QED and Fundamental Symmetries in Positronium Decays

TL;DR

This paper reviews how positronium decay experiments can test fundamental symmetries and constrain new physics, emphasizing the importance of precision measurements involving electrons and photons.

Contribution

It provides a comprehensive analysis of positronium decay processes in relation to fundamental symmetry tests and existing precision constraints.

Findings

Positronium decay measurements can test charge-parity-time symmetry violations.

Current experimental bounds limit possible new physics contributions.

Precision observables involving electrons and photons complement positronium decay tests.

Abstract

We discuss positronium decays with emphasis on tests of fundamental symmetries and the constraints from measurements of other precision observables involving electrons and photons.

Tables

Table 1. Table 1: Values of α 𝛼 \alpha extracted from different experiments.

Process	$1/\alpha$	Reference
$a_e^{\mathrm{SM}}$	137.035 999 1491 (331)	[41]
$h/m{(}^{133}\mathrm{Cs})$	137.035 999 046 (27)	[21]
$h/m{(}^{87}\mathrm{Rb})$	137.035 998 995 (85)	[46]
QHE	137.036 003 7(33)	[45]

Table 2. Table 2: Operators for the o-Ps → 3 ​ γ → o-Ps 3 𝛾 \hbox{o-Ps}\to 3\gamma process, and their properties with respect to C 𝐶 C , P 𝑃 P , T 𝑇 T , C ​ P 𝐶 𝑃 CP and C ​ P ​ T 𝐶 𝑃 𝑇 CPT symmetries. Here k 1 → → subscript 𝑘 1 \vec{k_{1}} and k 2 → → subscript 𝑘 2 \vec{k_{2}} denote momentum vectors of photons ordered according to their magnitude, k 1 ≥ k 2 subscript 𝑘 1 subscript 𝑘 2 k_{1}\geq k_{2} , S → → 𝑆 \vec{S} is the spin vector for the o-Ps, and ϵ i → = k i → × k i → ′ / k i ​ k i ′ → subscript italic-ϵ 𝑖 → subscript 𝑘 𝑖 superscript → subscript 𝑘 𝑖 ′ subscript 𝑘 𝑖 superscript subscript 𝑘 𝑖 ′ \vec{\epsilon_{i}}=\vec{k_{i}}\times\vec{k_{i}}^{\prime}/k_{i}k_{i}^{\prime} where k i → → subscript 𝑘 𝑖 \vec{k_{i}} and k i → ′ superscript → subscript 𝑘 𝑖 ′ \vec{k_{i}}^{\prime} denote the momentum of the i th superscript 𝑖 th i^{\rm th} photon before and after Compton scattering in the detector  [ 1 ] .

Operator	$C$	$P$	$T$	$CP$	$CPT$
$\overrightarrow{S}\cdot {\overrightarrow{k}}_1$	+	-	+	-	-
$\overrightarrow{S}\cdot \left({\overrightarrow{k}}_1\times {\overrightarrow{k}}_2\right)$	+	+	-	+	-
$\left(\overrightarrow{S}\cdot {\overrightarrow{k}}_1\right)\left(\overrightarrow{S}\cdot \left({\overrightarrow{k}}_1\times {\overrightarrow{k}}_2\right)\right)$	+	-	-	-	+
${\overrightarrow{k}}_1.{\overrightarrow{ϵ}}_2$	+	-	-	-	+
$\overrightarrow{S}\cdot {\overrightarrow{ϵ}}_1$	+	+	-	+	-
$\overrightarrow{S}\cdot \left({\overrightarrow{k}}_2\times {\overrightarrow{ϵ}}_1\right)$	+	-	+	-	-

