# Positronium Physics and Biomedical Applications

Steven D. Bass<EMAIL_ADDRESS>Kitzbühel Centre for Physics, Kitzbühel,
Austria Marian Smoluchowski Institute of Physics, Jagiellonian University, PL
30-348 Krakow, Poland Sebastiano Mariazzi<EMAIL_ADDRESS>Department of Physics, University of Trento via Sommarive 14, 38123 Povo,
Trento, Italy, TIFPA/INFN, via Sommarive 14, 38123 Povo, Trento, Italy Pawel
Moskal<EMAIL_ADDRESS>Marian Smoluchowski Institute of Physics,
Jagiellonian University, PL 30-348 Krakow, Poland Center for Theranostics,
Jagiellonian University, Krakow, Poland Ewa Stepien<EMAIL_ADDRESS>Marian Smoluchowski Institute of Physics, Jagiellonian University, PL 30-348
Krakow, Poland Center for Theranostics, Jagiellonian University, Krakow,
Poland

###### Abstract

Positronium is the simplest bound state, built of an electron and a positron.
Studies of positronium in vacuum and its decays in medium tell us about
Quantum Electrodynamics, QED, and about the structure of matter and biological
processes of living organisms at the nanoscale, respectively. Spectroscopic
measurements constrain our understanding of QED bound state theory. Searches
for rare decays and measurements of the effect of gravitation on positronium
are used to look for new physics phenomena. In biological materials
positronium decays are sensitive to the inter- and intra-molecular structure
and to the metabolism of living organisms ranging from single cells to human
beings. This leads to new ideas of positronium imaging in medicine using the
fact that during positron emission tomography (PET) as much as 40% of positron
annihilation occurs through the production of positronium atoms inside the
patient’s body. A new generation of the high sensitivity and multi-photon
total-body PET systems opens perspectives for clinical applications of
positronium as a biomarker of tissue pathology and the degree of tissue
oxidation.

††preprint: APS/123-QED

##  I Introduction

Positronium “atoms” are special as short lived bound states of an electron
$e^{-}$ and its antiparticle, the positron $e^{+}$. They are at same time
their own “anti-atoms”. Positronium is topical in both fundamental physics
research as well as in applications in biology and medicine, with prime focus
here on its role in new positron emission tomography, PET, technologies.

The physics of positronium is expected to be described by Quantum
Electrodynamics, QED, which is our most accurately tested theory, up to one
part in $10^{12}$, with tiny radiative corrections from the strong and weak
interactions. Recent experiments have revealed some surprises pushing the
boundaries of QED bound state theory [3] with the observation of anomalies up
to 4.5 standard deviations at the precision of $10^{-4}$ between measurements
and theory in hyperfine splittings of positronium energy levels. Possible
couplings of positronium to new interactions are being probed through
precision symmetry tests and rare decay measurements. These experiments
promise to yield new understanding of charged lepton bound states.

While there are uncertainties at this level, positronium is sufficiently well
understood to enable its role in applications from fundamental physics
experiments involving the study of gravitation on antimatter to diagnostic
tests in medicine. About 40% of the positrons in PET scans go through
positronium formation and decay in the body. Building on this result,
positronium is being explored as a vital ingredient in next generation total-
body PET devices where two or more photons are detected simultaneously from
individual positronium decays using the new technique of multi-photon
tomography. Quantum entanglement of the emitted photons may further enhance
the diagnostic power.

In this Colloquium we explore this physics first with an introduction to
positronium and then covering the present status of precision positronium
measurements and current anomalies between data and bound state theory. We
explain the mechanisms of positronium formation and decays in materials and
then, with key focus on biological substances, the application in next
generation PET devices. New positronium imaging technologies have the promise
to revolutionise total body PET imaging with benefit to medical diagnostics.

Positronium comes in two ground states, ${}^{1}S_{0}$ para-positronium,
denoted p-Ps, where the spins of the electron and positron add up to zero and
${}^{3}S_{1}$ ortho-positronium, denoted o-Ps, where the spins of the electron
and positron add up to one. The binding energy,

$E_{B}\approx-m_{e}\alpha^{2}/4=-6.8\ {\rm eV},$ (1)

is much less than the electron mass $m_{e}=0.51$ MeV with
$\alpha\approx\frac{1}{137}$ the fine structure constant. p-Ps is slightly
lighter by 0.84 meV due to the interaction between the electron and positron
spins and also the existence of virtual annihilation processes [42].

Spin-zero p-Ps decays in vacuum to two photons with a lifetime of 125
picoseconds and spin-one o-Ps decays to three photons with a lifetime of 142
nanoseconds. The factor of more than a thousand times longer lifetime of o-Ps
enables efficient distinction of these two states. The main reason for the
difference in lifetimes comes from an extra factor of the fine structure
constant $\alpha$ that enters with the three photon decay compared to two
photon decays.

Positronium was first discovered by Deutsch [62] following the initial
prediction of positron antimatter by Dirac [63], the discovery of the positron
by Anderson [16] and prediction of the $e^{+}e^{-}$ bound state in Mohorovicic
[171].

The two positronium ground states o-Ps and p-Ps, being bound states of $e^{-}$
and $e^{+}$, are both odd under parity transformations P. Under charge
conjugation C o-Ps is odd and p-Ps is even. C symmetry conservation determines
the decays of o-Ps and p-Ps into an odd and even number of photons
respectively, with photons being C symmetry odd [27, 42]. Since positronium is
unstable with leading decay to two or three massless photons (for p-Ps and
o-Ps), it is not an eigenstate of time reversal transformations T. This
property has the consequence that final state interactions involving photon-
photon rescattering interactions at higher order in $\alpha$ can mimic a tiny
CP and CPT violation in positronium decays.

Positronium spectroscopy research is presently focused on precision
measurements of hyperfine transitions, HFS, and also Rydberg states, the
latter with the aim to determine the Rydberg constant based on positronium
[42]. Several few standard deviations discrepancies have been reported between
the precision HFS measurements and QED bound state calculations performed
using the simplifications of non-relativistic QED effective theory, with the
differences entering at precision of a few parts in 10,000 or less [88, 99,
127]. Positronium decay measurements are so far in agreement with QED bound
state theory at similar accuracy. An important ingredient in modelling is that
positronium should satisfy the fundamental symmetries of its constituents.
Rare decays are strongly constrained by precision measurements of the electron
anomalous magnetic moment and electric dipole moment with a prime topic being
the search for invisible decays in connection with possible dark matter
candidates called “mirror matter” particles [256]. In connection with
gravitation, positronium is also playing an important role in precision tests
of gravity on antimatter planned at CERN, the experiments AEgIS [66] and GBAR
[204, 68].

In materials, positronium formation and decay is sensitive to the immediate
chemical environment. This has interesting medical applications with
sensitivity to the healthiness of biological tissue where the positronium is
produced, and may serve as a hallmark telling one about the size of inter- and
intra-molecular voids and the concentration in them of bio-molecules such as,
e.g., oxygen O2 [175]. These properties of positronium suggest its role as a
biomarker, a characteristic that is objectively measured and evaluated as an
indicator of normal biological (healthy) or pathogenic (cancerous) processes.
This result has inspired new ideas for positronium imaging - a new technique
in medical diagnosis that enables imaging of positronium properties inside the
bodies of living organisms [185]. Electromagnetic decays of positronium should
exhibit quantum entanglement of the final state photons [2, 102], with ideas
how this may be exploited in positronium imaging and next generation PET
devices discussed in McNamara  _et al._ [161], Hiesmayr and Moskal [103].

The plan of this article is as follows. In Section II we discuss the status of
precision QED measurements and theory, which constrains detailed modelling of
the positronium system. In Section III we turn our attention to materials
systems where positronium production and decays depend on the chemical
environment. This leads to discussion of positronium in fundamental physics
experiments and in medical applications. Positronium spectroscopy, its role in
the AEgIS and GBAR antimatter experiments at CERN and Bose-Einstein
condensates as well as quantum entanglement in positronium decays are
summarized in Section IV. Biological and medical applications are discussed in
Section V and VI including new developments with positronium imaging and the
emerging application of positronium as a biomarker for assessing the tissue
pathology in vivo. Conclusions and an outlook to future opportunities are
given in Section VII.

Complementary reviews of positronium physics, each with a different emphasis
are given in Adkins  _et al._ [3], Cassidy [42] as well as Bass [25],
Nagashima [188], Goworek [84], Karshenboim [128, 127], Gninenko  _et al._
[83], Berko and Pendleton [27]. An introduction to applications in medicine
and biology is given in Moskal  _et al._ [175] and Harpen [95].

##  II Positronium in the Standard Model

As a bound state of an electron and positron with dynamics determined by QED,
the physics of positronium is strongly constrained by precision QED
observables. QED is a gauge theory invariant under local U(1) transformations
of the phase of the electron and other charged fermions. The QED Lagrangian
reads

${\cal L}=\bar{\psi}i\gamma^{\mu}(\partial_{\mu}+ieA_{\mu})\psi-
m_{e}{\bar{\psi}}\psi-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}.$ (2)

Here $\psi$ represents the electron field, $A_{\mu}$ is the photon, $e$ is the
electric charge and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
is the electromagnetic field tensor; $\alpha=e^{2}/4\pi$ is the fine structure
constant. Electrons and positrons interact through massless photon exchange.

Measurements of the electron’s anomalous magnetic moment $a_{e}=(g-2)/2$ and
atomic physics measurements of the fine structure constant using atom
interferometry with Caesium, Cs, and Rubidium, Rb, atoms are consistent with
each other and with QED theory to one part in $10^{12}$.

The electron’s anomalous magnetic moment $a_{e}$ is non vanishing, differing
from the Born term level Dirac value $a_{e}=0$ by a perturbative QED expansion
in $\alpha$ which is known to ${\cal O}(\alpha^{5})$ [18]. Precision
measurement of $a_{e}$ thus allows determination of the fine structure
constant. The atom interferometry measurements give a direct measurement of
$\alpha$. Any “beyond the Standard Model” effects involving new particles
active in radiative corrections would give an extra correction to $a_{e}$ but
not the direct Cs and Rb interferometry measurements. Thus, comparing these
different determinations of $\alpha$ gives a precision test of QED as well as
constraining possible new physics scenarios.

QED radiative corrections involving heavy muons and tau leptons as well as
hadronic corrections from Quantum Chromodynamics, QCD, each enter $a_{e}$ at
the level of $2\times 10^{-12}$ and weak interactions at the level of $3\times
10^{-14}$ so the anomalous magnetic moment is a very precise test of electron
photon interactions.

