\section{Nuclear Theory}\label{sec:Theory}

\subsection{Overview}
Perhaps the most complicated quantum systems in the universe, atomic nuclei are intimately connected to some of the most profound questions in science, such as the nature of neutrinos and dark
matter, the role of fundamental symmetries, the inner workings of neutron stars, and
the nucleosynthesis pathways that lead to the observed abundances of elements.
%from Holt
Striving to answer the most fundamental question in our Universe, theorists work on projects ranging from developing a predictive {\em ab-initio} theory of nuclear structure and nuclear reactions to phenomenological approaches guided by empirical data in close collaboration with experiment, and on everything in between.

Progress in nuclear theory, both from a fundamental point of view, and in its connection with experimental measurements, has therefore to proceed on several fronts at once, and it is thus imperative to maintain a vibrant and diverse theoretical program.

The main challenge of nuclear theory is a non-conformal nature of quantum chromodynamics (QCD), so the nuclear interaction appears to be quite different at different energy scales. At high energies, most QCD calculations are amenable to perturbative techniques. However, at the energy scale where the details of nuclear structure are relevant, QCD is non-perturbative. Lattice techniques, although greatly improved in the recent years, still can not solve all problems in hadronic physics. However, there has been remarkable recent progress in effective theories which preserve the important symmetries of the underlying fundamental theory, and yet are applicable in a given energy interval. However, for a large class of problems, phenomenological approaches to methods and modelling are required. 

Some of the world leaders in nuclear theory, in fields such as {\em ab-initio} nuclear structure, 
relativistic heavy ion collisions, and precision calculations for low-energy measurements are already based in Canada, and there is a great opportunity to further strengthen and grow these programs with strategic investment in HQP who can accelerate the efforts of these groups and take full advantage of collaboration with Canadian experimentalists at new or recently-upgraded facilities. 


\subsection{The Canadian program}


Canadian theorists have kept at the forefront of the rapidly developing trends in nuclear and hadronic theory with many examples of close involvement in the major breakthroughs of the field. The Canadian effort is distributed in several groups coast-to-coast, from British Columbia to Newfoundland and Labrador, building intellectual capacity in their regions and pursuing a broad range of initiatives.
The advancement of nuclear physics is strongly dependent on interplay between theory and experiment, and many Canadian research programs are the excellent examples of such successful collaboration. Theory input is indispensable for many experimental programs, especially for the quickly-developing precision frontier which can reach for new physics at TeV scale. Canadian theorists identify promising future directions for the experimental programs in Canada and off-shore, participate in experimental proposals, develop new computational methodology, help to interpret the experimental data, and educate the future generation of researchers in both theory and experiment. 
There is a very active Canadian community pursuing a plethora of research topics on the theoretical aspects, especially hot and cold QCD and hadronic systems. Importantly, calculations involving QCD appear not only in nuclear physics, but in all computations and estimates that rely on the Standard Model such as calculating decay modes of the Higgs boson. In several cases, the overall precision of the measurements is often determined by our knowledge of the strong interaction or relevant radiative corrections. 
The Canadian community is also spearheading developments for the capability to theoretically describe light- and medium-mass nuclei as systems of nucleons interacting by forces rooted in the fundamental theory of strong interactions. Using a low-energy expansion of QCD, namely chiral effective field theory ($\chi$EFT), one can derive forces among nucleons and their interactions with external probes in a consistent way.  Studies in light- and medium-mass nuclei and nucleon polarizabilities are crucial to test such a theory and these initiatives are pursued hand-in-hand with experiments performed at TRIUMF-ISAC, JLab and elsewhere.
Nuclear Theory research represents a great opportunity to reinforce and advance an established Canadian strength at the forefront of modern research. In this section we highlight some Canadian activity in the theory of atomic, nuclear and particle physics.

%Further strengthening and growth of this
%program is highly-desirable through strategic investment into Highly-Qualified Personnel who can
%accelerate the efforts of these recognized world leaders at the forefront of 
%an exciting and fast-moving discipline which is intimately linked to Canadian
%experimental efforts in nuclear physics.

%====================================================
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=8cm]{./Figs/QCD_Phase_diag.pdf}
\caption{The conjectured phase diagram of quantum chromodynamics. Details are given in the text. This figure is from [B. V. Jacak and B. M\"uller, Science 337, 310 (2012).]}
\label{fig:phase_diag}
\end{center}
\end{figure}
%====================================================

\subsubsection{QCD under extreme conditions}
The collision of large nuclei at high energy is a practical way of creating strongly-interacting systems at high temperatures and at high densities. These conditions prevailed a few microseconds after the Big Bang, and several experimental facilities around the world have been devoted to the study of ``extreme QCD'', with vigorous experimental programs at RHIC and at the LHC, at FAIR/GSI, in Darmstadt, Germany, and at NICA, in Dubna, Russia. Apart from the reproduction of the physical environment that existed in the early Universe, the many-body nature of finite-temperature QCD is still not that well known and many questions remain. For example: What is the nature of the QCD phase diagram? Are there critical points? What are the bulk properties of the quark-gluon plasma? Some of those questions have immediate relevance not only for hadronic physics, and answering them  may also very well influence our understanding of the physics of dense stellar objects such as neutron stars, for the equilibrium aspects, and also supernovae explosions, for the non-equilibrium features. 

A prominent research program at McGill University aims to understand QCD under extreme conditions, in and out of equilibrium. The success of modelling heavy-ion collisions with relativistic hydrodynamics was spectacularly realized at RHIC and 
finally  confirmed by heavy-ion runs at the LHC \footnote{
\href{ https://doi.org/10.1017/9781108651998}{P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge, UK: Univ. Pr. (2019).}}. A part of McGill research program is a quantitative characterization of the quark-gluon plasma, and a search for its transport coefficients. Those can be obtained by taking the static, long wavelength limits of spectral densities, obtained from current-current correlators. Those correlators can be  estimated in models, or obtained from calculations made using lattice QCD.  In parallel, one may extract those from phenomenological studies of flow variables measured in heavy-ion collisions, interpreted through fluid dynamical simulations; and McGill group is pursuing both strategies \footnote{
\href{ https://journals.aps.org/prc/50th}{B. Schenke, S. Jeon, and C. Gale, Phys. Rev. C 82, 014903 (2010).}}.

