##  Chapter 0 Fundamental Physics in Small Experiments

T. Blum, P. Winter
T. Bhattacharya, T.Y. Chen, V. Cirigliano, D. DeMille, A. Geraci, N.R.
Hutzler, T.M. Ito, D. Kaplan, O. Kim, R. Lehnert, W.M. Morse, Y.K. Semertzidis

###  1 Introduction

High energy physics aims to understand the fundamental laws of particles and
their interactions at both the largest and smallest scales of the universe.
This typically means probing very high energies or large distances or using
high-intensity beams, which often requires large-scale experiments on earth or
in space. Relevant projects for this approach, like particle colliders,
telescopes, or other large-volume detectors have provided tremendous insights
but come with increasingly large associated costs.

A complementary approach to probe high energy scales is offered through high-
precision measurements in small- and mid-scale size experiments, often at
lower energies. The field of such high-precision experiments has seen
tremendous progress and importance for particle physics for at least two
reasons. First, they exploit synergies to adjacent areas of particle physics
and benefit by many recent advances in experimental techniques and
technologies. Together with intensified phenomenological explorations, these
advances have led to the realization that challenges associated with weak
couplings or the expected suppression factors from the mass scale of new
physics can be overcome with such methods while maintaining a large degree of
experimental control. Second, many of these measurements add a new set of
particle physics phenomena and observables that can be reached compared to the
more conventional methodologies using high energies. Combining high-precision
measurements of smaller-scale experiments with the large-scale efforts
described above therefore casts both a wider and tighter net for possible
effects originating from physics beyond the Standard Model (BSM).

The efforts described in this section are focused on providing crucial
insights into possible BSM physics by studying areas such as novel short-range
interactions, the small-scale structure of spacetime and in particular the
fate of Lorentz, translation, CPT, CP, T, and P symmetries, the gravitational
interaction of antimatter, certain quantum aspects of gravity, millicharged
particles, gravitational-wave measurements, dark matter and dark energy. The
growing impact of these high-precision studies in high energy physics and the
complementary input they provide compared to large-scale efforts warrants
strong support over the next decades. This section presents a broad set of
small-scale research projects that could provide key new precision
measurements in the areas of electric dipole moments (EDMs) (see Sec. 1),
magnetic dipole moments (see Sec. 2), fermion flavor violation (see Sec. 3),
and tests of spacetime symmetries (see Sec. 4) and tests with gravity (see
Sec. 5).

An area particularly ripe for discovery is the search for EDMs of elementary
particles which provide strong constraints on BSM theories like SUSY and, more
generally, any with complex phases. Rapid progress in this field is
anticipated due to new ideas and techniques from the traditional HEP community
as well as adjacent communities in nuclear and AMO physics. For example the
storage ring proton EDM experiment [1], with its detailed plans innovated off
decades of muon magnetic dipole moment experiments, is expected to increase
the sensitivity for the proton EDM by four orders of magnitude over the
current limit. Likewise, AMO experiments may achieve up to an incredible six
orders of magnitude improvement in two decades. EDMs are one of a very small
class of experiments capable of probing the PeV scale presently. The discovery
of an EDM would set an upper bound on the BSM scale of new physics and serve
as a guide to new experiments at the energy frontier. Finally, many EDM
experiments (like the srEDM) are capable of sensitive searches of dark matter
and dark energy as well.

###  2 Science opportunities

The rare processes and precision experiments covered in our topical group are
driven by the search for the unknown physics, dark matter and dark energy. In
this section we summarize several white papers covering the relevant
experimental proposals and corresponding theory needed to interpret them: EDMs
including the proton srEDM experiment, AMO and neutron EDM searches; the
theory of hadronic contributions to the muon anomalous magnetic moment and
related experimental data; and small, precision experiments to test
fundamental spacetime symmetries.

####  1 Electric Dipole Moments

The following has been condensed and lightly edited from the executive summary
of the white paper entitled “Electric dipole moments and the search for new
physics” [2].

Observation of a static, ground-state electric dipole moment (EDM) in any
experimental system (electron, neutron, proton, atom, molecule) at current or
near-future sensitivity would yield exciting new physics. EDMs can probe BSM
mass scales well beyond those probed at high energy colliders. To discriminate
between the many viable BSM theories, rule out baryogenesis scenarios for the
observed matter-antimatter asymmetry of the Universe, or tell whether CP
symmetry is spontaneously or explicitly broken in Nature, a well coordinated
program of complementary EDM searches in AMO, nuclear, and particle physics
experiments is needed (see Fig. 1). In Ref. [2] a compelling suite of such
experiments and an encompassing theoretical framework are proposed to discover
and establish the next fundamental theory of physics.

Figure 1: Timelines for the major current and planned EDM searches with their
sensitivity to the important parameters of the effective field theory (see
Fig. 2 for details). Solid (shaded) symbols indicate each experiment’s primary
(secondary) sensitivities. Measurement goals indicated by the black arrows are
based on current plans of the various groups.

Searches for fundamental EDMs have a long history, beginning with the neutron
EDM (nEDM) beam experiments of Purcell and Ramsey in 1949. Modern experiments
use ultracold neutrons (UCN) that can be polarized and stored in room-
temperature bottles for hundreds of seconds, leading to very precise
measurements. The best measurement of the nEDM stands at $(0.0\pm 1.1_{\rm
stat}\pm 0.2_{\rm sys})\times 10^{-26}\,e\cdot\rm cm$ [3] while several UCN
experiments are being developed around the world with the goal of reaching
slightly above $10^{-27}$ within the next 5–10 years and a few $\times
10^{-28}$ in 10–15 years.

Like neutrons, atoms and molecules have been sensitive platforms for precision
measurements of symmetry violations for decades, including CP-violation (CPV)
through EDMs [4, 5]. These experiments now set the best limits on the electron
EDM, semileptonic CPV interactions, and quark chromo-EDMs, and are competitive
with the nEDM for sensitivity to quark EDMs and $\theta_{QCD}$ [6, 7],
providing an excellent check on both types of experiments. This class of EDM
searches uses techniques from atomic, molecular, and optical (AMO) physics,
including very advanced methods for quantum control, to achieve their goals.
Improvements in sensitivity of one, two–three, and four–six orders of
magnitude appear to be realistic on the few, 5–10, and 15–20 year time scales,
respectively, by leveraging major advancements in quantum science techniques
and the increasing availability of exotic species with extreme sensitivity.

A whole new class of EDM experiments is possible in storage ring experiments.
A ring at BNL or Fermilab, using off-the-shelf technology and a design based
on the successful muon $g\\!-\\!2$ experiments can be operational and produce
first results for the proton in less than 10 years [1, 2]. The proton storage
ring is expected to reach $10^{-29}e\cdot\rm cm$ sensitivity in 10 years, and
the same level in an additional five years for the deuteron. The storage ring
experiment can also search for wave-like dark matter and energy. EDM storage
ring experiments for the electron and muon have been proposed at JLab [8] and
PSI [9], respectively. The latter aims for a sensitivity of $6\times
10^{-23}e\cdot$cm, about four orders of magnitude better than the limit set by
BNL E821. While not a conventional storage ring, the J-PARC g-2/EDM experiment
will probe the sensitivity of the muon EDM to $1.5\times 10^{-21}e\cdot$cm
[10].

Finally, with a beam polarization upgrade at SuperKEKB, the Belle II
experiment projects to improve the tau EDM measurement by two orders of
magnitude over existing bounds [11].

Fermion EDMs originate at a high mass scale through new complex CP-violating
phases and feed down to lower energy scales in a Standard Model effective
theory as summarized in Fig. 2. These elementary particle EDMs then manifest
in bound states like the proton and neutron, and even atoms and molecules. At
the quark-nucleon level, lattice QCD plays a crucial role in this matching
between energy scales. At lower energies still, nucleon chiral perturbation
theory, and finally, nuclear and atomic theories are needed. The reverse holds
as well: if a nucleon, atomic, or molecular EDM is measured, the underlying
BSM physics is tested and can be diagnosed only if the theoretical
calculations of the matrix elements reduce their current uncertainties by
factors of 2–3. This remarkable effective theory framework, encompassing tens-
of-orders of magnitude in energy, is in place today, but the calculations will
need substantial effort and computing resources over the next decade.
Crucially, the low-energy theory is continually improved and forms a basis for
any new high energy models that are invented.

