${}^{3}\rm He^{+}$ ion formed
after the decay of neutral Tritium. Such long lifetimes require that either
the atom be well insulated from graphene, for example by a thin layer of a
wide gap insulator, or $\prescript{A}{Z+1}{X}^{Q+1}$ is an equilibrium charge
state. The discussion of strategies for making $\prescript{A}{Z+1}{X}^{Q+1}$ a
long-lived state is outside the scope of the present work and it will be
addressed elsewhere. It is worth noting that in a situation where the charge
state of the daughter $\prescript{A}{Z+1}{X}^{Q+1}$ is metastable there always
exists a direct matrix element for the process in which the daughter atom is
created in the stable state $\prescript{A}{Z+1}{X}^{Q^{\prime}}$ and the
charge balance is taken care of by creation of elementary excitations in
graphene. Due to insufficient kinematic constraints, such a few body process
does not result in a sharp neutrino absorption peak. Rather, it forms a broad
feature at the tail of the Lorentzian distribution of the absorption resonance
defined by the metastable state.

Figure 2: Sketch of the beta emitter and graphene sheet.
$\prescript{A}{Z+1}{X}^{Q+1}$ is the production of the beta decay process, and
it introduces a sudden localized perturbation in the electron system of
graphene. This sudden localized perturbation causes an X-ray edge, which
changes the beta decay spectrum. The thin green layer is a spacer inserted
between the beta emitter and the graphene sheet to isolate the daughter
isotope $\prescript{A}{Z+1}{X}^{Q+1}$ from the graphene sheet. Thus it
improves the lifetime of $\prescript{A}{Z+1}{X}^{Q+1}$.

We can now state the main assumptions about the system that will enable us to
focus on the Fermi sea shakeup effect
(1) The recoil of the daughter ion from the beta decay will be neglected. This
effect and its mitigation strategies have been addressed in recent literature
[17, 27, 28, 18]. Although significant on the $10\,\rm meV$ scale, this effect
is not directly related to the Fermi sea shakeup discussed here.
(2) We neglect the electron-electron scattering in the graphene sheet. We
include the Coulomb interaction between electrons only in the form of the
Random Phase Approximation (RPA) [29, 30] corrections to the dielectric
response of graphene.
(3) We only focus on the processes which do not result in the excitation of
internal degrees of freedom of the daughter ion. Only such processes are
relevant to the neutrino capture experiment.
(4) We assume that the graphene sample spatially uniform. Where we analyse the
effect of disorder, we assume that the disorder is spatially uniform.
(5) We neglect crystalline lattice effects arising from the the lateral
position of the beta-decayer relative to the graphene sheet. This is because
the X-ray edge singularity is an infrared phenomenon, arising from the long-
range Coulomb interaction, which is insensitive to the exact position of the
Coulomb centre inside the unit cell.
(6) We assume for simplicity that the initial charge state of the beta-decayer
is neutral.
(7) We assume that the beta-decay of the nucleus produces a long-lived
daughter ion. The quantitative meaning of long-lived will be elaborated later.
(8) We assume that the temperature of the system is less than the resolution
of the detector, which is about 100K. We thus neglect thermal effects and
consider our model at zero temperature.

Based on the above assumptions, we formulate the Hamiltonian

$H=H_{0}+H_{w}+H_{D-G}$ (4)

which comprises the kinetic part, weak interaction part and the coupling
between the daughter ion and graphene. The kinetic part is trivial, and it is
the total relativistic energy of free motion of all particles involved in the
process

$H_{0}=H_{\nu}+H_{D}+H_{M}+H_{e}+\sum_{s}\sum_{k}s\hbar
v_{F}kC^{\dagger}_{ks}C_{k^{\prime}s}$ (5)

Here $\nu$ stands for the neutrino, $e$ for the electron, $M$ and $D$ for the
mother and the daughter isotopes respectively, $v_{F}$ is the Fermi velocity
$C^{\dagger}_{ks}$ and $C_{ks}$ are the creation and annihilation operators
for the electrons in the graphene respectively, and s is the band index of the
graphene. The spin and valley indices have been absorbed into the band index.

The term

$H_{w}=\int G_{F}\nu MD^{\dagger}e^{\dagger}d^{3}x,$ (6)

is the effective Hamiltonian of the beta-decay process, where $\nu^{\dagger}$
is the creation operator for the neutrino, $D^{\dagger}$ is the creation
operator for the daughter ion, and $M$ is the annihilation operator for the
mother isotope atom, $e^{\dagger}$ is the creation operator for the outgoing
electron, and $G_{F}$ is the effective beta decay interaction constant
absorbing the details of ultrafast electroweak processes inside the nucleus
[10, 31],
The final term in the Hamiltonian, $H_{D-G}$ describes the interaction between
the electrons in graphene and the charged daughter ion, which suddenly emerges
due to the beta-decay process.  Since the the daughter ion is very heavy and
therefore can be treated as a localised object 111In fact, the departure from
the local approximation due to the quantum zero-point motion of the beta-
decayer is another source of the uncertainty of the emitted electron
energy[17]. It could, in principle, be suppressed by choosing heavier beta-
decayers, e.g. rare earth atoms. We do not address the zero-point motion
effect in this work assuming that it is less important than the shakeup effect
analysed here. , we can write its density operator as
$D_{0}^{\dagger}D_{0}\delta(\mathbf{r})$, where the origin $\mathbf{r}=0$
coincides with the equilibrium position of the mother atom and
$D_{0}^{\dagger}D_{0}$ is the daughter ion counting operator. Thus we write
this interaction term as follows

$H_{D-G}=-\int\rho(x)V(x)D^{\dagger}_{0}D_{0}d^{2}x.$ (7)

where

$V(x)=\frac{e^{2}}{4\pi\epsilon\epsilon_{0}\sqrt{d^{2}+x^{2}}}$ (8)

is the Coulomb pontential of the ion in the graphene plane, and

$\rho(x)=:\psi^{\dagger}(x)\psi(x):,$ (9)

is the two-dimensional density of electrons in graphene. We have denoted the
dielectric constant of the graphene as $\epsilon$ and the distance between the
daughter ion and the graphene as $d$. The field operator of an electron in
graphene $\psi(x)$ admits for the standard plane wave decomposition

$\psi(x)=\sum_{k,s}C_{ks}\psi_{ks}e^{-i\vec{k}\vec{x}}$ (10)

where $\psi_{ks}$ is the spinor of the dirac electrons in the graphene,

$\psi_{ks}=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{-i\theta_{k}}\\\
s\end{pmatrix}$ (11)

with $\theta_{k}=\arctan(k_{y}/k_{x})$.

