# On the Financing of Climate Change Adaptation
in Developing Countries

Francis X. Diebold
University of Pennsylvania

(October 2, 2022)

Abstract: I offer reflections on adaptation to climate change, with emphasis
on developing areas.

Acknowledgments: Thanks to Aaron Mora, Glenn Rudebusch and participants at the
Wharton Climate Center / Perry World House Global Climate Finance Workshop.
The usual disclaimer applies.

Key words: Mitigation, Climate capture and storage, Sequestration

Contact<EMAIL_ADDRESS>
Let’s start with emissions mitigation rather than adaptation, and let’s first
consider mitigation in the developed world. Developed-world mitigation efforts
may fail significantly, for a variety of reasons, ranging from never-ending
arrivals of seemingly “more important” situations requiring immediate
attention and arguably requiring the postponing mitigation efforts, to the
ever-present and powerful incentive to free-ride on others. Indeed developed-
world mitigation is already failing significantly (e.g., United Nations
Environment Programme (2021)), a reality that underscores the importance of
developed-world adaptation, in addition to mitigation, strategies.

Now move to emissions mitigation in the developing world. Much of developing-
world mitigation may fail not just significantly, but quite spectacularly as
mitigation efforts are overwhelmed by the push for higher living standards and
further population growth (e.g., Cui et al. (2020)). The rapidly-developing
world will require substantial amounts of new energy, and inexpensive fossil
fuels will remain highly attractive there111Moreover, the developing world
understandably tends to frown on the rich for now condemning the burning of
fossil fuels, given that the rich got rich precisely by burning fossil fuels..
And in the stagnating part of the developing world, mitigation is a moot
issue, as there are few emissions to mitigate. The entire continent of Africa,
for example, contributes little to global emissions.

The bottom line as regards mitigation in the developing world: Mitigation is
likely to be sidelined in the rapidly-developing part of the developing world,
and is not of much relevance in the stagnating part of the developing world.
But simultaneously, both parts of the developing world will be forced to
adapt.

So adaptation is now emerging as crucially important for both the developed
and developing worlds. Unfortunately, however, climate economics has focused
almost exclusively on mitigation issues, such as what Pigouvian tax would
price carbon correctly, or whether capital markets penalize brown firms
sufficiently such that they are incented to go green, and related, how to
define and credibly measure “brown” or “green”. But carbon-tax proposals are
increasingly dead on arrival, and brown-firm capital market penalties alone
won’t solve our problems.

Interestingly and fortunately given its increasing role, adaptation has
several attractive intrinsic features (e.g., Koonin (2021)), some of which are
particularly relevant for the developing world: (1) Adaptation is agnostic. We
don’t need to know and target the source of the climate change. Whatever the
cause, it’s happening, and we can attempt to adapt. (2) Adaptation isn’t just
about policy. It is aided in significant part by the natural actions of
private agents acting in their own self-interest, and it has been working
successfully for millennia. (3) Adaptation project spending, typically for
things visible and conveying potential benefits “here and now”, may be more
politically palatable than mitigation spending that provides potential
benefits to distant future generations. (4) Adaptation is local
geographically. Different areas have different exposures to different climate
risks, and adaptation efforts can themselves be adapted to those differences.
(5) Adaptation is local not only geographically, but also temporally. It can
be scaled up or down as climate risks evolve and climate events unfold.

There are two key, and very different, aspects of climate change that
adaptation must confront, one chronic and one acute. The chronic aspect is
associated with gradually-evolving central tendencies. Front and center is of
course average temperature (and closely related, sea level), which is slowly
increasing, with concomitant slowly-evolving longer-run issues of increased
mortality and illness, reduced labor and agricultural productivity, long
draughts, forced migration, conflict, etc.

The acute aspect of climate change that adaptation must confront is not about
the central tendencies of climate indicator distributions, but rather about
their extreme tails, in particular the frequency and severity of “tail events”
like heatwaves, floods, wildfires, and hurricanes. And even if central
tendencies typically evolve linearly and gradually, they can occasionally
shift non-linearly and abruptly if “tipping points” are reached, so that
chronic aspects of the evolution of central tendencies can themselves feed
importantly into acute tail events.

Several developing-world adaptation questions arise. First, what tools are
needed? One key need is for better data to help with assessing adaptation
costs and benefits. In particular, the developing world needs sustained
availability of reliable climate data and damage estimates at a highly-
disaggregated level, as well as associated reliable and highly-disaggregated
climate model projections, not platitudes like “the sea level will continue to
rise”. The Climate Impact Lab (impactlab.org) is making important advances in
that direction (e.g., Carleton et al. (2022)).

Second, what should appear in developing-country adaptation budgets? They
should be sharply targeted, including climate-resistant infrastructure; heat-
resistant, drought-resistant, and flood-resistant seeds and varietals;
improved risk-management and risk-transfer tools, ranging from better crop
diversification to climate derivatives and insurance; improved climate event
early-warning systems using real-time data relayed from drones and satellites;
and not at all least, services for displaced people. It will be crucial,
moreover, to add adaptation expenditures without removing other well-targeted
items from existing budgets, like vaccines, antibiotics, fertilizer, and
medical personnel.

Third, how can developing-country adaptation be financed? There are many
issues: bureaucratic, political, and fiscal. Much of local project-based
adaptation has a public-good aspect, rendering purely private-market solutions
suboptimal and creating a role for governmental oversight. And quite apart
from the public-good aspect, the scale of adaptation infrastructure projects
(e.g., sea walls) promotes natural monopolies, which again create a role for
governmental oversight.

“Adaptation as public expenditure” would therefore seem to involve little that
is new in terms of financing, as the government budget constraint remains
inescapable: countries can inflate, tax, borrow, or accept aid. The problem is
that while rich countries can often afford to pay the “adaptation tax”,
developing countries often can’t. Hence aid may be an important part of
developing-country adaptation finance, at least in the short run, whether
directly via foreign aid, or indirectly via subsidized loans from development
banks or specialized vehicles like the Green Climate Fund (GCF)222Whether this
aid should be considered a form of reparations for past carbon pollution
remains a contentious issue..

In the medium and long runs, however, the success record of aid is mixed at
best (Easterly (2017)). Among other things, it creates undesirable incentives,
and it may be appropriated by corrupt bureaucrats. Hence, in the longer run,
successful adaptation will benefit from non-aid financing.

Venture capital finance of new adaptation technologies may prove especially
useful. Markets for venture capital, like all markets, function best in
environments of capitalism, democracy, and law (that is, well-defined and
well-enforced property rights), which are often lacking in developing
countries. But an important twist arises: Certain adaptation technologies may
effectively be financed and implemented in developed countries, but with all
countries – developed, rapidly-developing, and stagnating – nevertheless
benefiting equally.

A leading example is carbon capture and storage (CCS), which blends aspects of
both adaptation and mitigation. CO2 mixes very quickly in the earth’s
atmosphere, so that an emissions burst in Houston, for example, will quickly
contribute to atmospheric CO2 concentration globally and equally. Conversely,
CO2 capture and storage in Houston will reduce atmospheric CO2 concentration
globally and equally.

The massive positive externality associated with successful CCS is certainly
appealing, but there remains the issue of large-scale feasibility. The news
there is quite good: CCS technology is improving, and attractive high-capacity
geological storage locations are available (e.g., Aydin et al. (2010); Ball
(2022)). Leading candidates include deep saline aquifers and depleted fossil-
fuel reservoirs.

Investor-led venture capital funds like Breakthrough Energy Ventures
(breakthroughenergy.org) may play a major role in financing CCS and other
emerging technologies, which hold promise not only for reducing the annual
global carbon flow, but also for reducing the global carbon stock, which is
what ultimately drives warming333See also Gates (2021)..

## References

  * (1)
  * Aydin et al. (2010) Aydin, G., I. Karakurt, and K. Aydiner (2010), “Evaluation of Geologic Storage Options of CO2: Applicability, Cost, Storage Capacity and Safety,” Energy Policy, 38, 5072–5080.
  * Ball (2022) Ball, J. (2022), “The Big Business of Burying Carbon,” Wired, 59-69, September, https://www.wired.com/story/big-business-burying-carbon-dioxide-capture-storage/.
  * Carleton et al. (2022) Carleton, T., A. Jina, M. Delgado, M. Greenstone, T. Houser, S. Hsiang, A. Hultgren, R.E. Kopp, K.E. McCusker, I. Nath, J. Rising, A. Rode, H. K. Seo, A. Viaene, J. Yuan, and A. T. Zhang (2022), “Valuing the Global Mortality Consequences of Climate Change Accounting for Adaptation Costs and Benefits,” The Quarterly Journal of Economics, 137, 2037–2105.
  * Cui et al. (2020) Cui, C., D. Guan, D. Wang, V. Chemutai, and P. Brenton (2020), “Emerging Emitters and Global Carbon Mitigation Efforts,” World Bank Working Paper, https://openknowledge.worldbank.org/handle/10986/35845.
  * Easterly (2017) Easterly, W. (2017), The White Man’s Burden: Why the West’s Efforts to Aid the Rest Have Done so Much Ill and so Little Good, Penguin Books.
  * Gates (2021) Gates, B. (2021), How to Avoid a Climate Disaster: the Solutions We Have and the Breakthroughs We Need, Alfred A. Knopf.
  * Koonin (2021) Koonin, S.E. (2021), Unsettled: What Climate Science Tells Us, What it Doesn’t, and Why it Matters, BenBella Books.
  * United Nations Environment Programme (2021) United Nations Environment Programme (2021), Emissions Gap Report 2021: The Heat Is On – A World of Climate Promises Not Yet Delivered, Nairobi, https://www.unep.org/resources/emissions-gap-report-2021.


# Axion Searches Go Deep Underground

Carlo Broggini<EMAIL_ADDRESS>Istituto Nazionale di Fisica
Nucleare, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy Giuseppe Di
Carlo<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare,
Laboratori Nazionali del Gran Sasso, 67100 Assergi (AQ), Italy Luca Di Luzio
<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare, Sezione di
Padova, Via F. Marzolo 8, 35131 Padova, Italy Claudio Toni
<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia ‘G. Galilei’,
Università di Padova,
Via F. Marzolo 8, 35131 Padova, Italy Istituto Nazionale di Fisica Nucleare,
Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy

###### Abstract

We propose to investigate the time modulation of radioisotope decays deep
underground as a method to explore axion dark matter. In this work, we focus
on the $\alpha$-decay of heavy isotopes and develop a theoretical description
for the $\theta$-dependence of $\alpha$-decay half-lives, which enables us to
predict the time variation of $\alpha$-radioactivity in response to an
oscillating axion dark matter background. To probe this scenario, we have
recently constructed and installed a setup deep underground at the Gran Sasso
Laboratory, based on the $\alpha$-decay of Americium-241. This prototype
experiment, named RadioAxion-$\alpha$, will allow us to explore a broad range
of oscillation’s periods, from a few micro-seconds up to one year, thus
providing competitive limits on the axion decay constant across 13 orders of
magnitude in the axion mass, ranging from $10^{-9}$ eV to $10^{-20}$ eV after
one month of data collection, and down to $10^{-22}$ eV after three years.