The effective field theory captures the most important effects of the physics
at the high-energy scale in terms of a hierarchy of effective ‘operators’ at a
lower energy scale. These operators are organized by a dimension that
quantifies the expected power suppression of its effects by the ratio of the
low to high scales. Considering only the electric dipole moments of the light
charged leptons, nucleons, nuclei, atoms and molecules, the leading effects at
the scale characterizing nonperturbative aspects of QCD, i.e., 300 MeV–1 GeV,
are the QCD vacuum angle ($\theta$), the electric dipole moments of the
leptons ($\mu$ EDM, and $d_{e}$) and the quarks ($d_{q}$), the chromo-electric
dipole moments of the quark ($\tilde{d}_{q}$) and the gluon ($w$), and some
four-fermi interactions between the quarks and the electron ($C_{qe}$), or
involving only quarks ($C_{qq}$). The relation between these and the
fundamental theory is given in terms of Wilson coefficients that can be
calculated perturbatively. At the nuclear scale of a few MeV, these operators,
in turn, give rise to the CP-violating interactions of the electron with the
nucleons ($C_{S,P,T}$), the CPV pion-nucleon coupling $g_{\pi NN}$, and the
EDMs of the nucleons. Obtaining all of these requires the calculation of non-
perturbative matrix elements. It is anticipated that only lattice QCD will
provide them with the required precision to make full use of the expected
experimental constraints in the next decade. The atomic and nuclear EDMs, at
eV to MeV scales, can be expressed in terms of these more fundamental
quantities but need quantum mechanical calculations involving nuclear
structure as well as the electronic structure of atoms and molecules to relate
them. Figure 2 shows these relationships in the effective field theory between
the various scales. In the figure solid arrows depict the major determinants
at a higher scale for the lower scale physics, whereas the dashed lines
indicate less strong influences.

Figure 2: Flowdown diagram from the fundamental physics at high energy scales,
to the Wilson coefficients of the effective field theory, low energy
parameters, and the experimental CPV observables. Color outlines of the
various boxes inidcate the different energy scales. Solid arrows between the
boxes indicate strong connection, whereas dashed arrows indicate weaker
influence onto the lower lying parameter. Experimental systems shown in red
have already been used in EDM searches; those shown in black (as well as many
of those in red) are being developed for future searches. This figure was
adapted from [12].

##### The proton storage Ring EDM experiment

The storage ring EDM (srEDM) collaboration has proposed a proton storage ring
experiment [1, 13] based on off-the-shelf technology that can be sited in the
805 m AGS ring tunnel at BNL with an electric field strength of 4.4 MV/m. The
expected sensitivity, $10^{-29}e\cdot$cm, could be reached in less than 10
years from the start of construction and represents a three orders of
magnitude enhancement for the QCD $\theta$-term EDM over the current limit set
by neutron EDM experiments. The experiment has reach to new physics scales as
high as 1000 TeV and can probe models of vector dark matter and energy. The
various experiments are complemntary, the sensitivity of the proton EDM to CP-
violation in the Higgs sector is 30$\times$ that of the electron EDM [14], for
example, due to the smallness of the electron mass.

The BNL AGS produces $10^{11}$ highly polarized protons per fill. One fill of
the EDM ring per twenty minutes is required. As with the anomalous magnetic
moment of the muon experiments at CERN, BNL, and FNAL, the key is running at
the magic momentum (the proton magic momentum is 0.7 GeV/c.) where the
$g\\!-\\!2$ spin process in an electric field is zero. This provided a
2000-fold reduction in systematic uncertainty from CERN II to FNAL E989.

The muon magnetic moment experiments have used magnetic bending and electric
focusing. The electric dipole moment experiment will use electric bending with
magnetic focusing. The distortion of the closed orbit then automatically
compensates spin precession from radial magnetic fields, to first order. With
the following symmetries, the systematic uncertainties are estimated to be
below $10^{-29}e\cdot$cm [1]: storage of clockwise and counter-clockwise
bunches, storage of longitudinally polarized bunches with positive and
negative helicities and radially polarized bunches, 24-fold symmetric lattice
to support the ring, sign change of the focusing/defocusing quadrupoles within
0.1% of the ideal current setting per flip, beam planarity better than 0.1 mm,
split between counter rotating beams less than 0.01 mm, closed orbit BPM
resolution integrated over one second of one $\mu$m for the traditional linear
cut BPM and much better for SQUID BPMs.

With both the BNL E821 and FNAL E989 experiments, new effects were found at
the level of sensitivity, but were understood, and mitigated. A similar
experience is expected for the pEDM experiment. For the magic momentum in the
AGS tunnel, for example, the bending electric field is 4.4MV/m, which is
smaller than the Tevatron pbar-p separator electric field.

The time for construction is estimated at three to five years, with three
years of data collection and analysis.

##### Atomic and Molecular EDMs

The basic idea of AMO-based EDM searches is that CP-violating electromagnetic
moments of atomic and molecular constituents (i.e., electrons and nuclei) can
be amplified by the extreme internal electromagnetic environment. The CPV
interactions of electrons and nuclei in these “internal fields” can be orders
of magnitude larger than those directly realizable in laboratory fields. Due
to the presence of both electrons and nuclei, AMO-based searches are sensitive
to a wide range of underlying fundamental effects, such as the electron EDM,
new CPV nuclear forces, quark-chromo EDMs, $\theta_{QCD}$, and more. The
relative sensitivity to these effects depends on the electronic and nuclear
structure of the atom or molecule species studied.

It was long understood that the relevant “internal fields” in polar molecules
can be 3-4 orders of magnitude larger than in atomic systems [15, 16].
However, only recently have the methods of quantum control used for atoms
started to become available for increasingly complex systems such as
molecules. New molecule-based experiments using such methods led to a 100-fold
improvement in sensitivity to the electron EDM over the past decade. In the
meantime, atom-based experiments have achieved steady improvements in
sensitivity to CPV signals.

Figure 3: Electron EDM limits versus time, along with new physics reach for
one-loop and two-loop effects (see [17]). The solid line indicates the most
sensitive experimental limit, including the species used. The shaded area
indicates potential future improvements discussed in the text.

Improvements of existing experiments. Several AMO-based CPV searches have
yielded limits in the past decade, including experiments using YbF [18], HfF+
[19], and ThO [20, 6] molecules for the electron EDM, and using Hg [7], Ra
[21, 22], and Xe [23, 24] atoms for nuclear CPV via nuclear Schiff moments.
Most of these experiments are being upgraded to yield improvements, by factors
from 2 to 30, within a few years. Several also have longer-term plans for
improvements by several orders of magnitude beyond that, by using some of the
advances described below. Each of these experiments use considerably different
technologies with correspondingly different systematic errors, and use species
with different and complementary sensitivities to the underlying physical
sources of CPV. Hence, their mutual success is important to advance the
overall scientific goals.

New and extended methods for cooling and trapping. Laser cooling and trapping,
along with the quantum control enabled by ultracold temperatures, has been a
main driver of quantum science advances with atoms. Extending these techniques
and applying them to species with high sensitivity to CPV effects promises to
yield orders of magnitude improved sensitivities, by enabling very long spin
coherence times and/or enhanced statistics in such systems. The atomic Ra EDM
experiment has provided a preliminary demonstration of the power of this
approach [21, 22]. Major advances are now being made in the laser cooling and
trapping of neutral polar molecules, opening the possibility to also leverage
their enhanced sensitivity to CPV effects [25, 26, 27]. Broad advances in
molecular ion trapping have resulted in new, successful methods for CPV
measurement [19], quantum control [28], and the trapping and control of exotic
species with extreme sensitivity to CPV effects [29, 30]. New approaches also
are being developed to embed EDM-sensitive species in noble gas matrices, to
realize extremely high statistics with potentially long coherence times [31,
32, 33, 34].

Advances in quantum control. AMO-based CPV experiments rely on creation of
quantum superpositions that are closely analogous to those used in quantum
information science (QIS). While the techniques used so far are typically
primitive compared to those used in modern QIS experiments, advances in the
QIS field are a powerful resource for future dramatic improvements in
sensitivity to EDM-like CPV signals [35]. Applying certain types of many-
particle entanglement developed for quantum metrology applications—such as
spin squeezing—to experiments aimed at detecting CPV signals could provide
gains in sensitivity of up to a factor of $\sim\sqrt{N}$ (where $N$ is the
number of particles in a single measurement) [36]. Since typically $N\gg 1$ in
AMO-based EDM experiments, this could in principle enable improved sensitivity
by orders of magnitude; already in atomic clock-like systems a factor of 10
beyond the standard quantum limit has been demonstrated [37].

