\section{Fundamental Symmetries}\label{sec:funsym}% Gerald
%

\subsection{Overview}

Low energy tests of fundamental symmetries in nuclei and atoms have traditionally played an important role in the search for `new’ physics beyond the Standard Model (SM). The field is more active than ever, particularly in Canada, where it is represented by the working group ``Fundamental Symmetries'' within the Canadian Institute of Nuclear Physics (CINP).
The study of symmetries in subatomic physics is of key importance for two reasons. On one hand, the fundamental forces and conservation laws of nature are intimately linked to corresponding symmetries; the investigation of those symmetries and their violations give unique insights. In addition, from a practical point of view, symmetries can be exploited to single out vanishingly small signatures of new physics in the presence of much larger `conventional’ interactions, giving low-energy experiments a physics reach to energy scales orders of magnitude higher, and keeping them competitive with direct searches conducted at colliders. As an example, at the $Z$-resonance, the neutral current weak interaction dominates over electromagnetism as real $Z$ bosons are readily observed in $e^+ - e^-$ collisions; in an ordinary atom with binding energies on the order of electron volts, the $Z$-boson exchange amplitude between electrons and quarks is 12 or more orders of magnitude suppressed relative to the prevalent electromagnetic photon exchange, yet with help of the violation of the parity symmetry in the $Z$-exchange, this amplitude has been measured to 0.3\%, providing an important test of electroweak physics.

Our current understanding of the fundamental interactions and symmetries is reflected in the Standard Model; constructed half a century ago, it is still in agreement with experimental findings. It is a quantum field theory founded on the assumptions Lorentz symmetry and invariance under the combined transformation of charge (C), parity (P), and time reversal (T), or CPT. Since the 1950’s and 60’s we know that C, P, CP, and T are violated separately. However, the Standard Model contains an uncomfortably large number of free parameters, and while the P and CP symmetry violations have been successfully incorporated, the Standard Model cannot explain their origin. In addition, no link exists between the Standard Model and gravity. 

Theories such as quantum gravity and string theory are pursued intensely as a unifying approach valid up to the Planck scale. While they yield the Standard Model and General Relativity in the low-energy regime, they frequently assume Lorentz and CPT violation. High precision, low-energy fundamental symmetry-type experiments in nuclei and atoms can probe for very faint remnants of these symmetry violations occurring at energies far beyond the current frontier of direct searches. 

In Canada, there is currently a strong community of researchers working on fundamental symmetry tests. The work covers many of the hot topics and also has a remarkable breadth in the experimental approaches, from electron scattering experiments at the 10 GeV level to beta decay in laser traps using atoms at neV temperatures, a span of 19 orders of magnitude in energy!



\subsection{The Canadian program}


\subsubsection{Time reversal and CP violation: Permanent electric dipole moments and related searches}
The CP violation in the Standard Model is 10 orders of magnitude smaller than
needed to generate the observed baryon asymmetry of the universe in the method
outlined by Sakharov. The observation of a non-vanishing permanent electric
dipole moment (EDM) in an atom, nucleus, or the neutron would directly violate time reversal symmetry (and also CP, assuming CPT symmetry holds), independent of any need for radiative corrections or theoretical interpretation. EDM searches are among the hottest topics for physics beyond the Standard Model worldwide and have made great progress over the years in ruling out theories beyond the SM, as illustrated in Figure~\ref{fig:fs-edm-scale1}.

Several Canadian groups are active in this field at the confluence of atomic, nuclear, and particle physics. Experiments are under development or in the planning stage at TRIUMF, exploiting a unique mix of capabilities: The cyclotron is the backbone of the high-density, ultra-cold neutron facility, enabling a competitive neutron EDM search. It is the driver for the actinium targets at ISAC, which produce record quantities of radioactive, heavy elements such as Rn, Fr, and Ra which are of particular interest for fundamental symmetries studies due to the large enhancement of T/CP (and also P) violating interactions. A possible search for an electron EDM in Fr and T-violating triple-correlation measurements in $^{45}$K (discussed in~\ref{subsec:betaneutrino}) and in radiative $\beta$ decay harness the world-leading radioactive neutral atom trap infrastructure at ISAC. ARIEL rounds out the synergies by enabling long, dedicated campaigns with actinide target beams.
%
\begin{figure}[ht]
	\begin{center}
		%\vskip 0.5cm
			\includegraphics[angle=0,width=0.7\textwidth]{./Figs/fs-edm-scale1.pdf}
	\end{center}
	\caption{Experimental limits on permanent electric dipole moments and a rather incomplete and imprecise selection of predicted ranges for various theories. Initial version inspired by~\href{https://doi.org/10.1016/S0168-9002(99)01023-2}{J.M Pendlebury and E.A. Hinds, Nucl. Meth. Instr. A 440, 471 (2000)}. For up-to-date, detailed information, see the review by \href{https://doi.org/10.1103/RevModPhys.91.015001} {T.E. Chupp {\em et al.} Rev. Mod. Phys. 91, 015001 (2019)}.
		\label{fig:fs-edm-scale1}
	}
\end{figure} 


\paragraph{TUCAN: towards an improved measurement of the neutron's permanent electric dipole moment}

(TRIUMF) TRIUMF, Winnipeg, Manitoba, SFU, UNBC; {\intl Japan, USA}

The TUCAN (TRIUMF Ultra-Cold Advanced Neutron) collaboration is currently building an advanced, spallation-driven superfluid helium (He-II)  source of ultra-cold neutrons at TRIUMF. The first physics goal is to carry out a measurement of the neutron electric dipole moment, with an envisioned improvement of about an order of magnitude ($\delta d_n \approx 1\times 10^{-27}$ e-cm), placing tighter constraints  on CP violation in the hadronic sector.  At this level, numerous theories beyond the Standard Model can be critically tested. Measurements of the neutron EDM are complementary in their sensitivity to new physics, compared to measurements of EDMs in other systems such as atoms and molecules~\footnote{\href{https://doi.org/10.1103/RevModPhys.91.015001} {T.E. Chupp {\em et al.} Rev. Mod. Phys. 91, 015001 (2019).}}, and have strong impact on the SUSY CP problem, baryogenesis scenarios, and CP violation in the strong sector.
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=0.85\columnwidth]{./Figs/fs-tucan-source.jpg}
		%\hspace*{0.2in}
	\end{center}
	\caption{TUCAN: Cross section of the future UCN source scheduled to be installed in 2021-22.
	}
	\label{fig:fs-tucan-source}
	\vspace*{-0.0in}
\end{figure*}
%

Over the past years, a beamline and a UCN source were successfully established and tested~\footnote{\href{https://doi.org/10.1103/PhysRevC.99.025503}{S. Ahmed {\em et al.}, Phys. Rev. C 99, 025503 (2019). }}.
A few of the research highlights are the first UCN production at TRIUMF, and the characterization of UCN production and UCN interactions with superfluid helium~\footnote{Ibid.}.
%~\footnote{\href{https://doi.org/10.1103/PhysRevC.99.025503}{S. Ahmed {\em et al.}, Phys. Rev. C 99, 025503 (2019). }.  
%	
A new fast kicker magnet was designed, built, and commissioned~\footnote{\href{https://doi.org/10.1103/PhysRevAccelBeams.22.102401}{S. Ahmed {\em et al.}, Phys. Rev. Accel. Beams 22, 102401 (2019).}} which feeds a new proton beamline and spallation target~\footnote{\href{https://doi.org/10.1016/j.nima.2019.01.074}{S Ahmed {\em et al.}, Nucl. Instrum. Meth. A 927, 108 (2019).}}. The facility was operated routinely from 2017-2019 and used for R\&D related to the UCN source upgrade and EDM experiment.
%Over the past years, a beamline and a UCN source were successfully established and tested~\footnote{\href{https://doi.org/10.1103/PhysRevC.99.025503}{S. Ahmed {\em et al.}, Phys. Rev. C 99, 025503 (2019). }}. 
%
A CFI-funded source upgrade is now under way, with a new He-II cryostat provided by the collaborators from Japan, currently being tested at KEK. Co-magnetometry, magnetic shielding and detectors are also being developed.