In the standard vein of beta decay theory, the transition probability is given
by Fermi’s golden rule

$W=\frac{2\pi}{\hbar}\sum_{f}\biggl{|}\left<0\right|_{t}\left<0\right|_{\nu}\left<k\right|_{e}\left<1\right|_{h}\left<\lambda\right|\int
G_{F}\nu dh^{\dagger}e^{\dagger}d^{3}x\\\
\left|1\right>_{t}\left|1\right>_{\nu}\left|0\right>_{e}\left|0\right>_{h}\left|FS\right>\biggr{|}^{2}\delta(E_{f}-E_{i})$
(12)

where $f$ refers to the final states of the process, and
$\left|\lambda\right>$ is the final eignstate of the graphene Hamiltonian with
the energy $E_{\lambda}$ and $\left|\rm FS\right>$ is the Fermi sea of the
graphene electrons characterised by the ground state energy $E_{0}$. The total
energy of final state in the process is
$m_{e}c^{2}+\frac{\hbar^{2}k^{2}}{2m_{e}}+E_{\lambda}+m_{D}c^{2}$, and the
energy of initial state is $m_{v}c^{2}+m_{M}c^{2}+E_{0}$. The final state is
the tensor product of the neutrino state, the outgoing electron state, and the
graphene electron state. We can sum over the normal beta decay part and
graphene part independently. We use index $f^{\prime}$ to denote the final
state of beta decay and use index $\lambda$ to denote the final state of
electrons in graphene. Moreover, we can express the total transition
probability as a convolution of the transition probability function of beta
decay and graphene spectral density function. Hence, the expression of the
total transition probability can be simplified to

$W=\frac{2\pi}{\hbar}\int
dE\sum_{f^{\prime}}\sum_{\lambda}|V_{if^{\prime}}|^{2}|\left<\lambda|FS\right>|^{2}\delta(E+E_{\lambda}-E_{0})\\\
\times\delta(E_{k}-Q-E)\\\ =(\Gamma*A)(E_{k}),\\\ $ (13)

where $V_{if^{\prime}}$ is the weak interaction matrix element,
$Q=m_{M}c^{2}+m_{v}c^{2}-m_{e}c^{2}-m_{D}c^{2}$ is the emission energy of beta
decay, $E_{k}$ is the kinetic energy of the beta electron, $\Gamma(E)$ is the
beta decay transition rate of a single atom in vacuum, and $A(E)$ is the
electron spectral density function in graphene. Note, that the energy
conservation law in Eq. (LABEL:fermi) neglects the kinetic energy of both the
heavy particles and the neutrino. The latter is due to the fact that we are
interested in the narrow vicinity of the endpoint of the beta decay spectrum,
furthermore, the temperature of the cosmic neutrino background, $1.95\,\rm K$,
is negligible compared to the neutrino’s mass. Therefore, eq. (LABEL:fermi)
both applies for beta decay background and the cosmic neutrino absorption
process. Following the convention, we can consider
${d\tilde{\Gamma}}/{dE_{k}}$ as observable beta decay rate at given electron
kinetic energy corrected by the electron shakeup process

$\frac{d\tilde{\Gamma}}{dE_{k}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}N}\frac{dW}{dE_{k}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}N}\left(\frac{d\Gamma}{dE_{k}}*A\right)(E_{k}),$
(14)

where $N$ is the number of beta-decayers deposited on graphene. The discussion
above also valid for the process of background neutron decay. We have found
out that the smeared beta decay spectrum is nothing but the convolution
between the spectral density function of graphene and the original beta decay
spectrum. This result is vital to us since it tells us the influence of the
X-ray edge in the PTOLEMY project. Later, we will show that the spectral
density function does have an X-ray edge.

##  III Linked Cluster Expansion

In this section we assume that daughter ion is screened instantaneously. This
assumption does not work well in graphene and we will revise it later, however
at this stage we would like to keep our discussion as simple as possible
focusing on the key physics of the problem. With the help of the RPA and
neglecting time retardation, we obtain the effective static dielectric
constant of graphene [33]

$\epsilon=\frac{1+\kappa}{2}+\frac{2\pi e^{2}}{q}\frac{g_{s}g_{v}q}{16\hbar
v_{F}}$ (15)

where $\kappa$ is the external dielectric constant of the substrate, the
degeneracy factor $g_{s}=2,g_{v}=2$, and $v_{F}\approx 10^{6}\ \mathrm{m/s}$.
If the graphene is suspended in vacuum, then $\epsilon\approx 4.4$.

The spectral density function $A(E)$ is the central object of interest, so we
will investigate its property further in this section. The expression for it
is

$\begin{split}A(E)&=\sum_{\lambda}\left|\left<\lambda|FS\right>\right|^{2}\delta(E+E_{\lambda}-E_{0})\\\
&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\frac{1}{2\pi\hbar}\sum_{\lambda}\int
dt\left<FS\right|e^{-iH_{g}t}\left|\lambda\right>\left<\lambda|FS\right>e^{-i\left(E-E_{0}\right)t/\hbar}}\\\
&=\frac{1}{2\pi\hbar}\int
dt\left<FS|e^{-iH_{g}t/\hbar}|FS\right>e^{iE_{0}t/\hbar}e^{-iEt/\hbar},\end{split}$
(16)

where $H_{g}$ is the Hamiltonian of the graphene after the beta decay. We can
express it in momentum space [33, 34],

$H_{g}=\sum_{s}\sum_{k}s\hbar v_{F}kC^{\dagger}_{ks}C_{k^{\prime}s}\\\
-\frac{1}{L^{2}}\sum_{ss^{\prime}}\sum_{kk^{\prime}}V(k,k^{\prime})F_{ss^{\prime}}(k,k^{\prime})C^{\dagger}_{ks}C_{k^{\prime}s^{\prime}}$
(17)

with

$F_{ss^{\prime}}(k,k^{\prime})=\frac{1}{2}\left[ss^{\prime}+\exp(i\theta_{k}-i\theta_{k^{\prime}})\right]$
(18)

and

$V(k,k^{\prime})=\frac{2\pi
e^{2}\exp\left(-|k^{\prime}-k|d\right)}{\epsilon|k-k^{\prime}|}$ (19)

To calculate the spectral density function $A(E)$, we first focus on the
density function, which is its Fourier transform

$\rho(t)=\left<\mathrm{FS}\left|\mathrm{e}^{-iH_{g}t/\hbar}\right|\mathrm{FS}\right>\mathrm{e}^{iE_{0}t/\hbar}.$
(20)

One can recognize $-i\Theta(t)\rho(t)$ as the core hole Green function in the
conventional X-ray singularity problem [35, 22, 23, 36, 29]. Using linked
cluster expansion method [29, 37, 38], we can express the density function
$\rho(t)$ in another way.

$\rho(t)=\exp\left[\sum_{l}F_{l}(t)\right]$ (21)

where $F_{l}(t)$ is the l-th connected diagrams,

$F_{l}(t)=\frac{(-i)^{l}}{l}\int_{0}^{t}dt_{1}\cdots\int_{0}^{t}dt_{l}\left<TV(t_{1})\cdots
V(t_{l})\right>_{\mathrm{connected}}.$ (22)

$V(t)$ is the interaction term in the Dirac picture.  We note that
$F_{1}(t)=-iE_{i}t$ where the constant $E_{i}$ is called the self energy. The
self energy is responsible for the overall energy shift of the beta spectrum
relative to the vacuum one. Assuming the interaction is weak, we can restrict
ourselves to the second term [29].