%Bjoern Schenke, Sangyong Jeon, and Charles Gale, Phys. Rev. C 82, 014903 (2010). This article was chosen as a “milestone paper” by Physical Review C, celebrating its 50th anniversary: https://journals.aps.org/prc/50th.

Another means to characterize the QGP is tomography: one uses a standard probe such as photons, real and virtual, and QCD jets, and then measure deviations from expected behaviour to be attributed to in-medium effects. 

 \textsc{martini} \footnote{
 \href{https://doi.org/10.1103/PhysRevC.80.054913}{
 B. Schenke, C. Gale and S. Jeon, Phys. Rev. C 80, 054913 (2009)}} is an approach which evolves entire parton showers in a relativistic hydrodynamics background (once the system has hydrodynamized). With this evolved tool which includes collisional and radiative energy loss, the quenching of QCD jets and the emission of photons can be treated consistently and thus represent complementary observables. A version of \textsc{martini} where the treatment of QCD jet energy-loss was reformulated in coordinate space through the Light Cone Path Integral formalism \footnote{
 \href{https://doi.org/10.1103/PhysRevC.82.064902}
{S. Caron-Huot and C. Gale, Phys. Rev. C 82, 064902 (2010)}} enables a finite size (or formation time) modification which is important at the LHC \footnote{
\href{https://doi.org/10.1016/j.nuclphysbps.2017.05.066}{
C. Park, C. Shen, S. Jeon and C. Gale, Nucl. Part. Phys. Proc. 289-290, 289 (2017)}}. The  improved theory will be used to calculate the effect of jet energy-loss, of jet sub-structure,  of jets fragmenting into photons, and of photon production by jets interacting with the medium. From this privileged position, the group can tackle correlation-observables such as photon- and dilepton-tagged jets  at the LHC, which will reveal additional facets of jet energy loss. Finally, much recent progress in jet morphology analyses \footnote{
\href{https://doi.org/10.1103/RevModPhys.90.025005}{M. Connors, C. Nattrass, R. Reed and S. Salur, Rev. Mod. Phys. 90, 025005 (2018)}} raise the tantalizing possibility of establishing a reference (through photon tagging) to test different scenarios of jet-medium interactions. The photon tagging constitutes a trigger and the differential jet shape $\rho (r)$ 
%\footnote{M. Connors, C. Nattrass, R. Reed and S. Salur, Rev. Mod. Phys. 90, 025005 (2018)} 
%\footnote{Ibid.}
of the tagged jets will be analyzed to reveal energy loss and other in-medium interactions. The exploration of jet physics is the main focus of sPHENIX, an experiment currently in construction at RHIC. 

A fundamental question in all studies of the many-body effects observed in high energy heavy ion collisions is whether the observed correlations develop dynamically or are already present in the initial states. A new approach has recently been constructed \footnote{
\href{https://doi.org/10.1016/j.nuclphysa.2018.08.014}{Scott McDonald, Sangyong Jeon, Charles Gale.
Nucl. Phys. A982 239 (2019)}} which properly takes into account the
the evolution in full 3-dimensional space.
An exploration of the consequences of this full 3+1D IP-Glasma model in conjunction
with the hydrodynamics model will involve the vast amount of new data being produced at the LHC, which will continue to run its heavy-ion program in parallel with its pp agenda. Importantly, the physics of the IP-Glasma relies on that of {\em saturation}: the scale at which the non-linearities of the gluon field manifest themselves. The exploration of the {\em  saturation regime} is a central theme of the recently approved EIC, which will push our study of QCD to new frontiers. 

\subsubsection{Lattice field theory for QCD and beyond} 
A great deal of theoretical effort is devoted to calculate bound states masses and transition matrix elements from a variety of approaches. Notably, Canadians have been pursuing the use of potential models and of lattice QCD techniques to calculate these quantities for states involving heavy quarks accessible to the Belle II, LHCb, and GlueX experiments. The meson results highlighted here were obtained with an approach which includes a relativistic kinetic energy term, together with a Lorentz vector one-gluon-exchange interaction, with a QCD-motivated running coupling constant, and a Lorentz scalar linear confining interaction. 

\begin{figure}[ht!]
\begin{center}
\includegraphics[height=7.5cm]{./Figs/chiral_vol_extrapol_slim.png}
\caption{Binding energies for the $ud\bar{b}\bar{b}$ and $us\bar{b}\bar{b}$ tetraquarks obtained from lattice QCD for a range of pion masses. The physical pion mass is toward the left of the graph, marked by a vertical line. This figure is from Francis, Hudspith, Lewis and Maltman, Physical Review Letters 118, 142001 (2017).}
\label{fig:nrqcd}
\end{center}
\end{figure}


Potential models represent a powerful and elegant non-perturbative technique to obtain bound state masses and decay transition rates. A first-principles approach to these results is that of lattice QCD. The Canadian lattice community is obtaining non-perturbative results to compare with experimental measurements, or even making predictions for the results of observations yet to be performed. During the past few years, lattice QCD has confirmed the existence of a tetraquark with valence content $ud\bar{b}\bar{b}$ that is stable under both strong and electromagnetic interactions, decaying only weakly.  The first direct lattice QCD calculation was performed in Canada \footnote{
\href{https://doi.org/10.1103/PhysRevLett.118.142001}{A. Francis, R.J. Hudspith, R. Lewis, K. Maltman, Phys. Rev. Lett. 118, 142001 (2017)}}.  For decades, textbooks have told us that hadrons exist as mesons (with two quarks) or baryons (with three quarks), so it is very exciting to discover that a tetraquark (with four quarks) also exists within QCD.  Several experiments around the world have been observing other tetraquark candidates, and future experiments will search for the $ud\bar{b}\bar{b}$ tetraquark which is special by having only weak decays.  Meanwhile, future lattice QCD calculations will address the list of other tetraquark candidates. Currently, the research group at York University is using lattice field theory to advance knowledge in three major directions.
First, building on their experience with conventional hadrons, the group is now performing calculations of exotic hadrons. Although exotic candidates have been observed at many experimental facilities around the world, there is no consensus on a theoretical understanding. Lattice QCD provides access to quantitative results from first-principles theory, at least for quantities that today’s lattice
infrastructure can handle. At York University, the recent emphasis has been on doubly heavy
tetraquarks \footnote{
\href{https://doi.org/10.1103/PhysRevLett.118.142001}{
A. Francis, R.J. Hudspith, R. Lewis, K. Maltman, Phys. Rev. Lett. 118, 142001 (2017)} and
\href{https://doi.org/10.1103/PhysRevD.99.054505}{Phys. Rev. D 99, 054505 (2019)}}.
In addition, the group is using lattice methods to study a theory of
dark particles with a dark force that resembles the known theory of quarks and gluons \footnote{
\href{https://doi.org/10.1007/JHEP12(2018)118}{A. Francis, R.J. Hudspith,, R. Lewis, S. Tulin, J. High Energ. Phys. 2018, 118 (2018)}}.
And, an exciting new development, now that prototype quantum computers have been built in several labs around the world, it
is time to learn how to write efficient algorithms for the research applications that have the greatest
need for a quantum computer, and lattice gauge theory is a prime candidate. On classical computers,
lattice code involves Monte Carlo calculations in Euclidean spacetime, which accurately describe
the properties of motionless particles but unfortunately cannot provide access to the movement of
particles through real (Minkowski) time. Avoiding the Monte Carlo approach is not feasible on
classical computers, but it might be possible on quantum computers, greatly advancing lattice gauge theory.