Introduction. By addressing the strong CP problem Peccei and Quinn (1977a, b);
Weinberg (1978); Wilczek (1978) and the dark matter puzzle Dine and Fischler
(1983); Abbott and Sikivie (1983); Preskill  _et al._ (1983), the Quantum
Chromodynamics (QCD) axion provides a compelling pathway beyond the Standard
Model of particle physics. In recent years, there has been a flourishing of
new experimental strategies for axion detection (for reviews, see e.g.
Irastorza and Redondo (2018); Di Luzio  _et al._ (2020); Sikivie (2021)).
While traditional approaches to axion searches rely on the model-dependent
axion coupling to photons, a remarkable prediction of the QCD axion stems from
the model-independent axion coupling to gluons. By promoting the topological
$\theta$-term of QCD (defined in Eq. (11)) to be a time-varying axion field,
one can test the axion-gluon coupling through the oscillating electric dipole
moment (EDM) of the neutron, induced by the axion dark matter background
Graham and Rajendran (2013); Budker  _et al._ (2014); Stadnik and Flambaum
(2014). Alternative approaches to constrain the axion-gluon coupling have been
discussed e.g. in Refs. Blum  _et al._ (2014); Abel  _et al._ (2017); Roussy
_et al._ (2021); Schulthess  _et al._ (2022); Zhang  _et al._ (2023a).

More recently, the authors of Ref. Zhang  _et al._ (2023b) have proposed to
look for the time variation of the decay rate of certain radioisotopes,
focussing on the $\theta$-dependence of $\beta$-decay, previously developed in
Lee  _et al._ (2020). This enabled them to set bounds on the axion coupling to
gluons from Tritium decay, based on data taken at the European Commission’s
Joint Research Centre Pommé  _et al._ (2016a).

The search for a time dependence of the nuclear decay rates started at the
birth of radioactivity science. As a matter of fact, Madame Curie, in her
Ph.D. thesis Curie (1903), reports on the experiment she conducted to
determine the radioactivity of Uranium at midday and midnight, finding no
difference between the two determinations. In recent years, several studies
have reported a modulation at the per mille level in the decay constants of
various nuclei, typically over periods of one year, but also spanning one
month or one day (see Ref. McDuffie  _et al._ (2020) and references therein).
Conversely, other researchers have found no evidence of such an effect Pommé
_et al._ (2016b, a, c).

To clarify this intricate scenario, we performed a few $\gamma$-spectroscopy
experiments at the underground Gran Sasso Laboratory Bellotti  _et al._ (2012,
2015a, 2015b, 2018). The choice of the underground laboratory is, in our
opinion, a key point. Specifically, the rock overburden suppresses the muon
and neutron flux by six and three orders of magnitude, respectively. This
reduction renders irrelevant the impact of the annual time variation of the
cosmic ray flux, which has an amplitude of a few percent Bellotti  _et al._
(2015a). Eventually, we were able to exclude modulations of the decay constant
of radioisotopes with amplitudes larger than a few parts per 105÷6 in
$\ce{{}^{137}Cs}$ Bellotti  _et al._ (2012), $\ce{{}^{222}Rn}$ Bellotti  _et
al._ (2015a), $\ce{{}^{232}Th}$ Bellotti  _et al._ (2015b), $\ce{{}^{40}K}$
and $\ce{{}^{226}Ra}$ Bellotti  _et al._ (2018), for periods between a few
hours and one year.

In this work, we propose to investigate the time modulation of radioisotope
decays deep underground as a method to test axion dark matter. Unlike the
approach of Ref. Zhang  _et al._ (2023b), we focus on the $\alpha$-decay of
heavy isotopes and employ a setup designed to explore a much broader range of
periods, from a few micro-seconds to one year. From the theoretical side, we
have developed a framework for the $\theta$-dependence of $\alpha$-decay half-
lives, allowing us to predict the time variation of $\alpha$-radioactivity in
response to an oscillating axion dark matter background. On the experimental
front, we have constructed and installed a prototype setup
(RadioAxion-$\alpha$) at the Gran Sasso Laboratory, based on the
$\alpha$-decay of Americium-241. The choice of $\ce{{}^{241}Am}$ is motivated
by several factors. This isotope has a relatively long half-life of about
432.2 yr (approximately stable on the timescale of the measurement) and it
predominantly decays by $\alpha$-emission, with a $\gamma$-ray byproduct,
$\ce{{}^{241}Am}\to\ce{{}^{237}Np}+\alpha+\gamma(59.5\,\text{keV})$. The
resulting $\gamma$-ray can be efficiently detected, using for instance a NaI
crystal. Moreover, $\ce{{}^{241}Am}$ has been produced in nuclear reactors for
decades and is easily accessible, often used in ionization-type smoke
detectors.

In the following, we present the theoretical framework for the
$\theta$-dependence of $\alpha$-decay and describe the experimental setup that
we have installed at the Gran Sasso Laboratory. We conclude with a sensitivity
estimate of RadioAxion-$\alpha$ on the axion parameter space and discuss
future prospects.

Microscopic theory of $\alpha$-decay. We consider a theory of $\alpha$-decay
of a heavy isotope, $\ce{{}^{A}_{Z}X}\to\ce{{}^{A-4}_{Z-2}X}+\alpha$, obtained
by computing the tunnelling probability of the $\alpha$-particle within a WKB
framework that employs a microscopic $\alpha$–daughter-nucleus potential
Chowdhury  _et al._ (2006); Samanta  _et al._ (2007); Sayahi  _et al._ (2019).
In the semi-classical approximation, the half-life is calculated as Gamow
(1928); Gurney and Condon (1929)111See Ref. Holstein (1996) for a critical
assessment of the WKB formula.

$T_{1/2}=\frac{\ln 2}{\nu_{0}}\text{exp}(K)\,,$ (1)

where

$K=\frac{2}{\hbar}\int_{r_{1}}^{r_{2}}dr\
\sqrt{2\mu[V_{\text{tot}}(r)-Q_{\alpha}]}$ (2)

is the WKB integral, with $\mu=M_{\alpha}M_{d}/(M_{\alpha}+M_{d})\approx
M_{\alpha}$ the reduced mass of the $\alpha$–daughter-nucleus system,
$Q_{\alpha}=M(A,Z)-M_{d}-M_{\alpha}$ is the energy of the emitted
$\alpha$-particle and $r_{1,2}$ the turning points of the potential, defined
by the conditions $V_{\text{tot}}(r_{1})=V_{\text{tot}}(r_{2})=Q_{\alpha}$. In
Eq. (1), $\nu_{0}$ denotes the assault frequency, i.e. the frequency at which
the $\alpha$-particle collides against the wall of the potential, which is
given by (see e.g. Sayahi  _et al._ (2019))

$\nu_{0}=\frac{1}{2\mu}\left[\int_{0}^{r_{1}}\frac{dr}{\sqrt{2\mu|Q_{\alpha}-V_{\text{tot}}(r)|}}\right]^{-1}\,.$
(3)

In the limit of a square potential well of dept $-V_{0}$ this reads
$\nu_{0}=v/(2r_{1})$, with $v=\sqrt{2(Q_{\alpha}+V_{0})/\mu}$, which can be
interpreted as the frequency at which the $\alpha$-particle strikes the
barrier Holstein (1996).

The central potential among the $\alpha$-particle and daughter nucleus is the
sum of the nuclear potential, the Coulomb potential and the rotational term,
i.e.

$V_{\text{tot}}(\vec{R})=V_{N}(\vec{R})+V_{C}(\vec{R})+\frac{\ell(\ell+1)}{2\mu
R^{2}}\,,$ (4)

where $\ell$ denotes the angular momentum of the nuclear transition and
$R=|\vec{R}|$. The nuclear potential is obtained by double-folding the
densities of the $\alpha$ and daughter nucleus Satchler and Love (1979)

$\displaystyle V_{N}(\vec{R})$
$\displaystyle=\int\int\,d^{3}r_{\alpha}d^{3}r_{d}\,\rho_{\alpha}(\vec{r}_{\alpha})\rho_{d}(\vec{r}_{d})$
$\displaystyle\times\tilde{v}(\vec{r}=\vec{r}_{d}-\vec{r}_{\alpha}+\vec{R},\rho_{\alpha}(\vec{r}_{\alpha}),\rho_{d}(\vec{r}_{d}))\,,$
(5)

where
$\tilde{v}(\vec{r},\rho_{\alpha},\rho_{d})=v(\vec{r})\,g(\rho_{\alpha},\rho_{d})$
is the single-nucleon effective potential with a density-dependent correction.
Since for the $\alpha$-decay process only the iso-scalar component of the
potential contributes, we take as an input the iso-scalar term of the so-
called M3Y effective potential, supplemented by a zero-range potential for the
single-nucleon exchange Bertsch  _et al._ (1977); Satchler and Love (1979);
Kobos  _et al._ (1984); Gils (1987)

$\displaystyle v_{\text{M3Y}}(\vec{r})$
$\displaystyle=\Big{[}-2134\frac{\exp(-2.5r)}{2.5r}+7999\frac{\exp(-4r)}{4r}$
$\displaystyle-276\,\delta(\vec{r})\Big{]}\ \text{MeV}\,,$ (6)

with $r=|\vec{r}|$ in units of $1\,\text{fm}\approx 1/(198\,\text{MeV})$.
Different choices of the nuclear potential, such as the so-called Paris
version of the M3Y potential Anantaraman  _et al._ (1983), have a minor
impact, with differences at the per mille level in the final result of Eq.
(Axion Searches Go Deep Underground). For the density-dependent term, we
consider Chowdhury  _et al._ (2006)

$g(\rho_{\alpha},\rho_{d})=(1-\beta\rho_{\alpha}^{2/3})(1-\beta\rho_{d}^{2/3})\,,$
(7)

with $\beta=1.6\,\text{fm}^{2}$. Following Ref. Satchler and Love (1979), the
density distribution for the $\alpha$ particle has been taken to have the
Gaussian form

$\rho_{\alpha}(\vec{r})=0.4229\,\text{exp}(-0.7024r^{2})\,\text{fm}^{-3}\,,$
(8)

whose volume integral is equal to its mass number $A_{\alpha}=4$. The matter
distribution of the daughter nucleus can be instead described by a spherically
symmetric Fermi function Chowdhury  _et al._ (2006)

$\rho(\vec{r})=\frac{\rho_{0}}{1+\text{exp}\left(\frac{r-c_{d}}{a}\right)}\,,$
(9)

with $c_{d}=r_{d}(1-\pi^{2}a^{2}/(3r_{d}^{2}))$,
$r_{d}=1.13\,A_{d}^{1/3}\,\text{fm}$, $a=0.54\,\text{fm}$, while $\rho_{0}$ is
a normalization constant, taken so that the volume integral is equal to the
mass number of the daughter particle, $A_{d}=A-4$.

Finally, the Coulomb potential is given by

$V_{C}(\vec{R})=\begin{cases}\frac{Z_{\alpha}Z_{d}\alpha_{\text{QED}}}{2R_{c}}\left[3-\left(\frac{R}{R_{c}}\right)^{2}\right]\quad\text{for
$R<R_{c}$}\,,\\\ \\\
\frac{Z_{\alpha}Z_{d}\alpha_{\text{QED}}}{R}\quad\text{for
$R>R_{c}$}\,,\end{cases}$ (10)

where $R_{c}=c_{\alpha}+c_{d}$, with
$c_{\alpha}=r_{\alpha}(1-\pi^{2}a^{2}/(3r_{\alpha}^{2}))$ and
$r_{\alpha}=1.13\,A_{\alpha}^{1/3}\,\text{fm}$.

Within such a framework we are able to reproduce the $\alpha$-decay half-lives
of heavy isotopes at the order of magnitude level, in accordance with the
results of Refs. Chowdhury  _et al._ (2006); Samanta  _et al._ (2007); Sayahi
_et al._ (2019). Note that the $\alpha$-decay process is exponentially
sensitive to the WKB integral, so that the lifetimes span several orders of
magnitudes when varying the $Q_{\alpha}$ value for different nuclei.