Access to exotic species. Heavy nuclei with octupole deformations, such as
some isotopes of Fr, Ra, Th, Pa, and others, can have sensitivities to CPV
enhanced up to a thousand-fold compared to spherical nuclei [38, 39, 21, 40,
41, 42, 43]. Combined with molecular enhancements, heavy molecular species
with deformed nuclei can be up to $10^{7}$ times more intrinsically sensitive
[44, 41] to hadronic CPV than the current most sensitive experiment, which
uses atomic Hg [7]. With such systems, even the ability to trap a single
molecule with second-scale coherence would enable probes of new physics at the
frontiers of hadronic CP violation [45, 30, 29]. The short half-lives of these
species offer significant challenges, but several short-lived atomic species
have been laser-cooled and trapped. Experimental facilities such as GSI/FAIR,
FRIB, IGISOL, ISOLDE, and TRIUMF make it possible to perform experiments with
short-lived and exotic species; offline sources based on decay of heavier,
longer-lived isotopes [46, 47] may also be useful.

Community effort. As mentioned above, different atomic/molecular species offer
different relative sensitivities to the many potential underlying physical
sources of CPV; moreover, each experimental method offers different advantages
and different ways to diagnose and control systematic errors. Interpretation
of experimental results in terms of underlying fundamental physics, as well as
guidance to identify CPV-sensitive yet experimentally tractable species,
requires integrated efforts from a broad range of theorists bridging different
traditional communities including atomic, molecular, chemical, nuclear, and
particle physics. To continue the rapid pace of advances in this part of the
field and to continue opening new pathways, dedicated efforts are needed to
bring these communities together and support a broad portfolio of experimental
and theoretical work.

Summary. The new and upcoming developments in this field aim to probe a wide
range of CPV physics—with signals arising from underlying leptonic, hadronic,
and semileptonic CPV interactions—combining one or more of the new approaches
discussed above. Improvements by roughly an order of magnitude are anticipated
in the next few years, and potentially by several orders of magnitude in the
coming 5–10 years. This part of the field is exceptionally fast-moving at
present, and offers significant potential for discovery of physics beyond the
Standard Model.

##### Neutron EDM

The EDM of the neutron is, at the leading order, sensitive to
$\overline{\Theta}_{\rm QCD}$, the EDMs of quarks, the chromo-EDMs (cEDMs),
and the CP violating (CPV) four-quark interactions [48]. Unfortunately, the
coefficients of these are very poorly known [49, 50, 51, 52, 53, 54, 55, 56,
57, 58]:

$\displaystyle d_{n}=$ $\displaystyle-(1.5\pm 0.7)\cdot
10^{-3}\,\overline{\Theta}_{\rm QCD}\,e\,{\rm fm}$ $\displaystyle-(0.20\pm
0.01)\,d_{u}+(0.78\pm 0.03)\,d_{d}+(0.0027\pm 0.016)\,d_{s}$
$\displaystyle-(0.55\pm 0.28)\,\tilde{d}_{u}\,e-(1.1\pm
0.55\,)\tilde{d}_{d}\,e+(50\pm 40)\,\tilde{d}_{g}\,e\,{\rm Mev}\,,$ (1)

where all numbers are quoted in the $\overline{\rm MS}$-scheme at $2\ \rm
GeV$, $d$ and $\tilde{d}$ denote the EDMs and cEDMs of various particles, the
effects of 4-quark CPV interactions, about which even less is known, are not
shown. These large uncertainties dilute the impact of nEDM constraints on BSM
physics—in fact, studies of specific models [59] show that the matrix elements
must be known at the 10–25% level to reduce the possibility of accidental
cancellation between the various sources of CPV. This level of precision will
only be available in the foreseeable future from lattice QCD calculations,
which have seen rapid progress in recent years. Nevertheless, controlling the
systematics due to chiral extrapolation to the physical quark mass, light
multi-hadron excited state contamination of the nucleon state, the mixing with
lower dimensional operators, as well as reducing the statistical fluctuations
will need substantial theoretical and computational investment.

The first experiment to search for the nEDM was performed by Purcell and
Ramsey in 1949 using a cold neutron beam at the Oak Ridge reactor [60], giving
an upper bound of $5\times 10^{-20}$ $e\cdot$cm. Over the last 7 decades, the
limit has improved by more than six orders of magnitude, representing
significant improvements on the experimental method both to improve the
statistical sensitivity and to control the systematic effects at the
corresponding level. The current limit ($|d_{n}|<1.8\times 10^{-26}$
$e\cdot$cm, 90% C.L.) is given by the recent results from PSI [61]. This
result is dominated by statistical uncertainty. For this reason, there are
many efforts worldwide to build UCN sources to host new nEDM experiments: the
n2EDM experiment at Paul Scherrer Institute (PSI) in Switzerland [62], the
PanEDM experiment at Institute Laue-Langevin (ILL) in France [63], LANL nEDM
experiment at Los Alamos National Laboratory [64], and the Tucan experiment at
TRIUMF in Canada [65]. These experiments are based on Ramsey’s separated
oscillatory method with a comagnetometer and are to be performed at room
temperature. They are expected to produce results with sensitivity of
$\mathcal{O}(10^{-27})$ $e\cdot$cm in the next $\sim 5$ years. The nEDM@SNS
experiment [66], which employs a novel approach based on an innovative use of
superfluid liquid helium [67], is being developed with a sensitivity goal of
$\delta d_{n}=3\times 10^{-28}$ $e\cdot$cm. Like other nEDM experiments, this
experiment provides built-in methods to control the known systematic effects.
Finally, an experiment using a pulsed beam of cold neutrons, the Beam EDM
experiment [68, 69], is being developed at ILL and is eventually targeted for
the European Spallation Source (ESS) in Sweden. Because it has different
systematics, it will compliment the UCN experiments.

##### Tau EDM

The electric dipole moment of the $\tau$ lepton characterizes the T or CP
violation properties at the $\gamma\tau^{+}\tau^{-}$ vertex. The SM predicts
$d_{\tau}$ in the range $10^{-38}-10^{-37}$ $e\cdot\rm{cm}$ [70, 71, 72, 73],
many orders of magnitude below any experimental sensitivity. Best results come
from a recent Belle study [74], where the squared spin-density matrix of the
$\tau$ production vertex is extended to include contributions proportional to
the real and imaginary parts of the $\tau$ EDM. The expectation values of the
optimal observable were measured, yielding Re$(d_{\tau})=(-6.2\pm 6.3)\times
10^{-18}~{}{e}\cdot\rm{cm}$ and Im$(d_{\tau})=(-4.0\pm 3.2)\times
10^{-18}~{}{e}\cdot\rm{cm}$. Belle II is uniquely suited to test a large class
of new physics models, which predict enhanced contributions in EDM of the
$\tau$ lepton at observable levels of $10^{-19}$ $e\cdot\rm{cm}$ [75, 76]. The
proposed beam polarization upgrade at SuperKEKB will further increase
experimental sensitivity at Belle II [11], since the uncertainties in modeling
the forward-background asymmetry in the detector response are independent of
beam polarization and will largely cancel. Such an increase will enable
unambiguous discrimination between the new physics contributions to the $\tau$
EDM at the level of $10^{-20}~{}e\cdot\rm{cm}$, which is two orders of
magnitude below any other existing bounds [77, 78, 79].

##### Outlook

New techniques and technology innovations provide powerful opportunities for
elementary particle EDM searches. A proton storage ring experiment can improve
the sensitivity by three orders of magnitude within ten years while
simultaneously searching for wave-like dark matter and energy. Gains of three
to six orders on the AMO side are possible in 10–20 years. UCN experiments are
underway or being built that should see sensitivity improve by two orders of
magnitude in 10–15 years. Direct lepton EDM searches are expected to make
similar gains. Taken together with concomitant increase of precision in the
theoretical calulations, the shape of the new physics, should an EDM be
discovered, will be sharply constrained. And if an EDM is not discovered, the
tight constraints will provide important direction for where to look next.

####  2 Magnetic Dipole Moments

The prototypical precision experiment in our frontier is the measurement of
the muon $g\\!-\\!2$, or anomaly, $a_{\mu}$. Beginning in the late 1950’s at
CERN, then moving to the US at BNL, and finally to Fermilab, it has been
extremely successful in probing and constraining new physics beyond the
Standard Model. Interest in the current experiment, E989, is sky-high, and
could continue beyond the expected end of the experiment in a few years,
depending on the outcome of on-going theoretical calculations of the hadronic
contributions to $g\\!-\\!2$.