\subsubsection{Hadron structure with LFHQCD, ChPT and QCD sum rules}

Some of possible ways to tackling the challenges of QCD is with approximate analytical methods such as light-front holographic quantum chromodynamics (LFHQCD), Chiral Perturbation Theory (ChPT) and QCD sum-rule studies.
The Atlantic theory group, lead by four faculty members, at Acadia in Nova Scotia, Mount Allison in New Brunswick, and Memorial University of Newfoundland (MUN) in Newfoundland and Labrador, is developing an integrated approach to investigating hadron structure based LFHQCD and ChPT while proving joined training opportunities to HQP in their region. Dr. Barkanova’s move to MUN in 2017 with her position at Acadia filled by Dr. Sandapen not only helped to strengthen the graduate program in theoretical subatomic physics at MUN, the only one in the Atlantic Canada, but also to increase the diversity of the group and to maintain primary-undergraduate Acadia as an excellent “feeder” school for graduate programs in both theory and experiment, in nuclear, particle and astrophysics.
Immigrants from four different countries, the faculty members of the Atlantic Canada group can now combine their networks to recruit HQP in Mauritius, Iran, Pakistan, India, China, Eastern Europe, Russia and Germany, and train a significant number of undergraduate students while expanding a graduate program at MUN and contributing to science outreach for Black and Indigenous youth. 
The group works in both nuclear and particle physic theory, with a part of their combined expertise applicable to the low-energy hadronic physics such as calculations up to two-loops in perturbative expansion of specific quantum field framework with the development of corresponding computer algebra tools applicable to a broad spectrum of the processes \footnote{
\href{https://doi.org/10.1103/PhysRevD.101.053003}{A. Aleksejevs et al, Phys. Rev. D. 101, 053003 (2020)}; \href{https://doi.org/10.1103/PhysRevD.88.053008}{A. Afanasev et al, Phys. Rev. D.88.053008 (2013)}}.

LFHQCD, an approximate gravity dual to light-front QCD, is a realization of the Maldacena Conjecture which refers to a duality between a strongly-coupled QFT in physical spacetime and a weakly-coupled gravitational theory in a higher dimensional spacetime, and can be used to predict the elastic/transition form factors, radii and decay constants for vector and pseudoscalar mesons using the phenomenological extension of LFHQCD.
%(Brodsky et al, PRL 102, 081601 (2009)). 
The predictions for the pion TMDs  can be tested through pion-induced Drell-Yan process at COMPASS, and predictions for pion/kaon EM form factors (see Fig. \ref{RQCD}) can be accessible by the future experiments at JLab and the EIC.  

One of the recent group's finding, for example, is that light pseudoscalar and vector mesons share a universal holographic light-front wavefunction, modified by dynamical spin effects \footnote{
\href{https://doi.org/10.1103/PhysRevD.98.034010}{M. Ahmady et al, Phys. Rev. D 98, 034010 (2018)}}. The group also propose the spin structure to augment the LFWFs in order to achieve agreement with the experimental data using a universal confinement mass scale and provide the predictions for rare $B_s\to\phi\mu^+\mu^-$ decay rate  \footnote{
\href{https://doi.org/10.1103/PhysRevD.95.074008}{M. Ahmady et al, Phys. Rev. D 95, 074008 (2017)}; 
\href{https://doi.org/10.1103/PhysRevD.100.113005}{M. Ahmady et al, Phys. Rev, D 100, 113005 (2019)}}.  

Another set of projects include evaluation of the spin-independent electric and magnetic dynamical polarizabilities for the lowest in mass SU(3) octet of baryons \footnote{\href{https://doi.org/10.1088/0954-3899/38/3/035004}{A. Aleksejevs et al, J.Phys. G38, 035004 (2011)}} and spin-dependent dynamical polarizabilities \footnote{
\href{https://pos.sissa.it/174/470/pdf}{A. Aleksejevs et al, PoS ICHEP2012 469 (2013)}}, with work on including decuplet of resonances in all electric, magnetic and spin-dependent polarizabilities now in progress. Drs. Aleksejevs and Barkanova are also involved in feasibility studies of experiments and co-author proposals such as measuring the neutral pion polarizability proposal submitted to PAC 48 at JLab in 2020.

Both LFHQCD and ChPT can predict form factors for the proton and charged pion, so the group is also investigating the link between ChPT and LFHQCD in the low-energy range. 
With LFHQCD studies of the pion form factor, radius, decay constant and photon-to-pion transition amplitude extending into the kinematical regime of ChPT, it became possble to link LFHQCD and ChPT in SU(2) and SU(3) leading to a new range of opportunities in this challenging field.

Although the conventional quark
model represents the simplest combinations of quarks that can form colour singlets, there is a rich range
of other colour-singlet exotic structures such as gluonium (consisting entirely of gluons), hybrids (a
conventional hadron combined with gluonic content), and multiquark states such as four-quark 
mesons and pentaquark baryons. A related approach currently under investigation by Dr. Steele at the Subatomic Physics Institute at the University of Saskatchewan (SPIN) is theoretical studies of mass spectrum of exotic hadrons beyond the conventional quark model with QCD sum-rules \footnote{
\href{https://doi.org/10.1103/PhysRevD.100.034012}{J. Ho, R. Berg, T. G. Steele, Wei Chen, and D. Harnett, Phys. Rev. D 100, 034012 (2019)}} \footnote {
\href{https://doi.org/10.1103/PhysRevD.98.096020}{J. Ho, R. Berg, T. G. Steele, Wei Chen, and D. Harnett, Phys. Rev. D 98, 096020 (2018)}}. 