The most accurate measurement of $a_{e}$ is [93]

$a_{e}^{\rm exp}=0.00115965218073(28).$ (3)

If one substitutes the most recent direct $\alpha$ measurements from atom
interferometry measurements using both Cs [203]

$1/\alpha|_{\rm Cs}=137.035999046(27)$ (4)

and Rb [172]

$1/\alpha|_{\rm Rb}=137.035999206(11).$ (5)

into the perturbative QED expansion for $a_{e}$ one finds agreement to one
part in $10^{12}$ when comparing with $a_{e}^{\rm exp}$ in Eq. (3), viz.

$a_{e}^{\rm exp}-a_{e}^{\rm th}|_{\rm Cs}=(-88\pm 36)\times 10^{-14}$ (6)

and

$a_{e}^{\rm exp}-a_{e}^{\rm th}|_{\rm Rb}=(+44\pm 30)\times 10^{-14}$ (7)

when we substitute the $\alpha$ values in Eqs. (4,5) into the QED perturbative
expansion for $a_{e}$ to obtain the value $a_{e}^{\rm th}|_{\rm atom}$.

QED is working very well! For practical calculations of positronium
spectroscopy and decays one needs an extra step of QED bound state theory.

Bound state calculations are hard, even in QED. Some model simplifications are
needed to make the calculations tractable. The non-relativistic Schrödinger
equation for the $e^{-}e^{+}$ system gives the correct leading order
expression for the positronium binding energy, Eq. (1). With this in mind a
rigorous effective theory formalism has been developed for calculating
positronium spectroscopy and decays to multiple-photon final states. This is
called non-relativistic QED, NRQED [43]; for reviews see Karshenboim [127],
Kinoshita and Lepage [133], Labelle [142].

NRQED involves a perturbation expansion in $v/c\sim\alpha$ where $v$ is the
electron and positron velocities in the positronium, $c$ is the speed of light
and $\alpha$ the fine structure constant. This approximation allows for doable
calculations. One introduces a cut-off on relativistic effects from the
fundamental QED Lagrangian, Eq. (2). These are then implemented through adding
extra “correction terms” in the NRQED Lagrangian. The parameters are adjusted
to fit the results of experiments, and then the NRQED Lagrangian is used to
calculate new observables. One assumes that the incident electron positron
pair is non-relativistic with relativistic terms in the interactions taken
care of by the NRQED interactions. The fundamental discrete symmetries of QED
should carry over to the truncated NRQED.

Positronium energy levels have been calculated to order $m_{e}\alpha^{6}$ [59,
199], plus some contributions calculated to order $m_{e}\alpha^{7}$ – see
Cassidy [42] and references therein. Experiments in positronium spectroscopy
including anomalies at order $10^{-4}$ between precision measurements and
NRQED predictions are discussed in Section IV below.

For QED decays of positronium to photon final states, radiative corrections to
the tree level processes have been evaluated in NRQED calculations up to two
loop level [5]. The Born term level decay rates $\Gamma(\hbox{o-Ps}\to
3\gamma)=2(\pi^{2}-9)\alpha^{6}m_{e}/{9\pi}$ and $\Gamma(\hbox{p-Ps}\to
2\gamma)=\alpha^{5}m_{e}/2$ are multiplied by radiative correction terms of
the form $\\{1+c_{nm}\alpha^{n}\ln^{m}\alpha\\}$ where $n>m$ and $c_{nm}$ are
coefficients evaluated from the Feynman diagrams, presently up to $n=3$.
Branching ratios $\sim 10^{-6}$ for subleading decays of o-Ps to 5 photons and
p-Ps to 4 photons are suppressed relative to the leading 3 and 2 photon decays
by factors of $(\alpha/\pi)^{2}$. Radiative corrections from QCD and weak
interactions as well as QED radiative corrections involving heavy leptons are
tiny and presently beyond experimental accuracy.

The most accurate measurements of o-Ps decays are consistent with each other
and with NRQED theory. Working in vacuum, Vallery  _et al._ [248] found

$\Gamma=(7.0404\pm 0.0010\pm 0.0008)\times 10^{6}\ {\rm s}^{-1}.$ (8)

Kataoka  _et al._ [129] found

$\Gamma=(7.0401\pm 0.0007)\times 10^{6}\ {\rm s}^{-1}$ (9)

with the o-Ps produced in SiO2 powder. Including both the 3 and 5 photon
contributions NRQED gives the QED decay rate prediction $\Gamma=(7.039979\pm
0.000011)\times 10^{6}\ {\rm s}^{-1}$ [5]. The measurements are consistent
with the NRQED theory prediction with the caveat that the present experimental
uncertainties on the decay rate are about 100 times larger than the NRQED
theoretical error. The leading ${\cal O}(\alpha)$ correction to the decay
rates is needed to agree with the data. The ${\cal O}(\alpha^{2})$ terms are
of order the same size as the experimental error; ${\cal O}(\alpha^{3})$ terms
are well within the experimental uncertainties, as are QCD radiative
corrections.

For the p-Ps decay rate one finds $\Gamma_{p}=(7989.6178\pm 0.0002)\times
10^{6}\ {\rm s}^{-1}$ from NRQED theory with the 4 photon decay included [6],
which compares with the experimental result [10]

$\Gamma_{p}=(7990.9\pm 1.7)\times 10^{6}\ {\rm s}^{-1}$ (10)

with the experimental error 10,000 times the size of the theoretical error
within NRQED.

Going beyond QED decays to photon final states, branching ratios for possible
decays involving new particles beyond the Standard Model are strongly
constrained by precision measurements of the electron’s anomalous magnetic
moment with limits on couplings of any new particles to the electron. If a new
interaction were to couple to the electron with coupling $\alpha_{\rm eff}$ it
would give a leading contribution to $a_{e}$ of size $\alpha_{\rm eff}/2\pi$.
Taken alone, the $a_{e}$ measurements imply constraints on the branching
ratios of o-Ps decays to two photons plus a new light vector particle and to a
photon plus new light pseudoscalar of less than $10^{-9}$ and $10^{-6}$
respectively [25, 83]. Possible invisible decays of o-Ps have been looked for
in the context of mirror matter models of dark matter with the branching ratio
constraint from o-Ps decays in vacuum measured to be $<3\times 10^{-5}$ at 90%
confidence level [256].

As a bound state, positronium should inherit the symmetries of its
constituents. Fundamental QED interactions encoded in the Lagrangian, Eq.(2),
respect the discrete symmetries of P, C, T, and their combinations: CP and
CPT, with CPT a fundamental property of relativistic quantum field theories.

The precision confirmation of QED through the electron’s anomalous magnetic
moment $a_{e}$ implicitly implies CPT as a good symmetry for electrons,
positrons and photons. More directly, the symmetries of CPT and C have been
shown to work to the level of $2\times 10^{-12}$ through measurement of $g-2$
for both electrons and positrons [249],

$g(e^{-})/g(e^{+})=1+(0.5\pm 2.1)\times 10^{-12}.$ (11)

For spin-one o-Ps, CPT is tested in the 3 photon decay through measurement of
a CPT odd correlation
$A_{CPT}=\langle\vec{S}.(\vec{k}_{1}\times\vec{k}_{2})\rangle$ which measures
the T-odd integrated moments between the polarization vector $\vec{S}$ of the
o-Ps and the momenta of the emitted photons with magnitude $k_{1}\geq
k_{2}\geq k_{3}$. The most precise and recent measurement is consistent with
zero with the reached precision of $\pm 0.00095$ [186].

For CP symmetry the electron electric dipole moment, eEDM, puts strong
constraints on any new CP violating interactions coupling to the electron. The
eEDM has been precisely measured showing that any eEDM is tiny [17],

$|d_{e}|<1.1\times 10^{-29}e{\rm cm}.$ (12)

This measurement puts strong limits on new CP violating interactions coupling
to the electron. If interpreted in terms of possible CP violating new heavy
particle exchanges with couplings similar order of magnitude to Standard Model
ones, then one finds a constraint on the new physics scale of similar size to
constraints from the LHC high-energy experiments at CERN [17]. If instead the
eEDM is taken as due to the exchange of near massless particles, then one
finds a bound on their coupling to the electron of $\alpha_{\rm eff}\sim
5\times 10^{-9}$ [25]. Measurements of CP violating correlations involving the
polarisation of the o-Ps with the momentum vectors of the emitted photons are
consistent with zero at ${\cal O}(10^{-3})$ [263]. In future experiments up to
${\cal O}(10^{-5})$ precision in CP and CPT violating correlations is expected
from measurements with the J-PET tomograph in Krakow where new correlations
involving polarisation of the final state photons are also measured [181].

While the underlying QED conserves CP and CPT, finite values for CP and CPT
violating correlations in o-Ps decays at the level of ${\cal O}(10^{-9})-{\cal
O}(10^{-10})$ can occur associated with the fact that unstable Ps is not an
eigenstate of T symmetry. These non zero correlations are found in detailed
calculations of the final state interactions with the leading contribution
coming from light by light scattering of two of the three photons in the final
state [28].

Summarizing, precision studies of positronium decays are consistent with QED
theoretical predictions within the accuracy of present experiments.
Spectroscopic measurements discussed in Section IV.A reveal discrepancies
compared to theory that need to be understood. Positronium structure remains
an exciting topic of investigation. For these measurements one needs
development of precision methods and phenomenology of positronium production
and decay in materials discussed in the next Section. The present experimental
precision on the positronium decay rates in vacuum is however sufficient for
applications, e.g., to studies in biological materials relevant to medicine
discussed in Sections V and VI below.

##  III Positronium production and decay in materials

For experiments involving tests of fundamental physics with positronium as
well as in applications, e.g., to medicine, one first needs to produce the
positronium. How is it made? Positronium is produced via positron interactions
within materials and then emitted into the vacuum and employed in fundamental
physics experiments or its decays are directly studied in medium as a function
of the medium properties. In medicine one finds sensitivity to the healthiness
of the tissue that the positronium is produced in.

As a first step one needs $e^{+}$ production which proceeds either via pair
creation from $\gamma\to e^{-}e^{+}$ or through $\beta^{+}$ decays. The
positrons then interact with a material and annihilate directly with electrons
($e^{+}e^{-}\to{\rm photons}$) or make positronium which is either emitted and
used in experiments or its decay properties are studied directly in medium
with various processes at play, depicted in Fig. 1 and discussed in this
Section. When positronium is formed it may self-annihilate or, alternatively,
decay through annihilation of the positron with an electron in the medium or
via intermediate reactions with molecules in the system. One only considers
the leading 2 photon decays of p-Ps and 3 photon decays of o-Ps for
applications, with the branching ratios for two extra photons production
suppressed by a factor of $(\alpha/\pi)^{2}$ and safely taken as negligible.

