Table 1 presents three angular correlations measurable in the ortho-positronium three-photon annihilations. The correlations are represented as operators whose properties under the C, P and T transformations and their combinations follow from the respective behavior of positronium spin (

$\overrightarrow{S}$) and momentum vectors of the final state photons (

${\overrightarrow{k}}_{i}$ for

$i=1,2,3$ where the photons are labeled according to descending energy, i.e.,

$|{\overrightarrow{k}}_{1}|>|{\overrightarrow{k}}_{2}|>|{\overrightarrow{k}}_{3}|$ ) under these operations. In case of operators which are antisymmetric under a given transformation (marked with “–” in the table), expectation value of the operator must vanish if the respective transformation constitutes a good symmetry. Consequently, observation of a non-zero expectation value of such an operator would be an indication of violation of a given discrete symmetry [

5,

36]. The notion of testing discrete symmetries in the annihilations of ortho-positronium is therefore based on experimental determination of the expectation values of the angular correlation operators listed in

Table 1. Notably, only one experiment conducted to date attempted to probe a continuous distribution of such expectation values [

7] whereas all previous measurements were constrained to determination of an up-down asymmetry of the operators, a special case with significantly limited sensitivity [

6,

42,

43].

Operators 1 and 2 presented in

Table 1 are sensitive to CPT-violating effects if the annihilating positronium spin is known, e.g., due to vector polarization originating from positron polarization inherent to

${\beta}^{+}$ decay. While the third CP-odd operator appears as a simple product of the former two, its definition involves positronium spin twice in an anti-symmetric combination [

5,

44] and is only measurable in presence of more specific tensor polarization, calling for experimental approaches distinct from that available for operators 1 and 2, such as positronium polarization using an external magnetic field. In this work, we focus on the capabilities of the J-PET experiment in conducting precise measurements of the first two of the presented angular correlations, using a spin estimation technique without external magnetic field and probing full geometrically-allowed domains of the correlation operators for the first time.

#### 3.1. Estimation of Positronium Spin

An essential component of the angular correlations in positronium decays considered in this work is the knowledge of the positronium spin quantization axis. Former measurements either used a polarized positronium beam [

43], external magnetic field [

6,

42] or relied on the intrinsic linear polarization of positrons emitted in

${\beta}^{+}$ decay [

7]. The two former approaches exclusively allow for producing a degree of tensor polarization in the positronium sample, inevitable for conducting a test of the CP symmetry with operator no. 3 from

Table 1. However, setups required to convey the beam to the annihilation recording device and magnets providing sufficient

$\overrightarrow{B}$ field effectively prevent recording of the annihilation photons with a large angular acceptance.

Therefore, J-PET builds on the o-Ps polarization control scheme proposed in the best measurement of the

$\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}})$ operator to date [

7], in which longitudinally-polarized positrons from a point-like

${\beta}^{+}$ source of

^{68}Ge or

^{22}Na are allowed to form positronia only in a limited volume which defines a range of allowed

${e}^{+}$ spin quantization axes. As the positron polarization statistically translates to the formed ortho-positronium in 2/3 of cases, this allows for obtaining an estimate of the o-Ps spin direction with a finite uncertainty determined by the

${\beta}^{+}$ emission average energy and the applied geometry of positronium formation medium. In the original implementation of this idea, the latter uncertainty accounted for a reduction of statistical polarization by 0.686, in addition to the average polarization of

P ≈ 0.4 inherent to the method of producing ortho-positronium [

7]. On the other hand, it evaded the need for a acceptance-limiting hardware setup which allowed for the first measurement of a true distribution of an angular correlation in o-Ps annihilation, although limited in resolution by the coarse detector granularity.

In the measurements with J-PET proposed in this work, we extend the idea of estimating ortho-positronium spin without externally-induced polarization. While it limits the accessible symmetry violating operators to positions 1 and 2 from

Table 1 as measurement of the correlation

$(\overrightarrow{S}\xb7\overrightarrow{{k}_{1}})(\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}}))$ is only possible in the case of a tensor-polarized positronium sample [

45], this allows J-PET to observe an unprecedented spectrum of angular configurations of o-Ps decays and thus the full spectra of correlation operators 1 and 2.

To this end, we use positrons emitted from a point-like ${\beta}^{+}$ source which are characterized by linear polarization along their direction of emission to a degree of ${P}_{{e}^{+}}=\upsilon /c$ where $\upsilon $ is the positron velocity and c is the speed of light. The positrons are allowed to thermalize in a layer of porous medium enhancing positronium formation, which is spatially separated from the ${\beta}^{+}$ source by the volume of vacuum chamber ensuring free propagation of the positrons.