Equations

$$\Gamma (o-Ps\to 3\gamma ,5\gamma )$$

$$\Gamma =7.039979(11)\times 10^6{s}^{-1}$$

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

$$\Gamma =7.0404\pm 0.0010\pm 0.0008\times 10^6{s}^{-1}$$

$$Br(\hbox{o-Ps}\to 5\gamma)=\bigl{[}2.2^{+2.6}_{-1.6}\pm 0.5\bigr{]}\times 10^{-6}$$

$$Br(\hbox{o-Ps}\to 5\gamma)=\bigl{[}1.67\pm 0.99\pm 0.37\bigr{]}\times 10^{-6}.$$

$$\Gamma (p-Ps\to 2\gamma )$$

$${\Gamma }_p=7989.6178(2)\times 10^6{s}^{-1}$$

$${\Gamma }_p=7990.9(1.7)\times 10^6{s}^{-1}.$$

$$a_e^{\mathrm{QED},\mathrm{e}}$$

$$a_e(QED:\; mass-dependent)=2.7475719(13)\times 10^{-12}$$

$$a_e^{\mathrm{SM}}=a_e^{\mathrm{QED}}+0.03053(23)\times 10^{-12}(\mathrm{weak})+1.6927(120)\times 10^{-12}(\mathrm{hadronic}).$$

$$a_e^{\exp }=0.00115965218073(28).$$

$$1/\alpha {|}_{a_e^{\mathrm{SM}}}=137.0359991491(331).$$

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

$$1/\alpha {|}_{\mathrm{Rb}}=137.035998996(85)$$

$${\alpha }^2=\frac{2R_{\infty }}{c}\frac{m_{\mathrm{atom}}}{m_e}\frac{h}{m_{\mathrm{atom}}}.$$

$$1/\alpha {|}_{\mathrm{QHE}}=137.0360037(33).$$

$$a_e^{\exp }-a_e^{\mathrm{th}}{|}_{\mathrm{Cs}}=(-88\pm 36)\times 10^{-14}$$

$$|\Delta a_e^{\mathrm{New}\mathrm{Physics}}|\approx \frac{1}{2\pi }\frac{g_X^2}{4\pi }{m}_e^2/{\Lambda }^2$$

$$|\Delta a_e^{\mathrm{New}\mathrm{Physics}}|\approx \frac{1}{2\pi }\frac{g_X^2}{4\pi }.$$

$$|d_e|<1.1\times 10^{-29}e\mathrm{cm}.$$

$$BR(p-Ps\to 3\gamma /p-Ps\to 2\gamma )<2.8\times 10^{-6}\mathrm{at}68\%C.L.\text{cite}[cite][\text{@@bibref}X5]$$

$$BR(o-Ps\to 4\gamma /o-Ps\to 3\gamma )<2.6\times 10^{-6}\mathrm{at}90\%C.L.\text{cite}[cite][\text{@@bibref}X4]$$

$$BR(p-Ps\to 5\gamma /p-Ps\to 2\gamma )<2.7\times 10^{-7}\mathrm{at}90\%C.L.\text{cite}[cite][\text{@@bibref}X2]$$

$$A_{CP}$$

$$A_{CPT}$$

$$C_{CP}$$

$$C_{CPT}$$

$${\mathcal{L}}_a=-\frac{1}{2}{\partial }_{\mu }a{\partial }^{\mu }a+\frac{a}{M}\frac{{\alpha }_s}{8\pi }{G}_{\mu \nu }{\tilde{G}}^{\mu \nu }+\frac{a}{M}\frac{\alpha }{3\pi }{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }+\frac{i{f}_{\psi }}{M}{\partial }_{\mu }a\bar{\psi }{\gamma }^{\mu }{\gamma }_5\psi -...$$

$$\Gamma (o-Ps\to \gamma a)=\frac{8}{3}{g}_a^2\alpha |{\Psi }_n(0){|}^2\frac{m_{o-Ps}^2-m_a^2}{m_{o-Ps}^4}$$