$\theta$-dependence of $\alpha$-decay. The $\theta$-term of QCD is defined by
the operator

$\mathcal{L}_{\theta}=\frac{g_{s}^{2}\theta}{32\pi^{2}}G^{a}_{\mu\nu}\tilde{G}^{a\,\mu\nu}\,,$
(11)

where $|\theta|\lesssim 10^{-10}$ from the non-observation of the neutron EDM
Abel  _et al._ (2020). The smallness of $\theta$ constitutes the so-called
strong CP problem, which can be solved by promoting the $\theta$-term to be a
dynamical field, $\theta\to a(x)/f_{a}$, where $a(x)$ is the axion and $f_{a}$
a mass scale known as the axion decay constant. The axion field acquires a
potential in the background of QCD instantons and relaxes dynamically to zero,
thus explaining the absence of CP violation in strong interactions Peccei and
Quinn (1977a, b); Weinberg (1978); Wilczek (1978).

In the following, we will be interested in the $\theta$-dependence of nuclear
quantities, anticipating the fact that we will interpret $\theta(t)$ as a
time-varying background axion field, related to the dark matter of the
universe Dine and Fischler (1983); Abbott and Sikivie (1983); Preskill  _et
al._ (1983). The consequences of a non-zero $\theta$ in nuclear physics have
been previously investigated in Refs. Ubaldi (2010); Lee  _et al._ (2020),
also in connection with the idea of establishing an anthropic bound on
$\theta$.

There are various ways in which the $\theta$-dependence can manifest in
nuclear physics, the most prominent is through the pion mass Leutwyler and
Smilga (1992); Brower  _et al._ (2003)

$M^{2}_{\pi}(\theta)=M^{2}_{\pi}\cos{\frac{\theta}{2}}\sqrt{1+\varepsilon^{2}\tan^{2}\frac{\theta}{2}}\,,$
(12)

with $M_{\pi}=139.57$ MeV and $\varepsilon=(m_{d}-m_{u})/(m_{d}+m_{u})$. The
$\theta$-dependence of other low-lying resonances, including $\sigma(550)$,
$\rho(770)$ and $\omega(783)$ – which, along with the pion, are responsible
for the mediation of nuclear forces in the one-boson-exchange (OBE)
approximation – has been determined based on $\pi\pi$ scattering data in Ref.
Acharya  _et al._ (2015).

A key role for the binding energy of heavy nuclei is played by the $\sigma$
and $\omega$ channels, via the contact interactions Furnstahl and Serot (2000)

$H=G_{S}(\overline{N}N)(\overline{N}N)+G_{V}(\overline{N}\gamma_{\mu}N)(\overline{N}\gamma^{\mu}N)\,,$
(13)

which control, respectively, the scalar (attractive) and vector (repulsive)
part of the nucleon-nucleon interaction Donoghue (2006a); Damour and Donoghue
(2008). To describe their $\theta$-dependence we employ the following
parametrization

$\eta_{S}=\frac{G_{S}(\theta)}{G_{S}(\theta=0)}\,,\quad\eta_{V}=\frac{G_{V}(\theta)}{G_{V}(\theta=0)}\,.$
(14)

In Ref. Donoghue (2006b) it was found that the pion mass dependence of
$\omega$ exchange leads to subleading corrections compared to the effects
related to the $M^{2}_{\pi}$ sensitivity of the scalar channel. Hence, to a
good approximation, we can take $\eta_{V}=1$ and consider only the leading
$\theta$-dependence in the scalar channel, which is described by the following
fit Ubaldi (2010) to Fig. 2 in Damour and Donoghue (2008)

$\eta_{S}(\theta)=1.4-0.4\frac{M^{2}_{\pi}(\theta)}{M^{2}_{\pi}}\,.$ (15)

Moreover, based on the relativistic mean-field simulations of Furnstahl and
Serot (2000) for two specific nuclei, Ref. Damour and Donoghue (2008) finds
that the variation of the binding energy (BE) for a nucleus of mass number $A$
can be written as (keeping only the variation due to $\eta_{S}(\theta)$)

$\displaystyle\text{BE}(\theta)$ $\displaystyle=\text{BE}(\theta=0)$
$\displaystyle+(120A-97A^{2/3})(\eta_{S}(\theta)-1)\,\text{MeV}\,,$ (16)

where the terms proportional to $A$ and $A^{2/3}$ represent a volume and
surface contribution, in analogy to the semi-empirical mass formula Weizsacker
(1935).

Hence, substituting the expressions of the BEs above in the definition of
$Q_{\alpha}=\text{BE}(A-4,Z-2)+\text{BE}(4,2)-\text{BE}(A,Z)$, we find

$\displaystyle Q_{\alpha}(\theta)$
$\displaystyle=Q_{\alpha}(\theta=0)-97\text{ MeV }(\eta_{S}(\theta)-1)$
$\displaystyle\times((A-4)^{2/3}+4^{2/3}-A^{2/3})\,.$ (17)

It turns out that $Q_{\alpha}(\theta)$ provides, by far, the leading effect in
order to assess the $\theta$-dependence of $\alpha$-decay. Another possible
dependence from $\theta$ arises from the M3Y nuclear potential. This can be
implemented by interpreting the exponential terms in Eq. (Axion Searches Go
Deep Underground) as arising from $\sigma$ (attractive) and $\omega$
(repulsive) exchange in the OBE approximation. Focussing on the leading
$\theta$-dependence from $\sigma$ exchange, we can rescale the pre-exponential
and exponential factors respectively via $g^{2}_{\sigma NN}(\theta)$ and
$M^{2}_{\sigma}(\theta)$. The $\theta$-dependence of the $\sigma$ mass is
taken from Acharya  _et al._ (2015); Lee  _et al._ (2020), while the $\sigma$
coupling can be expressed in terms of $G_{S}(\theta)=-g^{2}_{\sigma
NN}(\theta)/M^{2}_{\sigma}(\theta)$. We find that the $\theta$-dependence
arising from the nuclear potential remains always subleading with respect to
that from $Q_{\alpha}(\theta)$, basically below the percent level in the final
result of Eq. (Axion Searches Go Deep Underground). The predominance of the
$\theta$-dependence of $Q_{\alpha}(\theta)$ with respect to that of $V_{\rm
tot}(\theta)$ in Eq. (2) can be understood by the fact that the WKB integral
is defined across the potential barrier, and the latter is dominated by the
Coulomb potential that is not affected by $\theta$.

Axion dark matter time modulation. Assuming an oscillating axion dark matter
field from misalignment Dine and Fischler (1983); Abbott and Sikivie (1983);
Preskill  _et al._ (1983), the time dependence of the $\theta$ angle can be
approximated as $\theta(t)=\theta_{0}\cos(m_{a}t)$, with

$\displaystyle\theta_{0}$ $\displaystyle=\frac{\sqrt{2\rho_{\rm
DM}}}{m_{a}f_{a}}\,,$ (18)

in terms of $\rho_{\rm DM}\approx 0.45\,\text{GeV/cm}^{3}$. For a standard QCD
axion, one has

$m_{a}f_{a}=\frac{\sqrt{m_{u}m_{d}}}{m_{u}+m_{d}}m_{\pi}f_{\pi}=(76\
\text{MeV})^{2}\,,$ (19)

corresponding to $\theta_{0}=5.5\times 10^{-19}$. In the following, we will
treat $m_{a}$ and $f_{a}$ as independent parameters and discuss the
sensitivity of $\alpha$-decay observables in the $(m_{a},1/f_{a})$ plane.

Following Ref. Zhang  _et al._ (2023b), we introduce the observable

$I(t)\equiv\frac{T^{-1}_{1/2}(\theta(t))-\langle T^{-1}_{1/2}\rangle}{\langle
T^{-1}_{1/2}\rangle}\,,$ (20)

where $\langle T^{-1}_{1/2}\rangle$ denotes a time average. Given that the
main $\theta$-dependence in Eq. (15) arises through the pion mass, we expect
that $T_{1/2}(\theta)$ is analytic in $\theta^{2}$ and admits the Taylor
expansion222This is also verified a posteriori by a numerical fit of the half-
life as a function of $\theta$.

$T_{1/2}(\theta)\approx T_{1/2}(0)+\mathring{T}_{1/2}(0)\theta^{2}\,,$ (21)

where we introduced the derivative symbol, $\mathring{f}\equiv
df/d\theta^{2}$. Since $\theta^{2}\ll 1$, Eq. (21) does provide an excellent
approximation to the full $\theta$-dependence, which is anyway taken into
account in our numerical analysis. Using $\langle\cos^{2}(m_{a}t)\rangle=1/2$
and expanding at the first non-trivial order in $\theta_{0}$, we find

$\displaystyle I(t)$
$\displaystyle\approx-\frac{1}{2}\frac{\mathring{T}_{1/2}(0)}{T_{1/2}(0)}\theta_{0}^{2}\cos(2m_{a}t)$
$\displaystyle=-4.3\times 10^{-6}\cos(2m_{a}t)\left(\frac{\rho_{\rm
DM}}{0.45\,\text{GeV/cm${}^{3}$}}\right)$
$\displaystyle\times\left(\frac{10^{-16}\,\text{eV}}{m_{a}}\right)^{2}\left(\frac{10^{8}\,\text{GeV}}{f_{a}}\right)^{2}\,,$
(22)

where $\mathring{T}_{1/2}(0)/T_{1/2}(0)\approx 125$ has been obtained by
fitting the numerical result of $T_{1/2}(\theta)$ at small $\theta$ values. In
Eq. (Axion Searches Go Deep Underground) we also used
$Q_{\alpha}(\theta=0)=5.486$ MeV, corresponding to the dominant $\alpha$-decay
transition of $\ce{{}^{241}Am}$, and substituted $\theta_{0}$ from Eq. (18).

Note that the large theoretical uncertainty in the prediction of $T_{1/2}(0)$,
stemming from its exponential dependence from the WKB integral $K$, is washed
out thanks to the normalization of Eq. (20). In fact, neglecting the small
$\theta$-dependence arising from $\nu_{0}$ in Eq. (3), amounting to an effect
below the per mille level in Eq. (Axion Searches Go Deep Underground), we have
$\mathring{T}_{1/2}(0)/T_{1/2}(0)\approx\mathring{K}(0)$.

Experimental setup. To study the time modulation of the $\alpha$-decay of
$\ce{{}^{241}Am}$, we built a prototype setup (RadioAxion-$\alpha$) which we
installed deep underground at the Gran Sasso Laboratory, in a dedicated
container. A $3"\times 3"$ NaI crystal detects the $\gamma$-rays due to the
$\alpha$-decay of 241Am, primarily ($85\%$ of the time) at 59.5 keV, and the
X-rays from 237Np atomic transitions. The signal from the photomultiplier is
processed by an Ortec digiBASE, a 14-pin photomultiplier tube base that is
directly connected to the photomultiplier. The digiBASE, the photomultiplier,
the crystal and the source, kept in a fixed position in front of the crystal,
are closed inside a parallelepiped made of polyethylene, completely surrounded
by a passive shielding of 5 cm of copper and 10 cm of lead, in order to
suppress the laboratory $\gamma$-ray background. Data acquisition operates in
list mode, i.e. each signal above the 10 keV threshold is converted to a
digital value which is transmitted to the computer along with the time of the
event. The time resolution is 1 $\mu$s. To mitigate the impact of the
digiBASE’s quartz aging, we also acquire a signal every second, generated by
an FS725 10 MHz Rb Frequency Standard which has a 20 year aging factor of less
than 5 $\times$ 10-9.

