11institutetext: Department of Engineering Science, The University of Electro-
Communications, Tokyo 182-8585, Japan22institutetext: Institut für
Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum,
Germany33institutetext: RIKEN Nishina Center for Accelerator-Based Science,
Wako, Saitama 351-0198, Japan44institutetext: Department of Physics, Tokyo
Institute of Technology, Ookayama, Meguro, Tokyo 152-8551,
Japan55institutetext: Department of Physics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan

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<EMAIL_ADDRESS>eYTe-mail<EMAIL_ADDRESS>
# Three-body forces and Efimov physics in nuclei and atoms

Shimpei EndoeSE,uec  Evgeny EpelbaumeEE,bochum  Pascal NaidonePN,riken  Yusuke
NishidaeYN,titech  Kimiko SekiguchieKS,titech  Yoshiro TakahashieYT,kyoto

###### Abstract

This review article presents historical developments and recent advances in
our understanding on the three-body forces and Efimov physics, from an
interdisciplinary viewpoint encompassing nuclear physics and cold atoms.
Theoretical attempts to elucidate the three-body force with the chiral
effective field theory are explained, followed by an overview of experiments
aimed at observing signatures of the nuclear three-body force. Some recent
experimental and theoretical works in the field of cold atoms devoted to
measuring and engineering three-body forces among atoms are also presented. As
a phenomenon arising from the three-body effect, Efimov physics in both cold
atoms and nuclear systems is reviewed.

††journal: Eur. Phys. J. A

##  1 Introduction

All visible matter in the universe is organized into resolution-dependent
hierarchical structures. A precise description of the interactions at each
hierarchical level starting from elementary quarks to composite systems like
hadrons, nuclei, atoms and molecules may help to improve our understanding of
the structure and dynamics of strongly interacting matter. Generally,
effective interactions at all hierarchical levels are dominated by pairwise
forces acting between two constituent particles. Meanwhile, recent advances in
computational, theoretical and experimental techniques allow one to go beyond
the two-body-force level and to quantitatively probe three-body interactions,
which are known to figure importantly in atomic and nuclear systems
Hammer:2012id .

Historically, the most well known three-body force model was proposed by
Fujita and Miyazawa in 1957 10.1143/PTP.17.360  to describe nuclear systems
using protons and neutrons, jointly called nucleons, as constituent particles.
The Fujita-Miyazawa type three-nucleon force ($3N$F), visualized by the
Feynmann diagram in Fig. 1, is driven by virtual pion-nucleon scattering with
an intermediate excitation of the nucleon into the $\Delta(1232)$ resonance.
Three-nucleon forces are thus intimately related to the inner structures of
the nucleons and their short-distance virtual excitations.

Figure 1:  Feynmann diagram of Fujita-Miyazawa type three-nucleon force.

Following the idea of Fujita and Miyazawa, one may expect three-body forces to
play an important role in quantum systems other than atomic nuclei. With their
high controllability, cold atoms have recently emerged as an excellent
platform to explore quantum systems of interests. It is possible to realize
cold-atom systems where the three-body forces among atoms appear from similar
exchanges of virtual excitations, thereby quantum-simulating the Fujita-
Miyazawa mechanism. It is even possible to engineer the strength of the three-
body force and thereby realize a system where the three-body force has
significant effects on few- and many-body properties, in stark contrast to
conventional systems where the two-body force dominates over the three-body
force.

The emergence of a three-body force from exchange processes can lead to an
even more exotic phenomenon, the Efimov effect. When three particles scatter
with each other via resonant two-body interactions, a long-range three-body
force emerges via multiple scatterings. The three-body force is attractive and
forms a series of weakly bound three-body states, known as the Efimov states.
The Efimov states have not only been observed in cold-atom experiments, but
they appear universally in various physical systems. Thanks to its universal
properties, Efimov physics provides a unified description of various classes
of three-body phenomena in cold atoms and nuclear physics.

In this review article, we discuss three-body forces in nuclear systems and in
cold atomic gases as a key aspect for bridging different hierarchies. We
present an overview of the three-body force in nuclear systems in Sec. 2.
After briefly presenting historical developments of the three-body forces in
Sec. 2.1, we show in Sec. 2.2 the effective field theory (EFT) description of
the three-body force in nuclei, followed by a review of the experimental
studies in Sec. 2.3. In Sec. 3, we present recent studies in the field of cold
atoms to realize and simulate the three-body forces. In Sec. 4, we show the
recent developments of Efimov physics in cold atoms and nuclear systems.

##  2 Three-body forces in nuclei

###  2.1 Three-nucleon force and its importance in nuclear phenomena

Since Yukawa’s meson theory proposed in 1935 Yukawa193548 , the nuclear force
has been modeled in terms of meson exchange interactions between nucleons.
Beside the dominant two-nucleon forces, the three nucleon forces ($3N$Fs) have
attracted an increasing attention in the last two decades. The development of
high-precision two-nucleon potentials in the 90s of the last century, coupled
with advances in ab initio few-body calculations based on these interactions,
have confirmed the important role played by $3N$Fs in various nuclear
phenomena. Following the seminal work by Fujita and Miyazawa, a number of the
$3N$F models utilizing the longest-range $2\pi$-exchange mechanism have been
developed such as, e.g., the Tucson-Melbourne 99 tm99 , the Urbana IX
PhysRevC.56.1720  $3N$F models. A new impetus to study $3N$Fs has come from
chiral perturbation theory, the low-energy effective field theory of QCD
Weinberg:1990rz ; kolk1994 ; Epelbaum:2002vt .

Historically, the first indication for a missing $3N$F came from the three-
nucleon bound states ${}^{3}\rm H$ and ${}^{3}\rm He$ PhysRevC.33.1740 ;
FewBodySys.1.3 . The binding energies of these nuclei were found to be not
reproduced by an exact solution of the three-nucleon Faddeev equation using
high-precision phenomenological $NN$ forces including the Argonne $V_{18}$
(AV18) AV18 , CD Bonn cdb  as well as Nijmegen I and II nijm  potentials. The
underbinding of 3H and 3He could be explained by adding a $2\pi$-exchange-type
$3N$F PhysRevC.33.1740 ; FewBodySys.1.3 ; PhysRevC.65.054003 . The importance
of $3N$Fs has also been noted in other instances. In particular, microscopic
calculations of light and medium-mass nuclei carried out using ab initio
methods such as, e.g., quantum Monte Carlo PhysRevC.66.044310 ;
Piarulli:2017dwd , no-core shell model PhysRevC.68.034305 , coupled cluster
theory Hagen:2012sh , self-consistent Green’s function method
Cipollone:2014hfa  and nuclear lattice simulations Lahde:2019npb  highlight
the important role of $3N$Fs in explaining the corresponding binding energies.
Furthermore, short-range repulsive $3N$Fs are considered key elements in
describing the nuclear equation of state and two-solar-mass neutron star
properties PhysRevC.58.1804 ; Gandolfi:2013baa ; PhysRevLett.116.062501 . In
the past two decades, low-energy nucleon-deuteron scattering, binding energies
of light and medium-mass nuclei as well as the equation of state of nuclear
matter have also been extensively studied in the framework of the chiral
effective field theory epelbaum2009 ; Kalantar-Nayestanaki_2012 ;
Hammer:2012id ; PhysRep21Hebeler . In all these investigations, it became
evident that $3N$Fs are key elements to understand various nuclear phenomena.

Discussions of three-body forces in nuclei currently extend to strange
baryonic systems, e.g. $NN\Lambda$ interactions, especially for the neutron
star properties Haidenbauer:2016vfq , which are needed to establish a
universal understanding of the forces acting in nuclear phenomena. Also, it is
notable that discussions of $3N$Fs stimulate the investigation of three-body
forces in different hierarchies. As described in Sec. 3, study of three-body
forces in the cold-atom systems are in progress not only from the theoretical
but also from the experimental point of view.

###  2.2 Three-nucleon forces in chiral effective field theory

####  2.2.1 Chiral perturbation theory

The interactions and dynamics of pions can be described using the most general
effective Lagrangian $\mathcal{L}_{\pi}$ that features the approximate chiral
symmetry of QCD. It includes all possible terms allowed by symmetry,
multiplied with coefficients that are commonly referred to as low-energy
constants (LECs). These LECs are not fixed by symmetry and carry information
about short-range QCD dynamics. It is convenient to classify terms in the
effective Lagrangian according to the number of derivatives and/or insertions
of the pion mass $M_{\pi}$:
$\mathcal{L}_{\pi}=\mathcal{L}_{\pi}^{(2)}+\mathcal{L}_{\pi}^{(4)}+\ldots$.
Multi-pion scattering amplitudes for the physically intersting case of $p\sim
M_{\pi}\neq 0$ can be calculated from the effective Lagrangian using chiral
perturbation theory (ChPT) Weinberg:1978kz ; Gasser:1983yg , i.e. via a
perturbative expansion in $Q\in\\{p/\Lambda_{b},\,M_{\pi}/\Lambda_{b}\\}$,
where $\Lambda_{b}$ is the breakdown scale of ChPT, which may be estimated by
the masses of lowest-lying resonances in the $\pi\pi$ system.

The perturbative approach outlined above can be straightforwardly generalized
to processes involving a single non-Goldstone-boson particle such as, e.g.,
the nucleon. The most general pion-nucleon effective Lagrangian
$\mathcal{L}_{\pi N}=\mathcal{L}_{\pi N}^{(1)}+\mathcal{L}_{\pi
N}^{(2)}+\mathcal{L}_{\pi N}^{(3)}+\ldots$ can be constructed using the
methods described in Refs. Weinberg:1968de ; Coleman:1969sm ; Callan:1969sn .
The explicit form of $\mathcal{L}_{\pi N}^{(1)}$ and $\mathcal{L}_{\pi
N}^{(2)}$ can be found, e.g., in Ref. Bernard:1995dp , while the Lagrangians
$\mathcal{L}_{\pi N}^{(3)}$ and $\mathcal{L}_{\pi N}^{(4)}$ are given in Refs.
Fettes:1998ud  and Fettes:2000gb , respectively. Compared to the Goldstone-
boson sector, special care is required for processes involving a nucleon to
ensure that the hard scale set by the nucleon mass $m_{N}$ does not spoil the
chiral power counting when computing loop diagrams. The simplest way to
achieve this is to perform a non-relativistic expansion of the effective
Lagrangian $\mathcal{L}_{\pi N}$ Jenkins:1990jv ; Bernard:1992qa . This
ensures that $m_{N}$ appears in $\mathcal{L}_{\pi N}$ only in the form of
$1/m_{N}^{n}$-corrections with $n>0$, and the corresponding framework is
referred to as the heavy-baryon ChPT. It is also possible to perform
calculations in a manifestly Lorentz-invariant way by choosing the appropriate
renormalization conditions Becher:1999he ; Gegelia:1999gf ; Fuchs:2003qc .

####  2.2.2 Chiral EFT for nuclear systems

Clearly, a direct application of ChPT to systems involving two and more
nucleons is not possible due to the non-perturbative nature of the nuclear
interactions, as reflected in the existence of shallow bound states such as
2H, 3H, 3He, 4He, etc. These bound states manifest themselves as subthreshold
poles of the corresponding scattering amplitudes and signal the breakdown of
the perturbative expansion.

In the early 1990s, Weinberg came up with an approach that is based on the
effective Lagrangian and allows one to analyze low-energy few- and many-
nucleon systems in a model-independent and systematic fashion Weinberg:1990rz
; Weinberg:1991um , which is nowadays commonly referred to as chiral effective
field theory (ChEFT). He has attributed the failure of perturbation theory in
the few-nucleon sector to the appearance of enhanced few-nucleon-reducible
ladder-type diagrams and argued that they need to be resummed non-
perturbatively. He also noticed that a resummation of ladder-type diagrams is
performed automatically by solving the corresponding Lippmann-Schwinger-type
integral equations for the scattering amplitude. Thus, Weinberg’s ChEFT
approach to few-nucleon systems technically reduces to the conventional A-body
problem,

$\bigg{[}\bigg{(}\sum_{i=1}^{A}\frac{-\vec{\nabla}_{i}^{2}}{2m_{N}}+\mathcal{O}(m_{N}^{-3})\bigg{)}+V_{NN}+V_{3N}+\ldots\bigg{]}\big{|}\Psi\big{\rangle}=E\big{|}\Psi\big{\rangle},$
(1)

but it opens the possibility to derive nuclear interactions $V_{NN}$,
$V_{3N}$, $V_{4N}$, $\ldots$, via a systematically improvable ChPT expansion
in harmony with the symmetries of QCD. Here, nuclear potentials are defined by
means of all possible few-nucleon-irreducible contributions to the scattering
amplitude, which are not affected by the above-mentioned enhancement and can
be derived in the framework of ChPT using a variety of methods. Following
Weinberg’s original work Weinberg:1990rz ; Weinberg:1991um , time-ordered
perturbation theory was employed in Refs. Ordonez:1995rz ; Pastore:2008ui ;
Pastore:2009is ; Pastore:2011ip ; Baroni:2015uza  to derive nuclear forces and
electroweak currents. Another approach, the so-called method of unitary
transformation (MUT), makes use of a unitary transformation of the pion-
nucleon Hamiltonian to decouple the purely nucleonic subspace of the Fock
space from the rest. The MUT was applied to derive few-nucleon forces as well
as electroweak and scalar nuclear currents in Refs. Epelbaum:1998ka ;
Epelbaum:1999dj ; Epelbaum:2002gb ; Epelbaum:2005fd ; Epelbaum:2005bjv ;
Epelbaum:2007us ; Bernard:2007sp ; Bernard:2011zr ; Krebs:2012yv ;
Krebs:2013kha ; Kolling:2009iq ; Kolling:2011mt ; Krebs:2016rqz ;
Krebs:2019aka ; Krebs:2020plh . Yet another method to derive the $NN$ two- and
three-pion exchange potentials from matching to the scattering amplitude was
applied in Refs. Kaiser:1997mw ; Kaiser:1998wa ; Kaiser:1999ff ; Kaiser:1999jg
; Kaiser:2001dm ; Kaiser:2001pc ; Kaiser:2001at ; Entem:2015xwa . For a
detailed discussion of these techniques and applications the reader is
referred to the review articles Epelbaum:2005pn ; epelbaum2009 ;
Machleidt:2011zz ; Epelbaum:2019kcf ; Krebs:2020pii ; deVries:2020iea .