$$|{\Psi }_n(0){|}^2=(\alpha m_e{)}^3/8\pi n^3$$

Full text

1 Introduction

Precision studies of positronium decays are a sensitive probe of QED and allow new tests of fundamental symmetries involving charged leptons. Prime topics for experimental investigation include decay rates, tests of discrete symmetries and looking for possible rare and invisible decays of positronium atoms. With a new generation of positronium experiments coming on-line [1, 2, 3] it is valuable to review the present status of theory and constraints from other processes on the key observables. This we present here, with main emphasis on physics experiments [1, 4, 5] that can be performed using the new J-PET detector, Jagiellonian Positron Emission Tomograph, being built in Cracow [6]. The J-PET is a new PET device based on plastic scintillators designed for total body scanning in medicine as well as biological applications [7] and fundamental physics research [1] with detection of positronium and (from its decay products) Compton rescattered photons in the detector.

The physics of positronium (an “atom” consisting of an electron and a positron) is described by QED with small radiative corrections from QCD and weak interaction effects in the Standard Model. Positronium comes in two ground states, ${}^{1}S_{0}$ para-positronium, denoted p-Ps, where the spins of the electron and positron are antiparallel and ${}^{3}S_{1}$ ortho-positronium, denoted o-Ps, where the spins of the electron and positron are parallel. 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. Spin-zero p-Ps has a lifetime of 125 picoseconds and spin-one o-Ps has a lifetime of 142 nanoseconds. Reviews of positronium physics are given in [8, 9, 10].

Measurements of positronium decay rates are consistent with QED theoretical predictions although the present experimental uncertainties $\sim{\cal O}(10^{-4})$ are very much larger than the theoretical uncertainties on the QED calculations, by a factor of 100 for o-Ps and by 10,000 for p-Ps, calling for increased experimental precision.

Precision observables in positronium decays can be used to test discrete symmetries $C$, $CP$ and $CPT$ with charged leptons. Charge conjugation invariance here has been tested up to the level of $10^{-6}$ [11, 12, 13]. The symmetries $CP$ [14, 15] and $CPT$ [16, 17] have each been tested up to ${\cal O}(10^{-3})$. QED final state effects in o-Ps decays can mimic $CP$, $T$ and $CPT$ violation at the level of $10^{-9}$ to $10^{-10}$ [18, 16]. Possible invisible decays of positronium are also an interesting topic of investigation. Mirror matter models of dark matter allow a branching ratio for the invisible decay of o-Ps in vacuum to mirror particles up to about $2\times 10^{-7}$, below the present experimental bound of $5.9\times 10^{-4}$ [19].

Possible “new physics scenarios” are strongly constrained by studies of other (QED related) observables including precision measurements of the fine structure constant [20, 21] and the electron electric dipole moment (EDM) [22]. Here we discuss the status of this physics taking into account the latest from theory and experimental investigation. Section 2 describes positronium decays in QED. In Section 3 we discuss precision measurements of $\alpha$ and the electron EDM, and their constraints on possible positronium decays. In Section 4 we explore discrete symmetry tests as well as rare and exotic decays, e.g. involving axions and possible invisible decays. Conclusions are given in Section 5.

2 Positronium decays in QED

2.1 o-Ps decay rate

Positronium properties such as decay rates and energy levels can be calculated using the formalism of non-relativistic QED [23]. The o-Ps decay rate within QED has been evaluated to two-loop level. One finds [24]

[TABLE]

Here $A=-10.286606(10)$, $B=44.87(26)$ if just the 3 $\gamma$ decay is included and $B=45.06(26)$ if we also include the $5\gamma$ decay, $C=A/3-229/30+8\ln 2=-5.517025(03)$ and $D$ remains to be calculated. There is good convergence of the perturbation expansion. The terms in Eq. (1) evaluate as 7.211167, -0.172303, -0.000630, 0.001753 (11), -0.000032, 0.000024, 0.00000009 D, – see Table XVII of [24]. That is, the NLO (next-to-leading order) term proportional to $A$ is 2% of the leading Born term and the NNLO term proportional to $B$ is just 0.02%. (Expressions for $A$ and $C$ in closed analytic form are given in [25].) The $5\gamma$ decay contributes 0.19(1)\bigl{(}\frac{\alpha}{\pi}\bigr{)}^{2}\approx 10^{-6} through the $B$ parameter in Eq. (1) [26, 27]. QED light-by-light contributions to the positronium decays [28] contribute 0.350(4) to the $B$ parameter for o-Ps in units of \bigl{(}\frac{\alpha}{\pi}\bigr{)}^{2} times the respective lowest order rate.