In 2020-21, the new horizontal He-II cryostat will be shipped from Japan and installed and tested at TRIUMF (see Figure~\ref{fig:fs-tucan-source}). In 2022, first operation for UCN production will be conducted. The magnetically shielded room which will house the EDM experiment will be installed and characterized with precision magnetometers. In 2023, the EDM experiment will be assembled and commissioned, with running occurring for the next two to three years thereafter.

The EDM experiment will run into the 2027--36 period, and data analysis will be ongoing throughout this period. The data will be blinded and analyzed by multiple groups. Upgrades to the EDM experiment could be considered in this period, and will depend on what is learned during the years of operation prior to this point. The UCN source is designed to be a two-user facility, and other experiments are considered that may use the second experimental port, such as a neutron lifetime experiment ($V_{ud}$, neutron lifetime puzzle), and a neutron gravitational levels experiment (extra dimensions, chameleon fields, short-distance modifications to gravity). It is anticipated to develop TUCAN into a user facility where outside groups would submit experimental proposals.



\paragraph{Radioactive molecules as a sensitive probe for permanent electric dipole moments
\label{sec:funsym:rams}}

(TRIUMF) TRIUMF, Toronto, UBC; {\intl USA, Belgium, Germany, Netherlands, UK}

In certain molecules, parity­ and time-­reversal violation effects can be greatly enhanced compared to atomic systems~\footnote{\href{https://doi.org/10.1103/RevModPhys.90.025008}{M. Safronova {\em et al.}, Rev. Mod. Phys. 90, 025008 (2018).}}. As these symmetry-­violating effects scale with the atomic number, nuclear spin and nuclear deformation, measurements of molecular isotopologues containing heavy radioactive nuclei are predicted to provide unique and highly sensitive laboratories in these studies. However, experimental measurements of such radioactive molecules are scarce, and their study requires overcoming major experimental challenges. This research program at TRIUMF focuses on precision studies of radioactive molecules, which will offer new opportunities for the study of the nuclear electroweak structure and the violation of fundamental symmetries. The proximity of molecular states of different parity can enhance $P$ and $T$ violating sensitivity by orders of magnitude when compared to atomic systems. Moreover, certain radioactive molecules can contain heavy nuclei with octupole deformation, providing further sensitivity enhancements to explore $P$ and $T$ violation~\footnote{\href{https://doi.org/10.1103/PhysRevC.99.035501}{V.V. Flambaum {\em et al.}, Phys. Rev. C 99 035501 (2019).}}. The proposed program at ISAC/TRIUMF targets permanent electric dipole moments in systems with sensitivity to the electron's EDM and $P$,$T$-violating nuclear Schiff moments, as well as parity violation.

Radioactive ions will be produced at ISAC/ARIEL and sent into a cryogenic buffergas cell where molecules can be formed by interactions with the gas (see Fig.~\ref{fig:fs-radmol}). The molecules will be extracted, selected and transferred to a cryogenic radio­frequency quadrupole trap, where the internal and translational temperature of the molecular ions will be reduced. Ions will be released with a kinetic energy of a few eV, neutralized, and decelerated down to sub-­eV. Cold and slow molecular beams will be transported through a optical pumping region, where they can be transferred to a preferred molecular state.
Subsequently, the molecular beam will interact with electric and magnetic fields. Finally, the population of a particular molecular state can be studied for different configurations of the external fields, providing information on a specific symmetry-­violating effect.
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=0.85\columnwidth]{./Figs/fs-radmol.pdf}
		%\hspace*{0.2in}
		%\includegraphics[angle=0,width=0.35\columnwidth]{./Figs/fs-alpha-alphag-apparatus.jpg}
	\end{center}
	\caption{Schematic of a setup for precision spectroscopy with cold radioactive molecules (note that {\em RaB} stands for {\em radioactive barium} and {\em RaM} for {\em radioactive molecule}, e.g. RaF or RaO).
	}
	\label{fig:fs-radmol}
	\vspace*{-0.0in}
\end{figure*}
%

In a first stage, the formation and properties of RaF will be studied~\footnote{\href{https://doi.org/10.1038/s41586-020-2299-4}{R.F. Garcia Ruiz {\em et al.}, Nature 581, 396 (2020).}}, followed by precision spectroscopy (at the MHz level) of the hyperfine structure. Next would be optical pumping and radiofrequency spectroscopy with kHz resolution, and the production of slow, cold RaF beams. Finally, spectroscopy at the Hz level, in the presence of electric and magnetic fields, has to be demonstrated, to measure parity violating effects in a molecular beam. Subsequently, a magneto-optical trap for RaF would be attempted. Post 2025, such a setup could be used to carry about competitive EDM searches, with increasing precision over 3 to 5 years. Collaborators at MIT, Caltech, Edinburgh, and TRIUMF have started on developing the required tools and techniques.


Similarly, the broader AMO community has established techniques to construct trapped cold diatomic molecules from separate laser-cooled atoms, possibly providing very long coherence times for EDM experiments.
There is ongoing consideration in the community to form diatomic molecules
with octupole-deformed laser-cooled $^{223,225}$Ra and $^{223}$Fr as the
heavy atom. Since low-$Z$ alkali atom partners do not produce dimers with useful sensitivity to EDM effects, the challenge is to develop useful atomic partners which are inherently more difficult to laser cool.


% end of RadMol


\paragraph{Searching for a permanent electric dipole moment of the electron with an atomic fountain: FrEDM}

(TRIUMF) TRIUMF, Manitoba; {\intl USA}

The goal of the FrEDM effort is to competitively constrain a permanent electric dipole moment (EDM) of the electron, using a laser-cooled francium atom fountain supplied by an intense beam of $^{211}$Fr from TRIUMF. Such EDM's break time-reversal symmetry, a key component of models to explain the excess of matter over antimatter. This experiment would be complementary to a variety of EDM experiments in the neutron and in atoms with or without spin, along with a number of particle physics experiments. High-$Z$ francium atoms feature a high sensitivity to an EDM, low sensitivity to EDM mimicking systematic effects, and extraction of the subatomic time-reversal violating (TRV) physics to high accuracy. The experiment will be developed at LBL using stable Cs as a lower-EDM sensitivity and higher-systematic sensitivity surrogate. Following the Cs prototype, a fully developed Fr experiment can make efficient use of TRIUMF time, space and resources. The aim is a sensitivity to an electron EDM at $\approx 1 \times 10^{-29}$ e-cm, similar to the sensitivity published by ACME's ThO measurement, which appears attainable
~\footnote{\href{https://doi.org/10.1038/s41586-018-0599-8}{N.~Hutzler {\em et al.}, Nature 562, 359 (2018).}}. Complementarity with particle physics results make the case for improvements particularly compelling. Among experiments that search for an electron EDM, experiments using alkali atoms and thallium are the gold standard, because their sensitivity to the electron EDM has been calculated using field theory (as has the sensitivity of the neutron to quark EDMs). That puts these quantities on the same theoretical footing as the calculations that underlie the Standard Model.