$F_{2}(t)=-\int_{0}^{\infty}\frac{du}{u^{2}}R_{e}(u)(1-\mathrm{e}^{-iut})$
(23)

where

$R_{e}(u)=\frac{1}{\pi L^{2}}\sum_{q}|V(q)|^{2}\Lambda(q,u)$ (24)

and

$\Lambda(q,u)=\mathrm{Im}P^{(1)}(q,u).$ (25)

where $P^{(1)}(q,u)$ is the polarizability function. The epxression of the
polarizability is given in many references [29, 39], so we just quote the
result

$P^{(1)}(q,u)=\frac{-ig^{\prime}}{L^{2}}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\sum_{p,s,s^{\prime}}|F_{ss^{\prime}}(p,p+q)|^{2}\\\
\times G_{ss}(p,E)G_{s^{\prime}s^{\prime}}(p+q,E+u),$ (26)

where $G_{ss}(P,E)$ is the Green function for graphene, $g^{\prime}$ is the
degeneracy, and it is $4$ for intrinsic graphene. The polarizabiltiy function
of the intrinsic graphene was calculated in the references [40, 33, 41]. In
particular, its imaginary part gives

$\Lambda(q,u)=\frac{q^{2}}{4\sqrt{u^{2}-v_{F}^{2}q^{2}}}\Theta(u-v_{F}q).$
(27)

To calculate $R_{e}(u)$, one needs to substitute eq. (27) into eq. (24). To
simplify the calculation, we omit the exponential factor in the eq. (19), and
the potential becomes

$V(k,k^{\prime})=\frac{2\pi e^{2}}{\epsilon|k-k^{\prime}|}$ (28)

Although this crude simplification is only valid near the edge of the spectral
density function, it gives us a physics insight in the first step. Later, we
will recover the effect of the distance d in the next section. Then we have

$R_{e}(u)=gu,$ (29)

where

$g=\frac{e^{4}}{2\epsilon^{2}v_{F}^{2}}$ (30)

Using the effective dielectric constant we get in the previous step, we can
find $g\approx 0.125$. Substituting it into the expression of $F_{2}$, thus
one can get

$\begin{split}F_{2}(t)&=-g\int_{0}^{\xi_{0}}\frac{(1-e^{-iut})du}{u}\\\
&\approx-g\int_{1/it}^{\xi_{0}}\frac{du}{u}\approx-g\ln(1+it\xi_{0})\end{split}$
(31)

Note that due to the logarithmic divergence of the integral we had to use an
arbitrary ultraviolet cutoff $\xi_{0}$ to complete the calculation. This
cutoff parameter is important for the estimate of the visibility of the
$\mathrm{C}\nu\mathrm{B}$ peak, therefore it cannot be chosen arbitrarily. The
physical meaning and the value of $\xi_{0}$ will be clarified later.

The expression (31) is valid only for large enough $t$, that is $\xi_{0}t>>1$.
The expression for the density function is obtained from eq. (31),

$\rho(t)=\exp(-iE_{i}t-g\ln(1+it\xi_{0})),$ (32)

where the self energy term $E_{i}$ can be absorbed into the exponent $E$ in
the Fourier transformation in the later steps. It has no effect on shape of
the spectral density function, but shift it by $E_{i}$.

##  IV Spectral density function

In this section, we give an explicit formula of the spectral density function
of graphene. Furthermore, we investigate the influence of the height from the
daughter ion above the graphene sheet, as well as the influence of disorder
and the dynamical screening on the spectral density function.

The Fourier transform of eq. (32) gives the spectral density function of
graphene at the given energy, and it is exactly the gamma distribution
function.

$\begin{split}A(\Omega)&=\int_{-\infty}^{\infty}\frac{dt}{2\pi}e^{-iEt}\rho(t)\\\
&=\Theta(-\Omega)\frac{\exp(\Omega)}{\xi_{0}\Gamma(g)(-\Omega)^{(1-g)}}\end{split}$
(33)

with

$\Omega=(E+E_{i})/\xi_{0}$ (34)

The spectral density function obeys the gamma distribution, and the result
coincides with that for a metal with a local constant interaction [29, 37,
38]. The gamma distribution function has two parameters: one is the shape
parameter, which is the coupling constant in our problem, and another one is
the scale parameter, which has so far remained an arbitrary cutoff parameter.
The coupling constant determines how sharp the peak is near the edge, and the
cutoff energy determines how many events are lost in the long tail of the
spectral density function. Fig. 3 represents the spectral density function. As
we can see in Fig. 3, the function sharply diverges when $\Omega$ goes to
zero. In other words, the differential emission rate is the largest when no
electron excitations are created in graphene at the end of the process.

Figure 3: The spectral density function of graphene at a given energy.
$A(\Omega)$ describes the possibility of graphene absorbing the outgoing
electron’s energy. The energy is measured in the unit of cut-off energy
$\xi_{0}$.

###  IV.1 Influence of the Height

In the previous discussion, we have calculated the spectral density function
and obtained its shape parameter $g.$ However, the cutoff energy remains
unknown. Typically, one would expects the cutoff energy to be associated with
the electron bandwidth or the Fermi energy. In this section we find that the
cutoff energy is dictated by the height of daughter ion above the graphene
sheet. Indeed, for any given $d$ the expression of $R_{e}(u)$ is

$\begin{split}R_{e}(u)=e^{4}\int_{0}^{u/v_{F}}\frac{qe^{-2qd}dq}{2\epsilon^{2}\sqrt{u^{2}-v_{F}^{2}q^{2}}}\\\
=gu+\frac{\pi
gu}{2}\left[L_{1}\left(\frac{2du}{v_{F}}\right)-I_{1}\left(\frac{2du}{v_{F}}\right)\right]\end{split}$
(35)

where $L_{1}\left(\frac{2du}{v_{F}}\right)$ and
$I_{1}\left(\frac{2du}{v_{F}}\right)$ are modified Struve function and
modified Bessel function, respectively.
This equation may look slightly intimidating however its intuitive meaning is
simple. The X-ray edge singularity is an infrared phenomenon, so only long
distance interaction is significant for it. Thus we can eliminate the short-
distance structure of the Coulomb interaction. If $q<<1/(2d)$, the Yukawa type
potential in eq. (35) recovers to the normal Coulomb potential, thus giving a
natural cutoff energy $\xi=\hbar v_{F}/2d$. From asymptotic analysis (details
are in A1), we find that the spectral density function was the same form as in
Eq. (33) albeit with a cutoff energy $\xi^{\prime}$,

$\xi^{\prime}=\frac{\xi_{0}}{1+2d\xi_{0}/v_{F}}.$ (36)

$\xi_{0}$ is a microscopic parameter on the order of the bandwidth energy.
Assuming that $\xi_{0}\gg v_{F}/d$, one finds

$\xi=\lim_{\xi_{0}\to\infty}\xi^{\prime}=\frac{\hbar v_{F}}{2d}$ (37)

The expression (37) should replace $\xi_{0}$ as the scale factor in the line
shape equation Eq. (33). One can see that the scale factor depends on the
height $d,$ so one could potentially manipulate the line shape by adjusting
the height of the ion.

###  IV.2 Influence of disorder

Until now, we have only considered intrinsic graphene, uncontaminated and
ungated. However, a finite density of beta emitters in graphene inevitably
introduces disorder, which changes the mean free path and the density-density
response function of electrons. Intuitively, one should expect that if the
time scale associated with the inverse energy resolution of the beta detector
is greater than the mean free time in graphene, i.e.,
$\tau_{\mathrm{res}}\gg\tau$, then one needs to take into consideration the
effect of electron diffusion on the polarizability function and thus the
coupling constant.