In Fig. 1 we show the energy spectrum of the events collected in 24 hours,
with and without the $\ce{{}^{241}Am}$ source. With the $\ce{{}^{241}Am}$
source we have a rate (counts per second) of about 4 kHz, to be compared to a
background of 0.2 Hz. The background, i.e. the counts in the absence of the
source, is essentially due to the inner radioactivity of the NaI crystal,
while the background due to the cosmic ray flux is safely negligible.

Figure 1: $\gamma$-spectrum (counts per second per keV) of the 241Am source
(upper curve) compared to the background (lower curve). The dominant
contribution arises from the $\gamma$ at 59.5 keV.

Sensitivity estimate. The theoretical prediction in Eq. (Axion Searches Go
Deep Underground) can be compared with $I_{\rm exp}(t)\equiv(N(t)-\langle
N\rangle)/\langle N\rangle$, where $N(t)$ is the observed number of events in
a given interval of time and $\langle N\rangle$ its expected value, according
to the exponential decay law. Potential sources of systematic errors include
the detection of $\gamma$-rays and their time-stamping. The former is
mitigated by operating the NaI detector well-below the radiation damage
threshold and by the reduced background in the underground environment. The
latter is handled thanks to the precision of a Rb atomic clock. Hence, we
expect our uncertainties to be statistically dominated in the current setup.

We started data taking at the beginning of May 2024. With a rate of about 4
kHz events, we expect to reach a 2$\sigma$ error of
$2/\sqrt{4000/s\times\pi\times 10^{7}s}\approx 6\times 10^{-6}$ on $I_{\rm
exp}$ after one year of data taking. Given the 1 $\mu$s time resolution of our
setup and referring to the oscillation period as $\Delta t$, we consider two
realistic benchmarks corresponding to distinct experimental phases: $i)$ Phase
1: $3\,\mu\text{s}<\Delta t<10\ \text{days}$ and $I_{\rm exp}=2\times 10^{-5}$
at 2$\sigma$ with one month of data taking and $ii)$ Phase 2:
$3\,\mu\text{s}<\Delta t<1\,\text{yr}$ and $I_{\rm exp}=4\times 10^{-6}$ at
2$\sigma$ with three years of data taking.

The sensitivity of the present experiment is ultimately limited by the number
of detected events due to the $\ce{{}^{241}Am}$ source. By increasing the
source activity by a factor of 10, it would be possible to improve the
sensitivity by a factor of 3. Further improvements would require, in addition
to a more powerful $\ce{{}^{241}Am}$ source, a faster detector, for instance a
plastic scintillator, and a significantly upgraded data acquisition system.
All in all, an improvement of up to two orders of magnitude in the sensitivity
could be possible with a set-up similar to ours but with more cutting-edge
technologies.

Figure 2: Constraints on the axion dark matter coupling to gluons. The
projected sensitivities of RadioAxion-$\alpha$ are displayed for two
experimental phases (yellow-shaded areas). Limits from laboratory experiments
(red-shaded areas) and from SN 1987A (green) are shown as well for comparison.
Figure adapted from O’Hare (2020).

The results of our sensitivity estimate for the two experimental phases of
RadioAxion-$\alpha$ are shown in Fig. 2. For comparison, we also display
laboratory limits from EDM searches Abel  _et al._ (2017); Roussy  _et al._
(2021); Schulthess  _et al._ (2022), radio-frequency atomic transitions Zhang
_et al._ (2023a), and Tritium decay Zhang  _et al._ (2023b), as well the SN
1987A bound stemming from the induced axion-nucleon EDM coupling Graham and
Rajendran (2013); Lucente  _et al._ (2022). The latter limit is the only one
that does not depend on the hypothesis of axion dark matter, but for
$1/f_{a}\gtrsim 3.3\times 10^{-4}\,\text{GeV}^{-1}$ (indicated by a dashed
line in Fig. 2) axions enter the trapping regime and the cooling bound does
not apply Lucente  _et al._ (2022). The yellow, QCD axion line stems from the
relation in Eq. (19). There are, however, QCD axion models (still addressing
the strong CP problem) that can deviate sizeably from the canonical QCD axion
line, particularly by featuring a lighter axion than in the usual case Hook
(2018); Di Luzio  _et al._ (2021a, b); Banerjee  _et al._ (2023).

Conclusions. Our investigation into the time modulation of radioisotope decays
deep underground at the Gran Sasso Laboratory has successfully established the
RadioAxion-$\alpha$ experiment. This setup, centered on the $\alpha$-decay of
$\ce{{}^{241}Am}$, will allow us to cover a wide range of oscillation periods
from microseconds to a year. Based on realistic projected sensitivities, we
will provide with just few years of data competitive constraints on the axion
decay constant, spanning 13 orders of magnitude in axion mass, from from
$10^{-9}$ eV to $10^{-22}$ eV.

This work not only marks an additional step in axion dark matter research but
also lays the groundwork for a broader project aimed at optimizing the study
of the $\theta$-dependence on radioactivity. Future efforts will focus on
identifying the most effective decay types and isotopes to fully leverage the
unique underground environment for axion detection.

###### Acknowledgements.

Acknowledgments. We thank Francesco D’ Eramo and Massimo Pietroni for
discussions at the initial stages of this project, as well as Fabio Zwirner
for suggestions on $\alpha$-decay theory and Roberto Isocrate for the
improvements of the electronics. The work of LDL and CT is supported by the
European Union – Next Generation EU and by the Italian Ministry of University
and Research (MUR) via the PRIN 2022 project n. 2022K4B58X – AxionOrigins. LDL
is also supported by the European Union – NextGeneration EU and by the
University of Padua under the 2021 STARS Grants@Unipd programme (Acronym and
title of the project: CPV-Axion – Discovering the CP-violating axion).

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capbtabboxtable[][]

11institutetext:  CentraleSupélec, Université Paris-Saclay, 91190 Gif-sur-
Yvette, France 11email<EMAIL_ADDRESS>22institutetext: MICS
Laboratory, CentraleSupélec, Université Paris-Saclay, 91190 Gif-sur-Yvette,
France 22email<EMAIL_ADDRESS>
# Artifact Removal in Histopathology Images

Cameron Dahan  11 Stergios Christodoulidis  22 Maria Vakalopoulou  22 Joseph
Boyd  22

###### Abstract

In the clinical setting of histopathology, whole-slide image (WSI) artifacts
frequently arise, distorting regions of interest, and having a pernicious
impact on WSI analysis. Image-to-image translation networks such as CycleGANs
are in principle capable of learning an artifact removal function from
unpaired data. However, we identify a surjection problem with artifact
removal, and propose a weakly-supervised extension to CycleGAN to address
this. We assemble a pan-cancer dataset comprising artifact and clean tiles
from the TCGA database. Promising results highlight the soundness of our
method.

###### Keywords:

C

ycleGANs; Weakly-supervised; Histopathology; Artifacts

##  1 Introduction

Due to their handling and large size, histopathology slides are susceptible to
contaminants and other artifacts, leading to image artifacts in whole slide
images (WSI). Common artifacts include pen marker, ink, blur, air bubbles,
tissue folds, dust and filaments [12]. Such artifacts create noise and
outliers in WSI datasets, potentially undermining statistical analysis. An
automatic system capable of localising and removing artifacts is therefore of
great interest.

CycleGANs [13] are unpaired image-to-image translation models with several
existing applications in histopathology, notably in stain normalisation [10,
3, 2], stain transfer [4], and cell segmentation [7]. CycleGANs have been
applied to histopathology artifacts before [1], but this was restricted to the
relatively straightforward case of pen marker.

In Section 2 we propose an extension to the CycleGAN framework aimed at
accounting for the surjective nature of artifact removal. In Section 3 we
demonstrate the improvement of our proposed method over baselines.

##  2 Methods

Our artifact removal system is based on the CycleGAN framework, for which we
denote artifact and clean tiles as image domains $A$ and $B$. Accordingly, our
baseline objective function is,

$\displaystyle\mathcal{L}_{base}$
$\displaystyle=\mathcal{L}_{GAN}(G_{AB},D_{B})+\lambda_{ABA}\cdot\mathcal{L}_{CYC}(G_{BA}\circ
G_{AB})+\lambda_{A}\cdot\mathcal{L}_{ID}(G_{BA})$ (1)
$\displaystyle{}+\mathcal{L}_{GAN}(G_{BA},D_{A})+\lambda_{BAB}\cdot\mathcal{L}_{CYC}(G_{AB}\circ
G_{BA})+\lambda_{B}\cdot\mathcal{L}_{ID}(G_{AB}),$

where generators $G_{AB}$ and $G_{BA}$ translate between the image domains,
and $D_{B}$ and $D_{A}$ are their respective discriminators. The loss
$\mathcal{L}_{GAN}$ represents a least squares GAN loss [8];
$\mathcal{L}_{CYC}$ a cycle consistency loss; and $\mathcal{L}_{ID}$ an
identity function loss. The system is optimised through a minmax optimisation
process.

The key observation of our method is that artifact removal is a _surjective_
operation; many artifact images may correspond with the same clean image. If
truly clean, the image should not retain information about the artifact. This
creates a difficult task for generator $G_{BA}$, which is required to
reconstruct artifacts in $\lambda_{ABA}\cdot\mathcal{L}_{CYC}$ (Equation 1).
This creates strong pressure for $G_{AB}$ to only partially clean or otherwise
encode extraneous information within artifact images. This is clearly at odds
with minimising the adversarial loss $\mathcal{L}_{GAN}(G_{AB},D_{B})$
(Equation 1). We refer to this as the surjection problem and hypothesise that
this leads the CycleGAN to a poor local minimum. An initial idea is to simply
set $\lambda_{ABA}=0$, and to rely on the other half of the cycle to enforce
consistency.

###  2.1 Weakly supervised CycleGAN

[width=0.8]tikz/model

Figure 1: Weakly-supervised CycleGAN for artifact$\to$clean$\to$artifact
consistency.

To better address the problem of surjective artifact removal, we leverage the
label of artifact tiles in two alternative models. In the first model, the
label is used as a conditioning variable for both the $G_{BA}$ generator and
the $D_{A}$ discriminator. For each, the condition is encoded as a one-hot
tensor with $7$ class channels, where a channel is set to ones to indicate the
class label. This conditioning is concatenated with the input image channel-
wise. Thus, $G_{BA}$ can learn what class of artifact has been removed. We
refer to this model as $\mathcal{M}_{cond}$.

In the second model we introduce an attention network, $\alpha$. This is a
two-layer MLP applied to each pixel of the penultimate layer, $g_{AB}$, of
generator $G_{AB}$, followed by a sigmoid activation function. The $A\to B\to
A$ cycle-consistency is thus redefined to be,

$G_{BA}\circ
G_{AB}:=G_{BA}\Big{(}concat\big{(}\alpha(g_{AB}),proj_{RGB}(g_{AB})\big{)}\Big{)}$
(2)

where $proj_{RGB}$ is a linear projection followed by a _tanh_ activation, and
$concat$ is channel-wise concatenation. As such, $G_{BA}$ has an additional
channel of input. Since the attention map is only available when computing the
cycle-consistency loss, at all other times we substitute a dummy channel of
Gaussian noise. This model is referred to as $\mathcal{M}_{attn}$.