In 2020 the Muon g-2 Theory Initiative released a comprehensive report on the
status and value of $a_{\mu}$ in the Standard Model [80]. Taken together with
the 2021 Run-1 result from E989 [81] and the BNL E821 result [82], experiment
and theory differ by about 4.2 combined standard deviations. However, at the
same time as the Fermilab announcement, the BMW lattice collaboration
announced its calculation of the hadronic vacuum polarization (HVP)
contribution to $a_{\mu}$ [83], which reduces the discrepancy to about 1.5
standard deviations.

In the 2020 report, the HVP contribution is taken from data-driven theory,
using the $e^{+}e^{-}\to$ hadrons cross section and a dispersion relation. The
errors from all other collaborations besides BMW are about four times larger
than the dispersive ones, which has touched off an intense effort in the
lattice community to reduce errors by a factor of four, or more. Experiments
at Belle II and future tau-charm factories aim to improve the dispersive
result as well. These are the subjects of white papers submitted to our
topical group[84, 85, 86], which we now summarize.

##### The muon g-2 in the Standard Model

The dominant errors in the Standard Model value for the muon $g\\!-\\!2$ [84]
arise from the leading and next-to-leading order hadronic contributions: the
hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL)
scattering. Figure 4, reproduced from [84], summarizes both. The error on the
HVP contribution is about twice the HLbL error, but with more data and
improved analysis the HVP may be reduced to where the two are comparable. To
reach the expected level of precision of the E989 measurement ($16\times
10^{-11}$), the HVP error ($40\times 10^{-11}$) should be reduced by a factor
of three, at least, while the HLbL error ($17\times 10^{-11}$) which is
already at about that level should be reduced so the goal on the total error
can be reached.

To claim discovery of new physics or test the Standard Model to even better
precision it is mandatory to have consistency between dispersive and lattice
calculations. As emphasized in [84], the highest priority now is to scrutinize
the BMW HVP result [83] while improving the other lattice results to the same
level of precision. At the time of the original Muon g-2 Theory Initiative
[80] the lattice average for the HVP contribution had an error of about 2.6%
while the later BMW result [83] comes in around 0.75%, resulting in a Standard
Model value that is consistent with experiment within 1.5 sigma. However it is
only about two sigma away from the data-driven result quoted in Refs. [80,
84]. It should also be stressed that the BMW error is dominated by
systematics, and in particular, the continuum limit extrapolation.

Different lattice groups use different discrete versions of continuum QCD, so
results must agree only in the continuum limit, $i.e.$, after extrapolation to
zero lattice spacing. While the leading dependence is known analytically, it
is challenging numerically to reach the limit. An intermediate step is to chop
up the total into smaller chunks according to the Euclidean separation of the
two electromagnetic currents comprising the HVP correlation function [87]. The
“window” quantities, especially those that avoid both very short and long
distances, are already precise enough to detect important differences. For
example, for the so-called intermediate window between 0.4 and 1.0 fm
separation, Aubin, et al., first reported a value significantly above the
R-ratio value [88]. The BMW result is 3.7 sigma above the R-ratio (dispersive)
result. Other groups find similar discrepancies [88, 89, 90, 91, 92], while
the RBC/UKQCD result [87] is consistent with the R-ratio value. It is expected
that by the second half of 2022, several lattice groups will have published
window quantities that have errors well below the percent level and that a
precise comparison among all of groups will be made.

To tackle the long-distance part of the HVP contribution requires established
noise-reduction techniques [93, 94, 95, 96, 97, 98] and recent algorithmic
improvements, like two-pion exclusive state reconstruction [99, 100]. The two-
pion state dominates the long-distance part of the correlation function and
can be computed more accurately as a separate contribution. This more accurate
“tail” in turn leads to a more aggressive, improved “bounding method” where
$a_{\mu}$ is determined by the overlap of strict upper and lower bounds at
relatively short distance, and noisy, but exponentially small contributions
can be neglected. Again, several groups, including Aubin, $et\,al.$, BMW, ETM,
Fermilab Lattice-HPQCD-MILC, Mainz, and RBC/UKQCD are working on next-
generation results with sub-percent errors to match BMW that are expected
towards the end of 2022 or early 2023. These computations are not cheap, and
continued progress to achieve sub-percent and even permille precision requires
continued, sustained access to substantial HPC resources.

The dispersive/data-driven error for the HVP contribution is at the 0.5
percent level, but tensions between the most precise data sets from BaBar and
KLOE, in the dominant two-pion channel, are a longstanding issue [80]. New
data and covariance matrix from SND and BESIII, respectively, are not yet
included in the global consensus value. New $2\pi$ results from BaBar, CMD-3,
BESIII, and Belle II, are expected soon. If the differences between the BABAR
and KLOE experiments can be resolved with the upcoming 2$\pi$ new data and
analyses, a precision of 0.3% seems feasible by 2025 [84].

Then what remains is to resolve possible differences between data-driven and
lattice results, which is also likely, as the various calculations, with ever
increasing precision, are scrutinized and compared with each other. As implied
above, the window quantities can be directly compared to the R-ratio data, so
differences between lattice and data-driven values can be isolated to specific
distances. Finally, if all differences are resolved, data-driven and lattice
results will be combined to yield a result with better precision than either
separately.

Figure 4 shows the current world average for the HLbL scattering contribution
to the muon $g\\!-\\!2$ (left panel). Like the HVP, there are both lattice and
data-driven results, but unlike the HVP, the two are quite compatible.

Since the so-called Glasgow consensus value (2009), significant strides have
been made in computing the leading pole and pion-loop contributions from data
and dispersion relations [80] which reduced the error considerably. At the
same time, the first complete lattice calculation was also finished, and the
two are combined in the world average (see Fig. 4 with an error of about 18%
which is less than 1/2 the HVP error. It seems the HLbL contribution is now
under very good control, and by itself can not explain a difference between
the experimental value and the Standard Model.

On the data-driven side, work remains for the scalar, tensor, and axial-vector
channels, as well as short-distance quark-loop contributions. New data from
Belle II and a dedicated two-photon program by BES III [101] may help reduce
some of the uncertainties [102], especially for the axial-vectors. Though they
are sub-leading, they dominate the uncertainty. For the lattice, a sustained
effort is needed to reduce dominant statistical and systematic errors
associated with finite volume and non-zero lattice spacing, which in total
amount to about twice the data-driven error. The recent Mainz calculation
[103] quotes a smaller error but relies on a large chiral extrapolation to
reach the physical point. Both the RBC and Mainz groups are improving their
calculations and could achieve 10% precision by 2025, roughly a factor of two
improvement over the current data-driven error.

Figure 4: The hadronic contributions to the muon anomaly from hadronic light-
by-light scattering (left) and hadronic vacuum polarization (right). Both
figures are taken from Ref. [84]. Left: Comparison of HLbL evaluations quoted
in Ref. [80] to earlier estimates (orange) and a recent lattice calculation
[103]. Right: Comparison of theoretical predictions of $a_{\mu}$ with
experiment [81, 82] (orange band). Each data point represents a different
evaluation of the leading-order HVP contribution. Red squares show data-driven
results; filled blue circles indicate lattice-QCD calculations that were taken
into account in the WP20 lattice average (blue band), while the open ones show
results published after the deadline for inclusion in that average; the purple
triangle gives a hybrid of the RBC/UKQCD18 and J17 results. The SM prediction
of Ref. [80] is shown as the black square and gray band and includes only
data-driven results.

.

##### New R-ratio measurements

Both existing and new $e^{+}e^{-}$ collider experiments promise significant
error reductions in the data-driven, dispersive, values for $a_{\mu}$ through
more precise measurements of the R-ratio (ratio of $e^{+}e^{-}\to$ hadrons to
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ cross-sections). Analyses of high-statistics
$2\pi$ data collected by the BaBar, BESIII, and CMD-3 experiments are ongoing
and are expected to provide new, more precise results within the next one-to-
two years. In China a symmetric machine [85] running at 2–7 GeV center-of-mass
energy is projected to have errors at or below 2%. Meanwhile Belle II will
probe the all-important two-pion channel from threshold to 3 GeV [102], and
compared to Babar and KLOE, “Belle II will perform these measurements with
larger data sets, and at least comparable systematic uncertainty, to resolve
this discrepancy” [86]. The importance of resolving the difference between
BaBar and KLOE in the 2$\pi$ channel cannot be overstated: this resolution
alone could reduce the total uncertainty by almost a factor of two. Finally
new $\tau$ semileptonic decay measurements at Belle II and the super tau-charm
factory may provide important additional information to improve the
determination of the HVP [102] if better quantification of isospin breaking
effects, which may be possible using lattice QCD, is realized.