With two theorists (Drs. Steele and Bick) and three experimentalists (Drs. Boland, Pywell and Rangacharyulu), the subatomic physics research group at SPIN has the critical mass to recruit new members and to maintain expertise in advanced quantum mechanics, QED, accelerator physics, and physics relevant for the Canadian Light Source, and to provide excellent career opportunities for their students in the areas covered by SPIN.

 \begin{figure}
		\centering
			\subfigure{\includegraphics[width=.45\textwidth]{./Figs/pionEMFF.pdf} }
			\subfigure{\includegraphics[width=.45\textwidth]{./Figs/kaonEMFF.pdf} }
			\caption{Electromagnetic form factor for pion (Left graph) and kaon (right graph) predicted by holographic QCD given for three different values of the dynamical spin parameter (B) and compared with experimental data.}
\label{RQCD}
\end{figure}

\subsubsection{Precision studies with electromagnetic probes} \label{sec:precisionstudies}

One way to obtain access to physics at multi-TeV scales is with high-precision electroweak experiments such as parity-violating Moller scattering, $e^+ e^-$ collisions or electron-nucleon scattering. Although these low-energy experiments tend to be less expensive than experiments at the high-energy colliders, they are more model-dependent and require significant theoretical input, at two-loop (next-to-next leading order (NNLO) and in some cases beyond. The low-energy sector is not as well served as the LHC, and there are no ready-to-use routines available for complete calculation of electroweak radiative corrections. The main challenge is that, for electroweak processes, the full two-loop calculations face dramatic complications due to massive vector bosons in the two-loops integrals, and thus require a new set of analytical tools. 

At Memorial University of Newfoundland, the electroweak precision research program is focused in two major directions: development of advanced methods for calculations of extensive sets of Feynman diagrams at many-loop level and applications of these methods to the specific kinematic conditions of various experiments. In other words, the group works on building and applying computational models for multi-loop theory input specifically required by electroweak precision frontier in the searches of physics beyond the Standard Model. In many cases, the same methodology is applicable for low-energy hadronic physics calculations, discussed in Section X. The recently-developed dispersive sub-loop insertion approach \footnote{
\href{https://doi.org/10.1103/PhysRevD.98.036021}{A. Aleksejevs, Phys. Rev. D 98, 036021 (2018)},
\href{https://arxiv.org/abs/1905.07936}{A. Aleksejevs et al, arXiv:1905.07936}} can be used to effectively reduce two-loop calculations to one-loop with an additional dispersive propagator. The approach is algorithmic and can be applied to the processes with mixed masses in the loop integrals, which is an essential component of any Beyond the Standard Model (BSM) searchers, electroweak theories, and low-energy hadronic physics.

%Blunden

Probes involving electromagnetic and electroweak interactions can also be instrumental for understanding the fundamental properties of nucleons and of nuclei. Although in purely-leptonic processes such as Moller scattering and $e^+ e^-$ collisions the main challenge in multi-loop radiative corrections, for the electron-nucleon scattering these need to be complimented by the equally-vital one-loop corrections accounting for the hadronic effects.

In the electromagnetic sector, calculations of two-photon exchange (TPE) radiative corrections have been instrumental in resolving the discrepancy between measurements of electron-nucleon scattering form factors using Rosenbluth and polarization-transfer techniques, in making progress in understanding the proton radius problem, and in the interpretation of other precision measurements in electron scattering. Low-energy experiments in atomic parity-violation and in parity-violating electron-proton scattering have the potential to give constraints on new physics, provided that radiative corrections, especially the critical hadronic radiative corrections, are understood. A major component of the theoretical research program lead by Dr. Blunden at the University of Manitoba is aimed at unravelling these hadronic contributions and their associated uncertainties through the use of dispersion relations based on data, covering both the deep-inelastic regime as well as nucleon resonances. The dispersive analysis avoids off-shell uncertainties inherent in traditional approaches based on direct evaluation of loop diagrams, and guarantees the correct unitary behavior in the high energy limit. For example, \footnote{\href{https://doi.org/10.1103/PhysRevC.95.065209}{P.G. Blunden and W. Melnitchouk, Phys. Rev. C 95, 065209 (2017).}} examines the two-photon exchange corrections to elastic electron-proton scattering within a dispersive approach, including contributions from both nucleon and delta intermediate states, and \footnote{
\href{https://arxiv.org/abs/2006.12543}{J. Ahmed, P.G. Blunden, W. Melnitchouk, arXiv: 2006.12543}}
includes contributions from hadronic $J^P=1/2^{\pm}$ and $3/2^\pm$ resonant intermediate states below~1.8~GeV.

\subsubsection{Nuclear many-body problem and astrophysical signatures} \label{sec:theory:guelph}

With the addition of Dr. Caballero to Dr. Gezerlis, the nuclear theory effort is starting to form a hub at the University of Guelph, aiming to provide answers to overarching questions related to nuclear forces, novel states in nuclei and matter, as well as the behaviour of matter at thermodynamic extremes found in neutron stars, their mergers, accretion disks around black holes, and supernovae.

A major goal is to connect nuclear many-body theory with experiment, nuclear-force theories,
cold-atom studies, as well as the investigation of compact stars. The work is heavily computational
involving both qualitative and {\em ab-initio} approaches and is touching on few-
body physics, nuclear structure, nuclear astrophysics, and the interface with atomic physics. The aim is to predict multi-messenger signals that in juxtaposition with observations will shed light into the structure of nuclear matter, the nature of stellar explosions and the synthesis of elements via r- and rp-processes, as well as provide theoretical input to experimental nuclear searches. mental nuclear searches. 

In addition to traditional phenomenological many-body approaches, in the Guelph group employs several {\em ab-initio} many-body methods, most of which can be
described as Quantum Monte Carlo (QMC) non-perturbative simulations. 