To enhance $\mathcal{M}_{attn}$ we employ a similar trick as in [11], where
the attentions are combined with the image inputs by element-wise product, and
then fed into an auxiliary convolutional network $C$, which classifies the
artifact under a cross-entropy loss, $\mathcal{L}_{cls}$. The mask is
regularised with smoothness $\mathcal{L}_{smooth}$ and sparsity
$\mathcal{L}_{sparse}$ losses. This compels the attention model to produce
masks that highlight the artifact regions only. Our hypothesis is that this
will allow $G_{AB}$ to decouple the tissue from the artifact, and to offload
all extraneous information onto the attention map. This model is referred to
as $\mathcal{M}_{ws}$, and is depicted in Figure 1. With $C$ guiding the
attention map, the model becomes weakly-supervised, and the objective function
becomes,

$\mathcal{L}_{ws}=\mathcal{L}_{base}+\lambda_{cls}\cdot\mathcal{L}_{cls}+\lambda_{smooth}\cdot\mathcal{L}_{smooth}+\lambda_{sparse}\cdot\mathcal{L}_{sparse}$
(3)

###  2.2 Dataset

Our dataset consists of $28$ WSIs randomly selected from the TCGA database
111https://www.cancer.gov/tcga, and thus spans a wide range of tissue types.
$6556$ non-overlapping tiles of size $300\times 300$px were manually extracted
capturing typical artifact classes at both $40$x and $10$x magnification. For
each artifact tile, a clean equivalent was extracted in close proximity, as in
Supp. Figure 1, to control for sampling bias. Each artifact tile was labelled
into one of seven classes: pen marker, ink, blur, air bubble, tissue fold,
dust and filament, as shown in Supp. Figure 2.

###  2.3 Model training and evaluation

We train $\mathcal{L}_{base}$ with UNet and attention UNet [9] generators,
which we denote $\mathcal{M}_{base}$ and $\mathcal{M}_{dpa}$ respectively.
Apart from $\mathcal{M}_{\lambda_{ABA}=0}$, the hyperparameters
$\lambda_{ABA}=5$, $\lambda_{BAB}=5$, $\lambda_{A}=5$, and $\lambda_{B}=5$ are
fixed for all models. For model $\mathcal{M}_{ws}$ we set $\lambda_{cls}=1$,
$\lambda_{smooth}=1$, and $\lambda_{sparse}=0.1$.

We train all networks with the Adam optimiser [6], with $lr=0.001$,
$\beta_{1}=0.5$, $\beta_{2}=0.999$, and an exponential learning rate decay of
$\gamma=0.9975$. All models were trained for 30 epochs with batch size 16. To
regularise learning, we augment the data with random horizontal and vertical
flips, followed by taking a random $256\times 256$px crop and resizing to
$128\times 128$px. At test time the random crop is replaced by a center crop.
We apply label smoothing to the discriminators and a weight decay of
$1\text{\times}{10}^{-5}$ for all networks.

There is an inherent difficulty in evaluating unpaired datasets, as the
targets are unknown. We therefore compare the distributional output of each
model with the negative samples using Fréchet Inception Distance (FID) on both
train (2620 samples) and test (655 samples) sets to evaluate the results (i.e.
an $80$-$20$ split). Note that FID is conventionally evaluated over $50,000$
samples [5], but this is not possible in our dataset.

##  3 Results

As shown in Table 1, the modified CycleGANs generally perform better, in
particular $\mathcal{M}_{ws}$. The train FID scores are systematically lower
than the test scores, but this does not seem to be a sign of overfitting,
rather the effect of sample size. Controlling for sample size, our best model
achieves a FID of $60.24$ on $655$ training samples and $59.36$ on the test
data. We also note the baseline is slightly improved upon with dot product
attention.

Figure 2 shows examples of artifact tiles from the test set, cleaned tile
model outputs and corresponding attention maps. We note the localisation of
artifact regions in the attention maps, which has been achieved from weak
labels only. We include additional examples in Supp. Figure 3, as well as
failure cases in Supp. Figure 4. Currently, the model struggles with opaque
pen marker and underrepresented classes such as filament artifacts.

|  $\mathcal{M}_{base}$ |  $\mathcal{M}_{dpa}$ |  $\mathcal{M}_{cond}$ |  $\mathcal{M}_{\lambda_{ABA}=0}$ |  $\mathcal{M}_{attn}$ |  $\mathcal{M}_{ws}$
---|---|---|---|---|---|---
Tr. FID |  $45.09$ |  44.95 |  42.62 |  42.38 |  45.39 |  $\mathbf{34.43}$
Te. FID |  $70.71$ |  70.29 |  67.93 |  67.72 |  75.33 |  $\mathbf{59.36}$
Table 1: FID scores on train and test data. Best results in bold. Figure 2:
Model inputs, outputs, and attention (top to bottom) for unseen test images.

##  4 Conclusions

In this paper we have identified a limitation of CycleGANs for the surjective
task of artifact removal in histopathology images. We have presented a
mechanism for incorporating weakly-supervised data into a CycleGAN, allowing
it to decouple tissue from artifact, and improving over baselines in its
ability to remove artifacts. An approximate artifact segmentation is a
byproduct of the removal process. Future work could aim to expand the dataset,
as currently some artifact classes are underrepresented.

## References

  * [1] Ali, S., Alham, N.K., Verrill, C., Rittscher, J.: Ink removal from histopathology whole slide images by combining classification, detection and image generation models. In: 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019). pp. 928–932. IEEE (2019)
  * [2] de Bel, T., Hermsen, M., Kers, J., van der Laak, J., Litjens, G.: Stain-transforming cycle-consistent generative adversarial networks for improved segmentation of renal histopathology. In: International Conference on Medical Imaging with Deep Learning–Full Paper Track (2018)
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  * [4] Boyd, J., Villa, I., Mathieu, M.C., Deutsch, E., Paragios, N., Vakalopoulou, M., Christodoulidis, S.: Region-guided cyclegans for stain transfer in whole slide images. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 356–365. Springer (2022)
  * [5] Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., Hochreiter, S.: Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems 30 (2017)
  * [6] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
  * [7] Mahmood, F., Borders, D., Chen, R.J., McKay, G.N., Salimian, K.J., Baras, A., Durr, N.J.: Deep adversarial training for multi-organ nuclei segmentation in histopathology images. IEEE transactions on medical imaging 39(11), 3257–3267 (2019)
  * [8] Mao, X., Li, Q., Xie, H., Lau, R.Y., Wang, Z., Paul Smolley, S.: Least squares generative adversarial networks. In: Proceedings of the IEEE international conference on computer vision. pp. 2794–2802 (2017)
  * [9] Oktay, O., Schlemper, J., Folgoc, L.L., Lee, M., Heinrich, M., Misawa, K., Mori, K., McDonagh, S., Hammerla, N.Y., Kainz, B., et al.: Attention u-net: Learning where to look for the pancreas. arXiv preprint arXiv:1804.03999 (2018)
  * [10] Rana, A., Yauney, G., Lowe, A., Shah, P.: Computational histological staining and destaining of prostate core biopsy rgb images with generative adversarial neural networks. In: 2018 17th IEEE International Conference on Machine Learning and Applications (ICMLA). pp. 828–834. IEEE (2018)
  * [11] Sahasrabudhe, M., Christodoulidis, S., Salgado, R., Michiels, S., Loi, S., André, F., Paragios, N., Vakalopoulou, M.: Self-supervised nuclei segmentation in histopathological images using attention. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 393–402. Springer (2020)
  * [12] Smit, G., Ciompi, F., Cigéhn, M., Bodén, A., van der Laak, J., Mercan, C.: Quality control of whole-slide images through multi-class semantic segmentation of artifacts (2021)
  * [13] Zhu, J.Y., Park, T., Isola, P., Efros, A.A.: Unpaired image-to-image translation using cycle-consistent adversarial networks. In: Proceedings of the IEEE international conference on computer vision. pp. 2223–2232 (2017)

##  Appendix 0.A Supplementary Figures

(a) Air Bubble

(b) Dust

(c) Filament

(d) Out of focus

(e) Tissue fold

(f) Ink

(g) Pen marker

Figure 3: The seven categories of image artifact studied. Figure 4: Annotated
tissue region comprising positive and negative tiles at two scales. Figure 5:
Artifact removal samples. Figure 6: Artifact removal failure cases.


Hawking radiation of massive bosons via tunneling from black strings

Zhong-Wen Feng<EMAIL_ADDRESS>Shu-Zheng Yang

College of Physics and Space Science, China West Normal University, Nanchong
637009, China

Abstract: In the present paper, the Hawking radiation of massive bosons from
4-dimensional and 5-dimensional black strings are studied in quantum tunneling
formalism. Firstly, we derive the Hamilton-Jacobi equation set via the Proca
equation and WKB approximation. Then, the tunneling rates and Hawking
temperature of the black strings are obtained. Our calculations show that the
tunneling rates and Hawking temperatures are related to the properties of
black strings’ spacetime. When compare our results with those of scalars and
fermions cases, it finds that they are the same.

Keywords:Massive bosons; Hawking radiation; Black strings

##  1 Introduction

One common feature among black hole physics is that the existence of black
holes’ thermal properties. This idea was first proposed by Bekenstein, who
proved that the entropy of black holes have satisfies the relation
$S={A\mathord{\left/{\vphantom{A4}}\right.\kern-1.2pt}4}$, where $A$ is the
horizon area [1]. In 1976, inspired by this entropy theory of black holes,
Hawking discovered that black holes have thermal radiation (now people called
this theory as Hawking radiation), and the temperature of black holes (Hawking
temperature) can be expressed as
$T={\kappa\mathord{\left/{\vphantom{\kappa{2\pi}}}\right.\kern-1.2pt}{2\pi}}$
with the surface gravity of black holes $\kappa$ [2]. The Hawking radiation is
very for the foundations of physics, which was great influence on
gravitational theory, quantum mechanism, and thermodynamic. Therefore, the
Hawking radiation has received wide attention.

Based on the original theory of Hawking, the black hole radiation was
considered as a quantum tunneling effect near the horizon. So, people can
discuss the Hawking radiation and the temperature of black holes via the
quantum tunneling method. In Ref. [3], Parikh and Wilczek put forward the
first kind of quantum tunneling method, which is called as the Null Geodesic
method. Using this method, they calculate the tunneling behaviors of massless
scalar particles from Schwarzschild black hole as well as Reissner-Nordström
black hole and obtained those black holes’ temperature. The Hamilton-Jacobi
ansatz is another kind of quantum tunneling method. In Ref. [4], the authors
studied the massive scalar particles tunneling from spherically symmetric
spacetimes. Subsequently, Kerner and Mann extended Hamilton-Jacobi ansatz to
fermions case and studied fermion tunneling. They first applied the WKB
approximation to Dirac equation. Then, neglecting the higher-order terms of
$\mathcal{O(\hbar)}$, the resulting equation to leading order in $\hbar$
becomes the Hamilton-Jacobi equation. According the Hamilton-Jacobi equation,
Kerner and Mann investigated the Hawking radiation of fermions via quantum
tunneling from in the Rindler spacetime and a general non-rotating black hole
[5, 6]. Soon after that, Banerjee and Majhi developed the Hamilton-Jacobi
ansatz to solve the information loss paradox [7, 8]. Nowadays, the radiation
behavior in many complicated spacetimes has been investigated via quantum
tunneling method [9, 10, 11, 12, 13, 14, 15, 16].

It is a well-known issue that the bosons or vector particles (e.g. $Z$,
$W^{\pm}$) play a very fundamental role as in the standard model for
electroweak interaction and the high energy physics. Therefore, the quantum
tunneling of bosons from the black hole has recently attracted people’s
attention [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In Ref. [19, 20],
Kruglov first used the Proca equation to calculate the Hawking radiation of
vector from low dimensional spacetimes. Then, we extended Kruglov’s work and
studied the tunneling behavior of massive bosons from Kerr-de Sitter black
hole and 5-dimensional Schwarzschild-Tangherlini black hole [21]. On the other
hand, the black strings are important solutions of the Einstein-Maxwell
equations. Those cylindrically symmetric solutions are helpful for us to study
the AdS/CFT correspondence. Thus, in the paper we will use Proca equation to
study the vector particles tunneling from 4-dimensional rotating black string
and 5-dimensional black string.