A francium version of the experiment could run and achieve design sensitivity by 2026. On the 2027-2036 horizon, with motivation depending on other advancements in the field, considerable effort would be needed to improve certain limiting systematics (e.g. Johnson noise in the mu-metal shield) for the francium atom EDM. Furthermore, this effort could lead over into one using laser-cooled molecules, as briefly discussed in the previous section.


\subsubsection{Neutral current weak interactions}

The violation of parity symmetry provides for an extremely sensitive means to study the neutral current weak interaction, which is generally masked by the dominating electromagnetic processes. P-violating physics beyond the Standard Model at the TeV scale can be observed in low-energy experiments, keeping this field competitive in the LHC era (see Figure~\ref{fig:fs-pvintro}). For example, when new states are discovered at the LHC, it will be important to know their couplings to the first generation of particles. Electrons and muons can be distinguished in the LHC detectors, but up/down quark jets cannot be separated from jets of other generations. 
Low-energy experiments are in a unique position to assist with this question. There are three types of such “low-energy” weak neutral current measurements with complementary sensitivity.
%
\begin{figure*}[ht]
	\vspace{0.1in}
	\begin{center}
		\includegraphics[angle=0,width=1.00\textwidth]{./Figs/fs-pvintro2.pdf}
	\end{center}
	\vspace{-0.10in}
	\caption{Left: Past and future measurements of the Weinberg angle. Data points for future experiments are arbitrarily centered on the Standard Model prediction (solid black line). Figure adapted from \href{https://doi.org/10.1140/epja/i2016-16268-9}{A.~Accardi {\em et al.}, Eur. Phys. J. A 52, 268 (2016)}, Chiral Belle from \href{https://doi.org/10.22323/1.367.0109}{J.M.~Roney,  PoS LeptonPhoton2019, 109 (2019)}. Scenarios with dark {\em Z} bosons for $m_{Z_d}$ of (a) 50 MeV, (b,c) 15 GeV (where area (c) is in tension with existing constraints) from \href{http://dx.doi.org/10.1103/PhysRevD.92.055005}{H.~Davoudiasl {\em et al.}, Phys. Rev. D 89, 095006 (2014)} and \href{http://dx.doi.org/10.1103/PhysRevD.92.055005}{H.~Davoudiasl {\em et al.}, Phys. Rev. D 92, 055005 (2015)}.
		Right: Constraints of parity-violating electron-quark couplings provided by Qweak and atomic parity violation in cesium. The red square denotes the SM prediction. This plot by \href{https://doi.org/10.1038/s41586-018-0096-0}{D. Androi\'c {\em et al.}, Nature 557, 207 (2018)} highlights the complementarity of these measurements.
	}
	\label{fig:fs-pvintro}
	\vspace*{-0.0in}
\end{figure*}
%

Electron-electron scattering (MOLLER) is measuring the electron's weak charge, electron-proton scattering (Qweak) determines the proton's weak charge and atomic parity violation (FrPNC) is predominantly sensitive to the neutron's weak charge. Different types of new physics contribute differently to each of them. For example, the atomic weak charge is relatively insensitive to one-loop order corrections from all SUSY particles; Moeller scattering is purely leptonic and has no sensitivity to leptoquarks. The Canadian community has had a long, prominent involvement in parity-violating scattering experiments and is playing a major role in the Qweak and MOLLER collaborations at Jefferson Lab. In addition, the FrPNC collaboration aims to measure atomic parity violation in laser-trapped francium at TRIUMF, ultimately hoping to improve on the Cs result. The Canadian involvement in the PREX/CREX experiments (discussed in the astrophysics section) nicely ties into this work by providing critical input to the francium experiment via the determination of nuclear neutron radii. As shown in Fig.~\ref{fig:fs-pvintro}, the upcoming electron-ion collider EIC (discussed in the section on hadronic physics) will provide important data in the region between low-energy experiments and the $Z$-pole.


 Theoretical work on one-loop and two-loop electroweak radiative corrections of critical importance to the parity experiments is carried out by groups at Memorial and Manitoba, presented in detail in Sections~\ref{sec:precisionstudies} and \ref{sec:synergy}.

\paragraph{Parity violating electron scattering: MOLLER
\label{sec:funsym:moller}}

(JLab); Manitoba, Memorial, TRIUMF, UNBC, Winnipeg; {\intl France, Germany, Italy, Mexico, USA}.

Within the Standard Model, the combined electro-weak interaction strength of electrons is parameterized by the so-called electron weak charge, which is itself related to the weak mixing angle, a fundamental parameter of the Standard Model that sets the degree of mixing between the electromagnetic and weak interactions.
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=0.55\columnwidth]{./Figs/fs-pvasym2.pdf}
	\end{center}
	\caption{Parity violating electron scattering, experimental precision $\delta(A_{PV})$ shown versus the parity violating asymmetry $A_{PV}$ (\href{https://doi.org/10.1007/s11467-015-0482-0}{P. Souder and K.D. Paschke, Frontiers of Physics (Beijing) 11  no.1, 111301 (2016)}). Open circles are either not yet published or future experiments.  Efforts discussed in this report are shaded red.
	}
	\label{fig:fs-pvasym}
	\vspace*{-0.0in}
\end{figure*}


While design efforts for MOLLER have been ramping up over the past years, the predecessor Qweak, measuring the weak charge of the proton, saw data analysis finalized and the publication of the results~\footnote{\href{https://doi.org/10.1038/s41586-018-0096-0}{D. Androi\'c {\em et al.}, Nature 557, 207 (2018).}}. The weak charge of the proton was determined as $0.0719 \pm 0.0045$, derived from the parity-violating asymmetry in the scattering of polarized electrons on protons, measured at the 9 ppb level (see Fig.~\ref{fig:fs-qweak-extrap}). The result is in excellent agreement with the SM prediction, and sets multi-TeV-scale constraints on any semi-leptonic parity-violating physics not described within the SM.
%
\begin{figure*}[t]
	\begin{center}
		\includegraphics[angle=0,width=0.85\columnwidth]{./Figs/fs-moller-apparatus-3.jpg}
	\end{center}
	\caption{Layout of the MOLLER experiment in Hall A at Jefferson Lab: The beam enters from the
		right, traveling toward the left. The primary Canadian contribution are the CFI proposed 
		main integrating detectors, shown on the left end.
	}
	\label{fig:fs-moller-app}
	\vspace*{-0.0in}
\end{figure*}

The MOLLER experiment aims to make the world's most precise off-resonance measurement of the weak mixing angle, using polarized electron-electron scattering at Jefferson Laboratory to observe the parity violating asymmetry at the percent level (see Figure~\ref{fig:fs-pvasym}). The experiment will test the interaction of electrons with respect to a number of new physics models and will search for electron substructure at the $10^{-21}$ meter scale. The electron-electron interaction is the cleanest process that exists among fundamental particles, uncontaminated by nuclear interactions and is experimentally extremely well controlled, and therefore a prime candidate to search for new physics. MOLLER is an international effort, presently involving more than 120 collaborators from the USA and Canada (the lead countries), as well as Germany, Italy, France, and Mexico. The current Canadian group includes about 20 people, which makes it the second strongest contiguous group in the project, behind Jefferson Lab. MOLLER employs elastic scattering using the 11 GeV electron beam at Jefferson Lab, and it requires a large, highly efficient, and precise array of electron detectors. The Standard Model prediction for parity violating asymmetry $A_{PV}$, for the proposed experimental design, is $\approx 33 \times 10^{-9}$ and the 2.4 \% relative error goal corresponds to an overall precision of 0.7 ppb on the asymmetry measurement, which in turn corresponds to a $\approx 0.1$ \% determination of the weak mixing angle.