##### The anomalous magnetic moment of the $\tau$ lepton

Present deviations of the observed magnetic moment of the muon from its SM
prediction make tau lepton measurements compelling, as the contribution to the
anomalous magnetic moment of a lepton is enhanced by the lepton mass-squared
in Minimal Flavor Violation (MFV) scenarios [104, 105, 106, 107] or generic
enhancements in other well motivated models [108]. The experimental
determination of the anomalous magnetic moment relies on the determination of
the cross-section or partial widths for $\tau$-pair production, together with
spin matrices or angular distributions of the $\tau$-decay products [109,
110]. The SM prediction [111] for the magnetic moment form factor of the
$\tau$ lepton is $-2.7\times 10^{-4}$ at Belle II energies [112]. The current
experimental results give ${\mathcal{O}(10^{-2})}$ precision [113, 114], which
is still weaker than the leading term $\frac{\alpha}{2\pi}=1.6\times 10^{-3}$
from Schwinger’s famous prediction [115]. With 40 ab-1 of
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ data with polarized electron beams,
requiring both the $\tau^{+}$ and $\tau^{-}$ to decay hadronically and
assuming a 60% selection efficiency, the statistical error on $a_{\tau}$ would
be $10^{-5}$ [11]. As the measurements involve differences in the asymmetries
of left-right polarization states of the beam, the dominant detector
systematic uncertainties cancel. Consequently, a polarization-upgraded
SuperKEKB constitutes a promising way to precisely measure $a_{\tau}$ at Belle
II. The path towards eventually constraining BSM contributions to $a_{\tau}$
at the $10^{-6}$ level will require more statistics as well as higher
precision measurements of $m_{\tau}$ and $M_{\Upsilon(1S)}$ [11], as well as
improved modeling of the moments of the $\tau$ lepton in the event generators
KK2F [116, 117, 118], Tauola [119, 120] and Photos [121, 122].

##### Outlook

The muon $g\\!-\\!2$ has sustained intense interest for almost 20 years, from
the BNL E821 measurement to the announcement of new results from Fermilab
E989. The Standard Model prediction based on data-driven calculations of the
HVP contribution shows a 4.2 sigma disagreement. A five sigma or even larger
disagreement is possible after E989 releases its final value in a few years.
Meanwhile an-almost-as-precise lattice calculation of the HVP contribution,
which suggests there may not be a discrepancy, has provided additional
motivation to the world-wide lattice community to undertake next-generation
calculations with the same, or better, precision. In contrast, both lattice
and data-driven HLbL calculations are in good agreement, signaling that any
discrepancy between theory and experiment cannot be explained by this
notoriously difficult to calculate QCD effect.

A big hurdle on the experiment side was overcome when E989 released first
results that were quite consistent with the BNL experiment E821. While not an
issue for concern at this time, the systematics of the storage ring
experiments will be tested by a whole new experiment based on cold muons at
J-PARC in Japan with first results possibly by 2027. The first phase of the
muon g-2/EDM (E34) experiment aims for precision goals of 0.5 ppm for
$g\\!-\\!2$. An upgrade of SuperKEKB to enable polarized beams offers an
opportunity to dramatically improve the measurement of the anomaly for tau
leptons by several orders-of-magnitude.

####  3 Fermion Flavor Universality

##### Lepton flavor universality with $\tau$ decays at Belle II

The fundamental assumption that all three leptons have equal coupling to the
charged gauge bosons of the electroweak interaction is known as charged-
current lepton universality. Previous measurements of universality [123],
expressible in terms of the coupling strength ($g_{\ell}$) of lepton of flavor
$\ell$ to the charged gauge boson of the electroweak interaction, are in
agreement with the Standard Model (SM) where $g_{\tau}=g_{\mu}=g_{e}=1$. A
broad class of SM extensions violate this assumption, such as two-Higgs-
doublet model [124] or charged-scalar singlets [125], etc. making searches for
violation of lepton flavor universality attractive. Significant deviations of
this nature are unambiguous signatures of new physics that provide crucial but
complimentary information to the extended Higgs sector [126, 127, 128] and
other new physics models with leptoquarks [129].

Lepton universality is probed by a wide class of experiments, such as ${\cal
B}(W\to\tau\nu_{\tau})/{\cal B}(W\to\mu\nu_{\mu})$ at the LEP [130] and the
LHC [131, 132], the $R(D^{(*)})={\cal B}(B\to D^{(*)}\tau\nu_{\tau})/{\cal
B}(B\to D^{(*)}\ell\nu_{\ell})$ [133, 134, 135], where $\ell$ = $\mu$, $e$,
and $R(K^{(*)})={\cal B}(B\to K^{(*)}\mu^{+}\mu^{-})/{\cal B}(B\to
K^{(*)}e^{+}e^{-})$ [136, 137, 138] anomalies from the B-Factories and LHCb,
kaon decays from NA62 [139], pion decays from PIONEER [140] as well as in tau
decays from BELLE II [141]. New physics that couples primarily to the third
generation could be revealed through deviations from the SM in precision
universality. While LHC measurements of lepton universality via
$W^{\pm}$-boson decays are sensitive only to charged currents, tests of lepton
flavor universality via $\tau$ leptons offer additional sensitivity to non-SM
contributions to weak neutral currents [142, 143].

Measurements in $\tau$ decays typically consist in determining precisely
branching-fractions ratios, such as $R_{\mu}\equiv\frac{{\cal
B}(\tau^{-}\to\mu^{-}\overline{\nu}_{\mu}\nu_{\tau})}{{\cal B}(\tau^{-}\to
e^{-}\overline{\nu}_{e}\nu_{\tau})}$ to test $\mu$-$e$ charged-current lepton
universality $g_{\mu}/g_{e}$, or $\frac{{\cal
B}(\tau^{-}\to\pi^{-}\nu_{\tau})}{{\cal
B}(\pi^{-}\to\mu^{-}\overline{\nu}_{\mu})}$ and $\frac{{\cal B}(\tau^{-}\to
K^{-}\nu_{\tau})}{{\cal B}(K^{-}\to\mu^{-}\overline{\nu}_{\mu})}$ for
$\tau$–$\mu$ charged-current lepton universality $g_{\tau}/g_{\mu}$. Particle
identification is the key experimental challenge, since particle species
distinguish between the relevant $\tau$ decay modes. Given the large sample
sizes, precision is typically limited by the uncertainties in the corrections
needed to match particle-identification efficiencies determined from
simulation to those observed in data calibration-samples. For instance, the
most precise measurement of $R_{\mu}$ from the BaBar experiment [144] has 0.4%
precision, which propagates to 0.2% precision on $g_{\mu}/g_{e}$, dominated by
the systematic uncertainty associated with lepton identification. Early Belle
II data show that lepton-identification uncertainties comparable to those in
Ref. [144] are at reach, thereby enabling to improve upon the world average
values obtained by the HFLAV [123]. In this baseline scenario, Belle II would
match the current best results, yielding a significant improvement in global
precision obtainable by future HFLAV combinations. In an advanced scenario
where an improved understanding of the detector and availability of more
abundant and diverse calibration samples reduces lepton-identification
uncertainties by a further factor of two, Belle II would lead to improved
precision on the $g_{\mu}/g_{e}$ determination [141].

An important input to lepton-flavor universality tests are measurements of
$\tau$ lifetime. The global value is dominated by the Belle result based on
reconstructing both $\tau$ decays into three charged particles,
$\tau(\tau)=(290.17\pm 0.52{\rm(stat)}\pm 0.33{\rm(syst)})\times
10^{-15}~{}\rm{s}$ [145]. The size of the Belle II data set will significantly
reduce the statistical uncertainties. The superior control of the vertex-
detector alignment, demonstrated in recent charm lifetime measurements [146],
will reduce the dominant systematic uncertainty. The expected absolute
precision of $0.2\times 10^{-15}~{}\rm{s}$ or better, will further improve the
precision of $g_{\tau}/g_{e}$.

Belle II will significantly improve the precision on inputs to lepton-flavor-
universality-violating quantities, yielding some of the most stringent
constraints on non-SM deviations from charged and neutral current lepton
universality [141]. The fact that high precision tests of lepton universality
using $\tau$ decays agree well with the Standard Model are particularly
interesting in light of the experimental hints for universality violations
observed in semileptonic B decays, the anomalous magnetic moment of the muon,
and the Cabbibo angle anomaly [143].