In contrast to the previous measurement [

7], we do not assume the positronium production region to be point-like but use the information on the locations and times of the three photons’ interactions in the detector to reconstruct the o-Ps

$\to 3\gamma $ annihilation point with a trilateration-based approach [

46]. In consequence, we can estimate the direction of

${e}^{+}$ spin separately for each event, thus reducing the related decrease in statistical o-Ps polarization from 0.686 to about 0.98 determined by the spatial resolution of the o-Ps

$\to 3\gamma $ reconstruction.

In the currently performed measurements, J-PET implements the aforementioned spin estimation scheme with a cylindrical positronium production chamber mounted axially in the center and extending for the whole length of the detector. A 10 MBq

${\beta}^{+}$ source of

^{22}Na is installed in the center of the chamber, while its walls are coated with 3 mm of R60G porous silica, allowing practically all positrons reaching the chamber walls to thermalize and interact within this layer [

47]. The chamber walls are made of 3 mm polycarbonate so as to minimize absorption and scattering of annihilation photons. The chamber mounted inside the J-PET detector is presented in the left panel of

Figure 1. The right panel of the figure illustrates a future enhancement of the chamber geometry, i.e., replacement of the cylinder with a spherical vacuum chamber (with R = 10 cm), which allows for a more efficient utilization of positrons from the

${\beta}^{+}$ source for positronium formation, increases o-Ps

$\to 3\gamma $ registration efficiency for extreme values of certain correlations and reduces spurious asymmetries as demonstrated in the next sections.

#### 3.2. The Correlation between o-Ps Spin and Annihilation Plane

The 2nd operator from

Table 1 is sensitive to potential violations of CPT invariance and has been previously studied in two experiments with the most precise result of the CPT-violation parameter

${C}_{CPT}$ of

$(2.6\pm 3.1)\times {10}^{-3}$ [

7,

43]. In fact, a similarly-defined triple correlation has been studied in search for T violation in decays using

${Z}^{0}$ spin and momenta of the most energetic produced jets [

48].

As can be seen from

Table 1, the

$\overrightarrow{S}\xb7\overrightarrow{{k}_{1}}$ correlation is also odd under the CPT transformation. The choice of the ostensibly more complicated operator in the previous measurements was motivated by the fact that

$\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}})$ contains a simple correlation between the o-Ps spin and positronium annihilation plane spanned by the momentum vectors of the emitted photons. The definition using the two most energetic photons’ momenta is merely an experimentally-useful convention and does not introduce a significant correlation between detection efficiency and photon energy as is the case for the

$\overrightarrow{S}\xb7\overrightarrow{{k}_{1}}$ operator as argued later on in this work. In order to avoid any dependence on selected photon energies, it is convenient to normalize this operator to the magnitude of the cross product, leading to the following definition:

which expresses the pure angular correlation between o-Ps spin and its decay plane.

The best measurement to date was simultaneously the first measurement going beyond the up-down asymmetry of the operator and determining its continuous distribution. However, due to the geometry of the Gammasphere detector used therein and the positronium production setup, the measurement was only sensitive to the values of this operator in the range of about $(-0.4,0.4)$ out of the allowed region of [−1, 1].

Due to its high granularity of the detection modules in the transverse plane and continuous interaction position measurement in the longitudinal coordinate, in conjunction with the spin estimation scheme which does not impose a distinguished positronium spin quantization axis with respect to the detector, the J-PET setup is able to record a substantially broader range of kinematic configurations of o-Ps$\to 3\gamma $ events.

To demonstrate the sensitivity of J-PET to the distribution of the

${\mathcal{O}}_{CPT}$ operator, a toy Monte Carlo simulation of ortho-positronium annihilations in the experimental setup described above was prepared, featuring allowed angular and energy distributions of photons from o-Ps

$\to 3\gamma $ annihilations expected from QED [

49], the geometry of the positronium production setup as well as geometric arrangement of the detection modules. Compton interactions of the annihilation photons were simulated according to the Klein–Nishina formula and a photon registration threshold was set on the simulated deposition of energy by the scattered photon in a scintillator strip.