Non-perturbative resummations of reducible diagrams in ChEFT pose
complications as compared with ChPT in the Goldstone-boson or single-nucleon
sectors. Consider the longest-range $NN$ force due to the one-pion exchange

$V_{NN}^{1\pi}(\vec{q}\,)=-\frac{g_{A}^{2}}{4F_{\pi}^{2}}\frac{\vec{\sigma}_{1}\cdot\vec{q}\vec{\sigma}_{2}\cdot\vec{q}}{q^{2}+M_{\pi}^{2}}\mbox{\boldmath$\tau$}_{1}\cdot\mbox{\boldmath$\tau$}_{2}\,,$
(2)

where $g_{A}$ and $F_{\pi}$ are the nucleon axial vector coupling and the pion
decay constant, respectively. Further, $\vec{\sigma}_{i}$
($\mbox{\boldmath$\tau$}_{i}$) denote the spin (isospin) Pauli matrices of
nucleon $i$, while $\vec{q}=\vec{p}\,^{\prime}-\vec{p}$, with $\vec{p}$
($\vec{p}\,^{\prime}$) being the nucleon center-of-mass (CM) momentum in the
initial (final) state, is the nucleon momentum transfer. The one-pion exchange
potential (OPEP) contributes to the leading-order (i.e., order-$Q^{0}$)
nuclear force and thus needs to be resummed non-perturbatively. However, the
$1/r^{3}$ singularity in the tensor part of the OPEP implies the appearance of
ultraviolet divergences in all spin-triplet channels upon performing
iterations of the Lippmann-Schwinger (LS) equation. That is, removing
ultraviolet divergences from the iterative solution of the LS equation
requires the introduction of an infinite number of counter terms from the
Lagrangians $\mathcal{L}_{NN}^{(0)}$, $\mathcal{L}_{NN}^{(2)}$,
$\mathcal{L}_{NN}^{(4)}$, $\ldots$. This is in strong contrast to ChPT, where
a finite number of counter terms are required to remove ultraviolet
divergences at any given order.

Clearly, the singular short-distance nature of $V_{NN}^{1\pi}(\vec{q}\,)$ is
unphysical and represents an artifact of using the low-momentum approximation
in Eq. (2) beyond its validity range (i.e., at short distances or large values
of the momentum transfer). Renormalization of the Schrödinger equation in EFTs
with singular interactions like $V_{NN}^{1\pi}(\vec{q}\,)$ can be achieved in
the way compatible with the weak-interaction limit by introducing a finite
cutoff $\Lambda$ of the order of the pertinent hard scale,
$\Lambda\sim\Lambda_{b}$ Lepage:1997cs . At each order of ChEFT and for every
value of $\Lambda$, renormalization is carried out implicitly by expressing
bare LECs from $\mathcal{L}_{NN}$, $\mathcal{L}_{NNN}$, $\ldots$, in terms of
measurable quantities. In practice, this is achieved by tuning few-nucleon
contact interactions to experimental data. The residual dependence of the
renormalized results on the cutoff $\Lambda$ serves as a measure of the
neglected contributions of terms beyond the EFT truncation level. It is
expected to decrease with the chiral order and provides an a posteriori
consistency check. In Refs. Gasparyan:2021edy ; Gasparyan:2023rtj , the
finite-$\Lambda$ formulation of ChEFT was formally proven to be renormalizable
(in the EFT sense) up to next-to-leading order $Q^{2}$ (NLO). Here and in what
follows, we restrict ourselves to the finite-cutoff formulation of ChEFT as
described in detail in Ref. Epelbaum:2019kcf . A pedagogical introduction into
the considered framework can be found in Sec. 6.3 of Ref. Gross2022hyw , while
a discretized (lattice) formulation is described in Ref. Lahde:2019npb .

####  2.2.3 Current status of the $NN$ potentials

Presently, the $NN$ force has been worked out completely up through fifth
order $Q^{5}$ (i.e., N4LO) including isospin-breaking corrections due to
$m_{u}\neq m_{d}$ and QED effects. It involves the one- two- and three-pion
exchange potentials, supplemented by short-range NN interactions from
$\mathcal{L}_{NN}^{(0)}$, $\mathcal{L}_{NN}^{(2)}$, and
$\mathcal{L}_{NN}^{(4)}$, which depend on $2$, $7$ and $15$111The above
numbers refer to isospin-invariant contact interactions. LECs, respectively,
that need to be tuned to $NN$ data. Notice that $3$ out of $15$ contact
interactions at N4LO contribute only to the off-shell part of the $NN$
potential and thus cannot be determined from two-nucleon scattering data.

While semi-quantitative ChEFT potentials have a long history that goes back to
the beginning of 2000s, see Refs. Epelbaum:2004fk ; Entem:2003ft  for the
first-generation order-$Q^{4}$ (i.e., N3LO) $NN$ potentials constructed using
non-local regulators, the field has received new impetus in the last decade by
developing a local regularization scheme for pion-exchange contributions that
preserves the long-range behavior of the nuclear force, which allowed to
significantly improve the predictive power of ChEFT Epelbaum:2014efa ;
Epelbaum:2014sza . The state-of-the-art $NN$ N4LO+ potential of Ref.
Reinert:2017usi  employs a local momentum-space regulator for the OPEP and
two-pion exchange potential (TPEP) via

$\displaystyle V_{NN}^{1\pi}(\vec{q}\,)$ $\displaystyle\to$ $\displaystyle
V_{NN}^{1\pi}(\vec{q}\,)\,e^{-\frac{q^{2}+M_{\pi}^{2}}{\Lambda^{2}}}+\ldots\,,$
(3) $\displaystyle V_{NN}^{2\pi}(q\,)$ $\displaystyle\to$
$\displaystyle\frac{2}{\pi}\int_{2M_{\pi}}^{\infty}d\mu\mu\frac{\rho_{2\pi}(\mu)}{q^{2}+\mu^{2}}\,e^{-\frac{q^{2}+\mu^{2}}{2\Lambda^{2}}}+\ldots\,,$
(4)

where the ellipses denote subtractions in the form of $NN$ contact
interactions, chosen in such a way that the corresponding $r$-space potentials
vanish at the origin. In the second equation, we have used a dispersive
representation of the TPEP Kaiser:1997mw , where the spectral function
$\rho_{2\pi}(\mu)$ is obtained from an analytic continuation of the
unregularized potential $V_{NN}^{2\pi}(q)$ to imaginary values of $q$ via
$\rho_{2\pi}(\mu):=\Im\big{[}V_{NN}^{2\pi}(0^{+}-i\mu)\big{]}$. Equations (3)
and (4) show that all finite-$\Lambda$ artifacts stemming from the
$1/\Lambda$-expansion of the regularized OPEP and TPEP have the form of short-
range contact interactions. The latter are regularized in Ref. Reinert:2017usi
using a Gaussian non-local regulator $\exp[-(p^{2}+p^{\prime
2})/\Lambda^{2}]$. The resulting semi-local momentum-space regularized (SMS)
$NN$ potentials are available at orders LO through N4LO and cutoff values of
$\Lambda=400$, $450$, $500$ and $550$ MeV. The N4LO+ potentials lead to a
statistically perfect description of the Granada-2013 database of mutually
consistent neutron-proton and proton-proton scattering data below pion
production threshold NavarroPerez:2013usk  and show, as expected, a very weak
residual $\Lambda$-dependence. In Ref. Reinert:2020mcu , the SMS $NN$
potential of Ref. Reinert:2017usi  was updated to include all relevant
isospin-breaking contributions up through N4LO, and it was used for a
precision determination of the $\pi N$ coupling constants from $NN$ scattering
and to perform a full fledged partial wave analysis of $NN$ data (including a
selection of mutually compatible data), see Ref. Epelbaum:2022cyo  for details
and Fig. 2 for representative examples

Figure 2:  Proton-proton differential cross section at $E_{\rm lab}=144.1$ MeV
(left panel) and the analyzing power $P$ at $E_{\rm lab}=142$ MeV (right
panel) at N2LO (green bands), N3LO (blue bands) and N4LO+ (red bands) in
ChEFT. Dark- and light-shaded bands show $1\sigma$ and $2\sigma$ confidence
levels at the corresponding order, respectively, estimated using the Bayesian
model $\bar{\rm C}^{650}_{0.5-10}$ from Ref. Epelbaum:2019zqc . Experimental
data are shown by filled symbols and taken from Refs. Cox:1968jxz ;
Jarvis:1971fla ; TAYLOR1960320 . Open circles are the results of the Nijmegen
partial wave analysis Stoks:1993tb . See Ref. Epelbaum:2019kcf  for more
details.

These new developments allowed one to test the predictive power of ChEFT by
quantitatively addressing the impact of the TPEP. In Refs. Epelbaum:2014efa ;
Epelbaum:2014sza ; Reinert:2017usi , a clear evidence of the parameter-free
contributions of the TPEP was observed at orders $Q^{3}$ (N2LO) and $Q^{5}$
(N4LO), where no additional $NN$ contact interactions appear. A significantly
smaller number of adjustable parameters in the SMS potential of Refs.
Epelbaum:2014sza ; Reinert:2017usi  as compared to the phenomenological high-
precision potential models provides yet another indication of the importance
of the TPEP and demonstrates the predictive power of ChEFT. For related
earlier studies along this line see Refs. Rentmeester:1999vw ; Birse:2003nz .

Other notable recent additions to the ChEFT $NN$ potentials involve the
nonlocal N4LO+ potentials by the Idaho group Entem:2017gor  as well as the
(nearly) local N3LO potentials of Refs. Piarulli:2014bda ; Somasundaram2023sup
; Saha:2022oep .

####  2.2.4 Chiral expansion of the $3N$F

We now turn to the main subject of this article and review the applications of
ChEFT to the $3N$F. It is instructive to first discuss the most general
structure of a $3N$F. In the static limit of infinitely heavy nucleons, the
potentials mediated by the exchange of one or multiple pions take a local
form, i.e. they depend only on the momentum transfers $\vec{q}_{i}$ and not on
the individual momenta $\vec{p}_{i}$, $\vec{p}_{i}^{\,\prime}$ of the
nucleons. Assuming parity, time-reversal invariance and isospin symmetry, the
most general local $3N$F can be written as Phillips:2013rsa ; Epelbaum:2014sea

$V_{3N}=\sum_{\alpha=1}^{20}\hat{O}_{\alpha}\,f_{\alpha}(q_{1},q_{2},q_{3})\text{
+ permutations}\,,$ (5)

where $q_{i}\equiv|\vec{q}_{i}\,|$, $\hat{O}_{\alpha}$, are rotationally and
isospin-invariant Hermitian operators constructed out of $\vec{\sigma}_{i}$,
$\mbox{\boldmath$\tau$}_{i}$ and $\vec{q}_{i}$, while $f_{\alpha}$ are the
corresponding scalar functions. Upon performing the permutations, the $20$
operators $\hat{O}_{\alpha}$ give rise to $80$ different spin-isospin-momentum
structures. When relaxing the locality constraint, the structure of the $3N$F
becomes more involved, comprising $320$ spin-isospin-momentum operators
Topolnicki:2017rnt . This enormous complexity of three-nucleon interactions,
along with the significant computational effort needed to solve the three-body
Faddeev equations, make the development of high-precision $3N$F models a
challenging task that requires a guidance from theory to constrain the
structure and identify the dominant contributions. ChEFT is well suited to
tackle the $3N$F challenge by predicting its long-distance behavior in a
parameter-free and model independent way and offering a systematic scheme for
classifying short-range $3N$ interactions according to their importance. Based
on the effective Lagrangian for pions, nucleons and external sources, ChEFT
also naturally allows one to maintain off-shell consistency between $NN$
potentials, $3N$Fs and the corresponding current operators as discussed in
Sec. 2.2.2.

Up-to-and-including N4LO, the $3N$F is given by tree-level and one-loop
diagrams, which can be grouped into six distinct topologies depicted in the
left column of Fig. 3.

Figure 3:  Chiral expansion of the $3N$F. Solid and dashed lines refer to
nucleons and pions, respectively. Diagrams in the leftmost column visualize
the two-pion exchange (a), two-pion-one-pion exchange (b), ring (c), one-pion-
contact (d), two-pion-contact (e) and purely contact (f) $3N$F topologies.
Cyan circles and ellipses in these diagrams represent scattering amplitudes of
the corresponding sub-processes. The second, third and fourth columns show
examples of diagrams, which contribute to the individual $3N$F topologies at
increasing orders of the ChEFT expansion. Here, solid dots, filled circles and
filled squares denote vertices from the effective Lagrangian of increasing
chiral dimension $\Delta$ as explained in the text. Vertices shown in red
involve LECs that need to be determined from three- or more-nucleon data.

In this figure, solid dots, filled circles and filled squares denote vertices
from the effective Lagrangians of the dimension $\Delta=0$, $\Delta=1$ and
$\Delta=2$, respectively, defined as $\Delta=d+\frac{1}{2}n-2$ with $d$ being
the number of derivatives or $M_{\pi}$-insertions and $n$ the number of
nucleon fields Weinberg:1990rz .

The leading $3N$F contributions arise at N2LO from tree-level diagrams
contributing to the topologies (a), (d) and (f), which are made out of the
$\Delta=0$-vertices and a single insertion of a $\Delta=1$ interaction
Weinberg:1992yk ; kolk1994 . The short-range topologies (d) and (f) depend on
the LECs $c_{D}$ and $c_{E}$, respectively, which cannot be fixed in the $NN$
system.

The first corrections to the leading $3N$F are generated by one-loop graphs
made out of the lowest-order vertices with $\Delta=0$, see the third column in
Fig. 3 for representative examples. Here, the shown diagrams represent sets of
irreducible time-ordered-like graphs, whose precise meaning (and the
corresponding algebraic expressions) depend on the employed choice for the
off-shell part of the $NN$ and $3N$ potentials. For example, the $3N$F from
the first of the two three-pion exchange diagrams of type (c) depends on the
(ambiguous) choice made for the $1/m_{N}^{2}$-correction to the OPE $NN$
potential and the $1/m_{N}$-corrections to the tree-level two-pion exchange
$3N$F of type (a), which appear at the same order.