Including both the 3 and 5 photon contributions gives the QED decay rate prediction

[TABLE]

where we here neglect the ${\cal O}(\alpha^{3})$ term proportional to the unknown constant $D$ [24].

The calculations in [24] are pure QED. There will also be hadronic QCD light-by-light and photon self-energy corrections. Hadronic light-by-light corrections are known to be important in the muon $g-2$ puzzle [29]. In positronium decays these corrections will enter at ${\cal O}(\alpha^{2})$ in the decay rate with extra suppression factor $(m_{e}/\mu)^{2}\sim 10^{-6}$ in the $B$ parameter with $\mu$ a typical hadronic scale, and are well beyond present experimental reach.

The most accurate measurements of o-Ps decays are consistent with each other and with the theoretical prediction, Eq. (2). Kataoka et al. [30] found

[TABLE]

with the o-Ps produced in SiO2 powder, whereas Vallery et al. [31] found

[TABLE]

working in vacuum.

These results are consistent with the QED theory prediction, Eq.(1) with the caveat that the present experimental uncertainties on the decay rate are about 100 times greater than the theoretical error. The leading ${\cal O}(\alpha)$ correction in Eq. (1) is needed to agree with the data, the ${\cal O}(\alpha^{2})$ terms are of order the same size as the experimental error and the ${\cal O}(\alpha^{3})$ terms are well within the experimental uncertainties. Five photon decay measurements are consistent both with zero and with theoretical expectations. Matsumoto et al. [32] found

[TABLE]

and Vetter and Freedman [33] found

[TABLE]

The energy spectrum for the o-Ps to $3\gamma$ decay is discussed in [34, 35] with formulas presented at leading order in $\alpha$. A first test of the ${\cal O}(\alpha)$ correction for the energy spectrum for orthopositronium decay is discussed in [36], where the NLO term is needed at 92% C.L.

2.2 p-Ps decay rate

For p-Ps the QED prediction is [37, 38, 39]

[TABLE]

The $4\gamma$ decay contributes an extra 0.274290(8)\bigl{(}\frac{\alpha}{\pi}\bigr{)}^{2} in units of $\frac{\alpha^{5}m_{e}}{2}$ [39]. Summing these contributions gives

[TABLE]

where we neglect the term proportional to the unknown $D_{p}$ coefficient. This value compares with the experimental result [40]

[TABLE]

The experimental error is 10,000 times the size of the QED theoretical error.

While not needed by present data, new physics possibilities might be explored within the uncertainties on the total decay rates, presently corresponding to branching ratios at the ${\cal O}(10^{-4})$ level. There are strong constraints from other processes including precision measurements of the fine structure constant $\alpha$ and the electron EDM, which we discuss in Section 3.

3 QED tests and $\alpha$ measurements

The most accurate determinations of $\alpha$ come from precision measurements of the electron’s anomalous magnetic moment, atom interferometry measurements with Caesium, Cs, and Rubidium, Rb, and the Quantum Hall Effect.