We consider eq. (16), in the presence of many random impurities, focusing on
the impurity-average effect. The impurity-average polarizability function
$P_{\mathrm{imp}}(q,\omega)$ is [30, 42, 39]

$P_{\mathrm{imp}}(q,\omega)=\frac{P^{(1)}(q,\omega+\frac{i}{\tau})}{1+(1-i\omega\tau)^{-1}\left[\frac{P^{(1)}(q,\omega+\frac{i}{\tau})}{P^{(1)}(q,0)}-1\right]}$
(38)

where $\tau$ is the momentum relaxation time of a quasiparticle. The required
energy resolution of the detector is $\omega_{\mathrm{res}}$ is about 10 meV,
therefore the resolution time $\tau_{\rm res}$ is $\sim 10^{-13}\ s$. If
$\tau>\tau_{\rm res}$, then one can neglect the effect of impurities, and the
polarizability function recovers to the Lindhard function [30]. Otherwise, one
needs to consider both diffusive and ballistic regimes.

To get some insight into the degree of coverage at which it is necessary to
take the impurity scattering into consideration, we give an elementary
estimate based on the mean free time calculated within the midgap model which
is known to work reasonably well for hydrogen and other atoms with covalent
on-site bonding on graphene [43, 44, 45]. In this model, mobility electrons in
graphene see a carbon site with attached hydrogen as a vacancy. The potential
is profiled as

$U(r)=\begin{cases}\infty,&\mbox{if }0<r\leq R^{\prime}\\\ U_{0},&\mbox{if
}R^{\prime}<r\leq R\\\ 0,&\mbox{if }r>R,\end{cases}$ (39)

where $R^{\prime}$ is the the radius of a hydrogen atom, and $R$ is the radius
of a carbon atom. The mean free time is defined from the following expression

$\frac{\hbar}{\tau_{k}}=\frac{8n_{i}}{\pi\rho(E_{k})}\sin^{2}\delta(k),$ (40)

where $\delta(k)$ is the phase shift of electrons with momentum k, and
$\rho(E_{k})$ is the density of state, with the explicit form

$\rho(E_{k})=\frac{2E_{k}}{\pi\hbar^{2}v_{F}^{2}}.$ (41)

Using the method in the paper [44], we can obtain the phase shift at $k\approx
0$,

$\delta(k)=-\frac{\pi}{2}\frac{1}{\ln(2kR^{\prime})}.$ (42)

We expand the sine function in eq. (40) for $kR^{\prime}\ll 1$. Thus it gives
the mean free time

$\begin{split}\tau_{k}&=\frac{\hbar\rho(E_{k})}{2\pi n_{i}}(\ln
kR^{\prime})^{2}\\\ &=\frac{k}{\pi^{2}v_{F}n_{i}}(\ln
kR^{\prime})^{2},\end{split}$ (43)

where $n_{i}$ is the impurity concentration. The mean free path has a minimal
value at $kR^{\prime}=\mathrm{e^{-2}}$, and its minimum value is

$\tau_{min}=\frac{4\mathrm{e^{-2}}}{\pi^{2}v_{F}n_{i}R^{\prime}}$ (44)

If $\tau_{min}$ is much greater than the resolution time, we can neglect the
impurity effect. It means that

$\frac{4\mathrm{e^{-2}}}{\pi^{2}v_{F}n_{i}R^{\prime}}\gg 1/\omega\sim
10^{-13}\ s$ (45)

$R^{\prime}$ is about 1 $\mathrm{\SIUnitSymbolAngstrom}$, so we find the
condition for a maximum impurity concentration.

$n_{i}\ll\frac{4\mathrm{e^{-2}}\omega}{\pi^{2}v_{F}R^{\prime}}=5\times
10^{11}/\mathrm{cm^{2}}$ (46)

To conclude, for atoms in the onsite boding configuration the effect of
disorder is negligible for surface coverage of less than $5\times
10^{11}/\mathrm{cm^{2}}.$ For atoms in other configurations the critical
concentration could be different, even though we do not expect the difference
to be dramatic. If impurity atoms are separated from graphene by an thin layer
of insulator the critical coverage may be significantly greater.

Apart from impurities, phonons can also influences the mean free time of
graphene. However, one can keep the system in a very low temperature to reduce
the phonon scattering. For intrinsic graphene under liquid nitrogen
temperature, the momentum relaxation time is about one ps [46, 47], which is
one order of magnitude greater than the resolution time. Therefore, we can
treat electrons of the graphene ballistically within the resolution time.

###  IV.3 Influence of Dynamic Screening

The instantaneous screening assumption is not valid in the tail of the
spectral function of graphene. While, if the weight of the tail is too high,
it will cause the beta decay spectrum to broaden widely, and it may take many
years to observe a single event. To get a more rigorous result, we need to
consider the dynamic screening effect. The dynamic screening potential is [29]

$V(q,\omega)=\frac{V_{i}(q,\omega)}{\epsilon(q,\omega)}$ (47)

For a sudden external potential [48],

$V_{i}(q,\omega)=\frac{V(q)i}{\omega+i\delta}.$ (48)

$\delta$ is an infinitesimal quantity, and $V(q)={2\pi e^{2}}/{(\kappa q)}$ is
the bare Coulomb potential with $\kappa$ the external dielectric constant of
the substrate. We notice that $\epsilon^{-1}(q,\omega)-1$ is the
susceptibility function so we can use the Kramers-Kronig relations.

$\frac{1}{\epsilon(q,\omega)}-1=-\frac{1}{\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\mathrm{Im}\left(\frac{1}{\epsilon(q,\omega^{\prime})}-1\right)\frac{1}{\omega-\omega^{\prime}+i\delta}$
(49)

Substituting it into the expression of dynamic screening potential and
performing the Fourier transform, one can find the time-dependent screening
potential.

$\begin{split}V(q,t)&=\int_{-\infty}^{\infty}V(q,\omega)\mathrm{e}^{-i\omega
t}d\omega\\\
&=\frac{V(q)\theta(t)}{\pi}\int_{-\infty}^{\infty}\frac{1-\mathrm{e^{-i\omega^{\prime}t}}}{\omega^{\prime}}\mathrm{Im}\left(\frac{1}{\epsilon(q,\omega^{\prime})}-1\right)d\omega^{\prime}\\\
&\quad\quad\quad\quad\quad\quad+V(q)\theta(t)\\\ \end{split}$ (50)

In the large time limit, the oscillating term disappears, so the potential
becomes

$V(q,\infty)=\frac{V(q)}{\epsilon(q,0)}$ (51)

For graphene, $\epsilon(q,0)$ does not depend on q, so that we can denote
$\epsilon(q,0)$ as $\epsilon$. The first term in the last line vanishes in the
short time limit, so the potential turns into the bare potential.

$V(q,0)=V(q)$ (52)

For an arbitrary time, we can express $V(q,t)$ as

$V(q,t)=\frac{V(q)\theta(t)}{\epsilon(q,0)}-\frac{V(q)\theta(t)}{\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\frac{\mathrm{e^{-i\omega^{\prime}t}}}{\omega^{\prime}}\\\
\times\mathrm{Im}\left(\frac{1}{\epsilon(q,\omega^{\prime})}-1\right)$ (53)

From detailed analysis (see A2), we found out that the spectral density
function is the same as the previous result in eq. (33), but with a different
coupling constant

$g_{1}=g+\int_{1}^{\infty}\frac{2g\epsilon}{\pi}\frac{a\log(\omega+\sqrt{\omega^{2}-1})}{\omega^{2}\left(\omega^{2}-1+a^{2}\right)}d\omega,$
(54)

where $a=(\epsilon-1)={\pi e^{2}}/{2\hbar\kappa v_{F}}$, and this result is
quoted from the A2 eq. (78 ). It is similar in meaning to the fine structure
constant in QED.