In this Section we first outline positron production and then positronium
formation and decay processes in medium with a focus on its use in fundamental
physics experiments (Section IV) and medical applications (Section VI) where
key media are water and mesoporous materials, materials with intermediate
pores (inter-atomic voids) size in the range of 2–50 nm. Water constitutes the
largest percentage of cells and in general biological materials. Mesoporous
silica targets are used as efficient positron to positronium converters for
the production and emission into vacuum of positronium for physics
experiments.

###  III.1 Positron sources and positron thermalization

Positrons are routinely produced in physics and biomedical laboratories around
the world via two processes: $e^{-}e^{+}$ pair production in the electric
field of the nucleus and through use of $\beta^{+}$ radioactive sources [53,
107]. With $e^{-}e^{+}$ pair production, photons with energy larger than
around 1.2 MeV are implanted in materials with high atomic number Z such as
tungsten and platinum, and their energy is converted to the mass of
$e^{-}e^{+}$ pairs. The high-energy photons can be generated via
bremsstrahlung from decelerating electrons previously accelerated to
relativistic energy by employing an electron linear accelerator (LINACs), see
for example Howell  _et al._ [105], Wada  _et al._ [257], Charlton  _et al._
[49]. As an alternative, $\gamma$ rays can be released from nuclear processes
(see for example Schut  _et al._ [225], Hugenschmidt  _et al._ [108], Hawari
_et al._ [97], Sato  _et al._ [221]). With $\beta^{+}$ decays the starting
nuclei transform into daughter nuclei (with atomic number Z reduced by one)
through emission of a positron and a neutrino. A large variety of $\beta^{+}$
radio nuclides with a half-life ranging from less than a second up to several
years and maximum positron energy ranging between several hundreds of keV and
a few MeV are available. The most commonly used in physical laboratories is
22Na (half-life of 2.6 years with maximum positron energy 0.54 MeV [107])
while for biomedical applications there is a growing interest in 44Sc (half-
life of 4 hours with maximum positron energy of 1.47 MeV [159, 51, 101, 215]).
44Sc radioisotope can be obtained from the 44Ti/44Sc generator [80, 209], and
also by irradiation with protons or deuterons of an enriched 44Ca target [51,
162]. 44Ti transforms to 44Sc with the half-lifetime of 60 years via electron
capture [214]. Long lifetime of Titanium-44 makes the 44Ti/44Sc generator
convenient for the laboratory and clinical applications.

In the decay of 22Na and 44Sc radionuclides, excited daughter nuclei are
produced that then de-excites through the emission of a prompt photon via the
following reaction chains

${}^{22}\mbox{Na}~{}\to~{}^{22}\\!\mbox{Ne}^{*}+e^{+}~{}+~{}\nu~{}+~{}\gamma(1.27~{}\mbox{MeV},3.6~{}\mbox{ps}),$
(13)
${}^{44}\mbox{Sc}~{}\to~{}^{44}\\!\mbox{Ca}^{*}~{}+~{}e^{+}~{}+~{}\nu~{}+~{}\gamma(1.16~{}\mbox{MeV},2.61~{}\mbox{ps}).$
(14)

Here the energies of prompt photon and mean de-excitation times [159, 51, 123,
243] are given in brackets.

Figure 1:  Diagram of the $e^{+}e^{-}$ annihilation processes in matter. The
scheme indicates annihilations to 2$\gamma$ and 3$\gamma$ only; ${\rm f_{d}}$
and ${\rm f_{Ps}}$ indicate fraction of direct annihilations and annihilations
via positronium formation, respectively. In biological materials, a ${\rm
f_{d}}$/${\rm f_{Ps}}$ ratio ranging between about 3/2 [95, 135] and about 1/4
[30] is reported. ${\rm f_{d}}$/${\rm f_{Ps}}$ ratios smaller than 1 are
common in mesoporous materials (see for example Goworek [84]). The total decay
rate is due to the positronium self-annihilation ($\lambda_{0}$), pick-off
processes ($\lambda_{\rm pick-off}$), ortho-para conversion reactions
($\lambda_{\rm conv}$) and other chemical processes as e.g. oxidation
($\lambda_{\rm other}$): $\lambda$ = $\lambda_{0}~{}+~{}\lambda_{\rm pick-
off}~{}+~{}\lambda_{\rm conv}~{}+~{}\lambda_{\rm other}$. In water
$\lambda^{p}_{0}=7990.9{\rm\mu s}^{-1}\gg\lambda_{\rm pick-off}=512.8{\rm\mu
s}^{-1}\gg\lambda_{\rm conv}+\lambda_{\rm other}\approx 27{\rm\mu
s}^{-1}(\mbox{O}_{2}\,{\mbox{saturated}})>\lambda^{0}_{0}=7.0401{\rm\mu
s}^{-1}$. In mesoporous silica $\lambda^{p}_{0}=7990.9{\rm\mu
s}^{-1}\gg\lambda_{\rm conv}\approx 25{\rm\mu s}^{-1}$ (O2 at 1 atm.)
$>\lambda^{0}_{0}=7.0401{\rm\mu s}^{-1}>\lambda_{\rm pick-off}\approx 1{\rm\mu
s}^{-1}$. Explanation is given in the text.

When produced positrons are implanted in materials, they rapidly lose kinetic
energy [138] in a variety of interactions (bremsstrahlung, ionization,
electron excitation, phonon excitation, vibrational and rotational excitation,
positronium formation etc…) approaching the thermal energy [224, 210]. The
efficiency of the positron stopping process depends both on the positron
energy range and on the type of material, where the positron is implanted. For
energies of a few tens of MeV, the dominant energy loss mechanism for
positrons (as well as for electrons) is bremsstrahlung in which the positron
interacts with the Coulomb field of the nuclei and the atomic orbital
electrons emitting photons [224, 200]. At implantation energies lower than a
few MeV, this energy loss mechanism becomes less efficient for positrons than
for electrons, due to the different sign of the electric charge, e.g., with
positrons attracted and electrons repulsed by the electric charge of the
nucleus (and vice versa by the atomic electrons) [77, 132].

Below several hundreds of keV [94, 224] down to few electronvolts or few
tenths of electronvolt, in the case of metals, the most important energy loss
processes are ionization and electron excitation [246, 247, 223, 47]. In this
energy range the rate of energy transfer is very high (up to $10^{17}-10^{18}$
eV/s) and the positron energy can be reduced to a few tens of electronvolt
within some picoseconds [205, 224]. At lower energies, electron excitation
processes vanish and other mechanisms involving phonon scattering [205, 224,
193, 86] and vibrational and rotational excitation processes become dominant
[29, 30]. These last mechanisms are less efficient than the electron
excitation [72, 224]. However, the total complete thermalization time is
estimated to be roughly 3 ps for positrons implanted with 1 keV in aluminum at
the temperature of 600 K [193]. In semiconductors and insulators, where the
electron excitations stop when the positron energy decreases under the band
gap and a wider region of energy must be lost through phonon excitation, the
thermalization results longer than in metals [86, 192, 163, 224]. In the case
of positron implantation in water, the contribution of ionization vanishes
below around 50 eV, while the contribution given by electronic excitations
becomes negligible below around 7 eV [29, 30]. Vibrational and rotational
excitations are expected to overcome the contribution given by other energy
loss processes below around 10 eV [30]. In water the entire process of
thermalization takes about 5 ps to 10 ps [236] while the mean diffusion range
is 1.5 mm and 2.1 mm for positrons from 22Na and 44Sc, respectively [243]. In
rare-gas solids, the absence of an optical-phonon branch further reduces the
energy loss efficiency [224]. As a result, positrons can diffuse for lengths
of several $\mu$m retaining several eV of kinetic energy [168] or, in other
words, positrons retain an eV energy for a few tens of picoseconds.

###  III.2 Positronium formation mechanisms

In materials with a wide energy band gap, a positron with kinetic energy less
than the band gap can also lose energy in positronium formation [224]. Ps
formation is energetically possible if the positron energy is within the so-
called Ore gap [197], i.e., between the ionization threshold ($I$) of the
material and $I$-6.8 eV (where 6.8 eV is the Ps binding energy in vacuum).
This process has been extensively investigated in the case of ice at the
beginning of the 1980s [74, 250, 75].

In addition to this Ore mechanism with positronium formation during the
process of positron thermalization, even thermalized positrons can form
positronium in condensed molecular media (dielectric liquids, polymers,
molecular solids and ionic crystals). For details, see for instance Brandt and
Wilkenfeld [33], Eldrup  _et al._ [74], Sferlazzo  _et al._ [226], Wang  _et
al._ [259], Jean [116]. During the positron thermalization process, a number
of positive ions, free electrons, excited molecules and radicals are created.
Freed electrons have a typical average kinetic energy of 10-50 eV [170] that
thermalize traveling for a few tens of nanometers in the material. (In ice
this distance is roughly 30 nm [74].) Positronium can then be formed by the
recombination of a thermalized electron and the thermalized positron. Two
models of recombination have been introduced (the spur model by Mogensen [170]
and the blob model by Stepanov and Byakov [232]) and successfully applied to
study Ps formation in solid [74] and liquid water [234] and, more in general,
in molecular liquids and polymers [61, 235]. In bulk of crystalline metals or
semiconductors, this bulk formation is hindered by the presence of free
electrons that screen the positron-electron interaction destroying the Ps
binding [224]. In such materials, positronium formation can occur only at the
surface, where a thermalized positron reaching the surface picks-up an
electron forming Ps [165, 151]. For unpolarized electrons and positrons
forming positronium, each of four $e^{+}e^{-}$ spin states
$|\\!\\!\uparrow\uparrow\rangle$, $|\\!\\!\downarrow\downarrow\rangle$,
$|\\!\\!\uparrow\downarrow\rangle$, $|\\!\\!\downarrow\uparrow\rangle$ are
equally populated. Formation of
${\mbox{o-Ps}}=\left\\{|\\!\\!\uparrow\uparrow\rangle;\frac{1}{\sqrt{2}}(|\\!\\!\uparrow\downarrow\rangle+|\\!\\!\downarrow\uparrow\rangle);|\\!\\!\downarrow\downarrow\rangle\right\\}$
is three times more probable than p-Ps =
$\frac{1}{\sqrt{2}}(|\\!\\!\uparrow\downarrow\rangle-|\\!\\!\downarrow\uparrow\rangle)$
so that $f^{p}_{Ps}=\frac{1}{4}$ and $f^{o}_{Ps}=\frac{3}{4}$. In materials,
o-Ps undergoes pickoff and conversion processes discussed below that shorten
its lifetime relative to its decay in vacuum. In the presence of voids and
open volumes (see the right panel of Fig. 2), o-Ps formed in the material can
be emitted in the porosity with a kinetic energy typically corresponding to
its workfunction that is usually of the order of a few eVs [245, 190]. Thanks
to its relatively long lifetime, o-Ps can lose a fraction of its energy by
collisions with the walls of the pores [48, 248]. If the pores are
interconnected and open to the surface (as illustrated in Fig. 2(right)), o-Ps
can diffuse along the pore network and eventually be emitted into the vacuum
with a significantly lower energy [248, 113, 241, 98, 37, 152]. The Ps
emission energy into the vacuum depends on the rate and duration of the energy
transfer to the surrounding material that are determined by the pores shape,
dimension and characteristics of their surface [248, 189, 113, 241, 98, 156,
37, 152, 58, 157]. In silica Ps formation occurs with an overall positron to
positronium conversion efficiency up to 84% [251]. Thanks to this
characteristic, in recent years efficient sources of Ps have been synthesized
by exploiting either silica-based disordered porous systems [37, 148, 54] or
oxidized nanochanneled silicon targets [152, 153]. The left panel of Fig. 2
shows an image of the oxidized tunable nanochannel in silicon.