The main research findings can be categorized as follows: (a) the use
of local chiral EFT in light nuclei and neutron matter\footnote{\href{https://doi.org/10.1103/PhysRevLett.116.062501}{J.E. Lynn {\em et al.}, Phys. Rev. Lett. 116, 062501 (2016).}}, (b) novel developments in first-principles or mean-field techniques \footnote{\href{https://doi.org/10.1103/PhysRevC.99.014321}{E. Rrapaj, A. O. Macchiavelli, and A. Gezerlis, Phys. Rev. C 99, 014321 (2019).}}, (c) the use of {\em ab-initio} methods to constrain selected aspects of more phenomenological approaches\footnote{\href{https://doi.org/10.1103/PhysRevLett.116.152501}{M. Buraczynski and A. Gezerlis, Phys. Rev. Lett. 116, 152501 (2016)}; \href{https://doi.org/10.1103/PhysRevLett.122.152701}{M. Buraczynski, N. Ismail, and A. Gezerlis, Phys. Rev. Lett. 122, 152701 (2019).}}, and (d) the application of techniques developed for nuclear theory to systems of
ultracold atoms \footnote{\href{https://doi.org/10.1103/PhysRevA.101.033601}{T. Zielinski, B. Ross, and A. Gezerlis, Phys. Rev. A 101, 033601 (2020).}}. As seen
already in this list, a main strength of our work is the ability to work at interfaces, whether between
different theories ({\em ab-initio} vs phenomenology) or between different physical systems (neutron stars
vs cold atoms).

The work is complementary to the efforts of the TRIUMF Theory group, since both groups
employ chiral Effective Field Theory (EFT) interactions which are then used in different few- and
many-nucleon frameworks.


The goal of Guelph’s program dedicated to astrophysical signatures is 
to build a consistent theoretical framework to predict
multi-messenger observations. The predictions are planned to include, among others, elemental abundances and neutrino counts, and will help constrain nuclear interactions and the nuclear matter
equation of state, offer limits on the detection of neutrinos in the Galaxy and at cosmological
distances, and provide input to numerical relativity simulations and experimental searches. 

\begin{figure}[ht!]
\begin{center}
\includegraphics[height=11.0 cm]{./Figs/observer.pdf}
\caption{Electron antineutrino surfaces for a steady-state model Ca  with constant accretion rate $3M_\odot$/s (outer) and dynamical torus Ja (inner). The observer is on the symmetry axis of the disk. The upper-half of the Ca neutrino surface is shown with lines to allow visualization of the Ja neutrino surface. The color scale indicates the local temperature at the antineutrino surface (Schilbach, T.~S.~H., Caballero, O.~L., \& McLaughlin, G.~C. Phys. Rev. D 100, 043008 (2019)).}
\label{astrocab}
\end{center}
\end{figure}

This program is building on the previous work on the role of black hole spin and the observer's inclination, for distant detection of black hole accretion disks\footnote{\href{https://doi.org/10.1103/PhysRevD.93.123015}{O.L. Caballero, T. Zielinski, G.C. McLaughlin, and R. Surman, Phys. Rev. D 93, 123015 (2016).}}, the detection of the relic neutrino background from accretion disks around black holes \footnote{\href{https://doi.org/10.1103/PhysRevD.100.043008}{T.S.H. Schilbach, O.L. Caballero, and G.C. McLaughlin, Phys. Rev. D 100, 043008 (2019).}} (see Fig. \ref{astrocab} ), and multi-messenger signals from neutron star mergers \footnote{\href{https://doi.org/10.1103/PhysRevD.92.044045}{C. Palenzuela {\em et al.}, Phys. Rev. D 92, 044045 (2015)}; \href{https://doi.org/10.1088/0264-9381/33/18/184002}{L. Lehner {\em et al.}, Class. Quant. Grav. 33, 184002 (2016).}}. On the accreting neutron stars front, progress was made on the understanding of the impact of degeneracy on neutron capture rates in accreting neutron stars\footnote{
\href{https://doi.org/10.3390/universe5010036}{B. Knight and L. Caballero, Universe 5(1), 36 (2019)}; \href{https://doi.org/10.1088/1742-6596/1078/1/012012}{Knight {\em et al.}, J. Phys. Conf. Ser. 1078, 012012 (2018).}}.

The Guelph’s nuclear-astro efforts are complementary to experimental research carried out at TRIUMF, and the neutrino program is aligned with the research goals of SNOLAB. 


\subsubsection{Ab-initio nuclear theory and applications} \label{sec:theory:abinitio}

Weakly bound or even unbound exotic nuclei produced at rare isotopes facilities
like TRIUMF can only be understood using methods that unify the description of
both bound and unbound states. 

One of the goals of TRIUMF theory group is to develop a predictive {\it ab-initio} theory of nuclear structure and nuclear reactions for light and medium mass nuclei.
The research team lead by Dr. Navratil combines an {\it ab-initio} nuclear structure approach capable to describe bound states of nuclei with a microscopic cluster method developed to describe reactions of nuclear clusters. The resulting method called the no-core shell model continuum (NCSMC) \footnote{\href{https://doi.org/10.1103/PhysRevLett.110.022505}{S. Baroni, P. Navratil, and S. Quaglioni, Phys. Rev. Lett. 110, 022505 (2013)} and  \href{https://doi.org/10.1103/PhysRevC.87.034326}{Phys. Rev. C 87, 034326 (2013)}; \href{https://doi.org/10.1088/0031-8949/91/5/053002}
{P. Navratil, S. Quaglioni, G. Hupin, C. Romero-Redondo, A. Calci, Physica Scripta 91, 053002 (2016).}} provides a unified description of both bound and unbound nuclear states starting from inter-nucleon interactions among protons and neutrons with the only input coming from nucleon-nucleon and three-nucleon interactions derived within the chiral Effective Field Theory. The reacting system is described using a basis expansion with two key components: one describing all nucleons close together, forming the composite nucleus, and a second one describing the separated clusters. The goals include investigations of reactions important for astrophysics, including radiative capture reactions such as ($p,\gamma$), ($n,\gamma$) and ($\alpha,\gamma$), the charge exchange and transfer reactions involving $\alpha$ particles, i.e., ($\alpha,n$), ($\alpha,p$). 

Another focus of applications of {\it ab-initio} NCSMC and NCSM (a simplified version applicable to bound states only) is on interpretation of precision experiments testing fundamental symmetries and physics beyond the Standard Model. Examples include the determination of the $v_{ud}$ matrix element of the CKM matrix and its unitarity tests, neutrinoless double beta decay experiments, as well as searches for nuclear electric dipole moments. Similarly, atomic and molecular parity-violating experiments require support from nuclear theory in calculations of nuclear structure contributions to anapole moments. 