If graphene is suspended in vacuum, then $g=0.125$, and $g_{1}\approx 0.200$.
For graphene on ${\rm SiO}_{2}$, the coupling constant g is about 0.065, and
$g_{1}\approx$ 0.090 222 The coupling constant $g_{1}$ is calculated from eq.
(54), and the coupling constant g is from the eq. (30). As one can see in Fig.
4, the dynamic screening effects of intrinsic graphene significantly change
the coupling constant at a small dielectric constant, and this can be
understood intuitively. For intrinsic graphene, the Fermi energy is zero,
therefore there is no intrinsic time scale. For long wave-length scattering,
the screening time is about $1/(v_{F}q)$, which is also the time scale for the
Fermi sea shakeup effect at the energy scale $v_{F}q$, so we cannot separate
the screening of the Coulomb potential from the formation of the X-ray edge
singularity. Hence, the dynamical screening effects have a considerable
influence on the coupling constant. In contrast, the Fermi energy is non-zero
for normal metals and gated or doped graphene, so the static screening is a
good approximation near the edge in those cases. In the large dielectric
constant limit, one can see from Fig. 4(b), the ratio of the dynamic coupling
constant and static coupling constant approaches 1. It can be understood from
eq. (15) and eq. (54), when the external dielectric constant is big enough, it
makes the dominant contribution to the total dielectric constant.

To summarize, the dynamical screening effect has a considerable influence on
the coupling constant in intrinsic graphene, but it can be suppressed by
applying an external gate voltage or increasing the dielectric constant of the
substrate.

(a) (b)

Figure 4: Fig. 4(a) depicts the coupling constant with different external
dielectric constants. $\kappa$ describes the dielectric constant of the
substrate, and g is the graphene’s coupling constant. The blue curve
represents the coupling constant considering the dynamic screening effect, and
the orange curve representing the coupling constant in the static screening
approximation. Fig. 4(b) describes the ratio of the dynamic coupling constant
and the static coupling constant. In the large dielectric constant limit, the
ratio approaches 1.

###  IV.4 Constraints on the Daughter Ion’s Lifetime

In the previous discussion, we assumed that the lifetime of the daughter ion
is infinite, however in reality this may not be true. The daughter ion could
be chemically unstable and have propensity to, for example, capture an
electron from its chemical environment. By virtue of Heisenberg’s principle
shorter life times would introduce greater uncertainty into the measured
energy of the beta electron. In the conventional X-ray edge singularity
context the life time of an ion is limited by the process of recombination of
a conductance electron with the core hole. Such a process requires
accommodation of a large amount of energy which is usually achieved through
the Auger effect. In the case of an ion which is formed through beta decay,
the electron deficit occurs in the chemically active outer shell therefore
capture of an electron may be possible through a direct tunneling process. The
efficiency of such a process will vary depending on the way an atom is
deposited on the surface. Insertion of a dielectric spacer between the atom
and a graphene layer, or deposition of atoms in the form of self-assembled
metal-organic complexes [50] could be possible ways to make the life time of
the daughter ion longer. Ideally, one should aim to tune the chemical
environment so as to make the daughter ion chemically stable. Somehow or
other, the life time of an ion formed after the beta decay process is
determined by a range of environmental mechanisms and is a matter for the
experiment to measure and manipulate. Here, we do not attempt to estimate it.
Rather we investigate how the finite life time of the daughter ion influences
the visibility of the $\mathrm{C}\nu{\mathrm{B}}$ signal.

Semi-empirically, one absorbs the effects of a finite life time into the
Lorentzian broadening of the delta function representing the energy
conservation law. For the spectral density function that amounts to an extra
convolution [29].

$A(E)\mapsto\int_{-\infty}^{\infty}A(\mathcal{E})L(E-\mathcal{E})d\mathcal{E}$
(55)

where $L(E)$ is the Lorentzian distribution function,

$L(E)=\frac{\gamma/2}{\pi[E^{2}+(\gamma/2)^{2}]}.$ (56)

$\tau=\hbar/\gamma$ is the lifetime of the daughter ion. The thus deformed
spectral function should then be used in Eq. (14) to describe the observable
beta spectrum. In particular, inserting eq. (56) into eq. (55), one obtains
the shape of the spectral density function

$A(E)=\operatorname{Im}\left[\frac{\mathrm{e}^{\tilde{\Omega}}\Gamma(1-g,\tilde{\Omega})}{\pi\xi_{0}\tilde{\Omega}^{1-g}}\right],$
(57)

where $\Gamma(x,y)$ is the incomplete gamma function and

$\tilde{\Omega}=\frac{E-i\gamma/2}{\xi_{0}}=\Omega-\frac{i\gamma}{2\xi_{0}}.$
(58)

To illustrate such a shape, we plotted $A$ as a function of the dimensionless
eneregy $\Omega$ for $g=0.125$, $\xi_{0}=1\ \rm eV$, and $\gamma=0.2\ \rm eV$
in Fig. 5. As one can see, at finite $\gamma$ the X-ray edge power law
singularity gets deformed into an asymmetric bell-shaped distribution. The
longer the life time, the closer the absorption peak is to the Gamma
distribution (33).

While it is tempting to conclude that the main manifestation of a finite life
time of the daughter ion is just some degree of broadening the neutrino
absorption peak, which can be neglected for $\gamma\ll m_{\nu},$ on closer
inspection the effect turns out to be a lot more detrimental and the
conditions on $\gamma$ a lot more stringent. Moreover, the semi-empirical one-
parameter model encapsulated in equation (56) turns out to be unsuitable for
the quantitative analysis of the visibility of the $\rm C\nu\rm B$ peak.
Indeed, unlike the pure X-ray edge singularity (33), the semi-phenomenological
distribution (57) has a long power-law tail on the right-hand side of the peak

$A(E)\sim\frac{\gamma}{E^{2}},\qquad E\gg\gamma$ (59)

At the same time, the beta-decay background near the emission edge
$E_{0}\approx Q$ has the parabolic shape

$\frac{d\Gamma(E)}{dE_{k}}=\alpha\frac{N}{Q^{3}T}(E_{0}-E)^{2},\qquad
E<E_{0}.$ (60)

which remains a good approximation up to $E_{0}-E$ on the order of $Q.$ In
this expression $\alpha$ is a nucleus-specific numerical constant on the order
of 1 and $T$ is the half-life time. Due to the rapid increase of the
background intensity away from the termination point $E_{0}$, the convolution
(14) at energies slightly above the edge of the spectrum is completely
dominated by the bulk of the beta spectrum and to a good accuracy can be
expressed as

$\frac{d\tilde{\Gamma}_{\rm BG}}{dE_{k}}=C+o(E-E_{0})$ (61)

where

$C=\frac{\gamma}{2\pi}\int_{0}^{Q}\frac{1}{(Q-\mathcal{E})^{2}}\frac{d\Gamma}{dE_{k}}(\mathcal{E})d\mathcal{E}\sim\frac{\gamma
N}{Q^{2}}$ (62)