##### Precision electroweak physics with SuperKEKB polarization upgrade

Polarized $e^{-}$ beams open up new windows of exploration through measurement
of the left-right asymmetry $(A_{\it LR})$ of the fundamental electroweak SM
parameter $\sin^{2}\theta_{W}$. A beam polarization upgrade of the SuperKEKB
has been proposed to pursue this unique precision program, along with the rest
of the core physics program at the Belle II experiment. This is described in
more detail in the Snowmass 2021 Whitepaper [11]. $\tau$-lepton polarization
measurements would provide precise ($<0.5\%)$ determinations of beam
polarization, required for measurements of left-right asymmetry. With 20 ab-1
of beam polarized data, the precision on $\sin^{2}\theta_{W}$ measured with
$\tau$s, as well as with muons, electrons, b-quarks and c-quarks, will be
comparable or better than the current world average. Expected precision on
$\sin^{2}\theta_{W}$ with 40 ab-1 of beam polarized data from SuperKEKB is
0.0002 at Belle II.

When $\sin^{2}\theta_{W}$ is measured with multiple fermions, these studies
will produce neutral-current lepton universality measurements of unprecedented
precision as the beam-polarization dependence cancels in the measurements of
the ratio $A^{f_{1}}_{LR}/A^{f_{2}}_{LR}$, producing a measurement of the
vector part of the fermion couplings ($g^{f_{1}}_{V}/g^{f_{2}}_{V}$).
Contributions from hadronization, which were the dominant component of the
uncertainties of the forward-backward asymmetries of the b- and c-quarks at
LEP, are avoided with polarized beams. For example, $g_{V}^{b}/g_{V}^{c}$
would be measured with a relative uncertainty below 0.3% with 20 ab-1 of beam
polarized data, which is an order of magnitude lower than the current
uncertainty on this ratio [130]. In addition, right-handed $b$-quark couplings
to the $Z$ can be experimentally probed with high precision at Belle II with
polarized beams. No other experiment, currently running or planned, can
perform such precision tests of vector coupling universality in neutral
currents [11].

SuperKEKB will yield the unique possibility of probing “dark forces” that can
serve as portals between baryonic matter and dark matter. SuperKEKB with
polarization complements other measurements as it is uniquely sensitive to a
parity violating light neutral gauge boson in the dark sector ($Z^{\prime}$)
under various mass and coupling scenarios, including models where $Z^{\prime}$
couples more to the 3rd generation via mass-dependent couplings. For example,
a 15 GeV $Z^{\prime}$ would cause a shift in the measurement of
$\sin^{2}\theta_{W}$ in the energy region where SuperKEKB with polarized beams
may have the best potential for discovery [147]. With polarized beams,
SuperKEKB can also probe parity violating couplings of new heavy particles
that couple only to leptons, complementing electroweak studies at the LHC.
Thus, this “Chiral Belle” proposal probes parity violation both at very low
energies much below the Z-peak, and very high energies [11].

In addition to the precision measurements of the weak mixing angle at 10 GeV,
the Chiral Belle physics program with polarized beam also enables the
measurement of $(g\\!-\\!2)_{\tau}$ at an unprecedented and unrivaled level of
precision, as discussed in Section 2. Other physics uniquely enabled with
polarized electron beams includes precision measurements of the tau EDM, as
discussed in Section 1. In addition, searches for lepton flavor violation in
tau decays and dynamical mass generation hadronization studies will be
enhanced with polarized beams [11].

####  4 Spacetime symmetries

Lorentz and CPT symmetry are foundational principles within the boundaries of
established high-energy physics as well as key assumptions in many of the
Standard Model extensions. In many of these theoretical frameworks, small
departures from these symmetries are allowed in particle ground states.
Searches for violation of Lorentz and CPT invariance hence poses an important
effort within particle physics. Phenomenological and experimental Lorentz- and
CPT-symmetry studies therefore fall within the confines of high-energy
physics, are critical to the future of the community, and should be
intensified.

This section presents a series of ongoing and future efforts that focus on
testing spacetime symmetries. Neutron interactions with heavy nuclei are used
in the NOPTREX experiment to search for P-odd/T-odd interactions. A multitude
of different techniques are testing Lorentz and CPT, both of which are
fundamental ingredients for quantum field theories. These efforts include
techniques using antihydrogen, clocks, cold neutrons, matter-wave
interferometry, muons, resonant cavities, or short-interaction studies and are
poised to yield crucial insights into proposed BSM physics. While we give
short summaries of the individual efforts below, more details are found in
Ref. [148].

##### Time reversal violation searches with NOPTREX

A search for new sources of time reversal violation is possible using neutron
interactions with heavy nuclei. The principle is based on using certain p-wave
resonances in the heavy nuclei to search for P-odd and T-odd interactions. Due
to the highly excited states involved, the studied neutron interactions offers
a qualitatively different environment to electric dipole moment experiments
using ground state nucleons and nuclei. The method is based on a ratio
measurement rendering it quite insensitive to the resonant state properties
and also in principle to final state interaction effects [149, 150, 151].
Assuming about 4 months of data taking with 139La at a MW-class short pulse
neutron spallation source, an order of magnitude improvement in sensitivity to
P-odd and T-odd neutron-nucleus interactions can be achieved [150, 151, 152,
153, 154], comparable to proposed next-stage neutron EDM searches. Past and
ongoing R&D to polarize 139La and other heavy nuclei as well as improvements
to neutron optics can potentially provide an additional order of magnitude in
sensitivity in the future. Conversion of the NOPTREX apparatus into a spin-
spin interferometer with recently developed birefringent neutron optical
devices could further help to isolate the P-odd/T-odd signal from many
possible sources of systematic uncertainties in the measurement.

##### Lorentz and CPT tests with various low-energy experimental approaches

Cold antiprotons at CERN’s Antiproton Decelerator have been essential for
precision measurements with antihydrogen and pave the way for various CPT
tests. The ALPHA and ASACUSA experiments will perform antihydrogen hyperfine
spectroscopy for direct CPT tests through comparison with the same
measurements in hydrogen [155]. The anticipated precision in terms of the
mass-antimass difference will even exceed neutral-kaon interferometry, the
particle-physics standard for CPT tests. Other antihydrogen efforts like
AEgIS, ALPHA-g, or GBAR at CERN [155] or a proposed future experiment at
Fermilab [156], will study CPT symmetry via the interaction of antimatter with
gravity by measuring the free-fall of antihydrogen. Both spectroscopic and
gravitational methods will be able to provide qualitatively new Lorentz and
CPT tests.

Another route for probing Lorentz and CPT invariance involve Penning traps,
which enable precision studies of charged particles and their antiparticles.
Sidereal time variations of the cyclotron and anomaly frequencies of trapped
(anti)particles and the instantaneous comparison of the anomaly frequency
between a particle and its antiparticle are common approaches for Lorentz and
CPT tests. Future experimental upgrades [157, 158] as well as ongoing
phenomenological efforts [159] would lead to both substantial sensitivity
gains and access to a broader range of Lorentz- and CPT-violating effects.

Atomic clocks, atom magnetometers, and other precision spectroscopic
experiments provide some of the sharpest Lorentz-violation bounds for protons,
neutrons, electrons, and photons [160, 161, 162, 163, 164, 165, 166, 167, 168,
169, 170, 171]. Comparisons of at least two transitions in atomic clock can
probe Lorentz symmetry violation observed as differences in the clock
frequencies. Terrestrial experiments would rely on the Earth motion [172,
173], whereas space-bound experiments access additional forms of Lorentz
breaking as they governed by the satellites’ motion [174]. Sensitivities will
further increase in the next years as clock precision and comparison schemes
are improved and new clock technologies are being developed [160, 175].

Cold neutrons provide an indispensable tool in experimental high-energy
physics research including Lorentz and CPT tests [176, 177, 178]. Future
results from nEDM measurements around the world [179, 180, 181, 182] will give
up to about two orders of magnitude improved sensitivities to such tests. The
planned NNbar experiment at the ESS [183] will provide new sensitivity to
neutron-antineutron oscillations including the corresponding distinct class of
CPT- and Lorentz-violation searches.

Lorentz violation can also manifest itself as a modification to the
interaction of gravity with matter [184, 185, 186, 187] accessible in
superconducting gravimeters or space-based missions which continue to increase
their sensitivities [188, 189, 190]. Matter-wave interferometers are also able
to place bounds on Lorentz violation through induced gravitational phenomena
[191, 192]. Future, upgraded atom-interferometer using large wave-packet
separation or simultaneous multispecies operation promise strong advances in
the sensitivity, pushing them to the forefront of the Lorentz tests in the
gravitation-matter sector [193, 194, 195, 196, 197].