For fundamental physics experiments such mesoporous based-silica converters
are used for the production of positronium and its emission into vacuum [40,
39, 258, 56, 8, 13, 88, 14]; see also Section IV. Special meso-structured
based-silica thin films enable emission of positronium in transmission
(forward) and in reflection (backward) relative to the direction of the
positron beam [15, 155].

Figure 2:  Left: Scanning electron microscope picture of the surface of a
silicon positron to positronium converter with oxidized tunable nanochannels
[152]. Right: Pictorial illustration of positron thermalization in mesoporous
material. Ionization places on a thermalization path are shown. They are
composed of electrons (-), ions (+) and positron (blue line). In the upper
example a positron thermalizes producing free electrons that quickly
thermalize. Next, positronium is formed by recombination with a thermalized
electron and Ps is localized in the void. After bouncing between the walls in
the void (black arrows) it leaves the material. In the lower example a
thermalized positron scatters in the material and annihilates directly into
two photons (red arrows).

###  III.3 Direct positron annihilation in matter

A positron passing through matter may annihilate with electrons directly in
flight [262, 110, 271]. However, due to the fact that the cross-section of
annihilation is inversely proportional to the positron velocity, these
annihilations represent only about 1% [67, 95] of the total annihilation rate.
At the end of the positron thermalization path, when its energy is small (in
the order of tens of eV compared to its initial energy of MeV) the
annihilation rate becomes large. As explained in Section III.B and illustrated
in the right panel of Fig. 2, a positron may either form Ps, or it may diffuse
in the material until it is directly annihilated with an electron. The average
time elapsing to direct annihilation of a thermalized positron can be quite
long. For instance, in water it is much longer than the mean lifetime of p-Ps,
125 ps, and amounts to about 400-450 ps [73, 135]. The fraction of implanted
positrons directly annihilating (${\rm f_{d}}$) in water has been estimated to
range between ${\rm f_{d}}\sim 0.2$ [30] and ${\rm f_{d}}\sim 0.6$ [95, 135].
In silica, where as seen in the previous section a large amount of implanted
positrons form positronium, ${\rm f_{d}}$ can be smaller than 0.2 [251, 84].

###  III.4 Positronium annihilation in matter

Positronium created in matter may annihilate via the processes shown in Fig.
3: (i) self-annihilation in vacuum described by the decay constant
$\lambda^{o}_{0}=7.04{\rm\mu s}^{-1}$ for o-Ps and
$\lambda^{p}_{0}=7990.9{\rm\mu s}^{-1}$ for p-Ps, see Eqs. (8,9,10); (ii)
annihilation via pick-off process where a positron from positronium
annihilates with the electron from the surrounding atoms ($\lambda_{\rm pick-
off}$) – see for example Brandt  _et al._ [32], Wada  _et al._ [257]; (iii)
o-Ps$\leftrightarrow$p-Ps conversion via spin exchange reactions with para-
magnetic molecules such as, e.g., O2 ($\lambda_{\rm conv}$) – see for example
Ferrell [79], Kakimoto  _et al._ [122], Shinohara  _et al._ [230], Cassidy
_et al._ [41] – and (iv) other reactions such as oxidation ($\lambda_{\rm
other}$) [233].

The total decay rate is then expressed as
$\lambda^{p(o)}(t,C)=1/\tau^{p(o)}_{0}=\lambda^{p(o)}_{0}+\lambda_{\rm pick-
off}(t)+\lambda_{\rm conv}(C)+\lambda_{\rm other}(C)$, with dependence on time
$t$ and the concentration of dissolved molecules, $C$. The pick-off rate is
decreasing in time and conversion and other chemical reactions depend on the
concentration of dissolved molecules. In the biological samples, by analogy to
water, the self-annihilation rate of p-Ps ($\lambda^{p}_{0}=7990.9{\rm\mu
s}^{-1}$) is much larger than the pick-off rate ($\lambda_{\rm pick-
off}=512.8{\rm\mu s}^{-1}$ in water), that in turn is much larger than
conversion and other reactions rate ($\lambda_{\rm conv}+\lambda_{\rm
other}\approx 27{\rm\mu s}^{-1}$ in O2 saturated water), which is larger than
the o-Ps self-annihilation rate ($\lambda^{0}_{0}=7.0401{\rm\mu s}^{-1}$). In
porous materials the relation changes and $\lambda_{\rm conv}\approx 25{\rm\mu
s}^{-1}$ (O2 at 1 atm.) $>\lambda^{0}_{0}=7.0401{\rm\mu s}^{-1}>\lambda_{\rm
pick-off}\approx 1{\rm\mu s}^{-1}$. More details are given in the caption of
Fig. 1 and in the following sections.

####  III.4.1 Annihilation via pick-off process

Figure 3:  (a) Above, the decay of a 44Sc isotope, see Eq. (14). (b) Below,
illustration of the basic processes leading to the annihilation of a positron
in the intra-molecular voids of a hemoglobin molecule. The most probable ways
of annihilation are: direct annihilation into two photons (yellow solid
arrows) with positron originating from the 44Sc decay, self-annihilation of
p-Ps (green dotted arrows), self-annihilation of o-Ps (blue dashed arrows),
o-Ps pick-off process (violet dotted arrows) and o-Ps conversion on O2
molecule (red dashed arrows). The distances and size of atoms is shown to
scale with the diameter of positronium twice as large as hydrogen.

The annihilation rate via the pick-off process ($\lambda_{\rm pick-off}$) may
be treated as independent of the positronium spin, with the same value for
o-Ps and p-Ps ($\lambda^{o}_{\rm pick-off}$ = $\lambda^{p}_{\rm pick-off}$)
[71], and expressed as $\lambda_{\rm pick-off}$ =
$\xi$($\frac{1}{4}\lambda^{p}_{0}+\frac{3}{4}\lambda^{o}_{0}$) $\approx$ $\xi$
($\frac{1}{4}\lambda_{2\gamma}+\frac{3}{4}\lambda_{3\gamma}$). Here $\xi$
denotes the positron-electron contact density normalized with respect the
vacuum value. $\lambda^{p}_{0}$ and $\lambda^{o}_{0}$ denote self-annihilation
rate of p-Ps and o-Ps, respectively. The fractions $\frac{1}{4}$ and
$\frac{3}{4}$ originate from the spin projection combinations as explained
earlier. $\lambda_{2\gamma}$ and $\lambda_{3\gamma}$ denote the decay rate
into 2 and 3 photons, respectively. Assuming, to good approximation, that p-Ps
self-annihilates to $2\gamma$ and o-Ps to $3\gamma$ the relative rate of
$3\gamma$ and $2\gamma$ pick-off annihilations is
$3\lambda_{3\gamma}/\lambda_{2\gamma}\approx 3\tau_{{\rm p-Ps}}/\tau_{{\rm
o-Ps}}\approx\frac{1}{378}$, where $\tau_{{\rm p-Ps}}$ and $\tau_{{\rm o-Ps}}$
denote the mean lifetime of p-Ps and o-Ps, respectively. The rate constant for
pick-off annihilations ranges from a fraction of ${\rm\mu s}^{-1}$ in some
mesoporous materials [217, 114] to about $\lambda^{\rm water}_{\rm pick-
off}=513{\rm\mu s}^{-1}-550{\rm\mu s}^{-1}$ in water [231, 228]. The value of
$513\mu s^{-1}$ is small compared to the self-annihilation rate constant of
p-Ps, $\lambda^{p}_{0}=7990.9{\rm\mu s}^{-1}$. Therefore, the p-Ps mean
lifetime $\tau^{p}$ is shortened due to pick-off by only about a few
picoseconds ($\tau^{p}_{0}-\tau^{p}=\frac{1}{7990{\rm\mu
s}^{-1}}-\frac{1}{7990{\rm\mu s}^{-1}+513{\rm\mu s}^{-1}}\approx 7$ ps). In
contrast, the self-annihilation rate of o-Ps with
$\lambda^{o}_{0}=7.0401{\rm\mu s}^{-1}$ is much smaller than the pick-off rate
in liquids and the o-Ps mean lifetime is significantly shortened, e.g., down
to 1.8 ns in water compared to 142 ns in vacuum. In mesoporous materials the
pick-off rate constant depending on the structure is of the order of 1
${\rm\mu s}^{-1}$. It decreases the o-Ps mean lifetime by tens of nanoseconds.
For example, for the IS3100 aerogel with $\tau_{\rm o-Ps}=132$ ns [114], one
finds $\lambda^{o}_{\rm pick-off}=\frac{1}{\tau_{\rm
o-Ps}}-\lambda^{o}_{0}\approx 0.6{\rm\mu s}^{-1}$. It is important to
emphasize that the pick-off annihilation rate is not constant in time. As
mentioned in section III.B and illustrated in Fig. 2, positronium after
formation is bouncing between the void’s walls losing its energy and hence
slowing down. Therefore, the average time intervals between subsequent
positronium interactions with electrons from surrounding molecules are growing
and the pick-off rate decreases in time. Experimentally, this may be
controlled by determining the time dependence of the ratio of the 3$\gamma$
(self-annihilation) to 2$\gamma$ (pick-off annihilation).