The N3LO contribution to the longest-range topology (a) was derived in Ref.
Ishikawa:2007zz  based on the order-$Q^{3}$ $\pi N$ amplitude. These results
were confirmed using the MUT in Ref. Bernard:2007sp , where also the
expressions for the topologies (b) and (c) were derived. The shorter-range
$3N$Fs of type (d) and (e), along with the $1/m_{N}$-corrections to the
topologies (c) and (d), are worked out in Ref. Bernard:2011zr . All results
mentioned above are obtained using dimensional regularization (DimReg) to
evaluate divergent loop integrals and are parameter-free.222As already
mentioned in Sec. 2.2.3, the N3LO $NN$ potential in the CM system depends on
$3$ off-shell LECs. Two further off-shell LECs contribute to the $NN$
potential away from the CM system Girlanda:2020pqn . These LECs, being
redundant in the NN system, will generally affect $3N$ observables calculated
at N3LO Girlanda:2023znc . Their contributions to $3N$ observables can,
however, be absorbed into redefinitions of LECs accompanying short-range $3N$F
at N4LO and, therefore, can be ignored beyond the N3LO level. Finally, it is
worth mentioning that one can also draw tree-level $3N$ diagrams constructed
from the leading $\Delta=0$-vertices and a single insertion of a
$\Delta=2$-interaction from $\mathcal{L}_{\pi N}^{(3)}$ that could potentially
contribute to the $3N$F at N3LO. However, all irreducible contributions
generated by such diagrams either contribute to renormalization of the $\pi N$
coupling constant or vanish.

Subleading corrections to the $3N$F at N4LO are visualized in the last column
of Fig. 3 and comprise one-loop diagrams made out of the leading
$\Delta=0$-vertices and a single insertion of a $\Delta=1$-interaction from
$\mathcal{L}_{\pi N}^{(2)}$, as well as tree-level graphs from the lowest-
order interactions and a single insertion of a $\Delta=3$-vertex. The longest-
range two-pion exchange $3N$F and the intermediate-range N4LO contributions of
types (b), (c) are derived using the MUT in Refs. Krebs:2012yv  and
Krebs:2013kha , respectively. The corresponding potentials do not involve any
unknown LECs. Short-range $3N$F terms of type (f) are considered in Ref.
Girlanda:2011fh  and depend on $13$ unknown LECs. Finally, one-loop
contributions to the topologies (d) and (e) at N4LO are still to be worked out
(and involve further unknown LECs).

In addition to the isospin-invariant $3N$F contributions discussed above and
depicted in Fig. 3, one also has to account for isospin-breaking corrections
stemming from different masses of the up and down quarks and QED effects. The
expressions for isospin-violating $3N$F up through N4LO are worked out using
the MUT in Ref. Epelbaum:2004xf , see also Ref. Friar:2004ca  for a related
work. Charge-dependent $3N$Fs involving virtual photon exchange are considered
in Refs. Yang:1979zz ; Yang:1983pd  and found to be rather weak. Parity- and
time-reversal-violating $3N$Fs have also been studied, see the review article
deVries:2020iea  and references therein.

To demonstrate the predictive power of ChEFT we consider below the chiral
expansion of the longest-range two-pion exchange $3N$F topology (a) as a
representative example. We restrict ourselves to isospin-invariant
contributions in the static limit, whose most general structure is given by

$\displaystyle V^{(a)}_{3N}$ $\displaystyle=$
$\displaystyle\frac{\vec{\sigma}_{1}\cdot\vec{q}_{1}\,\vec{\sigma}_{3}\cdot\vec{q}_{3}}{(q_{1}^{2}+M_{\pi}^{2})\,(q_{3}^{2}+M_{\pi}^{2})}$
(6) $\displaystyle\times$
$\displaystyle\Big{[}\mbox{\boldmath$\tau$}_{1}\cdot\mbox{\boldmath$\tau$}_{3}\,{\cal
A}(q_{2})+\mbox{\boldmath$\tau$}_{1}\times\mbox{\boldmath$\tau$}_{3}\cdot\mbox{\boldmath$\tau$}_{2}\,\vec{q}_{1}\times\vec{q}_{3}\cdot\vec{\sigma}_{2}\,{\cal
B}(q_{2})\Big{]}$ $\displaystyle+$ $\displaystyle\text{ short-range
terms}\;+\;\text{permutations}\,,$

where the functions ${\cal A}$ and ${\cal B}$ depend on the momentum transfer
$q_{2}\equiv|\vec{q}_{2}|$ of the second nucleon. These functions govern the
long-distance behavior of the $3N$F and are to be determined by means of the
chiral expansion

$\displaystyle{\cal A}(q_{2})$ $\displaystyle=$ $\displaystyle{\cal
A}_{[Q^{3}]}(q_{2})+{\cal A}_{[Q^{4}]}(q_{2})+{\cal
A}_{[Q^{5}]}(q_{2})+\ldots,$ $\displaystyle{\cal B}(q_{2})$ $\displaystyle=$
$\displaystyle{\cal B}_{[Q^{3}]}(q_{2})+{\cal B}_{[Q^{4}]}(q_{2})+{\cal
B}_{[Q^{5}]}(q_{2})+\ldots.$ (7)

The dominant contributions to ${\cal A}(q_{2})$ and ${\cal B}(q_{2})$ at N2LO,
stemming from the tree diagram in the second column of Fig. 3, have the form
Epelbaum:2002vt ; kolk1994

$\displaystyle{\cal A}_{[Q^{3}]}$ $\displaystyle=$
$\displaystyle\frac{g_{A}^{2}}{8F_{\pi}^{4}}\Big{[}(2c_{3}-4c_{1})M_{\pi}^{2}+c_{3}q_{2}^{2}\Big{]},$
$\displaystyle{\cal B}_{[Q^{3}]}$ $\displaystyle=$
$\displaystyle\frac{g_{A}^{2}c_{4}}{8F_{\pi}^{4}},$ (8)

where $c_{i}$ denote the $\pi N$ LECs from the subleading Lagrangian
$\mathcal{L}_{\pi N}^{(2)}$.

The leading corrections to Eq. (2.2.4) are generated at N3LO by one-loop
diagrams shown in the first line and third column of Fig. 3. Using DimReg, one
obtains Bernard:2007sp ; Ishikawa:2007zz

$\displaystyle{\cal A}_{[Q^{4}]}$ $\displaystyle=$
$\displaystyle\frac{g_{A}^{4}}{256\pi
F_{\pi}^{6}}\Big{[}\left(4g_{A}^{2}+1\right)M_{\pi}^{3}+2\left(g_{A}^{2}+1\right)M_{\pi}q_{2}^{2}$
(9) $\displaystyle+$ $\displaystyle
A(q_{2})\left(2M_{\pi}^{4}+5M_{\pi}^{2}q_{2}^{2}+2q_{2}^{4}\right)\Big{]},$
$\displaystyle{\cal B}_{[Q^{4}]}$ $\displaystyle=$
$\displaystyle-\frac{g_{A}^{4}}{256\pi
F_{\pi}^{6}}\big{[}A(q_{2})\left(4M_{\pi}^{2}+q_{2}^{2}\right)+(2g_{A}^{2}+1)M_{\pi}\big{]},$

where we have introduced the loop function

$A(q_{2})=\frac{1}{2q_{2}}\arctan\frac{q_{2}}{2M_{\pi}}.$ (10)

The loop function $A(q_{2})$ possesses a left-hand cut with the branch point
at $q_{2}^{2}=-(2M_{\pi})^{2}$, which corresponds to the kinematics when both
pions inside the loop of, e.g., the first N3LO diagram can become on-shell. On
the other hand, contributions from diagrams like the second N3LO graph do not
have left-hand cuts and are polynomial in $q_{2}^{2}$. Notice further that
loop integrals at N3LO involve only linear divergences, which vanish in DimReg
(and would have been absorbed into the LEC $c_{D}$ when using momentum-
dependent regularization schemes). This is consistent with the already
mentioned absence of tree-level $3N$F contributions involving a single
insertion from $\mathcal{L}_{\pi N}^{(3)}$.

Finally, the N4LO result for the functions ${\cal A}(q_{2})$ and ${\cal
B}(q_{2})$, obtained using DimReg, has the form Krebs:2012yv

$\displaystyle{\cal A}_{[Q^{5}]}$ $\displaystyle=$
$\displaystyle\frac{g_{A}^{2}\bar{e}_{14}}{2F_{\pi}^{4}}\big{(}2M_{\pi}^{2}+q_{2}^{2}\big{)}^{2}+\frac{g_{A}^{2}\big{(}M_{\pi}^{2}+2q_{2}^{2}\big{)}}{4608\pi^{2}F_{\pi}^{6}}$
$\displaystyle\times$
$\displaystyle\Big{\\{}\big{[}6c_{1}-2c_{2}-3c_{3}-2(6c_{1}-c_{2}-3c_{3})L(q_{2})\big{]}$
$\displaystyle\times$ $\displaystyle
12M_{\pi}^{2}-q_{2}^{2}\big{[}5c_{2}+18c_{3}-6L(q_{2})(c_{2}+6c_{3})\big{]}\Big{\\}},$
$\displaystyle{\cal B}_{[Q^{5}]}$ $\displaystyle=$
$\displaystyle\frac{g_{A}^{2}\bar{e}_{17}}{2F_{\pi}^{4}}\big{(}2M_{\pi}^{2}+q_{2}^{2}\big{)}\;-\;\frac{g_{A}^{2}c_{4}}{2304\pi^{2}F_{\pi}^{6}}$
$\displaystyle\times$
$\displaystyle\Big{\\{}q_{2}^{2}\big{[}5-6L(q_{2})\big{]}+12M_{\pi}^{2}\big{[}2+9g_{A}^{2}-2L(q_{2})\big{]}\Big{\\}},$

where $\bar{e}_{14}$ and $\bar{e}_{17}$ are renormalized LECs from
$\mathcal{L}_{\pi N}^{(4)}$, evaluated in the $\overline{\rm MS}$ scheme with
the renormalization scale $\mu$ set to $\mu=M_{\pi}$. Further, the loop
function $L(q_{2})$ is given by

$L(q_{2})=\frac{\sqrt{q_{2}^{2}+4M_{\pi}^{2}}}{q_{2}}\log\frac{\sqrt{q_{2}^{2}+4M_{\pi}^{2}}+q_{2}}{2M_{\pi}},$
(12)

and it also possesses a left-hand cut that starts at
$q_{2}^{2}=-(2M_{\pi})^{2}$. Notice that in Eq. (2.2.4), we have applied the
shifts of the LECs $c_{i}$ specified in Eq. (16) of Ref. Siemens:2016hdi  to
eliminate certain redundant linear combinations of LECs.

Given that all LECs entering Eqs. (2.2.4)-(2.2.4) are known from $\pi N$
scattering, which serves as a sub-process for the $3N$F topology (a), the
above expressions for ${\cal A}(q_{2})$ and ${\cal B}(q_{2})$ are to be
regarded as parameter-free predictions of ChEFT. To assess the convergence of
the chiral expansion for the longest-range $3N$F, we employ the numerical
values for the pion mass and decay constant of $M_{\pi}=138.03$ MeV and
$F_{\pi}=92.2$ MeV. The nucleon axial coupling $g_{A}$ is set to the effective
value of $g_{A}=1.289$ that accounts for the Goldberger-Treiman discrepancy.
The most reliable values of the higher-order $\pi N$ LECs are obtained from
matching ChPT with the solution of the dispersive Roy-Steiner equations for
$\pi N$ scattering at the subthreshold kinematical point, see Refs.
Siemens:2016jwj ; Hoferichter:2015tha  for details. In the following, we
employ the central values from the order-$Q^{4}$ heavy-baryon-NN fit of Ref.
Siemens:2016jwj : $c_{1}=-1.11$, $c_{2}=3.61$, $c_{3}=-5.60$, $c_{4}=4.26$,
$\bar{e}_{14}=1.16$ and $\bar{e}_{17}=-0.17$. Here, the values of $c_{i}$ and
$\bar{e}_{i}$ are given in units of GeV-1 and GeV-3, respectively.

In Fig. 4, we show the predicted behavior of the functions ${\cal A}(q_{2})$
and ${\cal B}(q_{2})$ at different orders in ChEFT.

Figure 4:  ChEFT predictions for the functions ${\cal{A}}(q_{2})$ and
${\cal{B}}(q_{2})$ that parametrize the longest-range behavior of the $3N$F
according to Eq. (6). Green dashed, blue dashed-dotted and red solid lines
show the results at N2LO, N3LO and N4LO, respectively.

In both cases, the order-$Q^{4}$ corrections to the dominant order-$Q^{3}$
contributions amount to about $30\%$. The order-$Q^{5}$ correction is very
small for ${\cal{A}}(q_{2})$ and amounts to less than $15\%$ of the N2LO
result for the function ${\cal{B}}(q_{2})$. The observed convergence pattern
for ${\cal{A}}(q_{2})$ and ${\cal{B}}(q_{2})$ fits well with expectations
based on the power counting with the expansion parameter $Q\sim{\rm
max}(M_{\pi},q_{2})/\Lambda_{b}$, see also the discussion in Sec. 2.2.3, and
shows that the low-momentum structure of the $3N$F can be described in ChEFT
in a controlled and systematically improvable fashion. Notice that in addition
to the static contributions considered above, the $3N$F of type (a) at N3LO
receives non-local $1/m_{N}$ corrections of relativistic origin Bernard:2011zr
. These have a much richer operator structure than the static terms and also
do not involve unknown LECs.

Parameter-free predictions for the one-pion-two-pion exchange and ring $3N$F
topologies corresponding to diagrams (b) and (c) in Fig. 3 can be found in
Refs. Bernard:2007sp ; Krebs:2013kha  and follow a qualitatively similar
pattern. We also emphasize that while the static two-pion exchange $3N$F has a
rather restricted form described by just two functions, the long-and
intermediate-range topologies (a), (b) and (c) contribute to all $20$
operators $\hat{O}_{\alpha}$ that appear in the parametrization of the $3N$F
according to Eq. (5). Interestingly, the results for the corresponding
functions $f_{\alpha}$ appear to be qualitatively in line with estimations
based on the large-$N_{c}$ expansion in QCD Phillips:2013rsa ;
Epelbaum:2014sea .