Muons and muonic systems offer diverse approaches for both Lorentz and CPT
tests [198, 199]. Spin motion of positive muons in $g\\!-\\!2$ experiments
have and will provide access to constraining Lorentz violation in the muon
sector [200, 201, 202, 203]. Muonic systems such as the theoretically well
understood muonium offer another windows to testing Lorentz and CPT invariance
[204, 205, 206, 207, 208].

Sensitivity to Lorentz violation in the photon sector is achieved with
electromagnetic resonant cavities where one typically compares resonant
frequencies of two differently orientated cavities [209, 210, 211, 212, 213,
214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228,
229, 230, 215, 226, 227]. The trend of orders of magnitude improved
sensitivities in various type of cavities is expected to continue in future
experiments.

Tests of the inverse-square law and searches for novel interactions can often
also probe spacetime symmetries due to their geometrical setup of test bodies
[231]. Some of the best constraints in the gravity sector are obtained this
way [232, 233] and future experimental efforts are leading to improved
sensitivities [234]. New techniques like a spin-polarized torsion pendulum can
further push the frontier on setting better limits on spatial-anisotropy
coefficients [235, 236, 237, 238, 239, 240, 241].

####  5 Precision tests with gravity

Gravity and its full understanding remains a vivid and important field of
research. Efforts in this area include precision gravitational tests with
antimatter and especially the sign of the force between matter and antimatter.
A repulsion predicted in an “antigravity” scenario would explain several
cosmological observations without the need for cosmic inflation. Other
important active fields of research in the gravitational sector include the
probing of general relativity and the quantum nature of gravity, or searches
for short-range corrections to the laws of gravity. The efforts in this
section are detailed more in Ref. [148].

##### Antimatter gravity tests with MAGE

Muonium—the bound state of an antimuon and electron—provides access to direct
antimatter gravity tests with a theoretical interpretation that is free of
hadronic effects as in $\overline{\mathrm{H}}$ [242, 243]. The proposed
concept uses a high-quality muonium beam in a Mach-Zehnder-type interferometer
to measure a small, gravity-induced phase shift, with soft x-rays providing a
needed calibration [244]. Ongoing R&D for the high-quality muonium beam aims
at reducing the 6D emittance of a surface muon beam by ten orders of magnitude
in order to stop it in a thin superfluid helium (SFHe) layer for muonium
formation. A one month data taking at PSI, Switzerland, could lead to a
5$\sigma$ determination of the sign of $\overline{g}$ (i.e., a 40% measurement
of $\overline{g}$) by the LEMING experiment. An alternative approach using a
much thicker SFHe layer can avoid the need for the low intensity cooled muon
beam envisioned at PSI. It could be developed in parallel at Fermilab and
initially lead to a 10% measurement of $\overline{g}$. This could be improved
to 1% or even better at a future Fermilab facility that would provide
competitive muonium beams in the PIP-II era. The MAGE experiment could yield
the first gravitational measurement of leptonic matter, 2nd-generation matter,
and possibly of antimatter [244]. These developments are also synergistic with
possible efforts for precision measurements of the muonium spectrum and
searches for muonium-antimuonium oscillation. The needed cryogenic stopping-
target R&D could start almost immediately at the Fermilab Linac. The required
slow-muonium facility would consist of a muon production target (or targets)
illuminated by a proton beam from a linac: either the existing 400-MeV Linac,
or the PIP-II 800 MeV one. R&D at the 400 MeV Linac can use the Irradiation
Test Area (ITA), where an effort is already in progress [245] to produce low-
energy muons for other applications. The needed antimuons would be deflected
into the vertical before entering a small cryostat, cooled by a dilution
refrigerator to sub-Kelvin temperatures, in which a layer of superfluid helium
would serve as an efficient $\mu^{+}$-to-muonium (Mu) converter, as well as
deflecting the Mu beam into the horizontal, thus producing a quasi-
monoenergetic, quasi-parallel muonium beam in vacuum. This beam would enable
(especially at PIP-II) world-leading sensitivities for muonium gravity,
muonium spectroscopy, and Mu—$\overline{\rm Mu}$ oscillation experiments.

##### Gravitational effects on CPV

An indirect measurement of antimatter gravity with kaons is proposed via the
measurement of the magnitude of CP violation in different gravitational field
strengths [246, 247, 248, 249, 250, 148]. The approach is motivated as many
Standard Model extensions imply a large CP violation and antigravity. Gravity-
generated CP violation [251] could also help to understand the missing
antimatter in the universe. The Earth’s gravitational field strength at the
surface and the long enough mixing time of the $K^{0}-\overline{K}^{0}$ system
would yield a separation of the antimatter components of the $K$ meson [252].
Assuming linearity of the CP violation with gravitational field strength, the
same effect would be $\sim$97% smaller on the moon. The proposed concept is
based on a comparison of the ratio of $K_{L}$ decay into two and three pions
in either a low orbit near the Earth’s surface or on the moon. In the absence
of a particle accelerator in these lower gravity environments, the production
of $K_{L}$ would leverage incident cosmic ray protons. A possible discovery of
the CP violation dependence on the gravitational strength could then motivate
a dedicated laboratory in space to perform similar measurements in greater
detail.

##### Tests of general relativity with a 229Th nuclear clock

Precision time-keeping plays an important role in many areas of physics.
Further advances in the precision of clocks has the potential to reveal new
physics through tests of the constancy of fundamental constants or tests of
general relativity. A nuclear transition with an energy low enough for laser
excitation in 229Th is a promising candidate for a novel nuclear clock that
could push the boundaries of the clock precision by 2–3 orders of magnitude.
Much of the ongoing R&D to better understand properties of 229Th like the
half-life of its meta-stable state 229mTh or the exact transition energy needs
to continue before such a nuclear clock could even be realized that would open
the pathway to ultra-precise new tests in the gravitational sector.

##### Mechanical tests of the quantum-gravity interface

Tests of the quantum nature of gravity would require experiments at the Planck
energy scale. Low energy probes offer an alternative that is accessible today
[253]. One such possibility is to attempt the entanglement of two masses that
are prepared in a quantum state of their motion. Two classes of experiments
can probe such gravitational quantum entanglement: i) interferometric tests
that rely on preparing masses in a quantum superposition of their positions
[254, 255, 256], which would dramatically decohere when exposed to classical
gravity. New interferometric experiments with levitated nano-particles are
planned for the next decade. As they are ultimately limited by the free-fall-
time on Earth, they provide the test-bed for subsequent space-borne setups.
ii) Non-interferometric tests that hope to precisely account for and measure
the subtle effect of gravitational entanglement [257, 258]. Such tests using
mechanical oscillators prepared in quantum states are poised to enter the
regime where gravity can be sourced and sensed using quantum objects. This
state of advance is largely due to the recent progress in understanding the
operating principles and limits of quantum-noise-limited displacement
measurement and control of mechanical motion at the quantum level. A new
generation of table-top experiments are being planned to set stringent bounds
on gravity’s ability to mediate entanglement [259, 260, 261, 262, 263, 264,
265, 266, 267]. These new experiments of both types share the need to
understand and develop experimental techniques of broader impact such as low-
environmental noise, mitigation of thermodynamic noises (for example via low-
noise cryogenics, materials science, and engineering), and shaping of quantum
noises (for example, via quantum-enhanced metrology and control). A dedicated
low-noise underground user facility and the use of freely-falling platforms
would be beneficial for advancing progress in this area.

##### Searches for short-range corrections to gravity

To help understand the more than 16 orders of separation between the apparent
energy scale of quantum gravity and that of the Standard Model (electro-weak
scale), investigations of the behavior of gravity at sub-millimeter distances
has been theoretically motivated [268, 269]. Such low-energy experiments are
challenged by the weakness of gravity and hence require ultra-precise
measurements. A wide set of experimental approaches is available, among which
are torsion pendulums, slow neutrons, or optically levitated sensors.

Torsion pendulums remain one of the promising paths forward in studying exotic
short-range gravity [270, 271], equivalence-principle violations [272, 273],
or novel spin-dependent interactions [274]. Both vibrations and time-varying
gradients from the environment are often the limiting factors. Future progress
with this powerful tool could hence be driven by the development of a suitable
underground facility that would strongly suppress such environmental
backgrounds.