The Canadian team is leading the design of the magnetic spectrometer and the integrating detectors plus associated electronics (shown in Figure~\ref{fig:fs-moller-app}), and further established leadership roles in simulation and analysis software. Design completion and prototype testing will finalize in 2021, and construction is scheduled to begin in 2022, with commissioning around 2025. MOLLER will take data until 2030 with a few years of data analysis to follow.


The group is also involved in the P2 experiment at the MESA facility in Mainz, Germany. The synergies with MOLLER in terms of detector electronics are substantial, and the timeline is expected to be comparable to MOLLER.


\paragraph{Atomic parity violation in laser-trapped francium at TRIUMF}

(TRIUMF) Manitoba, TRIUMF; {\intl Mexcio, USA}

In atoms, extremely weak electric dipole transitions between states of the same parity are induced by the parity-violating exchange of Z-bosons between the electrons and the quarks in the nucleus, an effect known as atomic parity violation (APV). Measuring this amplitude, one can study neutral-current weak interactions with atomic physics methods and search for `new' physics such as extra gauge bosons and leptoquarks. APV is strongly enhanced in heavy atoms (18 times larger in Fr compared to Cs), but is presently only sufficiently calculable in alkalis. While this makes Fr an obvious choice, its lack of a stable isotope requires work at a radioactive beam facility such as ISAC. There, the FrPNC collaboration has established a laser trap facility that can capture $\approx 10^6$ atoms of a single Fr isotope in a volume of $\approx 1$ mm$^3$ at $\mu$K temperatures, an ideal environment for precision laser spectroscopy towards APV measurements.
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=1.00 \columnwidth]{./Figs/fs-frpnc.pdf}
		%\hspace*{0.2in}
		%\includegraphics[angle=0,width=0.35\columnwidth]{./Figs/fs-alpha-alphag-apparatus.jpg}
	\end{center}
	\caption{Left: Online francium atom trap for spectroscopy of highly forbidden, Stark-induced transitions. Right: Observation of $7s - 8s$ vector ($\beta$) Stark-induced transition.
	}
	\label{fig:fs-frpnc}
	\vspace*{-0.0in}
\end{figure*}
%

Following initial commissioning work~\footnote{\href{http://dx.doi.org/10.1103/PhysRevLett.115.042501}{J.~Zhang {\em et al.},Phys. Rev. Lett. 115, 042501 (2015).}}, focus has shifted on the observation of the highly forbidden $7s - 8s$ transition on which an APV measurement could be based. It was first excited via two-photon spectroscopy
~\footnote{\href{https://doi.org/10.1103/PhysRevA.97.042507}{M.~Kalita {\em et al.}, Phys. Rev. A 97, 042507 (2018).}}, and recently observed as a Stark-induced single photon transition with oscillator strength of $\approx 10^{-10}$ (see Figure~\ref{fig:fs-frpnc}). In the coming years, a program of spectroscopy of increasingly faint transition amplitudes will be carried out, in particular characterization of the relativistically induced M1, with the ultimate goal of observing the interference between the parity violating amplitude with the much larger, Stark-induced E1. By mid-decade, when ISAC/ARIEL will provide up to 3 beams simultaneously, statistics runs could take place for a competitive APV measurement. On the longer horizon, 2027-36, the Fr laser trap facility could merge into efforts with ultra-cold molecules, photo-associated from laser-cooled atoms, to which Fr would contribute its high sensitivity to new physics, wide range of isotopes, including deformed ones, and structural simplicity (see outlook at end of this section).


\paragraph{Fundamental symmetries at the EIC} (Brookhaven) Manitoba, Regina; {\intl USA and others}

In addition to the hadronic physics program, the EIC presents opportunities in the area of fundamental symmetries and neutral current weak interactions. To maintain the flow of the description of the EIC program, the parity violation and lepton flavor violation program and their discovery potential are described in \ref{sec:eic3}.


\subsubsection{CPT, Lorentz and weak equivalence principle violation}

CPT invariance and Lorentz symmetry are at the very foundation of our current description of nature, as quantum field theories are firmly based upon these principles. However, in string theory and also in quantum gravity, which unifies the standard model of particle physics with general relativity in a ‘theory of everything’, CPT and Lorentz violation are frequently assumed. Furthermore, the weak equivalence principle --- a fundamental assumption in Einstein’s general relativity --- may be violated at a minute level in a fundamental theory. As low-energy experiments are, relatively speaking, not that much further away from the Planck scale as are colliders, the former play a very significant role in this field, driven by the extreme precision that can be reached mostly with laser and microwave-based measurement techniques. Canada is strongly involved in the high-profile endeavor of anti-hydrogen trapping and antimatter spectroscopy, and it provides the largest contingent by country in the ALPHA collaboration working at CERN. 
%The competing ATRAP effort also has Canadian members, which are funded outside of the subatomic envelope. 

\paragraph{ALPHA: Testing fundamental symmetries via antihydrogen-hydrogen comparison} --- (CERN); Calgary, SFU, TRIUMF, UBC, York; {\intl Brazil, Denmark, Israel,
Sweden, UK, USA, Switzerland}

The ultimate goal of antihydrogen studies is to test fundamental symmetries between matter and antimatter with the highest possible precision. Precision tests of CPT and the Weak Equivalence Principle (WEP) will confront the foundations of modern physics -- quantum field theory and general relativity. The ALPHA experiment located at CERN has played the leading role in this endeavour over the past 15 years. The current configuration of the apparatus is shown in Fig.~\ref{fig:fs-alpha-app}.
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=0.95\columnwidth]{./Figs/fs-alpha-app.pdf}
		%\hspace*{0.2in}
		%\includegraphics[angle=0,width=0.35\columnwidth]{./Figs/fs-alpha-alphag-apparatus.jpg}
	\end{center}
	\caption{The ALPHA-2 trap for precision laser spectroscopy (left) and the vertical ALPHA-g trap for gravity and microwave studies (right) at the CERN AD facility.
	}
	\label{fig:fs-alpha-app}
	\vspace*{-0.0in}
\end{figure*}
%
ALPHA is now firmly into the precision measurement stage using the ALPHA-II atom trap (left part of Fig.~\ref{fig:fs-alpha-app}), and over the past five years has delivered a range of firsts: Accumulation of over one thousand antihydrogen atoms for a single run~\footnote{\href{https://doi.org/10.1038/s41467-017-00760-9}{M.~Ahmadi {\em et al.}, Nat. Comm. 8, 681 (2017).}}; test of charge neutrality of antihydrogen at the $10^{-9}$ level~\footnote{\href{https://doi.org/10.1038/nature16491}{M.~Ahmadi {\em et al.}, Nature 529, 373 (2016).}}; determination of the hyperfine splitting~\footnote{\href{https://doi.org/10.1038/nature23446}{M.~Ahmadi {\em et al.}, Nature 548, 66 (2017).}}; laser spectroscopy~\footnote{\href{https://doi.org/10.1038/nature21040}{M.~Ahmadi {\em et al.}, Nature 541, 506 (2017).}} and measurement of the $1S - 2S$ interval with $10^{-12}$ level precision~\footnote{\href{https://doi.org/10.1038/s41586-018-0017-2}{M.~Ahmadi {\em et al.}, Nature 557, 71 (2018).}}; observation of the $1S - 2P$ transitions 
~\footnote{\href{https://doi.org/10.1038/s41586-018-0435-1}{M.~Ahmadi {\em et al.}, Nature 561, 211 (2018).}}, as well as fine structure and the Lamb shift~\footnote{\href{https://doi.org/10.1038/s41586-020-2006-5}{The ALPHA Collaboration, Nature 578, 375 (2020).}} (shown in Fig.~\ref{fig:fs-alpha}). Recently, laser cooling of antihydrogen was demonstrated. These accomplishments open the door to a new chapter in antimatter physics where spectroscopic precisions achievable with antihydrogen will rival those of hydrogen. In terms of sensitive symmetry tests, this implies that likely not only antihydrogen, but also hydrogen spectroscopy has to increase in accuracy. The near term goal is to achieve sub-kHz (i.e., $10^{-13}$) precision.  ALPHA-Canada, which constitutes more than one third of the collaboration, continues to provide leadership in particle detection, spectroscopy and laser cooling.
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=0.70\columnwidth]{./Figs/fs-alpha-finestructure.jpg}
		%\hspace*{0.2in}
		%\includegraphics[angle=0,width=0.35\columnwidth]{./Figs/fs-alpha-alphag-apparatus.jpg}
	\end{center}
	\caption{The $1S-2P$ fine-structure spectrum of antihydrogen. Experimental data (filled circles) and fitted lineshapes for doubly spin-polarized antihydrogen samples. Taken from \href{https://doi.org/10.1038/s41586-020-2006-5}{The ALPHA Collaboration, Nature 578, 375 (2020)}.
	}
	\label{fig:fs-alpha}
	\vspace*{-0.0in}
\end{figure*}
%