As pointed out in the introduction, an intermediate excitation of the nucleon
into the $\Delta(1232)$ resonance was historically recognized as one of the
most important $3N$F mechanisms and is at the heart of the celebrated Fujita-
Miyazawa $3N$F model 10.1143/PTP.17.360  as depicted in Fig. 1. How can this
important phenomenological insight be reconciled with the framework of ChEFT?
All results discussed in this section are obtained using the ChEFT formulation
with pions and nucleons as the only explicit degrees of freedom in the
effective Lagrangian. In such a framework, the information about the $\Delta$
resonance is included implicitly through its contributions to various LECs. In
particular, the LECs $c_{i}$ are largely governed by the $\Delta$ resonance
Bernard:1996gq :

$c_{2}^{\Delta}=-c_{3}^{\Delta}=2c_{4}^{\Delta}=\frac{4h_{A}^{2}}{9(m_{\Delta}-m_{N})}\simeq
2.7\;\text{GeV}^{-1}\,,$ (13)

where $m_{\Delta}$ refers to the mass of the $\Delta$ resonance while
$h_{A}\sim 1.34$ is the $N\Delta$ axial coupling constant. Thus, the $\Delta$
largely saturates the LECs $c_{2}$ and $c_{4}$, and provides about a half of
the $c_{3}$-value, thereby offering an explanation of the somewhat large
numerical values of these LECs as compared with their expected size
$|c_{i}|\sim\Lambda_{b}^{-1}$. This confirms that the intermediate $\Delta$
excitation indeed provides the dominant mechanism of the two-pion exchange
$3N$F through Eqs. (2.2.4) and (13).

Given the strong coupling of the $\Delta$ isobar to the $\pi N$ system and the
smallness of the mass difference $m_{\Delta}-m_{N}$, which is numerically of
the order of $2M_{\pi}$, it may be advantageous to include the $\Delta$ isobar
as an explicit degree of freedom in the effective Lagrangian instead of
integrating it out assuming $m_{\Delta}-m_{N}\sim\Lambda_{b}$, as done in the
standard formulation of ChEFT. In the $\Delta$-full formulation of ChPT of
Ref. Hemmert:1997ye , the mass difference $m_{\Delta}-m_{N}$ is treated as an
additional soft scale $m_{\Delta}-m_{N}\sim M_{\pi}$333For an alternative
counting of $m_{\Delta}-m_{N}$ see Ref. Pascalutsa:2002pi . (in spite of the
fact that this quantity does not vanish in the chiral limit). Accordingly, the
$\Delta$ isobar is treated explicitly in the effective Lagrangian, and the new
expansion parameter is denoted as
$\epsilon\in\\{p/\Lambda_{b},M_{\pi}/\Lambda_{b},(m_{\Delta}-m_{N})/\Lambda_{b}\\}$.
Extensive studies in the single-nucleon sector using manifestly covariant
formulations of ChPT have revealed that the explicit treatment of the $\Delta$
isobar indeed often leads to an improved convergence of the EFT expansion, see
e.g. Refs. Siemens:2016hdi ; Yao:2016vbz ; HillerBlin:2014diw ;
Thurmann:2020mog , albeit at the cost of more involved calculations and a
larger number of LECs. Similar conclusions have been reached for the $NN$
force using the heavy-baryon formulation of $\Delta$-full ChEFT, for which the
expressions are presently available up through N2LO $\epsilon^{3}$
Ordonez:1995rz ; Kaiser:1998wa ; Krebs:2007rh . Isospin-breaking contributions
to the $NN$ potential have also been considered within the
$\epsilon$-expansion scheme Epelbaum:2008td .

The explicit treatment of the $\Delta$ isobar also has implications for the
$3N$F. In particular, the dominant Fujita-Miyazawa-type contribution is
shifted from N2LO to NLO in the $\Delta$-full scheme, since the diagram in
Fig. 1 counts as of order $\epsilon^{2}$. Interestingly, this is the only
contribution of the $\Delta$ to the $3N$F up-to-and-including the order N2LO
(i.e., $\epsilon^{3}$) Epelbaum:2007sq . Recently, the longest-range $3N$F of
type (a) in Fig. 3 has been worked out at order $\epsilon^{4}$ Krebs:2018jkc .
The resulting parameter-free predictions were found to agree well with the
order-$Q^{5}$ results of Ref. Krebs:2012yv , which shows that the effects of
the $\Delta$ are well captured by resonance saturation of the $\pi N$ LECs in
the $\Delta$-less ChEFT framework and suggests that convergence might have
been reached for this topology. The corresponding results for the
intermediate-range $3N$F of types (b) and (c) within the $\epsilon$-expansion
are not yet available. Notice that in the $\Delta$-less scheme, the first
information about the $\Delta$ isobar for these topologies appears only at
N4LO, since the N3LO expressions depend solely on the LO LECs, see e.g. Eq.
(9), which contain no information about the $\Delta$ isobar. The explicit
inclusion of the $\Delta$ might, therefore, be advantageous to achieve
convergence for these topologies.

So far we have left open the question of regularization of the $3N$F. As
already pointed out in Sec. 2.2.2, ChEFT allows one to derive low-momentum
approximations of the nuclear forces, but the method looses its validity for
large momenta $|\vec{p}\,|\gtrsim\Lambda_{b}$. For example, the ChEFT
predictions for the function ${\cal{A}}(q_{2})$ in Eqs. (2.2.4), (9) and
(2.2.4) behave at large $q_{2}$-values as ${\cal{A}}_{[Q^{3}]}(q_{2})\sim
q_{2}^{2}$, ${\cal{A}}_{[Q^{4}]}(q_{2})\sim q_{2}^{3}$ and
${\cal{A}}_{[Q^{5}]}(q_{2})\sim q_{2}^{4}\,\log(q_{2})$, respectively. While
the expansion for ${\cal{A}}(q_{2})$ appears to converge well within the
applicability range of ChEFT, with the order-$Q^{5}$ result providing a tiny
correction as shown in the left panel of Fig. 4, the N4LO contribution starts
exceeding the N3LO correction at $q_{2}\sim 780$ MeV, signalling the breakdown
of the expansion at large momenta. As explained in Sec. 2.2.2, the expressions
for nuclear forces derived in ChEFT have to be regularized by introducing a
finite cutoff $\Lambda\sim\Lambda_{b}$ to remove their uncontrolled behavior
at large momenta and make the $A$-body Schrödinger equation well defined. In
the $NN$ sector, this can be achieved, e.g., by simply multiplying the
unregularized potentials with some regulator functions. Unfortunately,
following this same strategy for the $3N$F was found in Refs. Epelbaum:2019kcf
; Epelbaum:2019jbv ; Epelbaum:2022cyo  to violate the chiral symmetry. The
problem can be best illustrated with the $3N$F of type (b) stemming from the
first of the two diagrams shown in the second line and third column of Fig. 3.
When calculating the scattering amplitude, the irreducible $3N$F is
supplemented with the corresponding reducible contributions stemming from the
iteration of the Faddeev equation. This reducible contribution involves a
linearly divergent piece $\propto\Lambda$, whose structure violates the chiral
symmetry. Accordingly, the amplitude cannot be renormalized by re-adjusting
the LEC $c_{D}$, and the whole ChEFT expansion for $3N$ scattering breaks
down. As explained in Refs. Epelbaum:2019kcf ; Epelbaum:2019jbv ;
Epelbaum:2022cyo , the problem is caused by mixing of two different
regularization schemes, namely DimReg when deriving the $3N$F and cutoff
regularization in the Schrödinger equation. The same issue affects loop
contributions to the nuclear current operators Krebs:2019ddp ; Krebs:2020pii .
This shows that a consistent regularization of the $3N$F and nuclear currents
cannot be achieved by simply multiplying the corresponding potentials derived
using DimReg with some cutoff functions. Rather, $3N$Fs, 4NFs and nuclear
currents starting from N3LO need to be re-derived using a cutoff instead of
DimReg.

The above conceptual issues have been the main obstacle for the applications
of ChEFT to three- and more-nucleon systems beyond N2LO. In contrast to
DimReg, maintaining the chiral and gauge symmetries in cutoff regularization
represents a non-trivial task, and the calculations become considerably more
involved due to the appearance of an additional mass scale $\Lambda$.
Recently, it was shown that the symmetry-preserving gradient flow method,
which has been successfully applied to Yang-Mills theories Luscher:2011bx ;
Luscher:2013cpa  and is nowadays widely employed in lattice-QCD simulations
Luscher:2013vga , can be merged with ChEFT Krebs:2023gge  and applied to
obtain consistently regularized nuclear forces and currents Krebs:2023ljo .
This is achieved by generalizing the pion fields to the (artificial) fifth
dimension, usually referred to as the flow “time” $\tau$. The flow-time
evolution of the pion fields is governed by the chirally covariant version of
the gradient flow equation and amounts to smothening of the pion field, i.e.,
a non-zero flow time $\tau$ acts as a regulator. Moreover, when applied to the
OPEP, the gradient flow method reduces to the SMS regulator of Ref.
Reinert:2017usi  specified in Eq. (3). The new developments in Refs.
Krebs:2023gge ; Krebs:2023ljo  lay down the foundation for deriving
consistently regularized $3N$F and nuclear currents beyond N2LO. Work along
these lines is in progress.

####  2.2.5 Selected applications of the lading $3N$F

We now turn to applications of the chiral nuclear forces to the 3N and heavier
systems, focusing especially on the role of $3N$Fs. Given the lack of
consistently regularized $3N$Fs at N3LO and N4LO as described in the previous
section, the accuracy level of the applications reviewed below is limited to
N2LO.

Nucleon-deuteron ($Nd$) elastic scattering and breakup observables can be
calculated, starting from a given nuclear Hamiltonian, by solving the Faddeev
equations in the partial wave basis as described in detail in the review
article Gloeckle:1995jg . $Nd$ scattering calculations described below are
carried out without explicit inclusion of the Coulomb interaction and
neglecting relativistic effects, which are known to be small al low and
intermediate energies Witala:2004pv ; Witala:2011yq . We also restrict
ourselves to studies based on the second-generation of chiral $NN$ potentials
introduced in Refs. Epelbaum:2014efa ; Epelbaum:2014sza ; Reinert:2017usi ,
see Refs. epelbaum2009 ; Epelbaum:2005pn ; Kalantar-Nayestanaki_2012 ;
Epelbaum:2012vx ; Hammer:2012id  and references therein for earlier studies
along this line. The novel semi-local regularization method employed in these
NN potentials allows one to avoid the appearance of noticeable artifacts in
elastic $Nd$ scattering at large energies reported in Ref. Witala:2013ioa  for
the first-generation of N3LO chiral $NN$ potentials of Refs. Epelbaum:2004fk ;
Entem:2003ft , which can be traced back to the artifacts in the deuteron wave
function. The novel semi-local potentials thus provide a very good starting
ground for systematic studies of $Nd$ scattering in ChEFT.

In a series of papers by the LENPIC Collaboration, the semi-local coordinate-
space regularized (SCS) $NN$ potentials of Refs. Epelbaum:2014efa ;
Epelbaum:2014sza  at all available orders from LO to N4LO were employed to
calculate $Nd$ scattering observables and properties of selected nuclei up to
48Ca LENPIC:2015qsz ; Maris:2016wrd ; LENPIC:2018lzt . While these
calculations did not include the $3N$F and thus should be regarded as
incomplete starting from N2LO, they have yielded a number of interesting
observations. In particular, calculations beyond the last complete order NLO
were found to differ from experimental data well outside the estimated
truncation uncertainties, thus providing unambiguous evidence for missing
$3N$Fs. Moreover, the amount of deviations between theory and experimental
data was found to be consistent with estimations based on the power counting.

In Ref. LENPIC:2018ewt , these studies have been extended to include the N2LO
$3N$F, regularized in coordinate space consistently with the SCS $NN$
potentials. The need to perform regularization of the $3N$F in coordinate
space was found to introduce significant computational overhead for its
numerical implementation, which was one of the motivations to reformulate the
SCS regularization scheme to momentum space Reinert:2017usi . Notice that
partial wave decomposition of a general $3N$F can be carried out in an
automated way by numerically performing the required angular integrations as
described in Refs. Golak:2009ri ; Hebeler:2015wxa , see also Ref.
PhysRep21Hebeler .

As detailed in Sec. 2.2.4, the $3N$F at N2LO depends on the LECs $c_{D}$ and
$c_{E}$ that need to be determined from few-nucleon data. It is customary to
fix the linear combination of these LECs to reproduce the 3H binding energy,
which determines $c_{E}$ as a function of $c_{D}$. To fix the second LECs,
different observables have been proposed in the literature including the Nd
doublet scattering length Epelbaum:2002vt ; Piarulli:2017dwd , 3H beta decay
Gazit:2008ma , 4He binding energy Nogga:2005hp , charge radii of the $A=3,4$
nuclei and properties of few- and many-nucleon systems Navratil:2007we ;
Ekstrom:2015rta ; Lynn:2017fxg . Clearly, to allow for the most stringent test
of the nuclear Hamiltonian, the LECs should ideally be fixed from $A\leq 3$
observables. In Ref. LENPIC:2018ewt , a variety of observables including the
Nd doublet scattering length as well as the Nd total and differential cross
sections at the energies of $E_{\rm lab}=70$, $108$ and $135$ MeV have been
considered. Taking into account both the experimental errors and the EFT
truncation uncertainty, the strongest constraint on $c_{D}$ was found to
result from the requirement to reproduce the proton-deuteron (pd) differential
cross section minimum using the data from Ref. sekiguchi2002 . The resulting
Hamiltonian was then used to calculate $Nd$ elastic scattering observables,
ground state energies and selected excitation energies of $p$-shell nuclei up
to 12C. For almost all considered nuclei, adding the $3N$F was found to
significantly improve the description of experimental data. A detailed
analysis of elastic $Nd$ scattering and breakup using the same Hamiltonian is
presented in Ref. Witala:2019ffj .

These studies were further refined in Ref. Maris:2020qne  by employing the
high-precision SMS $NN$ interactions of Ref. Reinert:2017usi  along with the
consistently regularized N2LO $3N$Fs, utilizing Bayesian methods for
quantifying EFT truncation errors and extending the range of considered
observables.

Figure 5:  ChEFT predictions for the differential cross section, deuteron
vector analyzing power $A_{y}^{d}$ and deuteron tensor analyzing powers
$A_{xz}$ and $A_{xx}$ in elastic neutron-deuteron scattering at $E_{\rm
lab}=135$ MeV. Dark-shaded orange and green bands show the NLO and N2LO
results at the $1\sigma$ confidence level, respectively, while the
corresponding light-shaded bands show the $2\sigma$-intervals. Experimental
data are pd elastic scattering data from Ref. sekiguchi2002 . Dashed lines in
the middle of green bands are the actual N2LO predictions. Dotted lines are
obtained using the NN interaction at the highest available order N4LO+,
supplemented with the N2LO $3N$F (with the appropriately re-adjusted LECs
$c_{D}$ and $c_{E}$). In all calculations, the cutoff is chosen to be
$\Lambda=450$ MeV.