####  III.4.2 Positronium conversion and oxidation

Positronium in matter takes part in chemical reactions with radiolytic
products (e.g., solved in water as aqueous electrons, H3O+, OH-radicals)
created by the positron during thermalization and in reactions with dissolved
substances [234, 231]. Interaction of positronium with dissolved para-magnetic
molecules possessing magnetic moment, e.g., molecular oxygen O2, may lead to
the spin exchange and conversion of p-Ps into o-Ps and o-Ps into p-Ps, e.g.,
via the ${\rm\mbox{o-Ps}+O_{2}\to\mbox{p-Ps}+O_{2}}$ reaction [79, 231, 122,
230]. (Non-paramagnetic molecules as N2 are not causing conversion reactions).
In addition the O2 molecule may also oxidize positronium via the ${\rm
Ps+O_{2}}\to e^{+}+{\rm O_{2}^{-}}$ process. Both processes, o-Ps conversion
and oxidation, contribute to the quenching of o-Ps. Conversion in some organic
liquids (cyclohexane, isooctane, isopropanol) is 5 to 10 times more frequent
than oxidation [231]. The conversion and oxidation rate constants depend on
the dissolved oxygen concentration $C_{{\rm O}_{2}}$:
$\lambda_{conv}+\lambda_{other}=k_{O_{2}}\cdot C_{O_{2}}$, with value
$k_{O_{2}}$ for water measured to be $0.0204\pm 0.0008\ {\rm\mu mol^{-1}\mu
s^{-1}L}$ [228] and $0.0186\pm 0.0010\ {\rm\mu mol^{-1}\mu s^{-1}L}$ [231].
This gives $\lambda_{conv}+\lambda_{other}\approx 27{\rm\mu s}^{-1}$ with
saturated O2 in water $\sim 1400\ {\rm\mu molL^{-1}}$. Thus the conversion
rate depends linearly on the dissolved O2 concentration. In mesoporous
materials at relatively low concentrations, $<0.05$ atm., it exceeds the pick-
off rate. On the other hand, at $0.25$ atm. it exceeds the self-annihilation
rate of o-Ps [269]. In water the conversion rate is much lower than that of
pick-off, but in some organic liquids with high oxygen solubility it may
become the dominant effect [231]. The main annihilation processes of the
thermalized positron in a molecule of hemoglobin, which is the most important
component of red blood cells (erythrocytes), an example relevant for medical
applications discussed in Section VI, is shown in Fig. 3.

##  IV Fundamental physics experiments with positronium

In the last few decades, the development of techniques for trapping many
positrons and forming bunches containing up to several $10^{7}$ positrons
[238, 187, 38, 60] and the development of efficient positron to positronium
converters (see Section III) is allowing a significant advancement in the
field of experimental positronium physics [42]. This includes experiments with
positronium spectroscopy, tests of gravity on antimatter and production of a
positronium Bose-Einstein condensate.

###  IV.1 Positronium Spectroscopy

Positronium spectroscopy presently focuses on precision measurements of
hyperfine transitions between singlet and triplet states [42]. Today hyperfine
splittings in positronium are measured to the MHz level whereas the
theoretical NRQED calculations are typically at the kHz level. There are
several interesting “discrepancies” at the few standard deviations level
between measurements of hyperfine splittings and NRQED predictions [3].

Early measurements of the frequency of the 1$S$ hyperfine interaction [213,
166] are about 3 standard deviations below the theoretical NRQED predictions
[4, 104, 59], or about one part in 105. More recent measurements are reported
in Ishida  _et al._ [112, 111] and Miyazaki  _et al._ [169] with the first
closer to the theoretical prediction, to within 1.2 standard deviations. These
results have raised discussion about possible systematics in the experiments
[99] and the theory and contributing Feynman diagrams [127].

Motivated by the situation with the 1$S$ splitting, the ETH Zürich group plan
an in vacuum precision measurement to look at the $2^{3}S_{1}\to 2^{1}S_{0}$
transition and to compare with NRQED predictions. This new experiment will be
free of systematic uncertainties inherent in previous 1$S$ HFS transition
measurements [99].

Most recently Gurung  _et al._ [88, 89] have measured the $2^{3}S_{1}\to
2^{3}P_{0}$ transition on Ps emitted into vacuum from mesoporous silica
targets and determined the transition frequency $\nu_{0}=18501.02\pm 0.61$
MHz. This value differs from the NRQED prediction $\nu_{0}=18498.25\pm 0.08$
MHz, where the quoted theoretical error includes an estimate of unknown higher
order NRQED corrections. The difference is about one part in $10^{4}$ – a
$4.5\sigma$ effect.

The intriguing status of these measurements and their relation to theory calls
for new experiments. If there are no underestimated systematics in the
experiments, given the constraints on possible new interactions coupling to
the electron discussed in Section II, then attention will turn to QED bound
state theory. What from QED might be missing in present NRQED calculations?
Assuming no large extra diagrams waiting to enter at next order, one might
consider the input assumptions to the NRQED bound state calculations. One
effect might be a slightly underestimated harder momentum distribution of
electron velocities in the positronium wavefunctions, e.g., on external lines.
Alternatively, one might consider enhanced Ps resonance contributions as
admixtures in the self-energy diagrams for excited states.

###  IV.2 Positronium in gravity tests and Bose-Einstein condensates

Going beyond positronium spectroscopic and decay measurements positronium also
plays an important role in other fundamental physics experiments: tests of the
equivalence principle through the effect of gravity on antimatter and possible
Bose-Einstein condensates involving antimatter.

To measure the effect of gravity on antimatter, on the one hand o-Ps in
excited states is being used by two experiments at CERN’s Antiproton
Decelerator, AEgIS [66] and GBAR [204, 68], as an intermediate tool to produce
antihydrogen via a charge-exchange reaction with antiprotons [14]. The goal of
these experiments is to measure the acceleration experienced by antihydrogen
in the gravitational field of the earth. The cross section of the charge-
exchange reaction, for high value of the principal quantum number of o-Ps, is
expected to scale with the fourth power of the principal quantum number itself
[136]. Consequently, production of antihydrogen via charge-exchange reaction
will take benefit from the efficient laser excitation to Rydberg states
demonstrated in the last decade on o-Ps emitted into vacuum from silica-based
converters [39, 258, 7]. On the other hand, long-lived positronium states have
been proposed as probes for direct measurements of gravity on a matter-
antimatter system [164, 196, 154]. Both o-Ps exited to Rydberg states [39] and
to the metastable $2^{3}S$ level [13] have been proposed for such
measurements.

Ortho-positronium has the potential to form a Bose-Einstein condensate, BEC,
at high densities [207] which, if observed, would be the first BEC involving
antimatter with the experiments also providing information about high density
o-Ps collisions and possible o-Ps molecule formation.

A recent suggestion [167] involves taking a hollow spherical bubble containing
thousands of spin-aligned o-Ps atoms in superfluid liquid 4He. The bubble
would be stable against breakup into smaller bubbles, and the Ps would form a
BEC with a number density of $\sim 10^{20}$ cm3 and a BEC critical temperature
$T_{c}\approx 300$ K. With present experimental methods bubbles might be
formable containing about $10^{5}$ o-Ps atoms.

For a BEC involving spin-flip from o-Ps to p-Ps, the spontaneous radiation of
positronium atoms in the BEC due to the two-photon collective annihilation
decay might be used as an intense $\gamma$-ray source [20]. Due to BEC
coherence the number of emitted photons will grow at every place in the
condensate. For laser production with direction focused radiation an elongated
condensate might be used. Spontaneously emitted entangled and opposite
directed photon pairs will be amplified, leading to an exponential buildup of
a macroscopic population into end-fire modes in the elongated condensate.

###  IV.3 Photon Entanglement in Positronium Decays

Quantum entanglement of the emitted photons in positronium decays is
interesting as a fundamental physics issue [2, 102, 195]. It also has
interesting applications in PET devices for medical diagnostics [161, 244,
182, 103, 36, 261]. Thus far the entanglement of photons from positronium,
though predicted by theory, has not been observed experimentally.
Implementation of such experiment is challenging since the polarisation of
photons in MeV energy range cannot be studied using optical methods.

Photons are spin-1 particles characterized by their momentum and polarization,
with two transverse polarization states for real photons. In the linear
polarization basis, the 2$\gamma$ state $|\psi\rangle$ originating from p-Ps
can be written as

$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|H\rangle_{1}\otimes|V\rangle_{2}-|V\rangle_{1}\otimes|H\rangle_{2}\right)$
(15)

where $|H\rangle$ and $|V\rangle$ denote the horizontal and vertical polarized
states, and the symbol $\otimes$ refers to a tensor product. The minus sign
between the two combinations reflects the parity -1 eigenvalue of ground state
positronium. The entanglement of the 2$\gamma$ state described in Eq. (15)
manifests itself in the fact that there is no choice of basis
($|A\rangle$,$|B\rangle$) in which the state could be described by the single
tensor product of $|A\rangle\otimes|B\rangle$ – this we call entanglement.
Moreover, Bose-symmetry and parity conservation in the decay of p-Ps imply
that the state of the resulting two photons is maximally entangled and that
the photons polarizations are orthogonal to each other,
$\vec{\epsilon}_{1}\perp\vec{\epsilon}_{2}$ [103].

Figure 4: Schematic illustration of Compton scattering of two photons
originating from p-Ps annihilation. Due to the momentum conservation
($\vec{k_{1}}=-\vec{k_{2}}$), the annihilation photons propagate back-to-back
along the same axis in the p-Ps rest-frame. $\theta_{1}$ and $\theta_{2}$
denote the scattering angles, $\eta_{1}$ and $\eta_{2}$ denote the angles
between the scattering planes and the polarisation directions
$\vec{\epsilon}_{1}$ and $\vec{\epsilon}_{2}$, respectively, and $\varphi$
indicates the relative angle between the scattering planes.

Photons in the energy range of MeV interact in matter predominantly with
electrons through the photoelectric and Compton effects. Compton scattering
(Fig. 4) may be used for the estimation of the linear polarization of the
incoming photon, since the scattering is not isotropic and it is most probable
in the plane perpendicular to the polarization of the incoming photon [134].
For the p-Ps $\to 2\gamma$ process (Fig. 4), when each $\gamma$ interacts via
Compton scattering with an electron one can estimate the angle between the
polarization directions of the photons $|\eta_{1}-\eta_{2}|$ by measurement of
the relative angle $\varphi$ between the scattering planes [182]. The
distribution of $\varphi$ may serve for studies of quantum entanglement [103].
The experimentally determined distributions [173, 261, 1] indeed peak for
$\varphi$ = 90∘ and are consistent with predictions obtained under the
assumption that photons are entangled. However, in order to prove the
entanglement of photons from p-Ps, measurements in at least two different
bases are required [103], e.g., $(|H\rangle,|V\rangle)$ and
$(|+45^{\circ}\rangle,|-45^{\circ}\rangle)$. Yet, the researcher has no
influence on the Compton scattering and an active basis choice cannot be
realized. Therefore, so far the experimental challenge of proving the
entanglement of photons from positronium decays remains open. However, the
strong correlation between the photon’s polarisation may be applied in medical
diagnostics as discussed in Section VI.4.