Slow neutrons offer another versatile tool due to its special properties (like
electric neutrality with small magnetic moments and electric polarizability)
that renders them insensitive to many electromagnetic backgrounds. This makes
them very suitable for unique types of precision measurements [275, 276, 277,
278, 279, 280, 281, 282, 283, 284, 285, 286, 287]. The ability of slow
neutrons to penetrate and interact in matter with little decoherence allows
for interferometric measurements of large phase shifts of their quantum motion
amplitudes. These advantages have been exploited in many searches of new
weakly coupled interactions and particles of various types. Furthermore, the
dominant s-wave interaction for slow neutrons is experimentally well measured
and hence making their use as a probe insensitive to strong nucleon-nucleus
interactions. As such, slow neutrons are also a good probe for short-range
modifications of gravity at the length scales of 100 microns corresponding to
the scale set by dark energy densities. Ongoing and future prospects using
ultracold neutrons show great potential for continued progress as they are
often not yet limited by the statistical accuracy available at current neutron
facilities. Further options will arise from neutrons at higher energies that
undergo a nuclear resonance reaction which provide an amplification of the
small effects by the much prolonged interaction time compared to low energy
scattering.

Optically levitated dielectric objects in ultra-high vacuum are well suited
for high precision sensing as they are mostly decoupled from their environment
[288]. For example, optically-trapped nanospheres are promising candidates to
sense very feeble forces, accelerations, torques etc. [289, 290, 291, 292,
293, 294, 295]. Levitated objects also serve as sensitive probes for
millicharged particles [296], gravitational waves [297], or dark matter [298].
Trapped, levitated spheres can be used as test masses to probe for non-
Newtonian gravity-like and other forces [299, 300]. Advances in sensitivity
made possible by pushing the sensitivity of these sensors into the quantum
regime along with improved understanding and mitigation of systematic effects
due to background electromagnetic interactions such as the Casimir effect and
patch potentials will enable several orders of magnitude of improvement in the
search for new physics beyond the Standard model.

###  3 Sensors

Many of the precision tests described in Sections 1–5 often require new
detector technologies to probe the unexplored regions of some parameter space
and increase sensitivity to symmetry violations and new physics. In more and
more cases in the recent past, experimental efforts and their detection
systems reach the regime in which the laws of quantum mechanics become
relevant. New quantum sensors have emerged and are actively being developed to
use such quantum effects to their advantage and even achieve enhancements in
the detection sensitivity. Ongoing efforts and current opportunities are
highlighted in a subset of sensor types, such as interferometry,
optomechanical sensors, clocks and trapped atoms/molecules. Many of these
specific tools have high relevance in particle and high energy physics and
other communities as can be seen from some of the explicit examples summarized
in Sections 4 and 5. More details can be found in [301] as only a short
summary is given here.

####  1 Atom Interferometers

Atom interferometers have acquired a growing number of applications in
fundamental physics. Applications range for example from gravitational wave
detection, to the dark matter and dark sector, precision determination of
fundamental constants (like the fine structure constant $\alpha$), and precise
tests of the SM. One specific growing area is the pursuit of long-baseline
atomic sensors for GW detection. Significant progress on their sensitivity is
expected with the next generation which are proposed for both terrestrial and
space-based setups.

Several types of interferometers exist that have their own applications.
Light-pulse atom interferometry uses laser pulses to coherently split,
redirect, and recombine matter waves. The gradiometer configuration uses two
identical atom interferometers to run simultaneously at each end of a long-
baseline. Comparison of both signals enables cancellation of common mode noise
sources, which will allow for the possibility of single baseline GW detection.
To use an atom interferometer for GW detection, two freely falling atoms on
each end would act simultaneously as inertial reference and precise clock.
Laser light traveling between the two atom ensembles drives an atomic
transition and encodes the light travel time as a phase shift between the two
systems. Differential phase measurements are then sensitive to baseline
changes induced by a passing GW. Atom interferometers for GW detection are
promising to access a frequency gap between LIGO/Virgo/KARGA and LISA. This
frequency band of about 30 mHz to 3 Hz offers a variety of compelling physics
questions to be addressed.

In addition to GW detection, long-baseline interferometers can also be
sensitive to dark matter and new forces. For example, dark matter can affect
fundamental constants and with it the energy levels of the involved quantum
states in the interferometers. Dual-species interferometers operated with
different isotopes are sensitive to acceleration changes that could be caused
by dark matter. Finally, a comparison of atomic sensors with different atomic
species could also reveal the existence of new forces.

A prominent example for smaller atom interferometers is the application for
the precise determination of fundamental constants, such as the fine structure
constant $\alpha$, a measurement highly motivated by the ongoing determination
of the muon’s magnetic anomaly. Other applications using optical cavities can
place stringent limits on dark energy candidates.

The selected examples showcase the versatility of atomic interferometers and
their relevance to high energy physics questions. Future improvements to the
technique will lead to significantly more sensitive detectors for precision
measurements or new limits on BSM physics.

####  2 Optomechanical

Mechanical sensors that are read out by optical or microwave light have
undergone significant developments and many of them are now operated in the
quantum regime. These mechanical devices are well suited to look for signals
acting coherently over the typical size of the mechanical system. Mechanical
sensors also come in a wide variety of sizes, operational frequencies, and
other configuration parameters.

Besides their prominent use in GW detection and precision measurements in
metrology, they have a widening use in particle and high energy physics such
as ultra-light and ultra-heavy dark matter particle searches, neutrino
detection, or fifth-force modifications to Newton’s law. Various types of
ongoing and future experiment include the use of for example torsion balances,
optomechanical interferometers, resonant mass detectors, nanomechanical
systems, or levitated particles.

Further progress in the next years is expected from several opportunities. One
such R&D example is the goal to pass the so-called Standard Quantum Limit to
achieve advanced quantum techniques. Many of the improvement opportunities are
in need of theoretical effort to be able to implement advanced sensors.

####  3 Clocks

Optical clocks and their precision have improved steadily and they play an
important role in many applications like precision tests of the constancy of
fundamental constants, dark matter searches, or tests of Lorentz invariance
and general relativity.

Atomic clocks are based on transitions between hyperfine substates or
electronic levels. Comparison of the frequencies of two optical clocks are
sensitive to variations in $\alpha$ and with it to SM dark matter couplings.
The sensitivities can experience enhancement factors depending on the
electronic structure of the clock and tends to increase for atoms with heavier
nuclei. Future development of novel clocks with larger enhancement factors can
hence provide stronger limits on dark matter. Several R&D directions are being
pursued to achieve this goal. One examples is the development of clock
networks with new precision levels, which will yield better comparisons with
spatially separated clocks. Another focus area is the increase in clock
performance to surpass the standard quantum limit by introducing highly
entangled quantum states. Use of highly charged ions or the development of a
nuclear clock will provide significant increase in the sensitivity to $\alpha$
through large enhancement factors and represent other areas of ongoing clock
R&D. Finally, molecular clocks with their rotational and vibrational degrees
of freedom offer direct sensitivity to $\frac{m_{p}}{m_{e}}$ and its
variation.

####  4 Trapped atoms and ions

Radioactive atoms and molecules offer extreme nuclear charge, mass, and
deformations advantageous for studying nuclear structure, symmetry violations,
and precision measurements. Sensitivity to symmetry violations typically
scales with the number of the nucleus as $Z^{2-5}$ favoring heavy, radioactive
nuclei. Other special properties can bring further enhancements making certain
species like Fr, Ra, Ac, Th, or Pa attractive for CP-violation studies.
Certain unstable isotopes are also candidates for improved clocks or allow
through precise studies of the nuclear decay to search for sterile neutrinos.
As these heavy, radioactive species of interest are difficult to produce in
large quantities and decay quickly, traps have been critical to make efficient
use of them. R&D for further improvement of ion and neutral traps offers an
important opportunity to make significant progress in the physics
measurements.

###  4 Summary

The HEP community has entered an era of unprecedented precision experiments.
While relatively small in size and cost compared to their energy frontier
cousins, they are large in reach and discovery potential. The traditional
storage ring experiments for the muon $g\\!-\\!2$ coupled with equally precise
theory calculations have put us on the cusp of discovery, while new
experiments and improvements in lattice QCD calculations offer exciting
possibilities for EDMs. The proton storage ring experiment is estimated to
improve sensitivity by over three-orders-of-magnitude and will also search for
wave-like dark matter and energy. Meanwhile adjacent nuclear and AMO physics
communities will provide critical complementary searches, the latter with up
to six orders-of-magnitude improvements in the next 10–15 years. Together
these efforts will provide a sharply focused framework for new physics, or if
no EDMs are found, will point our field in important new directions.

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