In Fig. 5, we show selected results for $Nd$ elastic scattering observables at
$E_{\rm lab}=135$ MeV, which may serve as representative examples. Given that
the LECs $c_{D}$ and $c_{E}$ are fixed from the 3H binding energy and the
differential cross section minimum at $E_{\rm lab}=70$ MeV, the shown results
are to be regarded as predictions. The experimental data from Ref.
sekiguchi2002  are mostly in agreement with the calculations (within errors),
but the N2LO truncation uncertainty at this moderate energy appears to be
rather large. The description of $Nd$ data at N2LO is qualitatively similar to
the one for proton-proton scattering as a comparable energy, shown in Fig. 2.
Based on the results in the $NN$ system, it is expected that taking into
account the $3N$F up through N4LO would allow one to achieve a precise
description of $Nd$ scattering data, comparable to that of the neutron-proton
and proton-proton data reported in Refs. Reinert:2017usi ; Reinert:2020mcu .

It is interesting to explore the impact of corrections to the $NN$ force
beyond N2LO. To this aim, a set of calculations based on the SMS NN potentials
up through N4LO+, supplemented with the N2LO $3N$F, has been performed in Ref.
LENPIC:2022cyu . In all cases, the LECs $c_{D}$ and $c_{E}$ have been fixed
following the standard LENPIC fitting protocol described above. For the
considered $Nd$ scattering observables, the inclusion of corrections to the NN
force beyond N2LO changes the central N2LO predictions, shown by the dashed
lines in Fig. 5, to the dotted lines. The results visualized by the dashed and
dotted lines differ by N3LO terms, and it is comforting to see that the
differences between these lines are within the estimated N2LO truncation
errors. While the higher-order corrections to the $NN$ force do appear to
noticeably improve the description of the tensor analyzing powers $A_{xz}$ and
$A_{xx}$ in the angular range of $\theta_{\rm CM}\in[45^{\circ},100^{\circ}]$,
they still leave room for improvement that should come from higher-order
contributions to the $3N$F.

Further, we mention a comprehensive study of the symmetric space-star deuteron
breakup configuration in Ref. Witala:2021zmb . This particular configuration
is known to exhibit large discrepancies between theory and data at energies
below $E_{\rm lab}\sim 25$ MeV that could so far not be resolved. Moreover,
the calculated cross section appears to be largely insensitive to the types of
$3N$F considered so far. It would be interesting to study the impact of $3N$F
contributions beyond N2LO on this observable. Finally, a detailed
investigation of the deuteron breakup reaction at $E_{\rm lab}\sim 130$ and
$200$ MeV using the chiral SMS $NN$ and $3N$-forces and covering the whole
kinematically allowed phase space has been carried out in Ref.
Skibinski:2023nnn .

Figure 6:  ChEFT predictions for the ground state energies of light nuclei
calculated using the No-Core Configuration Interaction (NCCI) method
Barrett:2013nh . Crosses, circles and diamonds show the NLO results,
incomplete N2LO predictions using the $NN$ interactions only and the complete
results at N2LO using both the $NN$ force and $3N$F. Orange, green dark-shaded
and green light-shaded error bars show the NLO $(1\sigma)$, N2LO $(1\sigma)$
and N2LO $(2\sigma)$ ChEFT truncation errors, respectively, while black error
bars give the numerical uncertainty of the NCCI method. All theoretical
predictions are obtained using the cutoff value of $\Lambda=450$ MeV.
Horizontal lines depict the experimental values for the binding energies. See
Ref. Maris:2020qne  for details.

In Fig. 6, we show the NLO and N2LO predictions for the ground-state energies
of light $p$-shell nuclei from Ref. Maris:2020qne . For all nuclei, the
predicted binding energies at both NLO and N2LO agree with experimental values
within truncation errors. To facilitate the quantification of $3N$F effects,
we also show the results based on the N2LO $NN$ potential without inclusion of
the $3N$F. For light nuclei up to 10B, the inclusion of the $3N$F leads to a
significant improvement. However, for both considered $A=12$-nuclei, the $3N$F
effects appear to be too large leading to overbinding. This overbinding was
shown in Ref. LENPIC:2022cyu  to be resolved by taking into account the
corrections to the $NN$ force beyond N2LO.

Last but not least, N4LO short-range contributions to the $3N$F have been
considered in the exploratory studies of $Nd$ scattering reported in Refs.
Girlanda:2018xrw ; Epelbaum:2019zqc ; Witala:2022rzl . While incomplete, these
studies demonstrate that the N4LO contact interactions of natural size have
the potential to both resolve the long-standing $A_{y}$-puzzle in low-energy
elastic Nd scattering and strongly improve the description of scattering
observables at high energies.

###  2.3 Experimental studies of three-nucleon forces

To uncover the structure of $3N$Fs one must utilize systems with more than two
nucleons ($A>3$). Few-nucleon scattering offers a unique opportunity to probe
dynamical aspects of $3N$Fs, which are momentum, spin and isospin dependent,
since it provides not only the cross sections but also a variety of spin
observables at different incident nucleon energies. A direct comparison
between experimental data and rigorous numerical calculations using the
Faddeev theory and based on the realistic nuclear potentials provides detailed
information on the structure of $3N$Fs.

The importance of $3N$Fs in the continuum spectrum was shown, for the first
time, in nucleon–deuteron ($Nd$) elastic scattering at the end of the 1990s
wit98 . $3N$Fs were found to lead to pronounced effects around the cross-
section minimum occurring at the values of the c.m. scattering angle of
$\theta_{c.m.}\approx 120^{\circ}$ for incident energies above $60~{}\rm
MeV/nucleon$. Since then, proton-deuteron($pd$) and neutron-deuteron($nd$)
scattering experiments at 60–300 MeV/nucleon have been performed at the
facilities, e.g. RIKEN, RCNP, KVI, IUCF, TSL, and LANSCE, providing precise
data of the cross sections as well as various types of spin observables
Kalantar-Nayestanaki_2012 .

Figure 7:  Differential cross section and deuteron analyzing powers $iT_{11}$,
$T_{22}$ for elastic nucleon-deuteron scattering. See text for descriptions
for the theoretical calculations.

Figure 7 shows some representative experimental results reported in Refs.
sekiguchi2002 ; PhysRevLett.95.162301 ; PhysRevC.76.014004 ;
PhysRevC.89.064007  ($dp$ or $pd$ in black open circles, $nd$ in black solid
circles). Data shown in blue are from Refs. PhysRevC.68.051001 ;
PhysRevC.78.014006 . The experimental data are compared with the Faddeev
calculations with and w/o $3N$Fs. The red (blue) bands are the calculations
with (without) Tucson-Melbourne 99 $3N$F based on the modern $NN$ potentials,
i.e. CD Bonn, AV18, Nijmegen I and II. The solid lines are the calculations
based on the AV18 potential with including the Urbana IX $3N$F. The $3N$Fs
considered here are 2$\pi$–exchange types. For most of the observables shown
in the figure, large differences are found at the backward angles between the
data and the calculations based on $NN$ forces only. These discrepancies
become larger with an increasing incident energy. For the cross section, the
$3N$Fs remove the discrepancies at lower energies. At higher energies,
however, the differences still remain even including the $3N$F potentials at
the angles $\theta_{\rm c.m.}\gtrsim 120^{\circ}$, which extent to the very
backward angles $\theta_{\rm c.m.}\sim 180^{\circ}$ at 250 MeV/nucleon. For
the vector analyzing power $iT_{11}$, the description of the experimental data
by the theoretical calculations is similar to that of the cross section.
However, for the tensor analyzing power $T_{22}$ a different pattern is
observed as the calculations including the $3N$Fs do not explain the data at
the lower two energies.

A direct comparison between the data and the Faddeev calculations in elastic
$Nd$ scattering led to the following conclusions so far: (1) the $3N$F is
definitely needed in elastic $dp$ scattering; (2) the $3N$F effects are
clearly seen at the angles where the cross section takes its minimum, and
their effects become larger with an increasing incident energy; (3) spin
dependent parts of the current $3N$F models are deficient; (4) the short-range
components of the $3N$F are probably required for high-momentum transfer
region (at the very backward angles). These results of comparison between the
data and the calculations based on the above phenomenological nuclear
potentials have been pushing into more detailed study of three-nucleon
scattering based on the $\chi$EFT nuclear potential as described in Sec. 2.2.

Figure 8:  Experimental data (solid circles) and the calculations from the
solutions of exact AGS equations of the differential cross section
$d\sigma/d\Omega$ at 65 MeV for $p$-${}^{3}\rm He$ elastic scattering
PhysRevC.103.044001 . Calculations based on the $NN$ potentials are shown with
magenta dash-dotted (AV18), black solid (CD Bonn), red dot-dot-dashed
(INOY04), green solid (SMS400), and green dashed (SMS500) lines. Black dashed
lines are calculations based on the CD Bonn+potential.

It is important to keep in mind that the $Nd$ system is dominated by the
isospin $T=1/2$ states. Thus, one needs other probes to constrain the
properties of the $3N$Fs with total isospin $T=3/2$, whose importance is
strongly suggested in the description of asymmetric nuclear matter, e.g.
neutron-rich nuclei and pure neutron matter. Such aspects could be studied in
four-nucleon systems like proton-$\rm{}^{3}He$. In recent years, remarkable
theoretical progress in solving the 4N scattering problem using realistic
Hamiltonians has been reported Deltuva_PRC76 ; lazauskas2009 ; viviani_PRL111
even above the 4N breakup threshold deltuva_prc87 ; fonseca_FBS2017 , opening
new possibilities for nuclear force study in the 4N system at intermediate
energies. In Fig. 8 the recent results of $p$-${}^{3}\rm He$ scattering at
intermediate energy are presented PhysRevC.103.044001 . The cross section data
at 65 MeV/nucleon are compared with the calculations from the solutions of the
exact AGS equations as given in Refs. deltuva_prc87 ; fonseca_FBS2017  using a
variety of $NN$ potentials: the AV18, the CD Bonn, and the INOY04 INOY04 , and
the chiral N4LO $NN$ potentials with the cutoff parameters $\Lambda$ = 400
${\rm MeV}/c$ (SMS400) and $\Lambda=500{\rm MeV}/c$ (SMS500) SMS . The
calculations based on the CD Bonn+$\Delta$ model CDBD , which allows an
excitation of a nucleon to an isobar, thereby providing effective $3N$Fs and
$4N$Fs, are also presented. Large contributions of the effective $3N$ and $4N$
forces have been found to be largely canceled by the dispersive isobar effect,
leading to rather small total contributions from the $\Delta$-isobar. The
results are in contrast to those in $dp$ scattering, where the cancellation is
less pronounced Nemoto1999 . The obtained results indicate that $p$-${}^{3}\rm
He$ elastic scattering at intermediate energies is an excellent tool to
explore nuclear interactions including $3N$Fs, which cannot be accessed in
$3N$ scattering. It would be interesting to see how the predictions with such
$3N$Fs explain the data for the $p$-${}^{3}\rm He$ elastic scattering, which
will enable us to perform detailed discussions of the effects of $3N$Fs
including the $T=3/2$ isospin channels.

##  3 Three-body forces in atoms

Cold atoms are systems of dilute atomic gases cooled down to nano-Kelvin
temperatures. While the interactions among atoms are dominated by two-body
forces and their intrinsic three-body forces are negligible, one can engineer
and realize cold atoms with an effective three-body force as strong as or even
stronger than the two-body force, capitalizing on the high controllability of
cold-atom experiments gross2017quantum ; schafer2020tools . In this section,
we overview recent experimental and theoretical attempts to observe, control,
and utilize three-body forces in cold atoms.

###  3.1 Experimental observations of three-body forces for cold atoms in an
optical lattice

A system of cold atoms such as a Bose-Einstein condensate (BEC) or a Fermi-
degenerate gas loaded into an optical lattice is known to be an ideal
experimental platform for the quantum simulation of strongly-correlated
quantum many-body systems bloch_many-body_2008 , such as the Hubbard model,
owing to the high controllability of its parameters like hopping energy and
on-site interaction, and so on. When cold atoms are trapped in a sufficiently
deep optical lattice potential, the hopping between neighboring lattice sites
is negligible. This gives us a novel possibility to simultaneously realize
various well-defined few-body systems with definite atom numbers. Under these
conditions, the trapping potential for the atoms in each lattice site is well
approximated by a harmonic potential. Therefore, a system of cold atoms in an
optical lattice is also a useful platform for studying few-body physics in a
trap.

Various spectroscopic techniques have been developed in cold atoms, which are
quite useful to probe the energy of these few-body systems. The first
occupancy-resolved high-resolution spectroscopy was reported for a radio-
frequency spectroscopy of the ground hyperfine states of rubidium atoms
Campbell2006 . The observed almost equi-distance between the neighboring
resonance frequencies is explained by the pairwise interactions alone.
However, slight deviation of the equi-distance between the neighboring
resonance frequencies was also observed, indicating that the simple pairwise
interaction is insufficient. The qualitative explanation in the microscopic
description of the system was given as the broadening of the Wannier function
due to the two-body interaction. Similar observations of the slight deviation
from the prediction based on the pairwise interaction were reported in the
experiments using various methods like matter-wave collapse and revival
measurement Will2010 , resonant lattice modulation PhysRevLett.107.175301 ,
and laser spectroscopy Franchi_2017 ; Goban2018 . This deviation is
successfully explained by introducing an effective three-body force between
the trapped atoms within perturbative treatments Johnson_2009 ; Johnson_2012 .
Interestingly, the microscopic origin of the effective three-body force, where
one of the three atoms in the lowest vibrational state is excited to the
higher vibrational state due to the inter-atomic interaction with the second
atom and then is returned to the lowest state via the inter-atomic interaction
with the third atom, has close analogy with the Fujita-Miyazawa type nuclear
three-body force discussed in Sec. 1 10.1143/PTP.17.360  (compare Fig. 1 with
Fig. 9 in the next subsection).

Here, we stress that the three-body forces in the nuclear and trapped-atom
systems share the same grounds in that they are the effective forces which are
considered in low-energy effective descriptions of the systems. The cold-atom
system can be a useful testbed to explore the three-body forces, firstly
because of its high controllability, and secondly because the intrinsic three-
body force between atoms is so small that we can directly investigate the
physical mechanism for the emergence of the effective three-body forces.