The electron anomalous magnetic moment $a_{e}=(g-2)/2$ is generated by radiative corrections, which have been evaluated to tenth-order in QED perturbation theory plus small QCD and weak contributions. The electron $a_{e}$ value gives a precision measurement of $\alpha$ (modulo any radiative corrections from new physics beyond the Standard Model). Atom interferometry experiments with Cs and Rb and Quantum Hall Effect measurements provide a more direct determination (less sensitive to details of radiative corrections) but also involve a combination of parameters measured in experiments. The Cs measurements are presently the most accurate. Comparing these different determinations of $\alpha$ gives a precision test of QED as well as constraining possible new physics scenarios. Any “beyond the Standard Model” effects involving new particles active in radiative corrections will enter $a_{e}$ but not the Cs, Rb and Quantum Hall Effect measurements.

The anomalous magnetic moment $a_{e}$ is related to $\alpha$ through [41]

[TABLE]

from Feynman diagrams involving electrons and photons. Contributions from heavy leptons sum to

[TABLE]

with extra electroweak and QCD corrections

[TABLE]

The most accurate measurement of $a_{e}$ comes from the Harvard group, Gabrielse et al. [20]

[TABLE]

Plans for new, even more accurate, measurements are discussed in [42]. Using the theoretical formulae (10–12), the $\alpha$ value extracted from $a_{e}^{\rm SM}$ is [41]

[TABLE]

For direct measurements of $\alpha$, the most accurate comes from Cs interferometry [21]

[TABLE]

with

[TABLE]

from 87Rb atom interferometry [43, 44] quoted by CODATA [45, 46]. Note that these Cs and Rb results rely on a number of other experimental quantities involving the Rydberg constant $R_{\infty}$, the ratio of the atom to electron mass $m_{\rm atom}/m_{e}$ and new precision measurements of the Cs or Rb masses from recoil of a Cs or Rb atom in an atomic lattice, viz.

[TABLE]

Here $c$ is the speed of light and $h$ is Planck’s constant. Quantum Hall Effect experiments yield the value (CODATA [45])

[TABLE]

The Hall conductivity of two dimensional electron systems is quantised in integral multiples of $e^{2}/h$ [47]. Interest in the ratio $e^{2}/h$ lies not only in its application as an “atomic” resistance standard based on fundamental constants but also as a method to measure $\alpha$. The different measurements of $1/\alpha$ are collected in Table 1.

The new most accurate Cs atomic physics measurement corresponds to

[TABLE]

when we substitute the $\alpha$ value in Eq. (15) into Eqs. (10–12) to obtain the value $a_{e}^{\rm th}|_{\rm Cs}$. Suppose we interpret this 2.5 $\sigma$ “discrepancy” as an upper limit on contributions coming from new physics, $\Delta a_{e}^{\rm New\ Physics}\equiv a_{e}^{\rm exp}-a_{e}^{\rm th}$. If this originates from new heavy exchanges with coupling constant $g_{X}$ and mass scale $\Lambda$, then

[TABLE]

fixes $\Lambda$ bigger than $\sim 40$ GeV (much below collider constraints) with $\frac{g_{X}^{2}}{4\pi}\sim 4/137\simeq 0.03$ (that is, taking coupling constants of order the Standard Model ones). If one instead assumes a new light particle with mass $m_{X}^{2}\ll m_{e}^{2}$, then [10]

[TABLE]

Taking the numbers in Eq. (19), the upper bound on $g_{X}^{2}/4\pi$ is about 6 $\times 10^{-12}$, a factor of 50 reduction from the earlier analysis in  [10]. This upper bound constrains the branching ratios for possible decays of o-Ps to a photon and very-light mass pseudoscalar and to two photons and a new light vector boson to be $<10^{-6}$ and $<10^{-9}$ respectively. Further constraints on pseudoscalar axion models come from astrophysics and laboratory experiments, see Section 4.3 below. In a recent paper [48] the difference in Eq.(19) and the muon $g-2$ anomaly [49] are interpreted together through introduction of a new scalar with mass bigger than about 250 MeV and couplings to the muon and electron of $\sim 10^{-3}$ and a few times $10^{-4}$.