The distribution of the

${\mathcal{O}}_{CPT}$ operator, i.e., cosine of the angle between the normal to the annihilation plane and positronium spin direction was simulated either to be uniform as expected in the absence of CPT-violating effects [

5] or with an assumed level of asymmetry quantified by a

${C}_{CPT}$ coefficient. Following the approach used in Ref. [

7], the simplest asymmetric form of the distribution as a function of

$cos\theta $ was used where the total probability distribution contains a term linear in

$cos\theta $ whose contribution given by

${C}_{CPT}\in [0;1]$.

A simulation of

${10}^{13}$ positrons emitted from a

${\beta}^{+}$ source without CPT-violating effects in ortho-positronium annihilations results in a distribution of the

${\mathcal{O}}_{CPT}$ operator presented in the hatched blue histogram presented in

Figure 2. Red histogram in the same figure corresponds to the distribution obtained if maximal violation of the symmetry (

${C}_{CPT}=1$) is assumed. While the existing measurements exclude values of

${C}_{CPT}$ beyond the

${10}^{-3}$ level, the exaggeration in

Figure 2 was used to visualize the effects detectable through determination of the distribution of this angular correlation. It is visible that while detection efficiency peaks for values corresponding to the decay plane normal being close to perpendicular to the spin quantization axis, it does not drop to zero close to the extreme values of the correlation, in contrast to the previous measurement [

7].

The factors determining the detection efficiency of o-Ps$\to 3\gamma $ events in J-PET comprise (i) probability of interaction of an annihilation photon in a plastic scintillator strip which on average amounts to about 20%, (ii) geometrical acceptance resulting from the sparse arrangement of detection modules and their length; modules cover about 0.21 of the solid angle around the center of the detector, (iii) the energy deposition threshold above which a Compton-scattered photon is registered by a detection module. Furthermore, the total efficiency of observing ortho-positronium annihilations as a function of the ${\beta}^{+}$ source activity also involves (iv) the fraction of positrons forming o-Ps in the region of the annihilation chamber where the three emitted photons can be recorded simultaneously.

Figure 3a presents the total o-Ps

$\to 3\gamma $ registration efficiency as a function of the angular correlation defined in Equation (

1) obtained in toy MC simulation of

${10}^{13}$ positrons from a

^{22}Na source in the setup described in

Section 3.1. The efficiency is presented for two geometries of the annihilation chamber: cylindrical (presently used) and spherical (in preparation). In each case, three values of the energy deposition threshold for a single annihilation photon were considered: 40 keV, 100 keV and 140 keV. Results of the simulation show that lowering this photon registration threshold is vital for the total efficiency and each increase of the threshold by about 50 keV results in a reduction of the

$3\gamma $ registration efficiency by an order of magnitude.

Presently, the detection threshold of the J-PET photomultiplier tubes and signal sampling electronics achievable without entering the noise level is estimated to be about 80 keV.

The MC-based evaluation of detection efficiency also demonstrates the enhancement expected with the spherically-shaped positronium production vacuum chamber instead of the currently used cylindrical one. The drop of efficiency close to the extreme values of

${\mathcal{O}}_{CPT}$ visible in

Figure 2 will be substantially reduced with the new source geometry, resulting in an efficiency more flat across the whole operator spectrum.

A subtle asymmetry in the distribution of a given angular correlation X may be detected by evaluation of the following figure, accounting for asymmetry between event counts

N in subsequent intervals of for positive and negative values of a given angular correlation operator

${\mathcal{O}}_{X}$:

Subsequently, a comparison of the

$A(|{\mathcal{O}}_{CPT}|)$ distribution with one obtained in case of the MC-simulated distribution assuming maximal violation (

${C}_{CPT}$ = 1) would allow for extraction of the CPT symmetry violation coefficient in a similar manner as done e.g., in Ref. [

7]. For such a procedure, good understanding of the detector efficiency as a function of the value of the measured operator is crucial in order to avoid artificial asymmetries arising from efficiency nonuniformities due to the setup geometry. In J-PET, thanks to the large granularity of detection modules and continuous measurement of interaction positions along them, the expected shape of such efficiency is smooth as demonstrated in the left panel of

Figure 3. While this is already an improvement with respect to the previous measurement of the

${\mathcal{O}}_{CPT}$ operator where coarse granularity of the detectors constituting the Gammasphere array caused strong periodic fluctuations of efficiency [

7], the impact of detector geometry on the measured asymmetry requires a careful treatment nonetheless.