For the coming 5 year period, emphasis will be on the new, CFI-funded, ALPHA-g apparatus (right side of Fig.~\ref{fig:fs-alpha-app}, currently being readied for deployment, for the measurement of gravitational free-fall of neutral anti-matter, probing the gravitational interaction of antimatter. This device will also allow 100-fold improved hyperfine spectroscopy. As this device moves online, a next generation effort will be launched in Canada, HAICU (Hydrogen-Antihydrogen Infrastructure at Canadian Universities), which will exploit the amazing progress being made in the field of quantum sensing.
In HAICU, the (anti)hydrogen atoms will be cooled to $\mu$K temperatures, several orders of magnitude colder than for the presently most precise measurements of the hydrogen spectrum, enabling substantial progress with hydrogen during offline commissioning in Canada, with the introduction of atomic fountain and atom interferometry techniques.

On the 2027-36 horizon HAICU would be deployed at CERN, where atomic fountain based measurement can be performed with antihydrogen and hydrogen in the same apparatus, greatly diminishing the potential for systematic errors. Ultimately, it might be possible to form antimatter molecules and perform a CPT test at the $10^{-17}$ level.

\subsubsection{Nuclear decay}


\paragraph{Unitarity of the Cabibbo-Kobayashi-Maskawa matrix}

(TRIUMF); Guelph, Queens, SFU, Toronto, UBC; {\intl UK, USA}.

Precision measurements of the $ft$ values for superallowed $0^{+} \rightarrow 0^{+}$ Fermi $\beta$ decays between nuclear isobaric analogue states provide demanding tests of the electroweak Standard Model. Such measurements have, for example, confirmed the conserved vector current (CVC) 
hypothesis at the level of $1.2 \times 10^{-4}$, set the most stringent limit on a fundamental weak scalar current coupling to left-handed neutrinos at $(0.09 \pm 0.11)$\% of the vector strength, and, together with the Fermi coupling constant $G_{F}$ from muon decay, provide the most precise determination of the $V_{ud} = G_{V}/G_{F}$ element of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix~\footnote{\href{https://arxiv.org/abs/1807.01146}{J.C. Hardy and I.S. Towner, arXiv:1807.01146[nucl-ex]~(2018).}}. To determine $G_{V}$ from the superallowed data, the experimental $ft$ values must be combined with theoretical calculations of transition-dependent ``outer'' radiative corrections $(\delta_R)$, a transition-independent ``inner'' radiative correction ($\Delta^V_{R}$), and nuclear structure dependent isospin-symmetry-breaking corrections $(\delta_C)$.
%
% Figure 
\begin{figure}[ht!]
	\centering
	\includegraphics[width=0.75\columnwidth]{./Figs/fs-superallowed.pdf}
	\caption{\label{superallowedFt}
		%
		Corrected $Ft$ values for the 15 precisely measured superallowed Fermi $\beta$ decays as of June 2020.  These data confirm the conserved vector current hypothesis at the level of $1.2 \times 10^{-4}$ and provide the most precise determination of the $V_{ud}$ element of the CKM quark mixing matrix.}
	\label{fig:fs-ckm}
\end{figure}
%

New calculations of radiative corrections to $V_{ud}$~\footnote{\href{https://doi.org/10.1103/PhysRevD.100.013001}{C.Y Seng {\em et al.}, Phys. Rev. D 100, 013001 (2019).}}~\footnote{\href{https://doi.org/10.1103/PhysRevD.100.073008}{A. Czarnecki {\em et al.}, Phys. Rev. D 100, 073008 (2019).}}~\footnote{\href{http://hdl.handle.net/1993/34646}{K. Shiells, Ph.D. thesis, University of Manitoba, 2020.}}
indicate a violation of CKM unitarity of about $1\times 10^{-3}$ at 2 to 3 $\sigma$
significance.
As a violation of CKM unitarity would require new physics beyond the current electroweak Standard Model, all inputs must be carefully scrutinized. Figure~\ref{fig:fs-ckm} shows corrected $Ft$ values for the 15 most precisely measured decays.

The TRIUMF-ISAC facility produces high-quality beams of many of the superallowed  emitters with world-record intensities and hosts a suite of state-of-the-art spectrometers capable of precision measurements of all of the experimental quantities of interest in superallowed decays. These include high-precision half-life measurements through both counting with the 4pi gas proportional counter at the ISAC-I GPS facility and gamma-ray counting with GRIFFIN, high-precision branching-ratio measurements with GRIFFIN and its auxiliary detectors, high-precision Q-value measurements
with the TITAN mass spectrometer, and charge-radii measurements through collinear laser spectroscopy required as input to the isospin-symmetry-breaking calculations. In superallowed half-life and branching ratio measurements, the ISAC gamma-ray group has made significant progress recently: The half-life of the superallowed emitter $^{18}$Ne was determined twice as precisely, demonstrating the capabilities of the new thick tape system for precision measurements of noble gas isotopes~\footnote{\href{http://dx.doi.org/10.1103/PhysRevC.92.025502}{A.T. Laffoley {\em et al.}, Phys. Rev. C 92, 025502 (2015).}}
and was also used to improve the world-average half-life for the superallowed emitter $^{22}$Mg by more than a factor of 3~\footnote{\href{https://doi.org/10.1103/PhysRevC.96.045502}{M.R. Dunlop {\em et al.}, Phys. Rev. C 96, 045502 (2017).}}. Finally, a first superallowed half-life measurement to better than $10^{-4}$ was achieved with $^{10}$C, with the added benefit that as the lightest such emitter, it has greatest sensitity to possible scalar current contributions~\footnote{\href{http://dx.doi.org/10.1103/PhysRevLett.116.172501}{M.R. Dunlop {\em et al.}, Phys. Rev. Lett. 116, 172501 (2016).}}

The factor of $300-500$ gain in $\gamma - \gamma$ coincidence
detection efficiency provided by GRIFFIN compared to the previous $8\pi$ spectrometer at ISAC-I has also allowed
major advances in the control of the so-called ``Pandemonium" problem that has plagued previous attempts at high-precision
branching-ratio measurements for the heavy $(A \geq 62)$ superallowed decays, as demonstrated in our recent measurement
of the $^{62}$Ga superallowed branching ratio to $\pm 0.0026$\%~\footnote{\href{https://doi.org/10.1103/PhysRevC.102.054325}{A.D. MacLean {\em et al.}, Phys. Rev. C 102, 054325 (2020).}}, a
factor of 4 improvement over the previous world-average.  