%The method primarily used in this project is the no-core shell model with continuum (NCSMC) \footnote{S. Baroni, P. Navratil, and S. Quaglioni, Phys. Rev. Lett. {\bf 110}, 022505 (2013); Phys. Rev. C {\bf 87}, 034326 (2013), P. Navratil, S. Quaglioni, G. Hupin, C. Romero-Redondo, A. Calci, Physica Scripta {\bf 91}, 053002 (2016)}. This approach describes the reacting system using a basis expansion with two key components: one describing all nucleons close together, forming the composite nucleus, and a second one describing the separated clusters. 

Recent highlights of advanced NCSMC applications with chiral 3N forces include a paper on the polarized Deuterium-Tritium fusion in Nature Communications \footnote{\href{https://doi.org/10.1038/s41467- 018-08052-6}{G. Hupin, S. Quaglioni, and P. Navratil, Nat. Comm. 10, 351 (2019).}}, a study of the halo nucleus $^{11}$Be \footnote{\href{https://doi.org/10.1103/PhysRevLett.117.242501}{A. Calci, P. Navratil, R. Roth, J. Dohet-Eraly, S. Quaglioni, G. Hupin, Phys. Rev. Lett. 117, 242501 (2016).}} and the $^{10}$C scattering on proton \footnote{\href{https://doi.org/10.1103/PhysRevLett.118.262502}{A. Kumar, R. Kanungo, A. Calci, P. Navratil, {\it et al.}, Phys. Rev. Lett. 118, 262502 (2017).}} measured at TRIUMF IRIS facility (see %Fig.~\ref{fig:pC10}
Fig.~\ref{fig:IRIS-10C-Highlight}). 

The project is aligned with TRIUMF ISAC/ARIEL program and relates to the TRIUMF astrophysics group project ``Nucleosynthesis and Nuclear Energy Release from the Early Universe to the Stars.'' It also complements the project by Dr. Holt (TRIUMF) on {\it ab-initio} investigations of medium mass nuclei within the In-Medium SRG method and, to some extent, projects by Dr. Gezerlis (Guelph). 

% figure is already in the key questions section
%\begin{figure}
%\includegraphics[width=14 cm]{./Figs/pC10_4p15.pdf}
%\caption{Measured differential cross section for $^{10}$C($p,p$)$^{10}$C$_{gs}$ %at $E{=}4.16$ MeV compared to
%{\it ab-initio} NCSMC calculations. The black, blue, and red solid curves are %with the chiral NN, NN+3N400, N$^2$LO$_{\rm sat}$ interactions, respectively. %The red dashed curve is the N$^2$LO$_{\rm sat}$ calculation scaled down by a %factor of 2.3. The experiment was performed at TRIUMF IRIS facility. For more %details, see Ref. [A. Kumar, R. Kanungo, A. Calci, P. Navratil, {\it et al.}, %Phys. Rev. Lett. {\bf 118}, 262502 (2017].}
%\label{fig:pC10}
%\end{figure}

For Dr. Holt, the central focus is to advance his many-body approach, the valence-space formulation of the in-medium similarity renormalization group (VS-IMSRG), which can be thought of as an {\em ab-initio} shell model approach to atoms and nuclei, using the latest two and three nucleon forces developed by the community. In terms of nuclear structure, this novel approach will soon allow TRIUMF team to reach the heaviest region of the nuclear chart and predict the neutron skin of $^{208}$Pb as well as explore the superheavy region and search for a potential island of stability. The plan is also to investigate the structure of exotic nuclei, extending the predictions for the nuclear driplines and evolution of magic numbers to the heavy region of nuclei. 

Among the fundamental questions related to nuclear-weak physics, the most prominent are calculations of the nuclear matrix element for neutrinoless double beta decay and WIMP-nucleus structure functions for dark matter direct detection searches. Both processes are now implemented into the VS-IMSRG framework with results up to Ge, and planned extension to the vitally important Xe region soon.
 
Of course, both atomic and subatomic theory are vital to interpret searches for violations of fundamental symmetries in the universe. The TRIUMF team is currently developing both atomic and nuclear theory needed for such experiments including anapole moments, Schiff moments, and EDM, as well as calculating the atomic theory inputs for measurements of isotope shifts, which will give absolute nuclear charge radii and can be connected to other interesting searches for BSM physics.

The most recent progress includes the first global {\em ab-initio} calculations of all nuclei from helium to nickel, finding an rms deviation from absolute experimental ground-state energies of approximately 3~MeV, a level obtained with phenomenological nuclear mass models. The first {\em ab-initio} predictions for the proton and neutron driplines in the light and medium-mass region at least through Ni are shown in Fig. \ref{drip_unc}. To investigate the evolution of shell structure, we have also calculated two-neutron separation energy and first excited 2+ systematics for new magic numbers throughout the sd and pf shell region to future efforts at next generation RIB facilities worldwide.

In nuclear-weak physics we have recently included the effects of two-body currents in calculations of Gamow-Teller (GT) transitions. In a joint effort with coupled-cluster collaborators at ORNL, we provided the first systematic {\em ab-initio} study of $g_A$ quenching up to the Sn isotopes, published in Nature Physics \footnote{\href{https://doi.org/10.1038/s41567-019-0450-7}{P. Gysbers {\em et al.},Nat. Phys. 15, 428 (2019).}}. 


\begin{figure*}[t]
\includegraphics[width=\textwidth]{./Figs/AllProbBound.pdf}
\caption{Calculated probabilities for given isotopes to be bound with respect to one- or two-neutron/proton removal. The gray region indicates nuclei that have been calculated, while the height of the blue boxes corresponds to the estimated probability that a given nucleus is bound with respect to one- or two-neutron (proton) removal in the neutron-rich (deficient) region of the chart. The inset shows the global agreement with experimental ground-state energies. Figure from Ref. [J.D. Holt, S.R. Stroberg, A. Schwenk and J. Simonis, arXiv:1905.10475 (2019)]}
\label{drip_unc}
\end{figure*}

\subsubsection{Nuclear Theory and Experiment -- Synergy}\label{sec:synergy}
%new section

In addition to the development of the first-principle approaches addressing the key questions of our universe, an important role of subatomic theory is to guide and interpret experiments. In spite of being funding-limited, our theory groups are still recognized as world leaders in many areas, which is in large part due to the close coordination and collaboration with the experiments at home and off-shore combined with the SAPES funding model allowing some degree of synergy. While some theory programs require steady progress and years of advance planning to meet experimental timelines, most theorists can provide an immediate and flexible response to new opportunities. We only aim to provide some examples of theory and experiment synergy in this section -  there are too many to list. In addition, due to a nimble nature of theory research, theorists are well-positioned to respond to new research opportunities anywhere in the world and often collaborate with experimental groups abroad, greatly contributing to cross-connectivity between different research groups and areas of subatomic physics.