New, most accurate, measurements of the electron EDM [22] give

[TABLE]

A finite value of $d_{e}$ would correspond to some new $CP$ violating interaction. Within typical extensions of the Standard Model involving new heavy particles, the electron EDM constraint puts limits on the mass scales of this new physics as 30 TeV in one-loop calculations and 3 TeV at two-loops, strong constraints on new physics models which are competitive with the constraints from the LHC [22]. (These numbers are obtained assuming similar size couplings to the Standard Model ones and $\sin\phi_{\rm CP}\sim 1$, where $\phi_{\rm CP}$ is the $CP$ violating phase). Changing from possible new heavy particles to exchanges involving new near-massless particles corresponds to an upper bound on their coupling to electrons of $g_{X}^{2}/4\pi\sim 8\times 10^{-18}$ in the leading-order calculation and $\sim 5\times 10^{-9}$ within two-loop calculations for $CP$ violating interactions (evaluated by rescaling the heavy mass scale in the calcuations to the electron mass and assuming no special phase cancellation in the EDM). The latter bound on $g_{X}^{2}/4\pi$ corresponds to an upper bound on the branching ratio for $CP$ violating o-Ps decays of about $10^{-9}$.

4 Rare and exotic decays and new physics

4.1 $C$ and $P$ violating decays

Experimental bounds on possible $C$-violating decays of positronium have been reported by earlier experiments

[TABLE]

J-PET will push these limits. With a 10 MBq positronium source and upgraded 4 layer detector geometry, one expects to measure $9.4\times 10^{10}$ o-Ps to 3 photon decays and $3\times 10^{11}$ p-Ps to two photon decays in 365 days of data taking [50].

QED forbidden decays can proceed through weak interactions but with very small branching ratios because of the massive W and Z boson propagators that appear in these reactions. For example, the three photon decay of p-Ps is forbidden in QED but can occur through weak interactions involving a W-boson loop with branching ratio $BR(\hbox{p-Ps}\to 3\gamma)=4.4\times 10^{-77}$ [51]. Tiny branching ratios are found for decays into a photon and two neutrinos [52], less than about $10^{-21}$ for o-Ps and $10^{-24}$ for p-Ps. Exotic decays to a single photon and possible light mass “dark photon” (dark matter candidate) have been postulated with branching ratio up to ${\cal O}(10^{-10})$ [53].

4.2 CP violation

After the electron EDM, Eq. (22), o-Ps decays are an experimentally clean system to look for $CP$ violation with charged leptons [54]. The observed matter antimatter asymmetry in the Universe requires some extra source of $CP$ violation beyond the quark mixing described by the Cabbibo-Kobayashi-Maskawa (CKM) matrix in the electroweak Standard Model. Recent measurements by the T2K Collaboration in Japan are consistent with $CP$ violation in the neutrino sector at the level of two standard deviations [55]. So far there is no hint from experiments for $CP$ violation with charged leptons. The electron EDM places strong constraints on any new effect. Searching for $CP$ violation in positronium decays is an active topic of investigation. To see a signal new effects will need to be much larger than Standard Model $CP$ violations, which are very much suppressed.

Spin-one o-Ps decays are sensitive to $CP$ and $CPT$ odd correlations through the spin vector of the o-Ps. Previous experiments have focused on the correlations

[TABLE]

measuring the $T$-odd integrated moments between the polarisation vector $\vec{S}$ of the o-Ps and the momenta of the emitted photons with magnitude $k_{1}\geq k_{2}\geq k_{3}$.