In other words, the number of beta decay events as a function of the energy
distance from the emission threshold increases so rapidly that it overwhelms
the slow $1/E^{2}$ decay of the Lorentzian and thus creates a massive
background in the region of the $\rm C\nu B$ peak. In reality, the fact that
the main contribution to noise comes from the tail of the Lorentzian
distribution invalidates the model (56) and the resulting estimate (62). To
obtain the accurate result one would have to replace $L(E)$ with the exact
ion’s spectral function. Although such a spectral function is impossible to
calculate in practice, one can still make an order of magnitude estimate of
the background intensity in the $\rm C\nu B$ peak region basing on the
observation that the tail on the right-hand side of the resonant peak
terminates at energies on the order of the ionisation energy of the recombined
atom. More precisely, the termination point of the support of the ion’s
spectral function is defined by the process of direct transition of the beta-
decayer into the neutral daughter atom and a positively charged quasihole on
graphene’s Fermi surface, which is the lowest energy out-state of the combined
graphene - daughter atom system. In order to incorporate this physics into the
semi-phenomenological model we introduce a truncated Lorentzian model in which
$L(E)$ is given by Eq. (56) up to the first ionization energy $E_{I}$ of the
daughter atom attached to graphene and vanishes for $E>E_{I}.$ Using such a
truncation, the background will be approximately given by

$\frac{d\tilde{\Gamma}_{\rm
BG}(E)}{dE_{k}}\sim\frac{\gamma}{2\pi}\int_{E-E_{I}}^{E_{0}}\frac{1}{(E-\mathcal{E})^{2}}\frac{d\Gamma(\mathcal{E})}{dE_{k}}d\mathcal{E}.$
(63)

Such a function describes of an $E_{I}$-wide tail of background events above
the edge $E_{0}$ containing total events per unit time

$\frac{dN_{\rm BG}}{dt}\sim\frac{E_{I}^{2}\gamma N}{TQ^{3}}.$ (64)

To get some feel for the magnitude of this effect, we apply the estimate (64)
to beta decay of Tritium neglecting for illustrative purposes the problems
arising from recoil. A 100 g sample of Tritium contains $N=2\times 10^{25}$
atoms, and it is predicted to experience only 4 neutrino capture events per
year. We take $E_{I}\sim 10\,\rm eV$ and $Q\sim 10\,\rm keV$ and $T\sim 10$
years and demand that $dN_{\rm BG}/dt$ be less than 1 event per year, which
would roughly correspond to a $3\sigma$ detection confidence if 5 events are
observed during the 1 year period. This results in the requirement
$\gamma<10^{-14}\,\rm eV,$ that is a life time on the order of a few hundred
milliseconds or longer. The condition on the relaxation time can in principle
be relaxed if the energy window $(E_{0},E_{0}+\Delta E)$ containing the
neutrino capture peak is known a priori. In that case, the number of
background events inside that window is estimated as the right hand side of
(64) times $\Delta E/E_{I}.$ For $\Delta E$ on the order of $100\,\rm meV$
that gives the lower bound on the ion’s life time of about $10\,\rm ms.$ This
estimate is further elaborated in our numerical calculation, illustrated in
Fig: 8, of the effective amount of the radioactive material required in order
to achieve the p-value of $5\sigma$ in a $\rm C\nu B$ detection experiment
using the energy bin of 100 $\rm meV$ above the spontaneous beta-emission
threshold of Tritium.

Figure 5: The spectral density function taking into account the influence of
the finite lifetime of the daughter ion (blue curve) is compared with the
spectral density function at infinite life time (red curve). As one can see,
at finite life time the the sharp Gamma distribution is broadened into a
smooth asymmetric peak. In this plot we take $g=0.125,$ $\xi_{0}=1\,\rm eV$
and $\gamma=0.2\,\rm eV.$

##  V Discussion

In this section, we have investigated how the results of the previous sections
translate into the visibility of the $\mathrm{C}\nu{\mathrm{B}}$ signal in the
PTOLEMY experiment. From Eq. (14), we know that we can get the correct beta
decay spectrum by performing a convolution between the spectral density
function and the beta decay spectrum in vacuum. Using Mathematica, we plotted
the beta decay spectrum with, $m_{\mathrm{lightest}}=50\,\rm meV$. As can be
seen from comparison of Fig. 6(a) and Fig. 6(b), the shakeup of the electron
system results in each cosmic neutrino background peak getting deformed into a
strongly asymmetric shape peaking near the actual neutrino mass and having a
long power-law tail stretching into the beta decay background.  Note that the
distortion of the beta-decay background is not as conspicuous as in the case
of the C$\nu$B peak. The reason for that is that the distortion is a result of
each individual beta-electron losing some part of its energy in order to
create a collective excitation of electrons in graphene. That is why the
distortion of the C$\nu$B peak results in signal appearing at lower energies
in the gap between the C$\nu$B signal and the beta-decay background. At the
same time the distortion of the beta-decay background just leads to a certain
amount of re-distribution of electrons within the beta-decay continuum,
without affecting the end point. The latter effect is difficult to see by the
unaided eye, especially on the logarithmic scale. Using the calculated
spectrum, we obtained the visibility, that is the number of events in the non-
overlaping region in the spectrum. For the calculation of visibility we assume
the amount of radioactive material equivalent to four capture events per year
(about $100\,\rm g$ in the case of Tritium).

From our analysis it follows that the visibility depends on four parameters,
the lightest neutrino mass, the coupling constant $g$, the distance from the
daughter ion to the graphene sheet, and the lifetime of the daughter ion. We
investigate the PTOLEMY project’s sensitivity to those parameters by making
the plots of the visibility in different parameter planes.

In Fig. 7 we show the dependence of the visibility on the lightest mass of
neutrino and the coupling constant $g$. As shown in Fig. 7, the smaller the
coupling constant and the larger lightest mass of the neutrino, the better the
visibility. This result can be explained by the fact that a smaller coupling
constant leads to a sharper distribution, which causes the beta decay spectrum
less broadening. The effect of the distance between the daughter ion and
graphene is that it affects the cutoff energy and therefore influences the
shape of the distorted beta-emission peak. Smaller cutoff energy means less
weight in the long tail of the gamma distribution, therefore fewer losses and
better visibility. Indeed, as one can see from Fig. 9, visibility increases
with the helium ion’s height. Note that compared with the previous three
parameters the influence of height on visibility is less pronounced, at least
for a small enough coupling constant. This is explained by the fact that for
small $g$ a lot of spectral weight is concentrated in the infrared region near
the X-ray edge therefore the visibility is less sensitive to the ultraviolet
cutoff.

Finally, a parameter that has a huge impact on the visibility of the $C\nu B$
signal is the lifetime of the daughter ion. In Fig. 8 we illustrate this point
on the example of Tritium, even though this isotope suffers from problems
arising from recoil. We see that in order to achieve the signal to noise ratio
ensuring $5\sigma$ confidence of detection of the $C\nu B$ signal with a
reasonable amount of the radioactive material the life time of the daughter
ion needs to be greater than 100 $\mu s.$ Such a stringent requirement to the
life time of the daughter ion poses a separate challenge to the experiment.
Detailed proposals as to how this challenge could be addressed will be
discussed elsewhere.