This highly successful, long-term, program of high-precision superallowed $\beta$ decay half-life and branching
ratio measurements will continue to be developed at ISAC throughout the 2022--2026 period, and beyond as new beams become available.  
These measurements will focus on the decays that differentiate between leading models of the isospin-symmetry-breaking corrections,
namely the $N = Z - 2$ superallowed emitters $^{14}$O, $^{22}$Mg, $^{34}$Ar, and $^{38}$Ca, and the heavy
$A \geq 62$ emitters between $^{62}$Ga and $^{74}$Rb. 




\paragraph{BL3: a more precise determination of the decay rate of neutrons to protons
\label{sec:funsym:neutron}}

(NIST); Manitoba; {\intl USA}

The BL3 experiment seeks an improvement on the precision of the neutron lifetime using the ``beam'' method that will resolve or confirm the long-standing discrepancy between the ``beam'' and ``bottle'' methods. The decay protons from a collimated beam of cold neutrons are trapped in the electrostatic and electromagnetic fields of a quasi-Penning trap. Periodically the fields are lowered and the protons are counted, allowing a measurement of the weak decay lifetime. The project infrastructure proposal is currently under review by the US National Science Foundation. The experiment intends to run by 2025. Near term focus of the Canadian contribution will be on simulations of the decay protons in the trap and on their way to the proton detector, and proton detector prototype testing in the University of Manitoba 30 keV refurbished mass spectrometer.


\subsubsection{Beta neutrino correlations with trapped atoms and ions, and cold neutrons}
\label{subsec:betaneutrino}

Particle traps are ideal tools for studying correlations in $\beta$ decay. The nuclear recoil escapes freely from the trap (i.e. there is no backing material), and its momentum and that of the beta can be measured, so the neutrino momentum can be determined. Furthermore, the nuclei can be spin-polarized by atomic optical pumping methods to an extremely high degree, and can be probed by atomic methods independent of the nuclear decay. Correlation experiments set stringent limits on potential scalar and tensor weak currents, and can also be used to constrain more exotic physics. Canada is a world leader in this line of research. The TRINAT facility at TRIUMF has pioneered the use of neutral atom traps to measure beta decay correlations. On the cold neutron side, two landmark measurements of parity violating asymmetries have seen their successful conclusion, {\em nPDGamma} and {\em n3H3}, while their successor {\em Nab}, a decay correlation experiment, is gearing up to run during the coming 5 year period.

\paragraph{The TRINAT neutral atom trap at TRIUMF}
	
(TRIUMF); TRIUMF, UBC, Manitoba; USA.

TRINAT uses laser trapping and cooling techniques to study the decays of short-lived isotopes produced at ISAC, to search for new physics. Laser forces suspend  the atoms  in a mm-sized cloud in the center of a UHV chamber. The recoiling daughter nucleus of very low kinetic energy can freely escape. Its momentum, as well as that of the emitted beta particle can be precisely measured, permitting the reconstruction of the neutrino's momentum, which cannot be measured directly.
The angular distribution of the products of nuclear beta decay is sensitive to the Lorentz structure of the semileptonic currents, with discovery potential complementary to particle physics.

A recent development at TRINAT is the ability to optically pump the trapped atoms to a high degree ($> 99$ \% nuclear spin polarization and known to better than 0.1 \%)~\footnote{\href{http://dx.doi.org/10.1088/1367-2630/18/7/073028}{B. Fenker {\em et al.}, New J. Phys. 18, 073028 (2016).}}. This enabled the measurement of the $\beta$ asymmetry in $^{37}$K to 0.3 \%, the best such measurement in any nucleus, and in agreement with the Standard Model~\footnote{\href{https://doi.org/10.1103/PhysRevLett.120.062502}{B. Fenker {\em et al.}, Phys. Rev. Lett. 120, 062502 (2018).}}.

Plans for the coming years include completing the measurement of the recoil asymmetry in spin-polarized $^{37}$K decay and a first time-reversal test in radiative $\beta$ decay. An asymmetry accuracy of $\approx 5$ \% is expected, which is much more sensitive than constraints from other measurements such as the neutron radiative $\beta$ decay branch. This period will also see the completion of a program in neutrino spectra of fission products. TRINAT has kinematically isolated the $0^- \rightarrow 0^+$ decay branch of $^{92}$Rb and measured a $\beta - \nu$ correlation, consistent with expectations. The measurement will indirectly determine whether the neutrino spectrum has non-allowed components. 
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=0.35\columnwidth]{./Figs/fs-trinat.pdf}
		%\hspace*{0.2in}
		%\includegraphics[angle=0,width=0.35\columnwidth]{./Figs/fs-alpha-alphag-apparatus.jpg}
	\end{center}
	\caption{Adding $\gamma$-ray detection to TRINAT is enabling time-reversal symmetry tests in radiative $\beta$ decay and isospin-breaking measurements.
	}
	\label{fig:fs-trinat}
	\vspace*{-0.0in}
\end{figure*}
%

A 5-fold improvement appears possible for TRINAT's previous $^{38m}$K $\beta - \nu$ correlation measurement, extending into the 2027-36 period. Isospin breaking in $^{36}$K will be probed, by measuring the recoil asymmetry with respect to the spin, providing a test of modern calculations needed to deduce $V_{ud}$ from $0^+ \rightarrow 0^+$ decay. 
Finally, sensitivity to parity conserving, charge-symmetry breaking time-reversal violating interactions is enhanced by $10\times$ in the isospin-suppressed mirror decay of $^{45}$K over other mirror decay measurements of $D \vec{I} \cdot \vec{v_\beta} \times \vec{p_{\nu}}$.

%Finally, sensitivity to time-reversal breaking is enhanced by $10-100\times$ in the isospin-suppressed mirror decay of $^{45}$K over other mirror decay measurements of $D \vec{I} \cdot \vec{v_\beta} \times \vec{p_{\nu}}$.


\paragraph{Nab, nPDGamma, and nHe3: decay correlation and parity-violating asymmetry experiments with cold neutrons}

(Oak Ridge); Manitoba, TRIUMF, Winnipeg; USA

The {\em Nab} experiment at the Spallation Neutron Source (SNS) at Oak Ridge is gearing up to follow in the footsteps of two efforts that recently have provided the most stringent constraints on the 6 weak nucleon-nucleon coupling constant to date. The {\em nPDGamma} experiment carried out the world's first measurement of parity violation in the neutron-proton system with a first nonzero observation of the corresponding asymmetry $A_{PV} = 3.0 \pm 1.4 {\rm (stat)} \pm 0.2{\rm(sys)} \times 10^{-8}$~\footnote{\href{https://doi.org/10.1103/PhysRevLett.121.242002}{D. Blyth {\em et al.}, Phys. Rev. Lett. 121, 242002 (2018).}}. 
The result confirms in a simpler system measurements in $^{18}$F of the smallness of the isovector versus isoscalar weak {\em NN} interaction, nicely consistent with several recent theory approaches.
Likewise, the
	{\em n3He} experiment carried out the first measurement of parity-violation in the neutron-$^3$He system,
	which is the next simplest nuclear system, aside from the technically very difficult measurements on deuterium. This measurement was a first and observed an asymmetry $A_{PV} = 1.58 \pm 0.97{\rm(stat)} \pm 0.24{\rm(sys)} \times 10^{-8}$~\footnote{\href{https://doi.org/10.1103/PhysRevLett.125.131803}{M.~Gericke {\em et al.}, Phys. Rev. Lett. 125, 131803 (2020).}}. Future improvements of these measurements will be technically very challenging and are a long way off, requiring the construction of new facilities, such as the European Neutron Spallation Source.
	%
	\begin{figure*}[ht]
		\begin{center}
			\includegraphics[angle=0,width=0.80\columnwidth]{./Figs/fs-nab-2.pdf}
		\end{center}
		\caption{
		Left: Schematic illustration of the 8 meter long {\em Nab} spectrometer and detector assembly with electric and magnetic field configuration. 
		Right: Assembly and testing of the {\em Nab} large area silicon detectors at the University of Manitoba. 	
%		Left: Photograph of the proton test facility at the University of Manitoba. Hydrogen gas is ionized at the far left of the picture and accelerated toward the electrostatic analyzer, which is the grey arc structure in the foreground. The beam then travels to the dipole mass separation magnet, shown in red, and 30 keV, ionized protons are allowed to continue onto the detector test region. Right: Schematic view of the detector test region with the Si detector installed. The electrostatic beam steer is located just upstream of the detector region vacuum chamber and the reference MCP detector is located just in front of the Si detector.
		}
		\label{fig:fs-nab}
		\vspace*{-0.0in}
	\end{figure*}
	%
	