Following this line of research direction, an occupancy-resolved high-
resolution laser spectroscopy has been performed in recent experiments to
investigate a new regime of three-body force honda2024evidence . In
particular, by working with the ultra-narrow optical transition between the
ground ${}^{1}S_{0}$ and metastable ${}^{3}P_{2}$ states of ytterbium (Yb)
atoms PhysRevLett.110.173201 , one can utilize both the high resolution in the
determination of the binding energy of the few-body atomic system and the high
controllability of the two-body interaction through an inter-orbital
anisotropy-induced Feshbach resonance Chin2010 . This enables us to study a
new regime of three-body force beyond the perturbative treatment in the
weakly-interacting regime. While the data at small scattering lengths far from
the Feshbach resonance are well explained by perturbative calculations, as in
the previous works, the results obtained around the Feshbach resonance show
significantly different behaviors from the equi-distance between the
neighboring resonance frequencies, indicating a strongly-interacting regime of
three-body forces, owing to the resonant control of the two-body interaction
honda2024evidence . These results obtained by a cold-atom quantum simulator
with tunable interactions can be a useful benchmark for developing the theory
of the three-body forces beyond the perturbative regime, and will give
insights into the nuclear three-body forces where the non-pertubative
treatments are generally difficult.

###  3.2 Three-body forces in low dimensions

As demonstrated experimentally, three-body forces naturally appear in physics
of cold atoms when they are confined into low dimensions in spite of their
interaction being purely pairwise in free space. While three-body forces are
discussed in the previous subsection for quasi-zero-dimensional systems
created by three-dimensional optical lattices, they are also possible under
two- or one-dimensional optical lattices where atoms have freedom to move in
one or two directions. Here we provide some theoretical accounts of three-body
forces in such low dimensions and their physical consequences. We set
$\hbar=k_{B}=1$ in this and next subsections.

As an illustrative example, let us consider weakly-interacting bosons
subjected to a two-dimensional optical lattice that confines them into quasi-
one-dimension. Such a system is described by

$\displaystyle\hat{H}_{\mathrm{3D}}$
$\displaystyle=\int\\!d^{3}\bm{r}\,\biggl{[}\hat{\Phi}^{\dagger}(\bm{r})\left\\{-\frac{\nabla^{2}}{2m}+\frac{m\omega_{\perp}^{2}(y^{2}+z^{2})}{2}\right\\}\hat{\Phi}(\bm{r})$
$\displaystyle\quad{}+\frac{g_{\mathrm{3D}}}{2}\hat{\Phi}^{\dagger}(\bm{r})\hat{\Phi}^{\dagger}(\bm{r})\hat{\Phi}(\bm{r})\hat{\Phi}(\bm{r})\biggr{]},$
(14)

where $\hat{\Phi}(\bm{r})$ is the annihilation operator of bosons and the two-
body coupling is related to the scattering length via $g_{\mathrm{3D}}=4\pi
a_{\mathrm{3D}}/m$. Due to the transverse confinement, the motion of bosons in
$y$ and $z$ directions is quantized by the excitation energy of
$\omega_{\perp}$. Therefore, as far as low-energy physics relative to
$\omega_{\perp}$ is concerned, the transverse motion cannot be excited so that
the motion of bosons is restricted to the $x$ direction only. Accordingly,
such low-energy physics of the system should be described by an effective one-
dimensional Hamiltonian in the form of

$\displaystyle\hat{H}_{\mathrm{1D}}$
$\displaystyle=\int\\!dx\,\biggl{[}-\hat{\phi}^{\dagger}(x)\frac{\partial_{x}^{2}}{2m}\hat{\phi}(x)+\frac{g_{2}}{2}\hat{\phi}^{\dagger}(x)\hat{\phi}^{\dagger}(x)\hat{\phi}(x)\hat{\phi}(x)$
$\displaystyle\quad{}+\frac{g_{3}}{6}\hat{\phi}^{\dagger}(x)\hat{\phi}^{\dagger}(x)\hat{\phi}^{\dagger}(x)\hat{\phi}(x)\hat{\phi}(x)\hat{\phi}(x)+\cdots\biggr{]}.$
(15)

Here $\hat{\phi}(x)$ is the annihilation operator of bosons in the transverse
ground state and is related to $\hat{\Phi}(\bm{r})$ via its expansion of

$\displaystyle\hat{\Phi}(\bm{r})=\sum_{\ell\in\mathbb{Z}}\sum_{n\geq
0}\hat{\phi}_{\ell n}(x)\varphi_{\ell n}(y,z),$ (16)

where $\hat{\phi}(x)=\hat{\phi}_{00}(x)$ and $\varphi_{\ell n}(y,z)$ is the
normalized eigenfunction of a two-dimensional harmonic potential with energy
$E_{\ell n}=(1+|\ell|+2n)\,\omega_{\perp}$.

Figure 9:  Three-body scattering process inducing an effective three-body
coupling in low dimensions. Solid and dashed lines represent bosons in
transverse ground and excited states, respectively.

Because the original three-dimensional Hamiltonian in Eq. (3.2) has a two-body
coupling, the resulting Eq. (3.2) also has a two-body coupling provided by

$\displaystyle
g_{2}=g_{\mathrm{3D}}\int\\!dydz\,[\varphi_{00}(y,z)]^{4}=\frac{2a_{\mathrm{3D}}}{ml_{\perp}^{2}}$
(17)

to the lowest order in $a_{\mathrm{3D}}$ with
$l_{\perp}=1/\sqrt{m\omega_{\perp}}$. Furthermore, Eq. (3.2) has effective
three-body and higher-body couplings induced by virtual excitation of bosons
to transverse excited states PhysRevLett.89.110401 ; PhysRevLett.96.030406 ;
PhysRevLett.100.210403 . In particular, the three-body coupling to the lowest
order in $a_{\mathrm{3D}}$ is induced by the three-body scattering process
depicted in Fig. 9 and provided by

$\displaystyle g_{3}$
$\displaystyle=6\sum_{n=1}^{\infty}\frac{[g_{\mathrm{3D}}\int\\!dydz\,\\{\varphi_{00}(y,z)\\}^{3}\varphi_{0n}(y,z)]^{2}}{E_{00}-E_{0n}}$
$\displaystyle=-12\ln\\!\left(\frac{4}{3}\right)\frac{a_{\mathrm{3D}}^{2}}{ml_{\perp}^{2}},$
(18)

where
$\varphi_{0n}(y,z)=e^{-(\rho/l_{\perp})^{2}/2}L_{n}[(\rho/l_{\perp})^{2}]/(\sqrt{\pi}\,l_{\perp})$
with $\rho=\sqrt{y^{2}+z^{2}}$ is employed. We note that $g_{3}$ presented in
Refs. PhysRevLett.96.030406 ; PhysRevLett.100.210403  was four times larger
than Eq. (18) but was later corrected in Refs. PhysRevLett.105.090404 ;
Mazets_2010 . Dots in Eq. (3.2) include higher-body couplings as well as
couplings involving derivatives such as effective-range corrections.

The two-body coupling in Eq. (17) is linear in $a_{\mathrm{3D}}$ and can be
repulsive or attractive depending on the positive or negative sign of
$a_{\mathrm{3D}}$. On the other hand, the three-body coupling in Eq. (18)
appears at the quadratic order in perturbation, so that it is always
attractive. Because the former dominates over the latter for weakly-
interacting bosons, it is generally expected that physics is essentially
determined by the two-body coupling and the three-body coupling only provides
quantitative corrections that may be negligible without spoiling essential
physics. However, this is not the case in one dimension because the two-body
and three-body couplings have distinct characters: while the two-body coupling
preserves the integrability, it is broken by the three-body coupling
PhysRevLett.89.110401 ; PhysRevLett.96.030406 ; PhysRevLett.100.210403 .
Therefore, even if the three-body coupling is quantitatively small, it is the
leading perturbation to break the integrability and may have some
qualitatively significant consequences in one dimension.

In particular, because two-body scatterings in one dimension do not change the
momentum distribution, it is three-body scatterings that cause thermalization
of a quasi-one-dimensional Bose gas. The thermalization rate due to the
effective three-body coupling was estimated in Refs. PhysRevLett.100.210403 ;
Mazets_2010  and was found to be consistent with the time needed for
evaporative cooling of a 87Rb gas Hofferberth2007 ; Hofferberth2008 .
Similarly, the thermal conductivity of a weakly-interacting Bose gas in quasi-
one-dimension was shown to be dominated by the three-body coupling rather than
the two-body coupling PhysRevE.106.064104 . Its expression was obtained as

$\displaystyle\kappa=\frac{\mathcal{N}}{m^{3}g_{3}^{2}}\tilde{\kappa}\\!\left(\frac{mT}{\mathcal{N}^{2}}\right),$
(19)

where $T$ is the temperature, $\mathcal{N}$ is the number density, and
$\tilde{\kappa}(*)$ is a dimensionless function determined numerically in Ref.
PhysRevE.106.064104  (see Fig. 6 therein).

The three-body coupling also has significant consequences on few-body physics
in one dimension. When the two-body coupling is attractive, $N$ bosons in the
absence of three-body coupling are known to form a single bound state
10.1063/1.1704156 , whose binding energy is provided by

$\displaystyle E_{N}^{\mathrm{(MG)}}=-\frac{N(N^{2}-1)}{24}mg_{2}^{2}.$ (20)

Such an $N$-body cluster has no interaction (reflection probability) with an
extra boson, being another manifestation of the integrability
PhysRevLett.96.163201 ; PhysRevA.85.062711 . Therefore, their interaction in
quasi-one-dimension is dominated by the effective three-body coupling as the
leading perturbation to break the integrability. The scattering length between
one boson and the $(N{-}1)$-body cluster now in the presence of three-body
coupling was computed in Ref. PhysRevA.97.061603  and was found to be
repulsive for $4\leq N\leq 38$ but interestingly turn attractive for $N=3$
(see also Refs. PhysRevA.97.061604 ; PhysRevA.97.061605 ) and $N\geq 39$.
Because infinitesimal pairwise attraction immediately leads to a bound state
in one dimension, the latter case exibits new $N$-body cluster formation
induced by none other than the three-body coupling. Its binding energy
measured from the dissociation threshold was predicted as

$\displaystyle\Delta
E_{N}^{*}=-\frac{N\beta_{1,N-1}^{2}}{8(N-1)}m^{3}g_{2}^{2}g_{3}^{2},$ (21)

where $\beta_{1,N-1}$ is an $N$-dependent number associated with the boson-
cluster scattering length PhysRevA.97.061603  (see Fig. 2 and Table I
therein).

###  3.3 Artificial control of three-body forces

Three-body forces not only naturally appear in low dimensions, but they can
also be controlled artificially with cold atoms. Accordingly, it is even
possible to make three-body forces dominate over two-body forces. While
several such schemes have been proposed theoretically PhysRevA.89.053619 ;
PhysRevLett.112.103201 ; PhysRevA.90.021601 ; PhysRevA.93.043616 , we here
introduce the simple and versatile one proposed in Ref. PhysRevA.90.021601 ,
which employs two hyperfine spin components of bosons in an optical lattice.
When the two components are coupled by a nearly resonant field, the system in
the tight-binding approximation is described by

$\displaystyle\hat{H}=-t\sum_{\sigma=\uparrow,\downarrow}\sum_{\langle
i,j\rangle}\hat{b}_{\sigma i}^{\dagger}\hat{b}_{\sigma
j}+\sum_{i}\hat{\mathcal{H}}_{i}$ (22)

with

$\displaystyle\hat{\mathcal{H}}_{i}$
$\displaystyle=-\frac{\Omega}{2}(\hat{b}_{\uparrow
i}^{\dagger}\hat{b}_{\downarrow i}+\hat{b}_{\downarrow
i}^{\dagger}\hat{b}_{\uparrow i})-\frac{\Delta}{2}(\hat{b}_{\uparrow
i}^{\dagger}\hat{b}_{\uparrow i}-\hat{b}_{\downarrow
i}^{\dagger}\hat{b}_{\downarrow i})$
$\displaystyle\quad+\sum_{\sigma,\sigma^{\prime}}\frac{u_{\sigma\sigma^{\prime}}}{2}\hat{b}_{\sigma
i}^{\dagger}\hat{b}_{\sigma^{\prime}i}^{\dagger}\hat{b}_{\sigma^{\prime}i}\hat{b}_{\sigma
i}.$ (23)

Here $t$ is the inter-site tunneling amplitude, $\Omega$ is the Rabi
frequency, $\Delta$ is the detuning, and $u_{\sigma\sigma^{\prime}}$ are the
on-site interaction energies. Due to the Rabi coupling, the eigenstates of the
spin part of Hamiltonian (first line) in Eq. (3.3) are not $\uparrow$ and
$\downarrow$ bosons but their superpositions and their energies are separated
by the spin gap of $\sqrt{\Omega^{2}+\Delta^{2}}$. Therefore, as far as low-
energy physics relative to the spin gap is concerned, the higher-energy state
cannot be excited so that bosons only occupy the lower-energy state.
Accordingly, such low-energy physics of the system should be described by an
effective single-component Hamiltonian in the form of

$\displaystyle\hat{H}_{\mathrm{eff}}=-t\sum_{\langle
i,j\rangle}\hat{b}_{i}^{\dagger}\hat{b}_{j}+\sum_{i}\sum_{n=1}^{\infty}\frac{U_{n}}{n!}\hat{b}_{i}^{\dagger
n}\hat{b}_{i}^{n},$ (24)

where $U_{n}$ is an effective $n$-body interaction energy induced by virtual
excitation of bosons to the higher-energy state.

When one site is occupied by $N$ bosons, their on-site energy resulting from
the second term of Eq. (24) reads

$\displaystyle\sum_{n=1}^{N}\frac{N!}{n!(N-n)!}U_{n}.$ (25)

In order for the effective Hamiltonian in Eq. (24) to be the correct low-
energy description of the original Hamiltonian in Eq. (22), Eq. (25) should
match the lowest on-site energy of $N$ bosons resulting from Eq. (3.3), whose
matrix elements are provided by

$\displaystyle\langle
n|\hat{\mathcal{H}}_{i}|n^{\prime}\rangle=\begin{cases}\,\frac{N-2n}{2}\Delta+\frac{n(n-1)}{2}u_{\uparrow\uparrow}+\frac{(N-n)(N-n-1)}{2}u_{\downarrow\downarrow}\\\
\quad{}+n(N-n)u_{\uparrow\downarrow}\qquad(n^{\prime}=n)\\\\[4.0pt]
\,-\frac{\sqrt{n(N-n+1)}}{2}\Omega\qquad(n^{\prime}=n-1)\\\\[4.0pt]
\,-\frac{\sqrt{(n+1)(N-n)}}{2}\Omega\qquad(n^{\prime}=n+1)\\\\[4.0pt] \
0\qquad(\mathrm{otherwise})\end{cases}$ (26)

with $|n\rangle=[1/\sqrt{n!(N-n)!\,}]\,\hat{b}_{\uparrow i}^{\dagger
n}\,\hat{b}_{\downarrow i}^{\dagger N-n}|\mathrm{vac}\rangle$ and
$n=0,1,\dots,N$. Equating its lowest eigenvalue with Eq. (25) determines
$U_{N}$ for $N=1,2,\dots$ non-perturbatively as a function of $\Omega$,
$\Delta$, and $u_{\sigma\sigma^{\prime}}$. Because these parameters are
tunable in cold-atom experiments, independent control of $U_{N}$ is possible
at least for several lowest $N$.