(a) (b)

Figure 6: Panel 6(a) shows the theoretical beta decay spectrum of mono-atomic
tritium in vacuum assuming the lightest neutrino mass of $50\,\rm meV$ The
solid red curve represents the background beta decay. The solid blue curve
represents the signal from the cosmic neutrino background. $d\Gamma/dE$
describes the possibility that the events happen at a given energy per year.
The area under the curve represents the number of events per year. The area
under the blue curve is only four, that is the calculation is done for the
amount of Tritium that can capture at most four cosmic neutrinos per year. In
the region where the beta decay background (red curve) overlaps with the
$\mathrm{C}\nu\mathrm{B}$ (blue curve) one cannot distinguish the cosmic
neutrino signal from noise. Panel 6(b) describes the beta decay spectrum
adjusted for the Fermi sea shakeup effect. We choose the coupling constant
$g=0.125$.  Figure 7: Contour plot of visibility as a function of the lightest
mass of neutrino and the coupling constant. The cut-off energy is assumed to
be $1\,\rm eV$. The maximum possible number of C$\nu$B events is 4.  Figure 8:
Contour plot of the effective amount of Tritium required in order to achieve
the p value of 5$\sigma$ or better as a function of the coupling constant and
the lifetime. The unit of the effective amount of tritium is $\rm 100\
gram\times year$, which means one can achieve the same necessary number of
events by either increasing the amount of Tritium or increasing the duration
of the experiment. The plot is calculated within the truncated Lorentzian
model assuming the ionization energy $E_{I}=10\;\rm eV.$ Figure 9:  Visibility
at different heights with coupling constant $g=0.1$, and
$m_{\mathrm{lightest}}=50\ \mathrm{meV}$. The curves with different coupling
constants and the neutrinos’ lightest mass share a similar feature.

##  VI Summary and outlook

In this work, we discussed how the shakeup of graphene’s Fermi sea may
influence the visibility of the cosmic neutrino signal in the PTOLEMY project.
In the first section, we used Fermi’s golden rule to find the beta decay
spectrum adjusted for the Fermi sea shakup effect. We showed that it is the
convolution between the spectral density function of a coulomb centre in
graphene and the beta spectrum of the decaying isotope in vacuum. Furthermore,
we applied the linked cluster expansion technique, to obtain the spectral
density function, which is the Gamma distribution function and controlled by
two parameters, the cutoff energy and the dimensionless coupling constant.

In the next section, we examined factors that influence the coupling constant
and the cutoff energy, and we also considered the influence of the daughter
ion’s lifetime on the spectral density function. We found that the distance
between the ion and graphene dictates the natural distance-dependent energy
cutoff scale. Also, the effects of disorder can be neglected if the mean free
time of an electron in graphene exceeds the resolution time of the experiment,
which is about $10^{-13}\,s.$ However, the dynamic screening effects of the
intrinsic graphene significantly increase the coupling constant, which is
detrimental to the visibility of the PTOLEMY project. Fortunately, if the
external dielectric constant of the substrate is large enough, the dynamic
screening effects can be suppressed. We further established that the lifetime
of the daughter ion may have a hugely detrimental impact on the discernibility
of $\rm C\nu B$ signal. To illustrate this point we show that for the decay of
a reasonable amount of Tritium, the life time of the Helium ion would need to
be at least $100\,\rm\mu s$ or longer.

Overall, we have three main conclusions.
(1) Our preliminary analysis indicates that despite large energy scales
associated with the shakeup of the Fermi sea, the visibility can still be
protected by the X ray edge singularity and screening effects. Even though we
use the parameters of Tritium for numerical illustration, our results are
universal and directly adaptable to any other atom.
(2) The visibility of the $\rm C\nu B$ signal can be improved in the following
ways:
1) use a substrate with a high dielectric constant;
2) deposit the radioactive atoms in such a way as to maximize their distance
from the graphene sheet, possibly with the help of a dielectric spacer
(3) Not all effects are included in our analysis, so experiments should be
conducted to test the validity range of our theory. Also, the parameters such
as the coupling constant $g$ and the lifetime of the daughter ion can only be
reliably determined in an experiment.

We finally remark that although we have considered some important solid state
effects that can affect the spectral density function, other effects require
further study. Those include phonon emission, inhomogeneous broadening in a
disordered sample, Coulomb interaction of the beta-electron with electrons in
graphene and others. While such effects are not necessarily important in the
traditional X-ray emission experiment, the extraordinary energy resolution
required by the PTOLEMY experiment puts unusually stringent constraints on the
admissible rate and magnitude of disruption due to various solid state
processes. We hope that further research will improve our understanding of
such effects and their mitigation.

###### Acknowledgements.

We wish to acknowledge Dr. Boyarksy to provide us the data. We also appreciate
the discussion with Yevheniia Cheipesh. The first author especially wants to
express his gratitude to his wife Yiping Deng for taking care of him when
writing this paper.

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##  VII Appendix

### A1

The details of obtaining the natural cutoff energy are presented here.
Substituting eq. (35) into the eq. (23), one can find that

$F_{2}^{\prime}(t)=-g\int_{0}^{\xi_{0}}\frac{(1-e^{-iut})du}{u}\\\ -\frac{\pi
g}{2}\int_{0}^{\xi_{0}}du\frac{(1-e^{-iut})}{u}\left[L_{1}(\frac{2du}{v_{F}})-I_{1}(\frac{2du}{v_{F}})\right]$
(65)

The first integration is the original term, labeled as $F_{2}^{0}(t)$ and the
second integration is the correction term, labeled as $F_{2}^{\prime}(t)$.
Directly evaluating the correction term is difficult, but we can express it as
a double integral.

$F_{2}^{\prime}(t)=g\int_{0}^{2d\xi_{0}/v_{F}}\int_{0}^{1}e^{-zx}\sqrt{1-x^{2}}(1-e^{-izv_{F}t/2d})dxdz\\\
=P(\xi_{0})+N(\xi_{0},t)$ (66)

where

$P(\xi_{0})=g\int_{0}^{1}dx\sqrt{1-x^{2}}\left(\frac{1-e^{-2dx\xi_{0}/v_{F}}}{x}\right),$
(67)

and

$N(\xi_{0},t)=g\int_{0}^{1}dx\sqrt{1-x^{2}}\left(\frac{-1+\exp(-it\xi_{0}-2dx\xi_{0}/v_{F})}{x+iv_{F}t/2d}\right)$
(68)

with $z=2du/v_{F}$. From the calculation above, we can find that the
correction term can be split into two terms. The first part is a constant
independent of time t, and the second part is a time-dependent function.
Hence, only the function $N(\xi_{0},t)$ is significant to our calculation.
Obviously, if $t=0$, it cancels with the first term $P(\xi_{0})$. However,
when t approaches infinity, the correction term $F_{2}^{\prime}(t)$ should
equal to $P(\xi_{0})$. We need to make some approximations to evaluate the
function $N(\xi_{0},t)$.
At a large time limit, roughly, we can use 1 to replace the ugly square root
$\sqrt{1-x^{2}}$ since most contributions to the integration in $P(\xi_{0})$
and $N(\xi_{0},t)$ come from the vicinity of x=0. Then, we can write the
correction term $F_{2}^{\prime}(t)$ in a simple way.