New $CP$ and $CPT$ observables enter if we can also measure observables related to Compton scattering of photons from the positronium decay in the detector [1]. The Compton scattering cross section is peaked perpendicular to the electric field and polarisation axis of the incident photon [56, 57]. This leads to defining the polarisation related quantities $\vec{\epsilon_{i}}=\vec{k_{i}}\times\vec{k_{i}}^{\prime}/k_{i}k_{i}^{\prime}$, where $\vec{k_{i}}$ and $\vec{k_{i}}^{\prime}$ are the momenta of a photon from the positronium decay and the rescattered photon from Compton scattering in the detector [1]. These $\vec{\epsilon_{i}}$ vectors are peaked to lie along the axis of the incident photon polarisation vector. They are even under $P$ and $T$ transformations. One can form new $CP$ and $CPT$ correlations between the o-Ps spin vector, the momenta of the radiated photons and the $\vec{\epsilon_{i}}$ vectors, see Table 2. The first three rows in Table 2 refer to correlations involving just the o-Ps decay and the second three rows involve correlations between the positronium system and Compton scattering processes in the detector.

We briefly comment on the relation between the $\vec{\epsilon_{i}}$ and the polarisation vectors $\varepsilon_{\mu}^{(j)}$ for circularly polarised photons [58, 59]. These polarisation vectors transform under $P$ as $\vec{\varepsilon}^{\ (3)}={\vec{k}}/|k|\to-\vec{\varepsilon}^{\ (3)}$ for longitudinally polarised photons and $\vec{\varepsilon}^{\ L}(\vec{k})=\vec{\varepsilon}^{\ R}(-\vec{k})$ for left- and right-handed circularly polarised photons with definite helicity. That is, parity transformations change the sense of rotation about the flipped momentum axis. Of the two transverse direction components, one is parity even, like $\vec{\epsilon_{i}}$, with the other parity odd.

The present most accurate measurements are

[TABLE]

J-PET aims to improve the accuracy here to ${\cal O}(10^{-5})$ for $CP$ and $CPT$ observables [1, 60]. Experiments to improve the accuracy on CP observables are also planned at MIT [2].

Standard Model QED final state interactions can mimic $CP$, $T$ and $CPT$ violation by inducing finite values of the correlations in Eqs. (26, 27) at the level of ${\cal O}(10^{-9})-{\cal O}(10^{-10})$ [18, 16]. $CPT$ symmetry is a fundamental property of relativistic quantum field theories like QED [58]. While QED interactions preserve $CPT$, o-Ps as an unstable state is not an eigenstate of time reversal symmetry $T$ and of $CPT$. (It is an eigenstate of $C$ and $CP$.) This leads to non-vanishing values of the correlations in Eqs.(26) and (27) 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 [18].

The mass scale for weak interference effects in positronium is given by $G_{F}m_{e}^{2}\sim 10^{-11}$, so one needs a dramatic enhancement to obtain an observable effect. Theoretical conditions needed for observation of $CP$ violation in positronium decays are discussed in [18]. The electron EDM measurements strongly constrain any new sources of $CP$ violation coupled to the electron. Any observation of $CP$ violation in positronium decays in the next generation of experiments would point to some cancellation of $CP$ phases in the EDM.

4.3 Visible exotic decays of o-Ps to a photon and

axion

A possible resolution of the strong $CP$ puzzle in QCD is to postulate the existence of a new very-light mass pseudoscalar called the axion [61] which couples through the Lagrangian term

[TABLE]

Here the second and third terms denote coupling to gluons and photons, and the term in $\psi$ denotes possible fermion couplings to the axion $a$ with $f_{\psi}\sim{\cal O}(1)$. The mass scale $M$ plays the role of the axion decay constant and sets the scale for this new physics.

One finds [62, 63]

[TABLE]

with $g_{a}=-2m_{e}f_{e}/M$. Here $\Psi_{n}(0)$ is the wave function at the origin,

[TABLE]

where $n=1$ for the ground state [64]; $m_{e}$, $m_{a}$ and $m_{\hbox{o-Ps}}$ are the electron, axion and positronium masses. Eq. (30) becomes

[TABLE]

Axions are possible dark matter candidates. Constraints from experiments tells us that $M$ must be very large. Laboratory based experiments together with astrophysics and cosmology constraints suggest a favoured QCD axion mass between $1\ \mu$eV and 3 meV [65, 66] corresponding to $M$ between $6\times 10^{9}$ and $6\times 10^{12}$ GeV. Taking $f_{e}\sim{\cal O}(1)$ and dividing by the leading order decay rate term from QED in Eq. (1) gives an expected branching ratio for the decay to a photon and axion between about $10^{-28}$ and $10^{-22}$.