Figure 10:  Contour plots of $U_{2}/2\pi$ [kHz] (blue solid curves) and
$U_{3}/2\pi$ [kHz] (red dashed curves) in the $\Omega$-$\Delta$ plane. The
black dot marks a point for $U_{2}=U_{3}=0$.

As a concrete application, Ref. PhysRevA.90.021601  considered 39K atoms
subjected to a magnetic field $\approx 58$ G in an optical lattice, where the
on-site interaction energies in the harmonic approximation were estimated as
$u_{\uparrow\uparrow}\approx 2\pi\times 0.55$ kHz,
$u_{\downarrow\downarrow}\approx 2\pi\times 3.05$ kHz, and
$u_{\uparrow\downarrow}\approx-2\pi\times 0.91$ kHz for hyperfine spin
components of $F=1$, $m_{F}=-1$ ($\sigma=\ \uparrow$) and $F=1$, $m_{F}=0$
($\sigma=\ \downarrow$) D'Errico_2007 ; PhysRevA.81.032702 . The resulting
$U_{2}$ and $U_{3}$ as functions of $\Omega$ and $\Delta$ are presented in
Fig. 10, where $U_{3}$ is found to be tunable in both magnitude and sign along
the curve corresponding to $U_{2}=0$. In particular, $U_{2}$ and $U_{3}$
simultaneously vanish at $(\Omega,\Delta)\approx 2\pi\times(1.70,1.38)$ kHz so
that $U_{4}\approx 2\pi\times 0.18$ kHz becomes the leading interaction energy
PhysRevA.90.021601 . Accordingly, it is possible to realize exotic systems
without two-body coupling but with a tunable three-body coupling, and even
more exotic systems without two-body nor three-body couplings but with a four-
body coupling in any dimensions.

Such exotic systems are expected to exhibit unique physics. Let us first
consider identical bosons in two dimensions without two-body but with a three-
body coupling. When the three-body attraction is tuned to a resonance where a
three-body bound state just appears, four such bosons were shown to form an
infinite tower of bound states with the universal scaling law of

$\displaystyle E_{n}\propto e^{-2(\pi n)^{2}/27}\qquad(n\gg 1)$ (27)

in the binding energy of $n$-th excited state PhysRevLett.118.230601 . This
newly discovered few-body phenomenon is a unique consequence of the three-body
coupling, which was termed the semisuper Efimov effect by analogy with the
Efimov effect efimov1970energy ; efimov1973energy  and the super Efimov effect
PhysRevLett.110.235301 ; PhysRevA.90.063631 . This trio of effects constitutes
universal classes of quantum halos whose spatial extensions can be arbitrarily
large compared to the range of interaction potentials (see Sec. 4 for more
details on the Efimov effect).

Turning to one dimension again, we note that a three-body coupling therein is
special not only because it breaks the integrability but also because it is
marginally relevant in the sense of the renormalization group if it is
attractive PhysRevA.97.011602 ; PhysRevLett.120.243002 . Because the latter
character is analogous to that of a two-body coupling in two dimension,
similar physics is expected to emerge. In particular, Ref. PhysRevA.97.011602
showed that one-dimensional bosons without two-body but with weak three-body
attraction form a many-body cluster stabilized by the quantum mechanical
effect, resembling that of two-dimensional bosons PhysRevLett.93.250408 ;
Bazak_2018 . Its ground-state energy normalized by that of three bosons is
universal, as long as the system remains dilute, and was predicted to grow
exponentially as

$\displaystyle\frac{E_{N}}{E_{3}}\to\exp\\!\left(\frac{8N^{2}}{\sqrt{3}\,\pi}\right)$
(28)

with increasing number of bosons $N\gg 1$ PhysRevA.97.011602 .

Furthermore, a four-body coupling in one dimension is analogous to a two-body
coupling in three dimensions in the sense that both of them are irrelevant but
have a fixed point at finite attraction corresponding to a resonance where a
bound state just appears PhysRevLett.101.170401 ; nishida2011liberating . As
discussed in Sec. 4, three-dimensional bosons at a two-body resonance are
known to exhibit the Efimov effect efimov1970energy ; efimov1973energy .
Similarly, one-dimensional bosons without two-body and three-body couplings
but at a four-body resonance were shown to exhibit the Efimov effect, where
five such bosons form an infinite tower of bound states with the universal
scaling law of

$\displaystyle E_{n}\propto(12.4)^{-2n}\qquad(n\gg 1)$ (29)

in the binding energy of $n$-th excited state PhysRevA.82.043606 . The
resulting Efimov effect in one dimension is a unique consequence of the four-
body coupling since the Efimov effect induced by the two-body coupling is
possible only in three dimensions nielsen2001three .

Table 1:  Ground-state energies of universal $N$-boson clusters for $N\gg 1$ with few-body attraction (columns) in various dimensions (rows). The (semisuper) Efimov effect indicates the universality not in the ground state but only in higher excited states. The other systems marked by dashes are considered to be non-universal. Dim.\Att. | Two-body | Three-body | Four-body
---|---|---|---
1D |  $N^{3}$ 10.1063/1.1704156  |  $e^{8N^{2}/\sqrt{3}\pi}$ PhysRevA.97.011602  | Efimov PhysRevA.82.043606
2D |  $e^{2.15N}$ PhysRevLett.93.250408  | semisuper PhysRevLett.118.230601  | —
3D | Efimov efimov1970energy  | — | —

Our perspective developed so far on the fates of bosons with two-body, three-
body, or four-body attraction in various dimensions is summarized in Table 1,
which may be useful to develop further insight into the universality in
quantum few-body and many-body physics.

##  4 Efimov physics in nuclei and atoms

###  4.1 Overview of the Efimov effect

Among the quantum clusters of few particles, a certain class is remarkable:
those which are very close to dissociation into smaller clusters or even all
the constituent particles. In quantum systems with short-range interactions,
there is indeed a minimum strength of the particles’ attraction that is
required for the particles to remain bound to each other. These clusters are
thus realised when the attraction strength is just above such critical point,
and are therefore relatively weakly bound. What is remarkable about these
loosely bound clusters is that they can be very large, much larger than the
range of the interactions, thanks to the ability of quantum systems to explore
classically forbidden regions. One often speaks of “quantum halos” Jensen2004
when referring to these states, to emphasise their large extent and
diluteness. Since a dominant part of their wave function is delocalised
outside the region of interaction, it depends upon the interactions only
through a few effective parameters. As a result, these states are said to be
universal in the sense that only these few parameters are enough to
characterise the wave function to good level of approximation, as well as many
other properties such as their energy. In other words, different systems with
very different interactions can nonetheless lead to the same universal states
if their effective interaction parameters are the same.

Among these universal quantum halo states, a more specific class is
particularly remarkable and has been a centre of attention of physicists for
many years: the so-called “Efimov states”, discovered by V. Efimov in the
1970s Efimov1970a . These three-body clusters occur for inter-particle
interactions that are close to the dissociation point of two particles. This
means that they can exist either when their two-body subsystems themselves
form loosely bound two-body halos ($1/a>0$ side of Fig. 11), or even when any
of their two-body subsystems is unbound ($1/a<0$ side of Fig. 11). In the
latter case, these three-body clusters are said to be “Borromean”, a fairly
common property of quantum halos. But the unique feature of Efimov states is
that they are bound by an effective long-range three-body force called the
“Efimov attraction” arising from the short-range two-body forces between the
particles. As a result of this long-range attraction, an infinite number of
three-body bound states exist before any pair of particles can bind. Moreover,
the Efimov attraction is scale invariant, since it decays as $1/R^{2}$, where
$R$ is the size of the three-particle system. As a result, the infinity of
three-body bound states near the two-body dissociation point is invariant by
scaling transformations with scaling factors that are multiples of a certain
number $\lambda_{0}$. This property is called “discrete scale invariance” and
is depicted in Fig. 11. The figure shows in particular that the energy $E_{n}$
of the $n$th excited state at the two-body dissociation point is related to
that of the next one by $E_{n}=\lambda_{0}^{2}E_{n+1}$, which gives the
scaling law for the excited states:

$E_{n}\propto\lambda_{0}^{-2n}\ \ (n\gg 1).$ (30)

Since each three-body bound state is related to the next one by a fixed
scaling transformation, it is enough to specify the energy of one particular
three-body bound state to determine the energies of all other bound states.
This single energy scale fixing the whole spectrum is referred to as the
“three-body parameter” and is conventionally defined to be the limit of
$E_{n}\lambda_{0}^{2n}$ for large $n$.

Figure 11:  Efimov plot: schematic plot of three-particle energy $E$ as a
function of the inverse of the scattering length $a$ between particles, which
is measure of the particles’ attraction strength indicated by the wide purple
arrow. The energy $E$ is rescaled into a wave number
$k=\text{sign}(E)\sqrt{m|E|}/\hbar$ (where $m$ is the mass of the particles
and $\hbar$ the reduced Planck constant) so that both coordinates are
homogeneous to an inverse length, and can be expressed in units of the inverse
of the range $b$ of the particles’ interactions. As the attraction strength is
increased, the cluster of two particles (black line) appears from the point
$1/a=0$ indicated by the black arrow. Conversely, this point can also be
regarded as the two-body dissociation point when the attraction strength is
decreased. There is an infinite number of three-particle clusters appearing
before that point. Their energies form a discrete-scale invariant pattern,
whereby any point on these energy curves can be mapped to another point by
scaling the coordinates ($k,1/a$) by a factor $\lambda_{0}$, as indicated by
the red arrows.

The occurrence of discrete-scale invariance in systems of particles with
short-range interactions is called the “Efimov effect”. There are certain
conditions for the Efimov effect to occur. First of all, the quantum
statistics and spin of the particles play an important role, because the Pauli
exclusion directly competes with the Efimov attraction efimov1973energy ;
PhysRevA.67.010703 ; kartavtsev2007low ; PhysRevLett.103.153202 ;
PhysRevA.86.062703 ; endo2011universal . While identical bosons are always
subject to the Efimov effect whenever they are close to their two-body
dissociation point, identical fermions with spin 1/2 or less (polarised
fermions) cannot exhibit the Efimov effect due to the Pauli exclusion.
Generally, for the Efimov effect to occur, at least two pairs of particles
must be able to interact in the $s$-wave, close to their dissociation point.
This means that their corresponding $s$-wave scattering lengths must be, in
absolute value, much larger (typically 10 times PhysRevLett.107.120401 ;
PhysRevLett.108.263001 ; pascaleno3BP1 ; pascaleno3BP2 ; johansen2017testing ;
PhysRevLett.111.053202 ) than the range of their interactions. This is a
rather stringent requirement, since most systems have scattering lengths of
the order of the interaction range, and therefore do not exhibit the Efimov
effect.

In systems where the Efimov effect occurs, the value of the scaling factor
$\lambda_{0}$ depends on the quantum statistics and masses of the particles.
For identical bosons, its value is $\lambda_{0}\approx 22.7$, which makes
successive states very different in size and energy. For systems of particles
with mass imbalance, the scaling factor can differ significantly and even
approach 1 in the case of a very light particle interacting with two heavy
particles efimov1973energy ; nielsen2001three ; naidon2017efimov . In this
case, the energy spectrum is denser than that of identical bosons and more
easily observable Pires2014 ; Tung2014 ; PhysRevLett.115.043201 .

The term “Efimov physics” naidon2017efimov  has been coined to loosely
designate the study of any physical situation where the Efimov effect plays a
role, or an Efimov-like effect occurs. For instance, the energy spectrum of a
larger number of particles, such as four bosons, may also exhibit a discrete-
scale invariant pattern with the same scale factor $\lambda_{0}$ as in the
three-particle spectrum, due to the influence of the 3-body Efimov effect
PhysRevD.7.2517 ; hammer2007universalEPJ ; von2009signatures ;
PhysRevA.70.052101 ; yamashita2006four ; PhysRevLett.107.135304 ;
deltuva2011shallow ; deltuva2013properties ; PhysRevLett.108.073201 ;
PhysRevLett.113.213201 . Generalisations of the Efimov effect for systems in
mixed dimensions also exhibit discrete-scale invariance PhysRevLett.101.170401
; nishida2011liberating ; PhysRevA.79.060701 ; PhysRevA.82.011605 ;
PhysRevA.84.052727 . Other systems, such as particles close to dissociation in
the $p$-wave, either in 3D PhysRevLett.97.023201 ; PhysRevA.86.012711 ;
PhysRevA.86.012710 ; PhysRevA.77.043611 ; PhysRevLett.96.050401 ;
PhysRevLett.99.210402 ; PhysRevA.106.023304 ; PhysRevA.107.033329  or 2D
PhysRevLett.110.235301 ; Volosniev_2014 ; PhysRevA.92.020504 ;
PhysRevA.78.063616 ; Gridnev_2014 ; PhysRevA.90.063631 , are not scale
invariant but exhibit an effective long-range attraction similar to the Efimov
attraction.