The remainders of this paper are outlined as follows. Using the Proca equation
and WKB approximation, the tunneling process of massive bosons from
4-dimensional rotating black string are investigated in Section 2. In Section
3, we extend our calculations and study massive bosons Hawing radiation from
5-dimensional static black string. In Section 4, some conclusions and
discussions are presented.

##  2 Massive bosons tunneling from the rotating black string

In Ref. [23], the authors analyzed the Einstein-Maxwell equations with a
cosmological constant, which in a stationary spacetime admitting an isometry
group of $R\times\rm{U(1)}$. In their work, the rotating black string metric
is given by

$\displaystyle ds^{2}$
$\displaystyle=-\left[{\alpha^{2}r^{2}-\frac{{4M}}{{\alpha
r}}\left({1-\frac{{a^{2}\alpha^{2}}}{2}}\right)}\right]dt^{2}+\left[{\alpha^{2}r^{2}-\frac{{4M}}{{\alpha
r}}\left({1-\frac{3}{2}a^{2}\alpha^{2}}\right)}\right]^{-1}dr^{2}$
$\displaystyle+\left({r^{2}-\frac{{4Mra^{2}}}{{\alpha
r}}}\right)d\phi^{2}+\alpha^{2}r^{2}dz^{2}-\frac{{8Ma\sqrt{1-\frac{{a^{2}\alpha^{2}}}{2}}}}{{\alpha
r}}dtd\phi,$ (1)

where $\alpha^{2}$ is the negative cosmological constant, which satisfy the
relation $\alpha=-{\Lambda\mathord{\left/{\vphantom{\Lambda
3}}\right.\kern-1.2pt}3}$. Meanwhile, we denote that
$a^{2}\alpha^{2}=1-{\varepsilon\mathord{\left/{\vphantom{\varepsilon
M}}\right.\kern-1.2pt}M}$,
$\varepsilon=\sqrt{M^{2}-{{8J^{2}\alpha^{2}}\mathord{\left/{\vphantom{{8J^{2}\alpha^{2}}9}}\right.\kern-1.2pt}9}}$,
$J=\left({{{3Ma}\mathord{\left/{\vphantom{{3Ma}2}}\right.\kern-1.2pt}2}}\right)\sqrt{1-{{a^{2}\alpha^{2}}\mathord{\left/{\vphantom{{a^{2}\alpha^{2}}2}}\right.\kern-1.2pt}2}}$,
where $M$ and $J$ are the mass and angular momentum densities of the rotating
black string, respectively. For simplicity, the metric (2) becomes to

$\displaystyle ds^{2}=-\Delta\left({\gamma
dt-\frac{\delta}{{\alpha^{2}}}d\phi}\right)^{2}+r^{2}\left({\gamma
d\phi-\delta dt}\right)^{2}+\frac{1}{\Delta}dr^{2}+\alpha^{2}r^{2}dz^{2},$ (2)

with $\Delta=a^{2}r^{2}-{b\mathord{\left/{\vphantom{b{\alpha
r}}}\right.\kern-1.2pt}{\alpha r}}$,
$b=4M\left({1-{{3a^{2}\alpha^{2}}\mathord{\left/{\vphantom{{3a^{2}\alpha^{2}}2}}\right.\kern-1.2pt}2}}\right)$,$\gamma=\sqrt{{{\left({2-a^{2}\alpha^{2}}\right)}\mathord{\left/{\vphantom{{\left({2-a^{2}\alpha^{2}}\right)}{\left({2-3a^{2}\alpha^{2}}\right)}}}\right.\kern-1.2pt}{\left({2-3a^{2}\alpha^{2}}\right)}}}$
,${{a\alpha^{2}}\mathord{\left/{\vphantom{{a\alpha^{2}}{\sqrt{1-{{3a^{2}\alpha^{2}}\mathord{\left/{\vphantom{{3a^{2}\alpha^{2}}2}}\right.\kern-1.2pt}2}}}}}\right.\kern-1.2pt}{\sqrt{1-{{3a^{2}\alpha^{2}}\mathord{\left/{\vphantom{{3a^{2}\alpha^{2}}2}}\right.\kern-1.2pt}2}}}}$,
respectively. The horizons are given by the null-hyper surface condition
$g^{\mu\nu}\left({\partial_{\mu}F}\right)\left({\partial_{\nu}F}\right)=0$,
where $F$ is the hyper-surface of black string. So, the black string has three
horizons when $\Delta$ vanishes, the outer event horizon is located at
$r_{+}=a^{-1}b^{{1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}}$. Now,
applying the dragging coordinate transformation

$\displaystyle d\varphi=d\phi+\Omega
dt=d\phi+\frac{{r^{2}\gamma\alpha^{4}-\Delta\gamma\delta\alpha^{2}}}{{r^{2}\gamma^{2}\alpha^{4}-\Delta\delta^{2}}}dt,$
(3)

one can rewrite the metric (2) as

$\displaystyle
ds^{2}=-A(r)dt^{2}+\frac{1}{B(r)}dr^{2}+C(r)d\varphi^{2}+D(r)dz^{2},$ (4)

where $A(r)={{\Delta
r^{2}\left({\alpha^{2}\gamma^{2}-\delta^{2}}\right)^{2}}\mathord{\left/{\vphantom{{\Delta
r^{2}\left({\alpha^{2}\gamma^{2}-\delta^{2}}\right)^{2}}{\left({\Delta\delta^{2}-\alpha^{4}r^{2}\gamma^{2}}\right)}}}\right.\kern-1.2pt}{\left({\Delta\delta^{2}-\alpha^{4}r^{2}\gamma^{2}}\right)}}$,
$B(r)=\Delta^{-1}$,
$C(r)=r^{2}\gamma^{2}-{{\Delta\delta^{4}}\mathord{\left/{\vphantom{{\Delta\delta^{4}}{\alpha^{4}}}}\right.\kern-1.2pt}{\alpha^{4}}}$,
$D(r)=\alpha^{2}r^{2}$. On the event horizon, the $B(r)$ can be expended as
$B(r)=B^{\prime}(r_{+})(r-r_{+})+\mathcal{O}[(r-r_{+})^{2}]$ [24]. According
to Ref. [19], in order to study the tunneling behavior of massive bosons on
the event horizon, one need to use the Proca equation

$\displaystyle D_{\mu}\psi^{\nu\mu}+\frac{{m^{2}}}{{\hbar^{2}}}\psi^{\nu}=0,$
(5)
$\displaystyle\psi_{\nu\mu}=D_{\nu}\psi_{\mu}-D_{\mu}\psi_{\nu}=\partial_{\nu}\psi_{\mu}-\partial_{\mu}\psi_{\nu},$
(6)

where $D_{\mu}$ are covariant derivatives, $\psi_{\nu}$ are depended on
$\psi_{t},\psi_{r},\psi_{\theta}$ and $\psi_{\varphi}$, and the mass of bosons
is $m$. The anti-symmetric tensor $\psi^{\mu\nu}$ obeys the relation
$\psi_{\mu\nu}=\partial_{\mu}\psi_{\nu}-\partial_{\nu}\psi_{\mu}$. Thus, with
the help of equation
$D_{\mu}\psi_{\mu\nu}={{\partial_{\mu}\left({\sqrt{-g}\psi^{\mu\nu}}\right)}\mathord{\left/{\vphantom{{\partial_{\mu}\left({\sqrt{-g}\psi^{\mu\nu}}\right)}{\sqrt{-g}}}}\right.\kern-1.2pt}{\sqrt{-g}}}$,
Eq. (5) becomes

$\displaystyle\frac{1}{{\sqrt{-g}}}\partial_{\mu}\left({\sqrt{-g}\psi^{\nu\mu}}\right)+\frac{{m^{2}}}{{\hbar^{2}}}\psi^{\nu}=0.$
(7)

where $\psi^{\nu}$ and $\psi^{\mu\nu}$ are given as follows

$\displaystyle\psi^{0}=-{A^{-1}}\psi_{0},\psi^{1}=B\psi_{1},\psi^{2}={C^{-1}}\psi_{2},\psi^{3}={D^{-1}}\psi_{3},\psi^{01}=-{B}{A^{-1}}\psi_{01},$
$\displaystyle\psi^{02}=-{{(AC)}^{-1}}\psi_{02},\psi^{03}=-{{(AD)}^{-1}}\psi_{03},\psi^{12}={B}{C^{-1}}\psi_{12},\psi^{13}={B}{D^{-1}}\psi_{13},$
$\displaystyle\psi^{23}={{(CD)}^{-1}}\psi_{23}.$ (8)

Putting Eq. (2) into Eq. (5), and then, considering the relationship in Eq.
(6), one obtains

$\displaystyle\frac{1}{{\sqrt{-g}}}\left\\{{\partial_{r}\left[{\sqrt{-g}\frac{B}{A}\left({\partial_{t}\psi_{1}-\partial_{r}\psi_{{}_{0}}}\right)}\right]+\partial_{\varphi}\left[{\sqrt{-g}\frac{1}{{AC}}\left({\partial_{t}\varphi_{2}-\partial_{\varphi}\psi_{{}_{0}}}\right)}\right]}\right.$
$\displaystyle\left.{+\partial_{z}\left[{\sqrt{-g}\frac{1}{{AD}}\left({\partial_{t}\psi_{3}-\partial_{z}\psi_{{}_{0}}}\right)}\right]}\right\\}+\frac{{m^{2}}}{{\hbar^{2}}}\frac{1}{A}\psi_{0}=0,$
(9)
$\displaystyle\frac{1}{{\sqrt{-g}}}\left\\{{\partial_{t}\left[{\sqrt{-g}\frac{B}{A}\left({\partial_{t}\psi_{{}_{1}}-\partial_{r}\psi_{0}}\right)}\right]+\partial_{\varphi}\left[{\sqrt{-g}\frac{B}{C}\left({\partial_{r}\psi_{{}_{2}}-\partial_{\varphi}\psi_{1}}\right)}\right]}\right.$
$\displaystyle\left.{+\partial_{z}\left[{\sqrt{-g}\frac{B}{D}\left({\partial_{r}\psi_{{}_{3}}-\partial_{z}\psi_{1}}\right)}\right]}\right\\}+\frac{{m^{2}}}{{\hbar^{2}}}B\psi_{1}=0,$
(10)
$\displaystyle\frac{1}{{\sqrt{-g}}}\left\\{{\partial_{t}\left[{\sqrt{-g}\frac{1}{{AC}}\left({\partial_{t}\psi_{{}_{2}}-\partial_{\varphi}\psi_{0}}\right)}\right]+\partial_{r}\left[{\sqrt{-g}\frac{B}{C}\left({\partial_{\varphi}\psi_{{}_{1}}-\partial_{r}\psi_{2}}\right)}\right]}\right.$
$\displaystyle\left.{+\partial_{z}\left[{\sqrt{-g}\frac{1}{{CD}}\left({\partial_{\varphi}\psi_{3}-\partial_{z}\psi_{2}}\right)}\right]}\right\\}+\frac{{m^{2}}}{{\hbar^{2}}}\frac{1}{C}\psi_{2}=0,$
(11)
$\displaystyle\frac{1}{{\sqrt{-g}}}\left\\{{\partial_{t}\left[{\sqrt{-g}\frac{1}{{AD}}\left({\partial_{t}\psi_{3}-\partial_{z}\psi_{0}}\right)}\right]+\partial_{r}\left[{\sqrt{-g}\frac{B}{D}\left({\partial_{z}\psi_{1}-\partial_{r}\psi_{3}}\right)}\right]}\right.$
$\displaystyle\left.{+\partial_{\varphi}\left[{\sqrt{-g}\frac{1}{{CD}}\left({\partial_{z}\psi_{2}-\partial_{\varphi}\psi_{3}}\right)}\right]}\right\\}+\frac{{m^{2}}}{{\hbar^{2}}}\frac{1}{D}\psi_{3}=0.$
(12)