##  V Development of positronium research in biology and medicine

As discussed in the previous sections, the predictions of positronium
properties based on NRQED theory are currently many orders of magnitude more
precise than the experimental results. Experimental precision is to large
extent limited because positronium is produced in medium, and its properties
in materials are altered with respect to the vacuum. Yet, the variation of
positronium properties as a function of the nano-structure of the material and
concentration in it of paramagnetic molecules constitute the basis for its
application in studies of materials as well as in studies of biological
processes in living organisms and potentially also in medicine.

Although positron emitting radionuclides have been used in diagnostic medicine
since Kuhl and Edwards [140] developed the foundations of medical positron
emission tomography (PET) in the late 1950s, the properties of the positronium
atom have never until recently been used in medicine. Only recent advances in
developing multi-photon tomography [186] and imaging of positronium properties
[185] opened realistic perspectives of making use of positronium as a
diagnostic indicator of the tissue pathology in clinics [179].

In this Section we explain how positronium can help in understanding the
structure of biological objects, how positronium is used in life-science and
why its properties should be translated to medicine.

The method used for the studies is called positron annihilation lifetime
spectroscopy (PALS) and it is based on the measurement of positron lifetime
spectrum in the investigated sample. The positron may be implemented to the
sample by using positron beam or by the application of $\beta^{+}$
radionuclides such as, e.g., 22Na or 44Sc. Typically in the PALS method two
scintillation detectors are used. One detector for determining time of the
annihilation photon originating from the electron-positron annihilation, and
the other for determining the time when positron enters the sample which is
established by the measurement of prompt gamma photon emitted by the excited
daughter nucleus as described in Eq. (14). The measured lifetime spectra
enables one to extract the intensities and mean lifetime distributions of
positrons undergoing annihilations due to various processes depicted in Fig.
1.

The first measurements of intensity ($I_{\rm o-Ps}$) and the mean lifetime
($\tau_{\rm o-Ps}$) of o-Ps were performed on samples containing organic
compounds, benzene derivatives. This experiment showed significant differences
in the signal intensity ($I$) in the presence of halogen atoms (F, Cl, Br, I),
which was explained by the increased electron density of such molecules [96].
An attempt in those studies was made to find a relation between the mean
lifetime and the dissociation energy of the molecular bonds in simple organic
compounds and no suggestion was then given that the functional groups, similar
to those organic compounds, might affect the average lifetime of o-Ps
($\tau_{\rm o-Ps}$) in biological samples [130, 34]. Immediately after finding
positronium properties in organic fluids, Gustafson [90] measured $\tau_{\rm
o-Ps}$ in a biological sample (muscle). However, he focused his attention on
the order and arrangement of tissue water and not on muscle structure. Further
studies on complex biological systems like biological membranes or proteins
moved positronium research towards structural biology and showed that this
approach was efficient in the study of biological reactions (such as electron
transfer [117]), phase transition of lipids [237, 52, 119] macromolecule
structure [91], hydratation [92, 85, 9] and porosity [202, 46] of biological
samples. Nowadays, after many years of focused research, PALS appears to be a
promising technique in the investigation of the structure of macromolecules
[50] and clinical samples [176, 185, 19, 267].

###  V.1 Positronium in biological materials and systems

The first applications of positronium properties in the life sciences were
made by Handel in late 70s [91, 92]. Having in mind that positron annihilation
lifetime is sensitive to changes in free volume caused by pressure (which is
rarely studied in biological systems) or by thermal expansion in the same
phase, he showed significant changes in positronium lifetime during phase
transitions of biological systems. Covalent bonds between carbon atoms in
organic molecules hold structure (architecture) of biological macromolecules
contributing to the creation of free volumes (so-called molecular voids). Such
a molecular structure (here we call it nanostructure [206]) is changing
dynamically with temperature [91, 237, 219], and is stabilized by hydrogen
bonds [92, 85, 131]. The specific feature of the shortening of $\tau_{\rm
o-Ps}$ in the pick-off process has been proposed for probing sub-nanometer-
sized local free volumes in solid materials and organic polymers to assess the
size and nature of chemical environment [206, 65]. Positronium mesurements in
biological samples have been performed with a liquid 22Na source (such as NaCl
solution) prepared in a thin-walled glass bead [91] sealed between thin mylar
films [52], Al foil [117, 119] sealed between a Kapton foil [237, 85, 202, 35,
176] or directly dissolved as an open source [219].

In life sciences two types of bio-materials, hydrogels and biomembranes, were
studied intensively to characterize molecular structure (porosity,
permeability and hydrophobicity) in the context of their biological activity
[202, 219]. Various aspects of PALS applied to life sciences are discussed in
the review by Chen  _et al._ [50], which summarizes recent knowledge about
possible application of o-Ps in biology.

Fluidity and regularity of biological membranes are changing depending on the
phospholipids (e.g., POPC - palmitoyl-oleoyl-glycero-phosphocholine) and
cholesterol content, which can be observed as changes in $\tau_{\rm o-Ps}$.
For example, if cholesterol content is at the high value of 40% and DPPC
(dipalmitoylphosphatidylcholine) is 60%, $\tau_{\rm o-Ps}$ reaches the lowest
value of $\sim$1.86 ns [219]. Admixture of ceramides and cholesterol ceramide
interactions in DPPC membranes also influence $\tau_{\rm o-Ps}$ by changing
the free volume void pattern [23, 82]. The slope of $\tau_{\rm o-Ps}$ rises
rapidly where the membrane undergoes a gel-fluid transformation, at the
transition temperature of the lipid systems [119, 219] and biological
membranes (red cell ghosts) [52]. In contrast to lipid systems, where membrane
permeability is regulated by lipid fluidity [219], in hydrogels, interaction
between polymers and water is a crucial process regulating their biological
activity [220]. This process can be successfully studied by means of PALS
[201]. The dehydration process in crystallized and amorphous state of
maromoleculess (trehalose) is marked significantly by sharp changes in the
mean $\tau_{\rm o-Ps}$ and intensity, which are related to changes in the
total free volume fraction [131].

###  V.2 Positronium in ex vivo research

The first experiment on a biological sample was performed in 1970 and
dedicated to studying semicrystaline structure of water in muscle cells [90,
50]. However, the pioneering experiments on $ex\,vivo$ samples showing that
small temperature variations cause detectable changes in free voids were done
on bovine liver and rabbit muscle [76]. To develop technical details of o-Ps
measurements, a number of experiments were performed on human and mice skin to
study differences in the mean $\tau_{o-Ps}$ of normal cells and cancer (basal
cell carcinoma and squamous cell carcinoma) [120, 118]. This approach appeared
to be promising and indicated a reduction of free volume at the molecular
level for the skin with cancer while the number of patients was the limitation
to conclude about usefulness of positronium imaging in cancer diagnostics
[150]. Extending these studies to skin melanoma showed that positrons
annihilate at a smaller rate with increase of melanoma cells, which confirmed
o-Ps utility in biomedical research [149]. In addition to human and animal
tissues, also unicellular organisms and multicellular tissue-like structures
(spheroids) were investigated giving promising results in positronium research
[139, 22, 125]. Positronium annihilation in tissues strongly depends on water
content. In highly hydrated organs (lens) or tissues (myoma), the mean o-Ps
lifetime is below or around $\sim$2 ns [218, 267]. In adipose tissue this time
is significantly increased [176, 185, 19] confirming the observation from
biological systems that structural characteristic and molecular composition
determine positronium annihilation.

##  VI Medical applications of positrons and positronium

Non invasive imaging of the interior of the body constitutes a powerful
diagnosis tool enabling personalized and targeted therapy. Here we briefly
report on tomographic methods based on positron and positronium annihilations
inside living organisms. We begin with the description of Positron Emission
Tomography (PET) that is a well established diagnostic method delivering
information about the metabolism rate of administered pharmaceuticals, and
about receptor expression on cell membranes [109, 55, 252, 12]. PET is based
on the administration to the living organism of pharmaceuticals labeled with
positron-emitting isotopes, e.g., Fluoro-Deoxy-Glucose (FDG) labeled with 18F
or Prostate Specific Membrane Antigen (PSMA) labeled with 68Ga for metabolic
and receptor imaging, respectively [178]. The image of the annihilation
density distribution is reconstructed based on the registration of $2\gamma$
events originating mostly from direct annihilations, p-Ps self-annihilations
and o-Ps pick-off processes (Fig. 1). The reconstructed annihilation density
distribution corresponds to the image of the metabolic rate (glucose taken by
a cell) or to the image of cancer cell expression (density of cancerous
receptor on a cell) depending on the administered radio-pharamaceuticals. In
2019 the first total-body PET systems were introduced to clinics which enable
dynamical imaging (filming) of all organs and tissues in the body
simultaneously [252, 24, 126, 239]. So far PET detectors reconstruct only
annihilation position distribution. Only recently the method of positronium
imaging was developed which enables one to image in the living organisms
properties of positronium such as a mean lifetime, production intensity, and
the 3$\gamma$ to 2$\gamma$ decay rate ratio [174, 183, 184]. Positronium
imaging requires multi-photon tomography systems enabling registration of
electron-positron annihilations not only into two photons (as in standard PET)
but also decays to three photons, as well as simultaneous registration of
annihilation photons and prompt photon from the radionuclide deexcitation.
Such tomography systems as well as three-photon and mean lifetime image
reconstruction methods were recently demonstrated by the J-PET collaboration
[186, 185]. The first ex-vivo positronium images of healthy and cancer tissues
were published by Moskal  _et al._ [185]. The o-Ps mean lifetime (Section III)
tells one about the size of intra-molecular voids (free volumes between
atoms), whereas its formation probability (production intensity) informs one
about the voids concentration. Both lifetime and production intensity depend
on the bio-active molecule concentration. Notably, positronium may serve as a
biomarker for the assessment of tissue pathology [175, 176, 185] and may be of
particular diagnostics relevance as a biomarker of the concentration of oxygen
[177] (section VI.3). It is important to stress that positronium mean lifetime
imaging delivers information complementary to the metabolic and receptor PET
images and is also complementary to anatomic (electron density distribution)
and morphological (hydrogen atom density distribution) images achievable with
Computed Tomography (CT) and Magnetic Resonance (MR) tomography, respectively.

###  VI.1 Positron Emission Tomography

The healthy and cancerous tissue differ, e.g., by the expression profile of
receptors at the cell membranes, by the angiogenesis resulting in different
concentrations of oxygen molecules, or by glucose metabolism rate. We next
explain how a cancer cell, which needs more glucose for its metabolism and
unlimited divisions or has the vastness of cancerous receptors on its surface,
can be distinguished using PET scans from a healthy cell which is rather
modest in its needs and behaviour.