The control over backgrounds is one such area where combining theory and experiment efforts is crucial. 
For example, two large experiments Mu2e  \footnote{\href{https://doi.org/10.3389/fphy.2019.00001}{R.~H. Bernstein, Front.~in Phys. 7, 1 (2019).}} and COMET  \footnote{\href{https://doi.org/10.1051/epjconf/201921201006}{D.~Shoukavy, EPJ Web Conf. 212, 01006 (2019).}}
are about to search for the very rare muon-electron conversion near
a nucleus, $\mu N\to eN$, extending the search for lepton-flavor non-conservation
currently conducted by MEG  \footnote{\href{https://doi.org/10.22323/1.369.0081}{S.~Mihara, PoS NuFact2019, 081 (2020).}}.
Whereas MEG is looking for $\mu\to e\gamma$ and aims for a sensitivity
of around one exotic process per $10^{14}$ normal muon decays, Mu2e
and COMET will be sensitive to a broader set of New Physics processes
at the level of $10^{-16}$ -- $10^{-18}$. It is a challenge to theorists to map the spectrum of produced electrons precisely, taking into account bound states, nuclear effects, and radiative corrections.
There will also be an abundance of data, especially in the high-energy
tail of the electron spectrum, where the Standard Model background
overlaps with the exotic signal region  \footnote{\href{https://doi.org/10.1103/PhysRevD.102.073001}{M.~J. Aslam, A.~Czarnecki, G.~Zhang, and A.~Morozova, Phys. Rev. D 102, 073001 (2020).}}.

The physics of pions and neutrons is another area where the theory support for the experiment is essential.
One example is a new motivation for an improved measurement of the pion
beta decay, $\pi^{+}\to\pi^{0}e^{+}\nu\left(\gamma\right)$ \footnote{\href{https://doi.org/10.1103/PhysRevD.101.091301}{A.~Czarnecki, W.~J. Marciano, and A.~Sirlin, Phys. Rev. D 101, 091301 (2020).}}; another case is the neutron beta decay  \footnote{\href{https://doi.org/10.1103/PhysRevD.100.073008}{A.~Czarnecki, W.~J. Marciano, and A.~Sirlin, Phys. Rev. D 100, 073008
  (2019).}}, and, in both processes, understanding the interplay of radiative effects and hadronic
structure is still a major challenge.

Although theory is often more nimble than experiment and can address new challenges more quickly, sometimes a whole new set of theory tools needs to be created to address the needs of the experiment. 
One such emerging field are multi-loop electroweak radiative corrections evaluated with unprecedented precision and reliability needed for the upcoming ultra-precision experiments like MOLLER and P2. These are challenging, time-intensive research programs spanning multiple years and requiring close coordination with the experiment. For example, the first publication on one-loop electroweak radiative corrections by the theorists in the Canadian MOLLER group was in 2010  \footnote{\href{https://doi.org/10.1103/PhysRevD.82.093013}{A. Aleksejevs {\em et al.}, Phys.Rev.D 82, 093013 (2010)}}, the group is still just about to complete the large-scale evaluation of two-loop corrections while developing a new methodology for the task  \footnote{\href{https://doi.org/10.1103/PhysRevD.98.036021}{A. Aleksejevs {\em et al.}, Phys.Rev.D 98, 036021 (2018)}}. On the other hand, the theory project which was initially inspired by the MOLLER experiment's ambitious precision, has grown into a research field of its own, with many potential applications for planned and yet-to-be imagined measurements. 

The {\it ab-initio} calculations of double beta decay nuclear matrix elements became a focus of a broad north-American collaboration with contributions by at least three different groups. TRIUMF theory participates in IM-SRG calculations of $^{48}$Ca, $^{76}$Ge and $^{136}$Xe and in benchmarking of the coupled-cluster and IM-SRG calculations in light nuclei. A research program relevant for the whole neutrinoless double beta decay community including nEXO, an exciting work on {\it ab-initio} NMEs calculations can push towards the heavier candidates like $^{136}$Xe\footnote{\href{https://arxiv.org/abs/2008.06588}{A. Belley {\em et al.}, arXiv:2008.06588 (2020)}}, and the study of the quenching factor (Figure \ref{fig:GT-quenching}) will impact the sensitivity of the next generation of double beta decay searches. 

Several experiments performed at the IRIS facility at TRIUMF ISAC benefited from theoretical interpretation provided by {\it ab-initio} nuclear structure and reaction calculations, such as the $^{11}$C-proton scattering experiment, for example. Nuclear theory also supports planned measurements of the nuclear spin-dependent parity-violating effects in triatomic molecules by evaluating relevant nuclear anapole moments \footnote{\href{https://}{Y. Hao {\em et al.}, Phys. Rev. A 102, 052828 (2020)}}.

%The investigation of the structure of exotic $^{20}$Mg \footnote{\href{https://journals.aps.org/prc/abstract/10.1103/PhysRevC.99.021301}{J.S. Randhawa {\em et al.}, Phys. Rev. C 99, 021301 (2019)}} is another example. 

%(Y. Hao, P. Navratil, E. B. Norrgard, M. Ilias, E. Eliav, R. G. E. Timmermans, V. V. Flambaum, and A. Borschevsky, Phys. Rev. A 102, 052828 (2020)).

%(J.S. Randhawa, R. Kanungo, M. Holl, J.D Holt, P. Navratil, S.R. Stroberg, G. Hagen, G.R. Jansen, M. Alcorta, C. Andreoiu et al., “Observation of excited states in 20Mg sheds light on nuclear forces and shell evolution,” Phys. Rev. C 99, 021301 (2019)) 

At the same time, while well-recognized and in high demand as collaborators internationally, Canadian theorists are sometimes stretched too thin, and, as a result, miss out on exciting research opportunities which would require timely results. Progress in theory is HQP-driven, but the majority of Canadian theorists are not funded at a sufficient level to include Postdoctoral Research Fellows in their groups. Postdocs have a major impact on the research capacity - not only can they work independently on important and timely projects, they also contribute to collaborative discussions and help to train students. An optimized NSERC funding model allowing access to postdocs would help our theory community better support the experimental programs in Canada and off-shore.