In order to solve Eq. (2)-Eq. (2), one can express the solutions to above
equation set in the following form

$\displaystyle\psi_{\nu}=\left({c_{0},c_{1},c_{2},c_{3}}\right)\exp\left\\{{\frac{i}{\hbar}\left[{S_{0}\left({t,r,\varphi,z}\right)+\sum\limits_{i}{\hbar^{i}}S_{i}\left({t,r,\varphi,z}\right)}\right]}\right\\},$
(13)

where $i=1,2,3,\cdots$. According to the WKB approximation and Eq. (13), Eq.
(2)-Eq. (2) can be reexpressed as

$\displaystyle\left\\{{B\left[{c_{0}\left({\partial_{r}S_{0}}\right)^{2}-c_{1}\left({\partial_{t}S_{0}}\right)\left({\partial_{r}S_{0}}\right)}\right]+C^{-1}\left[c_{0}\left({\partial_{\varphi}S_{0}}\right)^{2}-{c_{2}\left({\partial_{t}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)}\right]}\right.$
$\displaystyle\left.{+D^{-1}\left[c_{0}\left({\partial_{z}S_{0}}\right)^{2}-{c_{3}\left({\partial_{t}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]}\right\\}+m^{2}\psi_{0}=0,$
(14)
$\displaystyle\left\\{{A^{-1}\left[{c_{0}\left({\partial_{t}S_{0}}\right)\left({\partial_{r}S_{0}}\right)-c_{1}\left({\partial_{t}S_{0}}\right)^{2}}\right]+C^{-1}\left[{c_{1}\left({\partial_{\varphi}S_{0}}\right)^{2}-c_{2}\left({\partial_{t}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)}\right]}\right.$
$\displaystyle\left.{+D^{-1}\left[{c_{1}\left({\partial_{z}S_{0}}\right)^{2}-c_{3}\left({\partial_{r}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]}\right\\}+m^{2}\psi_{1}=0,$
(15)
$\displaystyle\left\\{{A^{-1}\left[{c_{0}\left({\partial_{t}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)-c_{2}\left({\partial_{t}S_{0}}\right)^{2}}\right]+B\left[{c_{2}\left({\partial_{r}S_{0}}\right)^{2}-c_{1}\left({\partial_{r}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)}\right]}\right.$
$\displaystyle\left.{+D^{-1}\left[{c_{2}\left({\partial_{z}S_{0}}\right)^{2}-c_{3}\left({\partial_{z}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)}\right]}\right\\}+m^{2}\psi_{2}=0,$
(16)
$\displaystyle\left\\{{A^{-1}\left[{c_{0}\left({\partial_{t}S_{0}}\right)\left({\partial_{z}S_{0}}\right)-c_{3}\left({\partial_{t}S_{0}}\right)^{2}}\right]+B\left[{c_{3}\left({\partial_{r}S_{0}}\right)^{2}-c_{1}\left({\partial_{r}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]}\right.$
$\displaystyle\left.{+C^{-1}\left[{c_{3}\left({\partial_{\varphi}S_{0}}\right)^{2}-c_{2}\left({\partial_{\varphi}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]}\right\\}+m^{2}\psi_{3}=0.$
(17)

By analysing the spacetime of the rotating black string, one can find 3
Killing vectors. Therefore, the action $S_{0}$ can be expressed as

$\displaystyle
S_{0}=-\left({\omega-j\Omega}\right)t+W\left(r\right)+j\varphi+\mathcal{Z}\left(z\right),$
(18)

with the angular momentum $\omega$ and the energy of massive bosons $j$,
respectively. Now, substituting Eq. (18) into Eq. (2)-Eq. (2), a $4\times 4$
matrix is obtained as

$\displaystyle\Lambda\left({c_{0},c_{1},c_{2},c_{3}}\right)^{T}=0.$ (19)

where $\Lambda$ is a $4\times 4$ matrix, which components are given as follows

$\displaystyle\begin{array}[]{l}\Lambda_{00}=-B\left({W^{\prime}}\right)^{2}-{C^{-1}}\left({\partial_{z}\mathcal{Z}}\right)^{2}-{D^{-1}}j^{2}-m^{2},\Lambda_{01}=-BW^{\prime}\left({\omega-j\Omega}\right),\\\
\\\
\Lambda_{02}=-{C^{-1}}\left({\omega-j\Omega}\right)\left({\partial_{z}\mathcal{Z}}\right),\Lambda_{03}=-{D^{-1}}\left({\omega-j\Omega}\right)j,\\\
\\\
\Lambda_{10}=-{A^{-1}}\left({\omega-j\Omega}\right)W^{\prime},\Lambda_{11}=m^{2}-{A^{-1}}\left({\omega-j\Omega}\right)^{2}+{C^{-1}}\left({\partial_{z}\mathcal{Z}}\right)^{2}-{D^{-1}}j^{2},\\\
\\\
\Lambda_{12}={C^{-1}}\left({\omega-j\Omega}\right)\left({\partial_{z}\mathcal{Z}}\right),\Lambda_{13}={D^{-1}}jW^{\prime},\Lambda_{20}=-{A^{-1}}\left({\omega-j\Omega}\right)\left({\partial_{z}\mathcal{Z}}\right),\\\
\\\
\Lambda_{21}=BW^{\prime}\left({\partial_{z}\mathcal{Z}}\right),\Lambda_{22}={A^{-1}}\left({\omega-j\Omega}\right)^{2}+B\left({W^{\prime}}\right)^{2}+{D^{-1}}j^{2}+m^{2},\\\
\\\
\Lambda_{23}=-{j}{D^{-1}}\left({\partial_{z}Z}\right),\Lambda_{30}=-{A^{-1}}\left({\omega-j\Omega}\right)j,\Lambda_{31}=-BW^{\prime}j,\\\
\\\
\Lambda_{32}=-{C^{-1}}\left({\partial_{z}\mathcal{Z}}\right)j,\Lambda_{33}=m^{2}-{A^{-1}}\left({\omega-j\Omega}\right)^{2}+B\left({W^{\prime}}\right)^{2}+{C^{-1}}\left({\partial_{z}\mathcal{Z}}\right)^{2}.\\\
\end{array}$ (33)

In above equation, we denote that $W^{\prime}=\partial_{r}W$. For obtaining
the nontrivial solution of Eq. (19), it requires the
${\rm{det}}\left(\Lambda\right)=0$. Thus, one has

$\displaystyle
W_{\pm}=\pm\int{\sqrt{\frac{{CD\left({\omega-j\Omega}\right)^{2}-ACj^{2}-ACDm^{2}-AD\left({\partial_{z}\mathcal{Z}}\right)^{2}}}{{ABCD}}}dr},$
(34)

and the above equation is computed as

$\displaystyle W_{\pm}=\pm
i\pi\left[{\frac{{\left({\omega-j\Omega_{r_{+}}}\right)\gamma\alpha^{4}r_{+}^{2}}}{{\left({2\alpha^{4}r_{+}^{3}+b}\right)\left({\alpha^{2}\gamma^{2}-\delta^{2}}\right)}}}\right]+\mathcal{O}(RealPart),$
(35)

where $+/-$ represents the outgoing/incoming solutions on the outer event
horizon. Here we only need the imaginary part since the real part of Eq. (35)
does not contribute to the tunneling rate. The tunneling rate of massive
bosons from 4-dimensional rotating black string is given by

$\displaystyle\Gamma=\frac{{\Gamma_{\left({emission}\right)}}}{{\Gamma_{\left({absorpation}\right)}}}=\frac{{\exp\left({-2{\mathop{\rm
Im}\nolimits}W_{+}-2{\mathop{\rm
Im}\nolimits}\Xi}\right)}}{{\exp\left({-2{\mathop{\rm
Im}\nolimits}W_{-}-2{\mathop{\rm
Im}\nolimits}\Xi}\right)}}=\exp\left[{-\frac{{4\pi\left({\omega-j\Omega_{r_{+}}}\right)\gamma\alpha^{4}r_{+}^{2}}}{{\left({2\alpha^{4}r_{+}^{3}+b}\right)\left({\alpha^{2}\gamma^{2}-\delta^{2}}\right)}}}\right].$
(36)

Comparing Eq. (36) with the expression of Boltzman factor, that is
$\Gamma=\exp\left({{E\mathord{\left/{\vphantom{ET}}\right.\kern-1.2pt}T}}\right)$
with the temperature $T$ and the energy of particles $\omega$ [ch21], the
Hawking temperature of 4-dimensional rotating black string is

$\displaystyle
T_{H}=\frac{{\left({2\alpha^{4}r_{+}^{3}+b}\right)\left({\alpha^{2}\gamma^{2}-\delta^{2}}\right)}}{{4\pi\gamma\alpha^{4}r_{+}^{2}}}.$
(37)

From Eq. (36) and Eq. (37), one can find that the tunneling rate and Hawking
temperature of massive bosons from 4-dimensional rotating black string are
related to the outer event horizon $r_{+}$, the mass $M$ and angular momentum
$J$ of the black string . When comparing our results with those of scalars and
fermions cases, it finds that they are the same [29].

##  3 Massive bosons tunneling from the 5-dimensional black string

In this section, the massive bosons tunneling behavior from 5-dimensional
black string will be investigated by using the Proca equation. The metric of
5-dimensional black string is given by [30]

$\displaystyle
ds^{2}=-A\left(r\right)dt^{2}+\frac{1}{{B\left(r\right)}}dr^{2}+C\left(r\right)d\theta^{2}+D\left(r\right)d\varphi^{2}+Edz^{2}.$
(38)

where
$F\left(r\right)=1-{{r_{H}}\mathord{\left/{\vphantom{{r_{H}}r}}\right.\kern-1.2pt}r}$,
$G\left(r\right)=(1-{{r_{H}}\mathord{\left/{\vphantom{{r_{H}}r}}\right.\kern-1.2pt}r})^{-1}$,
$C\left(r\right)=r^{2}$, $D\left(r\right)=r^{2}\sin^{2}\theta$, $E=1$. $r_{H}$
is the event horizon of the 5-dimensional black string, we can rewrite the
$B(r)$ as $B(r)=B^{\prime}(r_{+})(r-r_{+})+\mathcal{O}[(r-r_{+})^{2}]$.
According to the line element of the 5-dimensional black string, one yields
the following expression

$\displaystyle\psi^{0}=-{A^{-1}}\psi_{0},\psi^{1}=B\psi_{1},\psi^{2}={C^{-1}}\psi_{2},\psi^{3}={D^{-1}}\psi_{3},\psi^{4}={E^{-1}}\psi_{4},$
$\displaystyle\psi^{01}=-{B^{-1}}{A}\psi_{01},\psi^{02}=-{{(AC)^{-1}}}\psi_{02},\psi^{03}=-{{(AD)}^{-1}}\psi_{03},$
$\displaystyle\psi^{04}=-{{(AE)}^{-1}}\psi_{04},\psi^{12}={B^{-1}}{C}\psi_{12},\psi^{13}={B^{-1}}{D}\psi_{13},\psi^{14}={B^{-1}}{E}\psi_{14},$
$\displaystyle\psi^{23}={{(CD)}^{-1}}\psi_{23},\psi^{24}={{(CE)}^{-1}}\psi_{24},\psi^{34}={{(DE)}^{-1}}\psi_{34}.$
(39)