	
	The focus in the coming years will be the {\em Nab} experiment. It measures the correlation parameter $a$ and the Fierz interference term $b$ in un-polarized cold neutron beta decay. The heart of the apparatus is an $\approx 4$ m tall solenoidal spectrometer with segmented Si detectors on either end to measure the energy of the decayed electrons and momentum of the decayed protons, shown on the left in Figure~\ref{fig:fs-nab}. The Canadian group's primary contribution to the experiment is the development of a 30 keV proton accelerator at the University of Manitoba that will be used to characterize the large area Si detectors used in the experiment (shown on the right in Figure~\ref{fig:fs-nab}). {\em Nab} will begin collecting data in 2021 and continue until 2024, after which spin-polarized cold neutrons will be used with the same apparatus to measure the neutron beta decay correlation coefficients $A$ and $B$.





\subsubsection{Neutrinos}

\paragraph{Neutrinoless double $\beta$-decay: The search for Majorana neutrinos with nEXO}

(SNOLAB) Carleton, Laurentian, McGill, Sherbrooke, TRIUMF, UBC; {\intl China, Germany, Russia, South Africa, Switzerland, USA}.

A major open question is the nature of neutrinos and how they influence the evolution of the universe.  The discovery that neutrinos are not massless has been transformative in that a neutrino could have an astonishing property of being its own anti-particle. An extremely rare nuclear decay mode known as
neutrinoless double beta ($0\nu\beta\beta$) decay offers the most sensitive experimental method to test for this possibility. 
The observation of this most exotic decay mode would provide irrefutable evidence that neutrinos are their own anti-particle and correspondingly that the symmetry of lepton number conservation is violated. Its observation would also provide strong experimental guidance for theories that go beyond the Standard Model, yielding insights into the origin of neutrino mass. In particular, if neutrinos are indeed their own antiparticles, they could not gain their mass through the interactions with Higgs particles in the same way as all other elementary particles in the Standard Model. Neutrinos with this property could also be key players in generating the observed excess of matter over anti-matter in our universe.
%
\begin{figure*}[ht]
	\begin{center}
		\includegraphics[angle=0,width=0.80\columnwidth]{./Figs/fs-nexo.jpg}
	\end{center}
	\caption{Artist rendering of the nEXO TPC (right) and its installation at the SNOLAB cryopit (left). The cryostat is submerged in a water tank, which acts as active shielding. SiPMs will be mounted between field shaping rings and detector wall.}
	\label{fig:fs-nexo}
	\vspace*{-0.0in}
\end{figure*}
%

The nEXO experiment will search for $0\nu\beta\beta$ decay in the isotope $^{136}$Xe. It is the successor to EXO-200, which over the past decade observed the $2\nu\beta\beta$ in this isotope and carried out several searches for the neutrino-less mode.
The goal of nEXO is to push the sensitivity by a factor of 100 or more, reaching half-lives of $10^{28}$ years~\footnote{\href{https://doi.org/10.1103/PhysRevC.97.065503}{J.B.~Albert {\em et al.}, Phys. Rev. C 97, 065503 (2018).}}. The optimum location for nEXO is the SNOLAB underground laboratory in Sudbury, Ontario, shown as an artist's conception in Figure~\ref{fig:fs-nexo}. It has advantages of depth (equal very low cosmic background), extensive clean room facilities, and existing capabilities and expertise for the design, construction and operation of the experiment. With a final approval by DOE expected soon, the collaboration, which has a very substantial Canadian component with 6 involved institutions, has embarked on extensive R\&D for key detector technologies. Current Canadian contributions include the development of novel photon sensors, assembly and testing of the light-collection system, radioactive background control, an external calibration source deployment system, a water shield with active muon veto, a water purification and assaying system, and SNOLAB infrastructure, and development of low-background techniques for a future upgrade.

Commissioning is expected around 2027/28, contingent on the DOE selection process; nEXO will then take data for at least one decade to reach the $10^{28}$ year half-life goal.

%GH: text added Nov 24 from Thomas Brunner
Canadian groups constitute about 20-25\% of the EXO-200 and nEXO Collaborations, and take substantial responsibilities, including chair of the EXO-200 collaboration board for the period of the last 5-year plan, serving as one of two EXO-200 analysis coordinators, and  contributions to operations as part of the EXO-200 management team.
Within nEXO, two Canadian PIs are Level-2 sub-systems physicists out of 11 subsystems (with the two subsystems as full Canadian responsibility), and three Canadian PIs hold L3 leadership responsibilities for subsystems in other WBSes.
%end new text
In parallel to the deployment of the nEXO apparatus, the Canadian team is pursuing techniques that would greatly suppress background, such as Barium tagging. In this approach, a small volume surrounding  a $0\nu\beta\beta$ event is extracted from the detector and probed for the presence of a Ba-ion, an unambiguous tag for a true decay event.


%\paragraph{BeEST: A search for keV-scale neutrinos in the electron capture decay of $^7$Be using superconducting quantum sensors}
%
%(TRIUMF); TRIUMF; {\intl USA}
%
%The search for sterile neutrinos is among the most promising avenues in our quest for understanding the microscopic nature of dark matter in our universe. Sterile neutrinos - unlike the active neutrinos in the Standard Model (SM) - do not interact with normal matter as they move through space, and as such are best observed via their mass-generated effects that result from momentum conservation with SM particles. The BeEST experiment (Beryllium Electron-capture in Superconducting Tunnel junctions) aims to perform the highest-sensitivity search for keV-scale sterile neutrinos to date using the electron capture decay of $^7$Be implanted into superconducting quantum sensors. This work has been possible by leveraging existing state-of-the-art superconducting tunnel junction (STJ) detector technology developed at Lawrence Livermore National Laboratory (LLNL), as well as more than a decade of work at TRIUMF-ISAC for the low-energy implantation of pure $^7$Be beams into thin films. The proposed program employs momentum reconstruction in the electron capture (EC) decay of $^7$Be implanted in these STJs to search for missing momentum in the nuclear recoil spectrum that would be generated by heavy neutrinos. Since the neutrino escapes the thin detector without interacting, the spectrum consists of four peaks corresponding to the energy released by the recoiling $^7$Li daughter atom for different decay processes. The signature of heavy neutrinos is a small fraction of events whose recoil energy peaks are shifted to lower energies due to the missing momentum. The relative fraction of these events to the total indicates the mixing fraction with the electron neutrino.
%
%Over the coming 5 years, BeEST will scale the experiment using existing 36- and 112-pixel detector arrays, fabricate new 128-pixel Al-based STJ detector arrays deposited on thin membranes, and develop a new target and beam purification techniques at TRIUMF-ISAC and ARIEL to reach $^7$Be intensities approaching $10^{10}$ s$^{-1}$. The goal is to either find a candidate or provide exclusion limits 10,000 times more stringent than previous tests in the 5-860 keV mass region. The Canadian contribution towards optimal beryllium implantation will make use of the new CFI-funded CANREB facility, and electron-induced photofission at ARIEL.