\subsection{Beyond the next five years}

Owing to its very nature, theoretical research is less depended on material infrastructure that requires long-term advance planning than its experimental counterpart. However, some of the projects outlined above are both long-term and labor-intensive, and thus require planning for continued or additional personnel such as postdocs, especially if the theory input is required for the timely progress of experimental projects. 

Coast to coast, the Canadian researchers are at the forefront of the many developing trends in nuclear and hadronic theory with many examples of close involvement in the major breakthroughs of the field, and is clear that advances in experimental nuclear physics will be accompanied by comparable progress in theory. In addition to the many Canadian experimental endeavours receiving theoretical support, theorists -- during the coming years and beyond -- will continue their association with major off-shore laboratories such as EIC, GSI/FAIR, RIKEN, NSCL/FRIB, JLab, the LHC, and RHIC. Our community will continue to make significant contributions in all the major areas that define modern nuclear theory, in the time period relevant for this report.\\

%\noindent{\underline{\it Lattice QCD and Non-Perturbative Approaches}} 
\subsubsection{Lattice QCD and non-perturbative approaches} 
Lattice QCD provides a first-principles method to explore the possibility of bound states other than the standard set of hadrons. For instance, recent experiments report results for new states near the charmonium and bottomonium thresholds that may involve tetraquarks or some similar states. Lattice QCD has the potential to study those scenarios and to interpret the underlying physics. In this context and in the short- and medium-term, new lattice methods will continue to be refined in the context of conventional bottom-quark hadrons that could prove useful for lattice QCD studies of unconventional bound states. In the finite temperature and baryon density domain, new techniques will be perfected to extend calculations of the QCD equation of state to higher densities, where the QCD action acquires an imaginary value that challenges the probabilistic nature of lattice calculations. Canadians are also actively pursuing efforts to obtain a gravity dual to QCD, thereby enabling the use of string theory techniques to perform analytical, strong-coupled field theory calculations of quantities with experimental relevance.\\

%\noindent{\underline{\it Perturbative QCD}} 
\subsubsection{Perturbative QCD}
Much of our current knowledge of the substructure of hadrons is based on perturbative QCD, the foundation of which is factorization: the ability to theoretical separate short- and long-distance physics, and to therefore deal with them separately. In the coming years, cross section measurements, paired with precise calculations up to high perturbative order  -- with the implied sophisticated technology -- will continue to be a necessary requirement for obtaining a  precise map of the hadronic substructure, including that of the transition region between microscopic and emergent degrees of freedom. Those developments will include the creation of sophisticated numerical tools for the automatic computation of high-order amplitudes. \\

%\noindent{\underline{\it Effective Field Theory (EFT)}} 
\subsubsection{Effective field theory (EFT)}
EFTs are a powerful tool in cases where the physics at hand requires a separation of energy scales. This turns out to be the case in QCD, where the long distance phenomena are described in terms of non-perturbative hadrons, and the short-distance physics is formulated in terms of partons. At energies below the proton mass, chiral effective theory incorporates the spontaneous breaking of QCD's chiral symmetry. It has successfully been applied to mesons for some time, and recent breakthroughs have occurred in its use in few-nucleon systems. Canadians theorists will therefore continue to test this theory by performing studies in light- and medium-mass nuclei, in a continuing fruitful dialogue with their experimentalist colleagues. Concrete future plans  for calculations involving the NCSMC (using $\chi$EFT)  include the evaluation of $(p,\gamma)$, $(\alpha,\gamma)$ and $(n,\gamma)$ capture reactions in light nuclei that are relevant for astrophysics.  The ultimate goal for the next five years in that field  is to study reactions involving $^{4}$He, e.g. $^{8}$Be$(\alpha,\gamma)$$^{12}$C, $^{12}$C$(\alpha,\gamma)$$^{16}$O, as well as the neutron source reaction $^{13}$C$(\alpha,n)$$^{16}$O relevant for the $s$-process.\\

%\noindent{\underline{\it Phenomenology and Model-Building}} 
\subsubsection{Phenomenology and model-building}
The development  of phenomenological approaches guided by empirical data are an essential step that often pave to way to the development of more fundamental theories. They not only serve to guide intuition, but therefore fill the gap that often exists between first-principles approaches and experimental measurements. This category encompasses a large variety of different models and techniques that have been very successfully used by Canadian researchers. Good examples of continuing work are the cross-comparison of results from chiral perturbation theory with lattice calculations, potential model predictions of hadronic mass spectra, and the development of geometries in weakly-coupled gravity that translate to QCD-like field theories under a duality transformation.

\subsubsection{Nuclear structure and nuclear reactions}

Atomic nuclei are intimately connected to some of the most profound questions in science, such as the nature of neutrinos and dark matter, fundamental symmetries, and the inner-workings of neutron stars and the nucleosynthesis pathways.
Thus, a comprehensive first-principles, or {\em ab-initio}, picture of all atoms and nuclei, with robust uncertainty estimates, is required to provide reliable theoretical predictions to help answer such questions. 
Experimental efforts in these directions are a clear priority for virtually all major international players over the next decade, from the numerous next-generation rare isotope beam facilities, to neutrinoless double-beta decay and dark matter direct detection searches worldwide, and all will require accurate and quantifiable subatomic theory input to both guide and interpret.

\subsection{Summary}
It is essential to maintain a diverse program of research in theoretical nuclear and hadronic physics. 
A large portion of this theoretical work demands the features associated with HPC facilities; these aspects are discussed in section \ref{sec:hpc} of this document. In addition, much of the progress in theory research is linked to the mentoring of the next generation of theorists. A great opportunity therefore exists to further strengthen and grow this program with strategic investment into highly-qualified personnel who can accelerate the efforts of the recognized Canadian world leaders at the forefront of an exciting and fast-moving discipline that is intimately linked to Canadian experimental efforts in nuclear physics. With a wider regional representation, theorists are also an excellent resource for increasing diversity in the field by attracting women, visible minorities and Indigenous students to this fascinating field, in both theory and experiment.  