In order to see a signal with branching ratio of ${\cal O}(10^{-8})$ one would need $M$ close to the TeV scale. The axion mass $m_{a}\sim 1/M$. If the axion is too heavy it would lead to axions carrying too much energy out of supernova explosions, thereby observably shortening the neutrino arrival pulse length recorded on Earth in contradiction to Sn 1987a data [65].

4.4 Invisible decays

The search for invisible decays is an interesting topic of experimental investigation. Invisible decays can occur in mirror matter models [67]. Here, the o-Ps can oscillate into a virtual photon which then oscillates into an invisible “mirror photon” and “mirror positronium” (with no interaction with the detector). Mirror matter was first proposed in connection with parity violation. The idea is that under spatial inversion particles should transform into parity reflected new mirror states [68], which then restore parity symmetry in nature. The mirror particles would interact with normal particles mainly through gravity and, as such, are dark matter candidates. Oscillations between photons and their mirror partners could proceed through the interaction term [67]

[TABLE]

The upper limit on the mixing term deduced from successful prediction of the primordial 4He abundance by the Standard Model is [69]

[TABLE]

A value of $\epsilon$ between $10^{-10}$ and $4\times 10^{-9}$ has been suggested in models aiming to explain the DAMA anomaly in dark matter physics [70, 71].

The most accurate constraint on a possible invisible decay of o-Ps in vacuum comes from the ETH Zürich group [19]

[TABLE]

which is interpreted as a constraint on the mixing parameter, $\epsilon\lesssim{\cal O}(10^{-7})$. For experiments in medium one also has to take into account o-Ps collisions within the apparatus which act as a source of decoherence and dilute the accuracy of the measurement of $\epsilon$ [72]. Next generation measurements aim to probe possible branching ratios of ${\cal O}(10^{-8})$ corresponding to mixing with $\epsilon\sim 4\times 10^{-9}$ [3]. These experiments will use a higher positron flux with improved confinement cavity and tagging system.

5 Conclusions

Precision measurements of positronium decays test fundamental symmetries with charged leptons. The large relative error on the experimental decay rates in Eqs. (3, 4) for o-Ps and Eq. (9) for p-Ps compared to the QED theory uncertainties in Eqs. (2, 8) demands improved experimental accuracy. Constraints from other observables involving the QED fine structure constant tell us that any new physics effects will be small. The branching ratios for processes beyond the most simple QED decays of o-Ps and p-Ps most likely start at ${\cal O}(10^{-6})$ with the five photon decay of o-Ps and the four photon decay of p-Ps. QED final state effects can mimic $CP$, $T$ and $CPT$ violation in spin momentum correlations measured in o-Ps decays at the level of ${\cal O}(10^{-9})-{\cal O}(10^{-10})$. This effect is associated with the fact that o-Ps as an unstable state is not an eigenstate of $T$. The range is similar to the upper bound ${\cal O}(10^{-9})$ suggested by measurements of the electron EDM for the branching ratio for o-Ps decays involving possible new $CP$ violating interactions (with the EDM measurement interpreted assuming no special phase cancellation in the EDM). The branching ratio for o-Ps decays to an axion and single photon is expected to be below ${\cal O}(10^{-22})$. For testing models of possible mirror particles the aim is to be sensitive to branching ratios for possible invisible decays at or below ${\cal O}(10^{-8})$.

Acknowledgements

I thank C. Aidala, P. Crivelli, N. Krawczyk, W. Krzemien, P. Moskal, J. Raj and M. Silarski for stimulating discussions. This article is based on a lecture presented at the 3rd Symposium on Positron Emission Tomography, Cracow, September 10-13 2018.