$\begin{split}F_{2}^{\prime}(t)&=g\int_{0}^{1}\left(\frac{1-e^{-2dx\xi_{0}/v_{F}}}{x}\right)dx\\\
&\quad+g\int_{0}^{1}dx\left(\frac{-1+\exp(-it\xi_{0}-2dx\xi_{0}/v_{F})}{x+iv_{F}t/2d}\right)\\\
&=g\int_{0}^{1}\left(\frac{1-e^{-2dx\xi_{0}/v_{F}}}{x}\right)dx\\\
&\quad-g\int_{iv_{F}t/2d}^{1+iv_{F}t/2d}\left(\frac{1-\exp(-2dy\xi_{0}/v_{F})}{y}\right)dy\\\
&=g\int_{0}^{1}\left(\frac{1-e^{-2dx\xi_{0}/v_{F}}}{x}\right)dx\\\
&\quad+g\int_{0}^{iv_{F}t/2d}\left(\frac{1-\exp(-2dy\xi_{0}/v_{F})}{y}\right)dy\\\
&\quad-g\int_{0}^{1+iv_{F}t/2d}\left(\frac{1-\exp(-2dy\xi_{0}/v_{F})}{y}\right)dy\end{split}$
(69)

The factor $1-\exp(-2dy\xi_{0}/v_{F})$ behaves as $2dy\xi_{0}/v_{F}$ in the
limit $u\to 0$. We can use $\frac{v_{F}}{2d\xi_{0}}$ to replace the lower
limit of the integration, since $\frac{v_{F}}{2d\xi_{0}}\ll 1$. At very large
time t, we omit the factor $\exp(-2dy\xi_{0}/v_{F})$. Hence, the correction
term $F_{2}^{\prime}(t)$ can be approximately expressed as a very neat form.

$F_{2}^{\prime}(t)\approx
g\int_{v_{F}/(2d\xi_{0})}^{1}\frac{dx}{x}+g\int_{v_{F}/(2d\xi_{0})}^{iv_{F}t/2d}\frac{dy}{y}\\\
-g\int_{v_{F}/(2d\xi_{0})}^{1+iv_{F}t/2d}\frac{dy}{y}\\\ \approx
g\ln(1+\frac{2d\xi_{0}}{v_{F}})+g\ln(i\xi_{0}t+1)\\\
-g\ln(i\xi_{0}t+\frac{2d\xi_{0}}{v_{F}}+1)$ (70)

Substituting eq. (70) into equation (65), and using equation (21) and equation
((33)), one can obtain the expression of spectral density function of
graphene.

$\begin{split}A(\Omega^{\prime})&=\int_{-\infty}^{\infty}\frac{dt}{2\pi}e^{-iEt}\exp(F_{2}^{0}(t)+F_{2}^{\prime}(t)\\\
&=\Theta(-\Omega^{\prime})\frac{\exp(\Omega^{\prime})}{\Gamma(g)\xi^{\prime}(-\Omega^{\prime})^{(1-g)}}\end{split}$
(71)

where $\xi^{\prime}=\frac{\xi_{0}}{1+2d\xi_{0}/v_{F}}$, and
$\Omega^{\prime}=(E+E_{i})/\xi^{\prime}$.
The cut-off energy $\xi_{0}$ is entirely arbitrary, but we can get a natural
cut-off energy $\xi$ when we extend $\xi_{0}$ to infinity.

$\xi=\lim_{\xi_{0}\to\infty}\frac{\xi_{0}}{1+2d\xi_{0}/v_{F}}=\frac{\hbar
v_{F}}{2d}$ (72)

We get nearly the same result as the previous one, eq. (33), only with
different cutoff energy $\xi$.

### A2

For intrinsic graphene, the time-dependent potential is

$V(q,t)=\frac{V(q)\theta(t)}{\epsilon(q,0)}+V(q)\theta(t)T(v_{F}qt).\\\ $ (73)

$T(v_{F}qt)$ is a one-variable fast-decay function that only depends
$v_{F}qt$.

$T(v_{F}qt)=-\frac{2}{\pi}\int_{1}^{\infty}\frac{a\cos(\omega
v_{F}qt)\sqrt{\omega^{2}-1}}{\omega\left(\omega^{2}-1+a^{2}\right)}d\omega,$
(74)

where $a=(\epsilon(q,0)-1)=\frac{\pi e^{2}}{2\hbar\kappa v_{F}}$. When
$t<<1,T(v_{F}qt)\approx\frac{1}{\epsilon(q,0)}-1$, and when
$t>>1,T(v_{F}qt)\approx 0$
From the general expression of the $F_{2}(t)$, eq. (23), we can find that

$F_{2}(t)=\frac{1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}\int_{0}^{\infty}\mathrm{e^{-iu(t_{1}-t_{2})}}\frac{du}{\pi
V}\\\
\times\sum_{q}\biggl{[}\frac{1}{\epsilon^{2}(q,0)}+\frac{T(v_{F}qt_{1})}{\epsilon(q,0)}+\frac{T(v_{F}qt_{2})}{\epsilon(q,0)}+\\\
T(v_{F}qt_{1})T(v_{F}qt_{2})\biggr{]}\left|V(q)\right|^{2}\Lambda(q,u)$ (75)

When $t\ll 1$, then we can find $F_{2}(t)\approx 0$. It is trivial and has no
contribution to the spectral density function. Therefore, the most significant
part of the X-ray edge is in the long-time domain, and the function
$T(v_{F}qt)$ decays very fast, so $T(v_{F}qt_{1})T(v_{F}qt_{2})$ is negligible
compared to other terms in the bracket. Therefore, one can get

$F_{2}(t)\approx
F^{0}_{2}(t)-\frac{gu\epsilon}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}\int_{0}^{\infty}\mathrm{e^{-iu(t_{1}-t_{2})}}du\\\
\times\int_{1}^{\infty}\frac{aH_{-1}(u\omega
t_{1})\sqrt{\omega^{2}-1}}{\omega\left(\omega^{2}-1+a^{2}\right)}d\omega\\\
-\frac{gu\epsilon}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}\int_{0}^{\infty}\mathrm{e^{-iu(t_{1}-t_{2})}}du\\\
\times\int_{1}^{\infty}\frac{aH_{-1}(u\omega
t_{2})\sqrt{\omega^{2}-1}}{\omega\left(\omega^{2}-1+a^{2}\right)}d\omega,$
(76)

where $F^{0}_{2}(t)$ is the original term in eq.(23) without considering
dynamic screen effect. In the long time limit, we can extend t into infinity,
and then one can get

$\begin{split}F_{2}(t)&\approx
F^{0}_{2}(t)+\int_{i/t}^{\xi_{0}}\frac{2g\epsilon}{\pi
u}du\int_{1}^{\infty}\frac{a\log(\omega+\sqrt{\omega^{2}-1})}{\omega^{2}\left(\omega^{2}-1+a^{2}\right)}d\omega\\\
&\approx\log(1+g_{1}i\xi t)\end{split}$ (77)

with

$g_{1}=g+\int_{1}^{\infty}\frac{2g\epsilon}{\pi}\frac{a\log(\omega+\sqrt{\omega^{2}-1})}{\omega^{2}\left(\omega^{2}-1+a^{2}\right)}d\omega.$
(78)

Therefore, the spectral density function has the same form as the previous
result in eq. (33), but with a different coupling constant.