The number of PET scans has doubled within the last 10 years reaching around
2,000,000 PET scans per year in USA (2017) and 45,000 in Poland (2016) [45].
Typically and most commonly, 18F-Fluoro-Deoxy-Glucose (FDG) is used as a
positron-emitting compound (radiopharmaceutical) in PET for testing cancer and
brain metabolism. This radiotracer was developed and first tested in humans
for imaging cerebral glucose metabolism in 1976 at the University of
Pennsylvania [212, 11, 12], and is still used in around 90% of PET scan
examinations.

Figure 5:  Positronium imaging of HER2-positive cancer cells. A. The HER2
receptor (epidermal growth factor receptor - EGFR) is scarcely expressed on
the surface of healthy cells and significantly (100-times or more)
overexpressed on cancer cells (e.g. breast cancer Venter  _et al._ [255],
Rubin and Yarden [216]). Different colours represent combinations of different
units forming dimmers of the HER2 molecule. B. Trastuzumab (herceptin), a
humizied monoclonal antibody that binds to HER2, is labelled with 44Sc
isotope. C. 44Sc isotope emits a prompt $\gamma$ and a positron ($e^{+}$), to
form a positronium atom. D. Positronium atoms annihilate in cells (with
highest abundance in cancer), on their surface and within cell organelles
(cell membranes, cytosol $nuclei$). Environmental factors like temperature,
water content, and other specific tissue features like chemical and molecular
composition, determine the $\tau_{\rm o-Ps}$ in the diagnosed tissue.

FDG is a glucose analogue, where at the second carbon atom in a glucose ring
($C2$), the normal hydroxyl group (-OH) is substituted with the 18F isotope.
The half-lifetime of the 18F isotope is 110 min which makes 18F-FDG a useful
radiopharmaceutical in the diagnosis of disease processes characterized by
increased glucose consumption, primarily in neoplastic diseases, for the
assessment of brain metabolism or myocardial viability, in drug-resistant
epilepsy, diagnosis of Alzheimer’s disease spectrum (ADS), inflammatory
processes, and systemic diseases [147, 12]. After administration via
intravenous injection, FDG is distributed through the blood stream within
minutes and is actively transported into the cells by specific glucose
transporters - membrane proteins which contribute in glucose uptake (mostly
GLUT1 and GLUT3 Marom  _et al._ [158], Avril [21]). Normally, once
phosphorylated, a glucose molecule continues along the glycolytic pathway
(glycolysis) for energy production. However, FDG cannot undergo glycolysis
because the -OH group is substituted with the 18F atom. Only after 18F decays
radioactively, fluorine at the $C2$ position is converted to 18O. After
picking up a proton (H+) from a hydronium ion (H3O+) in its aqueous
environment, the FDG molecule becomes glucose-6-phosphate labeled with
harmless nonradioactive “heavy oxygen” in the -OH group at the $C2$ position,
ready to be metabolized in the cell [137]. Another approach used for PET
imaging applies a radiotracer for direct labelling of a target cell. In breast
cancers approximately 20-30% of cases overexpress the HER2 receptor (human
epidermal growth factor receptor family), which results from HER2-gene
amplification [216, 222]. In around 90% of HER2-positive cancer cells, up to
several hundred HER2-gene copies are generated to produce over 100 times more
protein receptor in a cancer cell relative to a healthy cell [255, 265]. This
makes the HER2 protein a suitable and ideal biomarker for treatments and
diagnosis of HER2-positive cancer not only in breast, but also in gastric,
bladder, pancreatic and ovarian cancers [222, 264]. Several groups of
molecules targeting HER2 have been developed for molecular imaging with
radiotracers used in PET. Among them, the designed humanized monoclonal
antibody against HER2 protein (trastuzumab) has been used in multiple clinical
trials [100]. Currently, clinical trials with HER2 targeting radiotracers use
radionuclides emitting additional prompt $\gamma$ during $\beta^{+}$-decay,
which would enable determination of positronium mean lifetime, as proposed
recently in Moskal and Stepien [178]. Using $\beta^{+}\gamma$ emitters for
targeting HER2 opens up new possibilities for positronium imaging in breast
cancer diagnostics and treatment (Fig. 5).

###  VI.2 Positronium Imaging

Figure 6:  (Left) Scheme of the example of total-body PET for the positronium
and quantum entanglement imaging showing an axial cross section of the
tomograph design composed of two detection layers [178]. The single detection
module consists of a scintillator and WLS strips read out by SiPM matrices.
Here elements are presented not to scale. Dashed red arrows indicate example
lines of response originating from $e^{+}e^{-}$ annihilation. $2\gamma$
originating from the brain scatter twice in the plastic scintillator and are
shown as an example of event usable for quantum entanglement tomography
discussed in section VI.4. Violet dotted arrows indicate $3\gamma$ decay and
dashed red arrows together with solid black arrow indicate annihilation and
the prompt photon useful for positronium imaging discussed in section VI.2.
Superimposed charts indicate the sensitivity (in arbitrary units) along the
axial field of view, AFOV. (Right) Sensitivity for the $2\gamma$ positronium
imaging (2 annihilation $\gamma$ plus prompt photon) compared to the
sensitivity for the standard $2\gamma$ PET imaging. Results for crystal (LYSO)
PET and plastic PET are shown as a function of the scanner’s AFOV [178]. The
sensitivity gain is shown with respect to the 20 cm AFOV LYSO PET (indicated
with horizontal blue dotted line).

During PET diagnosis of a living organism, annihilation of positrons proceeds
via formation of positronium in about as much as 40% of cases [95, 115, 135].
This makes newly invented positronium imaging [185] a promising method for the
$in\,vivo$ assessment of tissue pathology. Positronium imaging may be defined
as a spatially resolved reconstruction of positronium properties in living
organisms [174]. Information about positronium mean lifetime may be directly
inferred by the application of $\beta^{+}\gamma$ emitters such as 44Sc, which
enable one to determine the positronium lifetime in the organism by
measurement of the time of the emission of the prompt photon and the time of
annihilation (Fig. 6). Coincident detection of both prompt and annihilation
photons and registrations of their positions and times of interaction in the
tomograph allows one to reconstruct the position of annihilation and lifetime
of positronium in each image element (voxel) separately (on a voxel by voxel
basis). For the reconstruction of the annihilation position and time, both
$3\gamma$ self-annihilation of o-Ps [186, 183, 81] as well as $2\gamma$ pick-
off and conversion processes of o-Ps [185, 184] may be applied. A first multi-
photon PET detector enabling positronium imaging was constructed based on
plastic scinillators [180, 191, 70]. It recently provided $ex\,vivo$ $2\gamma$
positronium images of phantoms (objects designed to test the imaging
performance) built from cardiac myxoma cancer tissues and adipose tissues
[185], as well as $3\gamma$ images of the extended cylindrical phantoms [186].
Positronium mean lifetime imaging based on two photons is more than 300 times
more effective than that based on three photons because: (i) in the tissue,
due to the pick-off and conversion processes, o-Ps decays about 70 times, viz.
$(\frac{\tau^{o}_{0}}{\tau_{\rm tissue}}-1$), more frequently to $2\gamma$
than to $3\gamma$, (ii) the attenuation of $3\gamma$ events in the body is
much larger (more than 4 times) with respect to $2\gamma$, both due to higher
number of photons and their lower energies, and (iii) the efficiency for the
detection and selection of $3\gamma$ is lower than for $2\gamma$. The right
panel of Fig. 6 shows a comparison of sensitivity for standard PET imaging and
$2\gamma$ positronium imaging [178]. The sensitivity for positronium imaging
is lower since it requires registration of prompt photon in addition to two
annihilation photons. However, the sensitivity is increasing with the growth
of the axial field of view, and the figure indicates that total-body PET
systems (with 200 cm long scanner) will provide even higher sensitivity for
positronium imaging then current (20 cm long) scanners provide for standard
PET diagnostics. Using the standard whole-body PET protocol, total-body PET
sensitivity enables one to achieve determination of the positronium lifetime
with the precision of about 20 ps for the 2 cm $\times$ 2 cm $\times$ 2 cm
voxels [184, 185], and 2 ps when averaging over the whole organs [177] (see
section VI.3). The time resolution for determining mean o-Ps lifetime in the
tissue depends predominantly on the value of the mean o-Ps lifetime and may be
estimated as $\tau_{tissue}$/$\sqrt{N}$, where $N$ denotes the number of
events in a given voxel of the positronium image [184]. Assuming that
$\tau_{tissue}$ = 2 ns, it can be estimated that with 104 registered events
per $cm^{3}$ (as expected for the total-body PET systems [184]), a resolution
of $\sigma\approx 20$ ps is achievable. Interpretation of positronium
properties as a diagnostic parameter will still require systematic research.
The resolution of 20 ps obtained in first in-vitro images [185] and expected
for total-body PET systems [184] is sufficient to distinguish between the
healthy and cancer tissues for which differences larger than 50 ps (in the
range of 50ps - 200ps [115, 267]) or even 700 ps [185, 176] are observed.
Recently, both the new methods for precise analysis and decomposition of
positron annihilation lifetime spectra [69, 229], as well as new positronium
image reconstruction methods were developed using maximum likelihood image
estimation, the latter resulting in spatial resolution of the image better
than 4 mm [211, 270].

These results indicate that positronium imaging may be introduced in clinics
for the assessment of tissue alterations at the molecular level before they
lead to the functional and morphological changes [179]. The practical
diagnostic benefits of positronium imaging will be a subject of long-standing
research and are yet to be determined. Here we hypothesized that when applied
to brain diagnostics, positronium imaging with its potential for the in-vivo
assessment of the changes of the nano-structure of tissues may become an early
diagnostics indicator for neuro-degenerative diseases such as Dementia,
Alzheimer or Parkinson.

###  VI.3 Positronium as a biomarker of hypoxia

The decay rate of ortho-positronium in the tissue due to the conversion
processes on paramagnetic molecules is linearly proportional to the
concentration on these molecules (see discussion in section III.4.2).
Therefore, positronium may be used for oxygen concentration assessment in the
tissue [177, 228, 231, 266]. In this section the possibility of in-vivo
sensing of oxygen concentration by means of positronium mean lifetime
determination is considered.

A normal level of oxygen concentration in the cells is referred to as normoxia
while hypoxia is defined as a state or condition, in which oxygen supply is
not sufficient to support physiological processes in tissue and organs. Local
hypoxia is usually a result of vessels occlusion (arteries or arterioles) to
cause stroke, myocardial infarction or other organ injury leading to cell
death namely necrosis [160]. In solid tumors hypoxia is often observed and
leads to the development of an aggressive phenotype, acquired treatment
resistance, and is associated with a poor prognosis [253, 31, 137, 160].
Therefore, $in\,vivo$ assessment of the degree of hypoxia would be
advantageous for the personalised cancer therapy [57]. Recently, it was argued
that the possibilities of application of positronium imaging with total-body
PET scanners opens perspectives for the application of positronium as a
biomarker for in-vivo assessment of the degree of hypoxia [177]. Fig. 7
demonstrates that the partial pressure of oxygen (pO2) in cancer tissues is
significantly lower than in corresponding healthy tissue. The differences vary
between 10 mmHg (for the brain) and 50 mmHg (for the pancreas).