\paragraph{BeEST: A search for keV-scale neutrinos in the electron capture decay of $^7$Be using superconducting quantum sensors}

(TRIUMF); TRIUMF; {\intl USA}

The search for sterile neutrinos is among the most promising avenues in our quest for understanding the microscopic nature of dark matter in our universe. Sterile neutrinos - unlike the active neutrinos in the Standard Model (SM) - do not interact with normal matter as they move through space, and as such are best observed via their mass-generated effects that result from momentum conservation with SM particles. The BeEST experiment (Beryllium Electron-capture in Superconducting Tunnel junctions) aims to perform the highest-sensitivity search for keV-scale sterile neutrinos to date using the electron capture decay of $^7$Be implanted into superconducting quantum sensors. This work has been possible by leveraging existing state-of-the-art superconducting tunnel junction (STJ) detector technology developed at Lawrence Livermore National Laboratory (LLNL), as well as more than a decade of work at TRIUMF-ISAC for the low-energy implantation of pure $^7$Be beams into thin films~\footnote{\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.032701}{S.~Fretwell {\em et al.}, Phys. Rev. Lett. 125, 032701 (2020).}}. The proposed program employs momentum reconstruction in the electron capture (EC) decay of $^7$Be implanted in these STJs to search for missing momentum in the nuclear recoil spectrum that would be generated by heavy neutrinos ~\footnote{\href{https://arxiv.org/abs/2010.09603}{S.~Friedrich {\em et al.}, arXiv:2010.09603 (2020).}} . Since the neutrino escapes the thin detector without interacting, the spectrum consists of four peaks corresponding to the energy released by the recoiling $^7$Li daughter atom for different decay processes. The signature of heavy neutrinos is a small fraction of events whose recoil energy peaks are shifted to lower energies due to the missing momentum. The relative fraction of these events to the total indicates the mixing fraction with the electron neutrino.

Over the coming 5 years, BeEST will scale the experiment using existing 36- and 112-pixel detector arrays, fabricate new 128-pixel Al-based STJ detector arrays deposited on thin membranes, and develop a new target and beam purification techniques at TRIUMF-ISAC and ARIEL to reach $^7$Be intensities approaching $10^{10}$ s$^{-1}$. The goal is to either find a candidate or provide exclusion limits 10,000 times more stringent than previous tests in the 5-860 keV mass region. The Canadian contribution towards optimal beryllium implantation will make use of the new CFI-funded CANREB facility, and electron-induced photofission at ARIEL.

\paragraph{Synergies with theory}

Manitoba, Memorial, TRIUMF

Fundamental symmetry measurements critically depend on nuclear, atomic, and particle physics theory to extract the underlying physics from the data. As mentioned earlier in this section, groups at Memorial and Manitoba have been active in carrying out one-loop and two-loop electroweak radiative corrections crucial for parity violating electron scattering experiments. Alberta and Manitoba are also playing a leading role in CKM unitarity tests by advancing the state-of-the-art in radiative correction calculations for $V_{ud}$. More information is found in~\ref{sec:precisionstudies}

In addition, recent advances in {\em ab initio} nuclear structure calculations by theorists at TRIUMF are starting to have impact on fundamental symmetry work, such as neutrinoless double-beta decay, permanent electric dipole moment searches, atomic parity violation (anapole moments), and beta decay. Details are presented in the following theory section, in particular in~\ref{sec:theory:abinitio} and~\ref{sec:synergy}.




\subsection{Beyond the next five years}\label{sec:beyond-funsym}

A good mix of ongoing projects, efforts currently ramping up, and preparations underway for future experiments, Canada's program in fundamental symmetries is well prepared to remain world-leading beyond the immediate 5 year horizon, with a wide range of investigations leading us far into the 2030s.

The TUCAN neutron EDM experiment will run beyond 2026, with extensive data analysis, and possible upgrades. A second port at the UCN source can accommodate other measurements with ultra-cold neutrons, such as the neutron lifetime or a neutron gravitational levels experiment probing short range modifications to gravity. In this phase TUCAN will turn into a user facility seeking external proposals.

A major new effort at TRIUMF/ISAC will be precision spectroscopy with radioactive molecules, in particular as related to searches for permanent electric dipole moments. This project is currently taking shape, with the development of a beamline. After initial work on molecule formation and preparatory spectroscopy, beyond 2025, a competitive EDM search would unfold, with increasing precision over 5 years. A francium atomic fountain electron EDM experiment could be on the floor at ISAC within a few years, and reach design sensitivity by 2026, with improvements in the following years. By mid-decade, when ISAC/ARIEL have reached the capacity to deliver up to 3 radioactive beams simultaneously, the FrPNC experiment will have established the techniques to start an atomic parity violation production run. ISAC's ability to produce a wide range of Fr isotopes in large quantities and its unique expertise in Fr laser trapping naturally suggest a program based on cold molecules assembled from laser trapped Fr and other species such as Ag. 

For parity-violating electron scattering, the decade from 2026 to 35 will be a very busy and productive one. In the first half, MOLLER and P2 will take data and aim for 1 \% measurements of parity violating asymmetries, with analysis wrapping up in the second half, just in time for EIC-based programs in fundamental symmetries picking up.

In antimatter research, we will see the deployment of HAICU by the ALPHA collaboration, with precise, simultaneous spectroscopy of hydrogen and antihydrogen in an atomic fountain apparatus. Another goal will be the production of antimatter molecules, and prospects of CPT tests at the $10^{-17}$ level.

In beta decay, the long-standing program of super-allowed decay lifetime and branching ratio measurements at ISAC shows no sign of slowing down. As new beams get developed, additional cases can be measured, and systematic investigations, e.g. on isospin-symmetry breaking corrections, can be carried out. Especially in light of the recent tensions between theory and experiment for the unitarity of the CKM matrix, this will be a worthwhile endeavour. On the beta decay correlations front, the TRINAT laser trapping facility at ISAC will aim for a 5-fold improvement of the $^{38m}$K beta-neutrino correlation measurements, further tightening constraints on scalar interactions. Isospin breaking in $^{36}$K will help benchmark theory related to $V_{ud}$. At Oak Ridge, the {\em Nab} setup will reconfigure to measure that neutron beta decay correlation coefficients $A$ and $B$.

In the neutrino sector, an approved nEXO would see commissioning in the late 20s, running then for at least a decade, pushing the sensitivity for the $0\nu\beta\beta$ half-life of $^{136}$Xe to $10^{28}$ years, either observing $0\nu\beta\beta$ decay  or setting a new limit. In parallel, the Canadian contingent will push background suppression techniques such as barium tagging.




\subsection{Summary}

The Canadian nuclear physics community is involved, and in many cases is heading, world-leading efforts in the area of fundamental symmetries. The record over past decade has been excellent, and several major, exciting new initiatives will enter the stage over the next five years, ensuring Canadian leadership well beyond a 10-year horizon. The program is remarkably balanced in terms of work at Canada's own world-class facilities, TRIUMF and SNOLAB, versus off-shore projects. The scientific coverage is amazingly diverse, addressing the pressing questions in the field. Research in fundamental symmetries of nature is undoubtedly one of the jewels in Canadian science. 