Putting Eq. (3) and Eq. (13) into Eq. (5), and keeping the first order term of
$\hbar$, one has

$\displaystyle
B\left[{c_{1}\left({\partial_{t}S_{0}}\right)\left({\partial_{r}S_{0}}\right)-c_{0}\left({\partial_{r}S_{0}}\right)^{2}}\right]+{C^{-1}}\left[{c_{2}\left({\partial_{t}S_{0}}\right)\left({\partial_{\theta}S}\right)-c_{0}\left({\partial_{\theta}S_{0}}\right)^{2}}\right]$
$\displaystyle+{D^{-1}}\left[{c_{3}\left({\partial_{t}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)-c_{0}\left({\partial_{\varphi}S_{0}}\right)^{2}}\right]+{E^{-1}}\left[{c_{4}\left({\partial_{t}S_{0}}\right)\left({\partial_{z}S_{0}}\right)-c_{0}\left({\partial_{z}S_{0}}\right)^{2}}\right]$
$\displaystyle-m^{2}\psi_{0}=0,$ (40)
$\displaystyle{A^{-1}}\left[{c_{0}\left({\partial_{r}S_{0}}\right)\left({\partial_{t}S_{0}}\right)-c_{1}\left({\partial_{t}S_{0}}\right)^{2}}\right]+{C^{-1}}\left[{c_{1}\left({\partial_{\theta}S_{0}}\right)^{2}-c_{2}\left({\partial_{r}S_{0}}\right)\left({\partial_{\theta}S_{0}}\right)}\right]$
$\displaystyle+{D^{-1}}\left[{c_{1}\left({\partial_{\varphi}S_{0}}\right)^{2}-c_{3}\left({\partial_{r}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)}\right]+{E^{-1}}\left[{c_{1}\left({\partial_{z}S_{0}}\right)^{2}-c_{4}\left({\partial_{r}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]$
$\displaystyle+m^{2}\psi_{1}=0,$ (41)
$\displaystyle{A^{-1}}\left[{c_{0}\left({\partial_{\theta}S_{0}}\right)\left({\partial_{t}S_{0}}\right)-c_{2}\left({\partial_{t}S_{0}}\right)^{2}}\right]+B\left[{c_{2}\left({\partial_{r}S_{0}}\right)^{2}-c_{1}\left({\partial_{\theta}S_{0}}\right)\left({\partial_{r}S_{0}}\right)}\right]$
$\displaystyle+{D^{-1}}\left[{c_{2}\left({\partial_{\varphi}S_{0}}\right)^{2}-c_{3}\left({\partial_{\varphi}S_{0}}\right)\left({\partial_{\theta}S_{0}}\right)}\right]+{E^{-1}}\left[{c_{2}\left({\partial_{z}S_{0}}\right)^{2}-c_{4}\left({\partial_{\theta}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]$
$\displaystyle+m^{2}\psi_{2}=0,$ (42)
$\displaystyle{A^{-1}}\left[{c_{0}\left({\partial_{\varphi}S_{0}}\right)\left({\partial_{t}S_{0}}\right)-c_{3}\left({\partial_{t}S_{0}}\right)^{2}}\right]+B\left[{c_{3}\left({\partial_{r}S_{0}}\right)^{2}-c_{1}\left({\partial_{r}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)}\right]$
$\displaystyle+{C^{-1}}\left[{c_{3}\left({\partial_{\theta}S_{0}}\right)^{2}-c_{2}\left({\partial_{\theta}S_{0}}\right)\left({\partial_{\varphi}S_{0}}\right)}\right]+{E^{-1}}\left[{c_{3}\left({\partial_{z}S_{0}}\right)^{2}-c_{4}\left({\partial_{z}S}\right)\left({\partial_{\varphi}S}\right)}\right]$
$\displaystyle+m^{2}\psi_{3}=0,$ (43)
$\displaystyle{A^{-1}}\left[{c_{0}\left({\partial_{z}S_{0}}\right)\left({\partial_{t}S_{0}}\right)-c_{4}\left({\partial_{t}S_{0}}\right)^{2}}\right]+B\left[{c_{4}\left({\partial_{r}S_{0}}\right)^{2}-c_{1}\left({\partial_{r}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]$
$\displaystyle+{C^{-1}}\left[{c_{4}\left({\partial_{\theta}S_{0}}\right)^{2}-c_{2}\left({\partial_{\theta}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]+{D^{-1}}\left[{c_{4}\left({\partial_{\varphi}S_{0}}\right)^{2}-c_{3}\left({\partial_{\varphi}S_{0}}\right)\left({\partial_{z}S_{0}}\right)}\right]$
$\displaystyle+m^{2}\psi_{4}=0.$ (44)

Considering properties of spacetime, the action for 5-dimensional black string
can be expressed as

$\displaystyle S_{0}=-\omega
t+W\left(r\right)+\Theta\left(\theta\right)+j\varphi+\mathcal{Z}\left(z\right),$
(45)

where $\omega$ and $j$ are the energy and angular momentum of the massive
bosons. Inserting Eq. (45) into Eq. (3)-Eq. (3), the $5\times 5$ matrix
equation can be expressed as

$\displaystyle\Lambda\left({c_{0},c_{1},c_{2},c_{3},c_{4}}\right)^{T}=0,$ (46)

where d the elements of $\Lambda$ are

$\displaystyle\begin{array}[]{l}\Lambda_{00}=-B\left({W^{\prime}}\right)^{2}-{C^{-1}}\left({\partial_{\theta}\Theta}\right)^{2}-{D^{-1}}j^{2}-m^{2}+{E^{-1}}\left({\partial_{z}\mathcal{Z}}\right)^{2},\Lambda_{01}=-B\left({W^{\prime}}\right)\omega,\\\
\\\
\Lambda_{02}=-{C^{-1}}\omega\left({\partial_{\theta}\Theta}\right),\Lambda_{03}=-{D^{-1}}\omega
j,\Lambda_{04}=-{D^{-1}}\omega\left({\partial_{z}\mathcal{Z}}\right),\Lambda_{10}=-{A^{-1}}\omega
W^{\prime},\\\ \\\
\Lambda_{11}=m^{2}-{A^{-1}}\omega^{2}+{C^{-1}}\left({\partial_{\theta}\Theta}\right)^{2}-{D^{-1}}j^{2}+{K^{-1}}\left({\partial_{z}\mathcal{Z}}\right)^{2},\Lambda_{12}={C^{-1}}\omega\left({\partial_{\theta}\Theta}\right),\\\
\\\
\Lambda_{13}={D^{-1}}jW^{\prime},\Lambda_{14}=-{D^{-1}}\left({\partial_{z}\mathcal{Z}}\right)W^{\prime},\Lambda_{20}=-{A^{-1}}\omega\left({\partial_{\theta}\Theta}\right),\Lambda_{21}=BW^{\prime}\left({\partial_{\theta}\Theta}\right),\\\
\\\
\Lambda_{22}=-{A^{-1}}\omega^{2}+B\left({W^{\prime}}\right)^{2}+{D^{-1}}j^{2}+m^{2}+{E^{-1}}\left({\partial_{z}\mathcal{Z}}\right)^{2},\Lambda_{23}=-{D^{-1}}j\left({\partial_{\theta}\Theta}\right),\\\
\\\
\Lambda_{24}={E^{-1}}j\left({\partial_{\theta}\Theta}\right),\Lambda_{30}=-{A^{-1}}\omega
j,\Lambda_{31}=-BW^{\prime}j,\Lambda_{32}=-{C^{-1}}\left({\partial_{\theta}\Theta}\right)j,\\\
\\\
\Lambda_{33}=m^{2}-{A^{-1}}\omega^{2}+B\left({W^{\prime}}\right)^{2}+{C^{-1}}\left({\partial_{\theta}\Theta}\right)^{2}+{E^{-1}}\left({\partial_{z}\mathcal{Z}}\right)^{2},\Lambda_{34}=-{D^{-1}}\left({\partial_{z}\mathcal{Z}}\right)j,\\\
\\\
\Lambda_{40}=-{A^{-1}}\omega\left({\partial_{z}\mathcal{Z}}\right),\Lambda_{41}=-BW^{\prime}\left({\partial_{z}\mathcal{Z}}\right),\Lambda_{42}=-{C^{-1}}\left({\partial_{\theta}\Theta}\right)\left({\partial_{z}\mathcal{Z}}\right),\\\
\\\
\Lambda_{43}=-{D^{-1}}j\left({\partial_{z}\mathcal{Z}}\right),\Lambda_{44}=m^{2}-{A^{-1}}\omega^{2}+B\left({W^{\prime}}\right)^{2}+{C^{-1}}\left({\partial_{\theta}\Theta}\right)^{2}+{D^{-1}}j^{2},\\\
\end{array}$ (64)

For obtaining a nontrivial solution, the determinant of the matrix $\Lambda$
must equals to zero. Therefore, $\det\left(\Lambda\right)=0$ leads to the
following equation

$\displaystyle{\mathop{\rm Im}\nolimits}W_{\pm}^{\prime}$
$\displaystyle=\pm\rm{Im}\int{\sqrt{\frac{{C\left[{DE\omega^{2}-AEj^{2}-ADEm^{2}-AD\left({\partial_{z}\mathcal{Z}}\right)^{2}}\right]-ADE\left({\partial_{\theta}\Theta}\right)^{2}}}{{ABCDE}}}dr}$
$\displaystyle=\pm i\pi\omega r_{H},$ (65)

where the plus/minus denote the outgoing/incoming solutions of massive bosons.
Here, we need to neglect the real part of above equation. Finally, the
tunneling rate of the massive bosons is

$\displaystyle\Gamma=\frac{{\Gamma_{\left({emission}\right)}}}{{\Gamma_{\left({absorpation}\right)}}}=\frac{{\exp\left({-2{\mathop{\rm
Im}\nolimits}W_{+}-2{\mathop{\rm
Im}\nolimits}\Xi}\right)}}{{\exp\left({-2{\mathop{\rm
Im}\nolimits}W_{-}-2{\mathop{\rm
Im}\nolimits}\Xi}\right)}}=\exp\left({-{{4\pi}}\pi\omega r_{H}}\right).$ (66)

According to the the Boltzman factor, the Hawking temperature of 5-dimensional
black string becomes

$\displaystyle T_{H}^{\prime}=\frac{1}{{4\pi r_{H}}}.$ (67)

Obviously, the Hawking temperature of 5-dimensional black string is only
depended on the event horizon $r_{H}$. If people investigates the scalar
particles and fermions tunneling from the 5-dimensional black string, they
would obtain the same results.

##  4 Discussion and conclusion

In this paper, the tunneling behaviors of massive bosons from the event
horizon of 4-dimensional rotating black string and 5-dimensional black string
have been investigated via the Proca equation. By analyzing Eq. (36) and Eq.
(37), one can see that the tunneling rate and Hawking temperature of
4-dimensional rotating black string are related to the the event horizon, mass
and angular momentum line densities of the rotating black string, and the
negative cosmological constant. For the 5-dimensional black string case, it
finds that $\Gamma^{\prime}$ and $T_{H}^{\prime}$ are only dependent on the
the event horizon of the black string. Moreover, when comparing our results
with those of scalars and fermions tunneling cases, it finds that they are the
same. Meanwhile, according to the WKB approximation, we derived the Hamilton-
Jacobi ansatz from Proca equation, which indicates that the Hamilton-Jacobi
equation is a fundamental equation in the semiclassical theory. Therefore,
Hamilton-Jacobi ansatz can help us to investigate the semiclassical dynamic
behavior of emit particles on the black holes’ event horizon.

Acknowledgements This work is supported by the Natural Science Foundation of
China (Grant No. 11573022).

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