Figure 3b shows examples of asymmetries of the

${\mathcal{O}}_{CPT}$ operator defined as in Equation (

2) using the two positronium production chamber geometries considered in this work, for the cases of no asymmetry assumed in the MC simulations and of CPT violation at a level of 10% (

${C}_{CPT}=0.1$), exaggerated for better visibility. These results were obtained with a simulation of

${10}^{13}$ positrons from a

^{22}Na source. It is visible that the

$\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}})$ angular correlation is not sensitive to the geometry of the positronium annihilation region. Not only are the asymmetries detected using the cylindrical and spherical chambers in good agreement, but also the

$A({\mathcal{O}}_{CPT})$ distribution obtained in absence of simulated CPT violation does not reveal signs of a false asymmetry in any of the cases.

These simulations confirm the robustness of the $\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}})$ angular correlation as an observable of discrete symmetry tests. While potentially sensitive to genuine effects of CPT violation, its definition allows to cancel out many geometrical effects related to the measurement setup. For this reason, this correlation has been favored over the ostensibly simpler operator $\overrightarrow{S}\xb7\overrightarrow{{k}_{1}}$ in the past measurements, even though all of the previous experiments focusing on $\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}})$ were in principle capable of studying the $\overrightarrow{S}\xb7\overrightarrow{{k}_{1}}$ correlation as well. Later on, we discuss the experimental differences between these two correlations.

#### 3.3. The Correlation between o-Ps Spin and Most Energetic Photon

As discussed in the previous Section, out of the two angular correlations sensitive to discrete symmetries’ violation in absence of ortho-positronium tensor polarization (operators 1 and 2 in

Table 1), the operator

$\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}})$ has already been studied in several experiments. On the contrary, the 1st operator which is a simple projection of the most energetic photon momentum onto the direction of spin of the decaying ortho-positronium atom has never been measured to date despite its sensitivity to both CP and CPT-violating effects.

The reason lies in its simpler construction which makes its distribution prone to spurious effects and thus experimentally more challenging. While its usage as an observable of a CP and CPT test requires strict control of the impact of the measurement setup geometry on the observed asymmetry, here we argue that smooth efficiency curves offered by the J-PET detector in conjunction with detailed MC simulations may allow for the first measurement of the $\overrightarrow{S}\xb7\overrightarrow{{k}_{1}}$ operator.

Similarly as in Equation (

1), it is convenient to introduce normalization of the photon momentum into the definition of the angular correlation operator:

Figure 4a presents the efficiency of J-PET to o-Ps

$\to 3\gamma $ events with a given value of

${\mathcal{O}}_{CPT}^{\prime}$ evaluated with the toy MC simulation in a similar manner as described in

Section 3.2. A comparison with

Figure 3a immediately reveals the challenge posed by usage of this operator. In this case, the efficiency curves contain a modulation which is not symmetric as a function of

${\mathcal{O}}_{CPT}^{\prime}$. Moreover, this effect is magnified with increasing value of the energy deposition threshold for

$\gamma $ detection. This energy dependence originates from the choice of the most energetic photon which introduces a correlation with the kinematical configuration of a given o-Ps

$\to 3\gamma $ decay. This phenomenon is absent in case of

$\overrightarrow{S}\xb7(\overrightarrow{{k}_{1}}\times \overrightarrow{{k}_{2}})$ because the geometrical entity used therein (orientation of the decay plane) is agnostic of the kinematics of a particular annihilation event and thus also of the energy-based choice of photons.

Successful use of the ${\mathcal{O}}_{CPT}^{\prime}$ operator as a probe of CP and CPT violation therefore requires two factors: (i) maintaining the energy deposition threshold as low as possible, (ii) reducing the spurious asymmetries originating from asymmetric and energy-dependent efficiency to a low and well-understood level.

The latter can be achieved by manipulation of the geometry of positronium production medium. As displayed in the right panel of

Figure 4, although with both simulated setups the

$\overrightarrow{S}\xb7\widehat{{k}_{1}}$ a significant asymmetry appears even in case of no CPT violation assumed in the simulations (where possible violation is introduced in a similar manner as described in

Section 3.2), usage of the spherical vacuum chamber results in a simpler dependence of the false asymmetry on the value of

${\mathcal{O}}_{CPT}^{\prime}$ which is easier to parameterize. Additionally, two independent measurements with different chambers would allow for discrimination between the setup-specific false asymmetry and a possible genuine effect as well as for extraction of the latter. It is important to stress that while the results presented in this work are based on a toy MC simulation, the actual experiments will be augmented with simulations of the full setup based on the Geant4 package, which are currently being commissioned.