Figure 7: Comparison of partial pressure of oxygen molecules (pO2) in healthy
and cancer tissues. The horizontal axis indicates a tissue type (1-12)
explained below the graph. The right axes refers to water and indicate: the
concentration of oxygen ($C_{{\rm O}_{2}}$), the o-Ps annihilation rate
$\lambda^{o}$ in water, and the change of the o-Ps mean lifetime $\Delta t$
due the concentration of oxygen molecules dissolved in water. Normal pO2 for
healthy tissues (red circles) and hypoxia in cancer tissues (black squares)
are shown based on experimental data summarized and reviewed in medians which
do not show real patients variability [254, 253, 160, 240]. For head and neck
tumours, sarcoma and normal subcutaneous tissue data were compiled from the
studies [194, 26]. For a kidney and melanoma there were separate studies [143,
144]. The tissue type is numbered according to the increasing partial pressure
difference between healthy and cancer tissue, indicated by the blue dashed
line.

The experimentally established relationship (for water) between the partial
oxygen pressure pO2 and the o-Ps decay rate constant $\lambda^{o}$ reads
[228]: ${\rm pO}_{2}{\rm[mmHg]}=26.3(11)\cdot(\lambda^{o}-519.9(16){\rm\mu
s}^{-1})$ where 519.9${\rm\mu s}^{-1}$ accounts for o-Ps self-annihilation and
pick-off rate in water. This relation indicates (as shown in Fig. 7) that the
differences of pO2 in the range from 10 mmHg to 50 mmHg result in the change
of ortho-positronium mean lifetime in water by about 2 ps to 7 ps. Estimation
for water is the most pessimistic since for organic liquids (e.g. cyclohexane,
isooctane, isopropanol) these mean oPs lifetime changes are larger [228, 231,
236]. These estimations indicate that in order to apply positronium as a
biomarker for hypoxia an extremaly high (few ps) mean lifetime resolution
determination is required. Resolution of 20 ps was already obtained in the
first experimental positronium images for $ex\,vivo$ studies of cardiac myxoma
tumors [185, 176] with about $10^{4}$ registered o-Ps annihilations. With 100
times more registered o-Ps annihilations, $10^{6}$, 2 ps resolution would be
achievable. Such number of annihilations can be collected by means of the
total-body PET system for organs with the volume larger than 100 cm3 (e.g.
pancreas or liver). Therefore, identification of hypoxia (organ averaged)
using positronium as a biomarker may become feasible with total-body PET
systems

###  VI.4 Quantum Entanglement Tomography

Figure 8:  Distribution of the angle $\varphi$ between scattering planes of
the photons emitted in p-Ps$\,\to 2\gamma$ when the photons are each scattered
at $\theta\sim 82^{\circ}$; for definition of these angles see Fig. 4. The
black solid curve corresponds to the photon pair being entangled, the dashed
blue curve to independent Compton interaction of the two photons, and the red
dotted line indicates when the photon polarizations are uncorrelated, e.g.,
when the photons originate from two different p-Ps decays.

Photons originating from the decay of positronium are expected to be quantum
entangled in polarization and exhibit non-local correlations as discussed in
subsection IV.3. These correlations may be used for the improvement of the
quality of PET image reconstruction [161, 244], and for the elaboration of new
quantum biomarkers by using entanglement witnesses as a diagnostic indicators
[102, 103]. The latter may work provided that the type and degree of quantum
entanglement of photons from the decay of positronium is affected by the
molecular environment in cells. This is a topic of current investigation [102,
36, 227] requiring new experimental input.

Fig. 8 compares the distribution of the angle $\varphi$ between scattering
planes calculated under the assumption that photons from the p-Ps $\to
2\gamma$ process are entangled (black solid curve), for the case when
scattering of one photon is completely independent of the scattering of the
other photon (blue dashed curve), and for the case when photons originate from
different annihilation processes (red dotted curve). Recently it was shown
experimentally by Moskal [173], Watts  _et al._ [261], Abdurashitov  _et al._
[1] that indeed the $\varphi$ distribution for $2\gamma$ annihilations is
enhanced at $\varphi=90^{\circ}$ as expected for the quantum entangled state
of $2\gamma$.

The image quality of standard $2\gamma$ PET may be improved by reduction of
the fraction of events for which any of the photons was scattered in the
patient, or for which photons originate from different annihilation events.
This may be achieved by selecting events for which the angle between
polarization direction of the two photons is close to $\varphi=90^{\circ}$
[161, 244, 182, 261]. Application of the selection based on the relative angle
between the scattering planes will decrease the fraction of unwanted scatter
and random events [244] relative to the fraction of useful events. This will
challenge the designs of PET systems, especially since the visibility of the
quantum correlation is maximal for scattering angles around
$\theta_{1}=\theta_{2}$ $\sim$ 82∘ while the scattering cross sections is the
highest for forward scattered photons ($\theta_{1}=\theta_{2}$ $\sim$ 0∘)
[134].

###  VI.5 Roadmap for multi-photon total-body positron and positronium
tomography

Positron emission tomography is presently experiencing a quantitative change
in diagnosis paradigm [12, 252, 239, 178, 137, 64]. With the advent of total-
body PET systems [24, 126, 208, 106, 191], covering the whole human body (with
the detector length of about 2 m), the simultaneous imaging of the metabolism
rate of all organs and tissues in the body becomes possible. This opens
possibilities for studies (in physiology, biochemistry and pharmacology) of
the kinetics of administered pharmaceuticals in the whole body, and in
determining pharmaceuticals’ uptake correlations in close and distance organs
[24, 268]. High sensitivity of total-body PET systems [252, 239] (up to factor
of $\sim$40 higher with respect to standard 20 cm long PET, see Fig. 6)
enables also dynamic and kinetic model based parametric imaging [78, 260], and
therefore increases the diagnosis specificity in differentiating between the
healthy and cancer tissues. In parallel, recent development in PET technology
resulted in the first multi-photon (multi-gamma) PET system [186] capable of
positronium imaging [185] based on registration of two [184] and three photons
[183] from positronium annihilations and the prompt photon from deexcitation
of isotopes attached to the pharamaceuticals. Moreover, there is a continuous
development of new materials [145, 146] and new systems and techniques [141,
198, 87, 121, 242, 44] for improving the time and spatial resolution to the
point where imaging by direct determination of the density distribution of
annihilation points will become possible. The direct PET image of the 2D brain
phantom was experimentally demonstrated with the spatial resolution of 4.8 mm
[141]. The new generation of total-body PET systems will combine high
sensitivity with multi-photon imaging, and next also with high timing
resolution. The technology for the multi-photon total-body imaging is known
and it is at the stage of translation into clinics [179]. The annihilation
photons’ detection technology for PET is also developing towards a more cost-
effective solution focusing on plastic scintillators [178] and sparse detector
configurations [124]. It is important to stress that the total-body multi-
photon PET systems will enable also high precision studies of fundamental
positronium decays [181] by increasing efficiency for the studies of 3-photon
positronium decays by more than order of magnitude with respect to present
detectors [186].

##  VII Conclusions and Perspectives

Positronium, the bound state of $e^{-}e^{+}$, is interesting both in
fundamental physics and in applications from antimatter research through to
biology and medicine. The properties of positronium formation and decay in
medium depend on the chemical environment and open new windows of opportunity
in biological and medical science. Effective mean decay rates are sensitive to
the health of biological tissue with exciting prospects to revolutionize next
generation total-body PET scanning through positronium imaging as a new tool
in medical diagnosis.

Traditional PET is based on the parameters of the concentration of the
radiopharmaceutical and does not take into account changes in the positronium
annihilation mechanism due to the chemical environment. The average lifetime
of positronium, due to its sensitivity to changes in the nanostructure, will
allow to take into account an additional parameter in the reconstruction of
the histopathological image.

In this Colloquium we have surveyed prime topics in positronium physics and
new prospects for medical applications. We conclude with a summary of key
issues and open questions where next generation experiments should yield vital
information:

  * •

Pushing limits of QED bound state theory, anomalies between data in
spectroscopic measurements and theory call for new precision measurements of
positronium. Are there missed systematics or might the data be pointing to new
(bound state) physics waiting to be understood?

  * •

Studies of gravity on antimatter will provide new tests of the equivalence
principle. Does gravity couple equally to matter as to antimatter?

  * •

Might next generation experiments searching for invisible decays of
positronium help in understanding dark matter?

  * •

About 40% of the positrons in conventional PET scans are formed via
positronium intermediate states. Can this be developed through positronium
imaging as a practical tool for medical diagnostics? Might imaging diagnostics
be further enhanced by sensing quantum entanglement of emitted photons?

  * •

Traditional histopathological imaging requires punctual tissue sampling, which
is always somewhat invasive for the patient. More traditional liquid biopsy,
which is based on taking a sample of blood or other body fluid, does not give
the possibility of locating the lesion. Might virtual biopsy with positronium
imaging be able to tell whether the tissue is cancerous or not without need
for invasive incision?

  * •

Possible differences expected in mean o-Ps lifetime between healthy and cancer
tissues are predominantly due to the structural changes caused by the
increased over-expression of receptors, cell-cycle controlling molecules and
other changes in metabolic pathways (due to inherited or acquired mutations)
and to some extent by the changes of the concentration of the molecular oxygen
dissolved in cells. The influence of the oxygen concentration on the mean o-Ps
lifetime may enable organ-averaged identification of hypoxia with positronium
as a biomarker. How might this be translated to clinics?

## Acknowledgments

We thank E. Czerwiński, W. Krzemień, C. Malbrunot and E. Perez del Rio for
helpful discussions, and M. Durak-Kozica, E. Kubicz, D. Kumar, S. Parzych and
Shivani for help with preparation of the figures. We acknowledge support by
the Foundation for Polish Science through the TEAM POIR.04.04.00-00-4204/17
programme, the National Science Centre of Poland through grants no.
2019/35/B/ST2/03562, 2019/33/B/NZ3/01004, 2021/42/A/ST2/00423 and
2021/43/B/ST2/02150, the Jagiellonian University via project
CRP/0641.221.2020, as well as the SciMat and qLife Priority Research Area
budget under the program Excellence Initiative - Research University at
Jagiellonian University.

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