###  4.2 Geometry of Efimov states

Although the main feature of the Efimov states is the discrete scaling
invariance around the two-body dissociation point, they are also characterised
by universal geometric properties. For instance, at the two-body dissociation
point, the three-body bound states close to zero energy (a.k.a Efimov trimers)
have the same probabilistic distribution of triangular configurations, up to a
global scaling by the universal factor $\lambda_{0}$. This distribution
favours elongated configurations where one particle remains away from the
other two. Quite counterintuitively, these typical configurations get more and
more elongated as the attraction between particles gets stronger, until the
three-body bound state dissociates into a two-body bound state and a free
particle, as shown by the merging of the blue curves with the black curve in
Fig. 11. On the opposite side (Borromean region), where the attraction between
particles is weaker, the configurations become more equilateral.
Interestingly, near the three-body dissociation, they conform to another
universal pattern known as “halo universality” Fedorov1994a ; Jensen2004 ;
naidon2023universal . Halo universality is a generic feature of few-body
systems close to their full dissociation threshold and is independent of the
Efimov effect itself. Unlike Efimov universality, which is characterised by a
scaling factor between consecutive states (and thus difficult to demonstrate),
halo universality is characterised by a universal geometry: all distances in
the three-body system diverge when the binding energy approaches zero, thereby
turning the system into a halo, but their ratios have well defined values.
Remarkably, this universal halo geometry generally applies to states close to
their three-body dissociation point, including the ground state, in sharp
contrast with the Efimov universality which is accurate only for excited
states.

In the case of three identical bosons, the universal halo geometry close to
the three-body threshold is characterised by an equiprobability of all the
triangular configurations of the three bosons. In the case of two identical
particles and another particle, the universal halo geometry depends on the
scattering length $a\gg b$ between the two identical particles and the binding
energy $\kappa=\sqrt{m|E|}/\hbar\ll b$ of the system: when $\kappa a\ll 1$, it
is the same universal halo geometry as that of three bosons, but when $\kappa
a\gg 1$ it goes to a different geometry. These universal geometric properties
can be derived analytically as a function of $\kappa a$ PhysRevLett.128.212501
; naidon2023universal .

###  4.3 Efimov states in nuclear physics

The major requirement for the conventional Efimov effect to occur is having
pairs of particles with scattering lengths much larger than the range of their
interactions. It turns out that the scattering lengths for nucleons are
relatively large, with both the triplet scattering length $a_{t}\approx 5$ fm
and the singlet scattering length $a_{s}\approx-20$ fm being larger than the
range of nuclear forces, which is of the order of 1 fm. However, the Coulomb
repulsion between protons, which is a long-range force and introduces an
additional scale, can easily spoil the Efimov effect; for example, the Hoyle
state hoyle1954nuclear ; efimov1970energy , the excited state of 12C nuclei,
argued to be analogous to the Efimov state of three $\alpha$ particles, is
significantly affected by the Coulomb interaction and cannot be described by
the universal Efimov theory hammer2008model ; PhysRevC.91.014004 ;
otsuka2022alpha . Therefore, the Efimov states investigated so far in nuclear
physics mostly involve two neutrons interacting with a nucleus.

##### Triton.

The simplest example is the triton, the system composed of two neutrons and
one proton efimov1970energy ; naidon2017efimov . This system fits
qualitatively into the Efimov plot of Fig. 11, but for a positive scattering
length $a=a_{t}$ that is not so large compared to the interaction range $b$,
thus on the right-hand side of Fig. 11 where only one three-body bound state
exists. Indeed, there is no excited bound state of the triton. It is thus
impossible to check the discrete-scale invariance, which is a hallmark of the
Efimov effect. Yet, the universal features of the Efimov effect can readily
explain some theoretical properties of the triton, such as the Phillips line
phillips1968consistency ; efimov1985explanation ; efimov1988correlation ,
which is a correlation between the triton energy and the neutron-deuteron
spin-doublet scattering length.

##### Halo nuclei.

Systems of two neutrons interacting with a nucleus are more promising
candidates for Efimov states. Although a single neutron tends to be absorbed
by a nucleus due to the strong force, the situation changes when the number of
neutrons becomes large and approaches the neutron dripline where neutrons is
no longer tighty bound around the nucleus. Around this dripline, it may happen
for certain nuclides that they could almost but not quite bind an extra
neutron. If one regards the nucleus and the neutron as two distinct particles,
this situation corresponds to the two-body dissociation point discussed
earlier, around which the Efimov effect can occur. It is thus expected that,
in the presence of two neutrons, such a nucleus could form a three-body Efimov
state, where the two neutrons would remain at large distance forming a halo
around the nucleus. Indeed, nuclei with two-neutron halo structures have been
studied both experimentally and theoretically since the 1980s, such as 6He,
11Li, 12Be, 17B, 19B, 20C, 22C, 62Ca, and 72Ca tanihata1985measurements ;
AlKhalili2004 ; Tanihata2013 ; Fedorov1994 ; Amorim1997 ; AnnRev_HamPlatt ;
naidon2017efimov ; FREDERICO2012939 ; Canham2008 ; Hammer_2017 ;
ACHARYA2013196 ; PhysRevLett.111.132501 ; PhysRevC.105.024310 ;
PhysRevLett.120.052502 . Compared to the triton, these systems stand as better
candidates for Efimov states Fedorov1994 ; Amorim1997 ; Canham2008  because
some of them would correspond to the Borromean regime (left-hand side of Fig.
11) where the universal description in terms of Efimov state is more
effective. It is indeed widely believed that these two-neutron halo nuclei are
examples of Efimov states. Three-body models of some nuclei seem to reproduce
observations and confirm this view.

However, the Efimov scenario as shown in Fig. 11 is difficult to evidence in
two-neutron halo nuclei because of several factors. First, the $s$-wave
scattering length between the core nucleus and a neutron may not be large
enough. Typical values for the neutron-rich nuclei are 5-10 fm. This limits
the number of Efimov states as
$\dfrac{1}{\pi}\mathrm{ln}\left(\frac{|a|}{b}\right)\lesssim 1$, which
precludes the possibility of observing two adjacent trimers and testing the
discrete scale invariance. As the synthesis of the neutron-rich nuclei is
currently limited to light nuclei, the good candidates to see the excited
Efimov state with current experimental techniques are likely be 19B and 22C.
In particular, a large $s$-wave scattering length is reported in 17B-n as
$\gtrsim 100$ fm SPYROU2010129 ; carbonell2020low ; PhysRevLett.121.262502 ,
which may currently be the best candidate to see the elusive excited Efimov
state in neutron-rich nuclei PhysRevC.100.011603 . Second, the size of the
trimer is not large enough and the core may not be regarded as a featureless
inert point particle. Rather, the size and internal structure of the core
nucleus may play a significant role. In such cases, the collective excitations
of the core may couple with the halo neutrons. The halo neutrons may also have
significant correlations with the neutrons inside the core nucleus, in which
case the two neutrons are better described as forming a BCS-like pair induced
by the Fermi sea of the nuclei, rather than the three-body Efimov picture.
Even when the three-body picture is valid, contributions from higher partial
waves may play a significant role, as in 6He, 11Li PhysRevC.50.R550 ;
PhysRevC.90.044004 . Finally, the effective range (i.e. finite radius of the
nucleus) may not be negligible, so that finite-range corrections to the zero-
range universal Efimov theory must be taken into account. Further experimental
observations and comparison with three-body models are thus necessary to
confirm whether two-neutron halo nuclei do indeed conform to the geometry of
Efimov states.

Interestingly, the geometry of Efimov states near three-body dissociation
tends to the universal halo geometry mentioned in Sec. 4.2. Thus, if two-
neutron halo nuclei are indeed Borromean Efimov states, they should approach
this halo universality. However, the neutron-neutron scattering length is
somewhat too small with respect to $b$ to fully reach the universal halo
regime, so that they only constitute approximations of such halos. Among the
known two-neutron halo nuclei, 19B and 22C are good candidates to exhibit the
halo universality naidon2023universal . This could be experimentally confirmed
by improving the measurements of their various mean square radii and binding
energies.

##### Excited nuclei.

Recently, stable nuclei excited around their neutron breakup thresholds have
been proposed as novel nuclear candidates exhibiting an enormous $s$-wave
scattering length and the Efimov states endotanaka2023 . While the stable
nuclei well inside the valley of stability of the nuclear chart do not show
any halo structure in their ground states, they may form an $s$-wave halo of a
core nucleus and a neutron if they are excited in the vicinity of the one-
neutron separation threshold, around which the neutron may be barely bound by
the core nucleus. From the thermal neutron capture cross section data
shusterman2019surprisingly , it has been argued that ${}^{88}\mathrm{Zr}$ and
${}^{157}\mathrm{Gd}$ should have very large $s$-wave halo, and hence large
$s$-wave scattering lengths $|a|\gtrsim 10^{4}$ fm, which is 1-2 orders of
magnitude larger than those in the neutron-rich nuclei. It is predicted that
there are at least one and possibly two Efimov trimers made of the core plus
two neutrons in ${}^{90}\mathrm{Zr}$ and ${}^{159}\mathrm{Gd}$, which appear
as sub-threshold excited state around the two-neutron separation threshold. It
has also been conjectured that the $s$-wave halo and the Efimov states may
universally appear in the stripe regions of the nuclear chart when the new
excitation energy axis is added above the nuclear chart, presenting a global
picture to encompass the Efimov physics in the neutron-rich nuclei and the
Efimov physics in the excited nuclei around the neutron separation threshold.
This new perspective needs to be confirmed by further theoretical and
experimental studies.

###  4.4 Efimov states in atomic physics

####  4.4.1 Helium-4

Atomic physics has been more favourable for the demonstration of the Efimov
effect. Soon after its prediction, it was realised that helium-4 atoms are
good candidates for the observation of Efimov states, since they are bosonic
and their $s$-wave scattering length happens to be about 20 times larger than
their interaction range. This allows the existence of two trimer states, a
ground state and an excited state Lim1977 . There was a long experimental
effort to observe these states, which involved diffracting beams of helium
clusters through a grating, selecting the deflecting beam corresponding to
trimers and analysing their geometry. Although the ground state was observed
from the 1990s Schoellkopf1994 ; Bruehl2005 , it does not fully conform to the
Efimov zero-range theory. Observation of the excited state was finally
reported in 2015 Kunitski2015 , thanks to the Coulomb explosion imaging
technique. It revealed that its geometry excellently conform to the Efimov
theory, although the universal scaling ratio $\lambda_{0}$ could not be
directly confirmed.

####  4.4.2 Cold atoms

Although helium-4 was for a long time the only candidate for the observation
of Efimov states, it turned out that the atomic species used in cold-atom
experiments are even better for this purpose. Although their natural
scattering lengths are usually not large enough to expect the existence of
Efimov states in these atomic systems, it is possible to alter the scattering
length at will by applying a magnetic field onto these atoms, thanks to the
so-called magnetic Feshbach resonances Tiesinga1993 ; Chin2010 . This allows
to experimentally probe the Efimov plot of Fig. 11, not only at a single
scattering length value as in the case of helium-4 or halo nuclei, but over a
whole range of scattering lengths.

##### Observing Efimov states via three-body loss.

The simplest way to reveal the presence of Efimov states in this range is to
monitor the losses occurring among atoms through three-body collisions and
recombinations into deeply bound dimers PhysRevLett.83.1751 ;
PhysRevLett.85.908 . Indeed, nearby the scattering lengths at which an Efimov
state appears at the three-body zero-energy threshold, these losses are
significantly enhanced by the three-body resonances above threshold associated
to these Efimov states Kraemer2006 ; PhysRevLett.103.163202 ;
zaccanti2009observation ; pollack2009universality . Since these scattering
lengths are related to each other by a scaling with the universal scaling
ratio $\lambda_{0}$, it makes possible the experimental determination of this
ratio. The discrete scale invariance was thus confirmed in both systems of
identical bosons Huang2014  and systems of heavy bosons interacting with a
lighter boson Tung2014 ; Pires2014 . The measured scaling ratios were found to
be consistent with the universal predictions, within the experimental
uncertainties and corrections due to the finite-range of interactions. To
minimize the effects of finite temperatures, trap size, and other aspects
deteriorating Efimov signatures, there is an on-going attempt to observe
Efimov states in the microgravity environment aboard the International Space
Station (ISS) elliott2018nasa ; oudrhiri2023nasa ; frye2021bose .

##### Direct association of Efimov states.

In addition to observing the Efimov states through losses, there has also been
efforts to directly probe the Efimov states LompeScience11 ;
PhysRevLett.106.143201 ; PhysRevLett.108.210406 . This can be done by shining
a radio-frequency (RF) pulse to a mixture of atoms and Feshbach dimers. The
atoms are initially prepared in a hyperfine state $|0\rangle$, while the
dimers composed of two atoms in the $|1\rangle$ and $|2\rangle$ hyperfine
states. A certain frequency of the RF pulse can drive the atoms in hyperfine
state $|0\rangle$ to a hyperfine state $|3\rangle$. If that state has a
simultaneous Feshbach resonant interaction with the atoms remaining in states
$|1\rangle$ and $|2\rangle$, there are Efimov states composed of atoms in
$|1\rangle$, $|2\rangle$, and $|3\rangle$ states just below this threshold. It
is thus possible to form these Efimov states from the initial mixture by
shining a slightly reduced RF frequency. By measuring the frequency
difference, one can directly obtain the binding energies of the trimers. Such
a system can be prepared with 6Li atoms, for which there is a magnetic field
region where the $s$-wave scattering length between the three hyperfine states
can be simultaneously large. In Refs. LompeScience11 ; PhysRevLett.106.143201
, such a direct photo-association of an Efimov trimer has been performed, and
its binding energy has been directly measured. This RF association can also be
performed without the Feshbach dimer; in Ref. PhysRevLett.108.210406 , the RF
association of an Efimov trimer has been successfully performed with 7Li atoms
from an initial state where all atoms are unbound. This method, while not
requiring three resonant hyperfine states nor Feshbach dimers, relies on a
large spatial overlap between the initial three-atom state and the final
Efimov state and therefore can only be performed around the dissociation point
of the Efimov trimer. More recently, Efimov trimers have also been created
from a low-temperature Bose-Einstein condensate of atoms by rapidly sweeping a
magnetic field, hence varying the $s$-wave scattering length from small
positive values to large positive values for which Efimov trimers exist
PhysRevLett.119.143401 .

##### Coherent association of Efimov states.

While the above methods can directly create and probe Efimov trimers, the
associated trimers undergo a three-body recombination and soon get lost from
the trap. Recently, the group of Lev Khaykovich has observed coherent
signatures of Efimov trimers in a manner insensitive to their recombination
loss PhysRevLett.122.200402 ; yudkin2023reshape ; they initially prepare cold
7Li atoms on the positive $a>0$ side of the Feshbach resonance, and shine a
short RF pulse which is tuned to induce a transition into a superposition of
an Efimov trimer state and a Feshbach dimer + atom state. The superposition
state undergoes a time evolution with a phase factor which differs by
$E_{t}-E_{d}$ where $E_{t}$ and $E_{d}$ are the energies of the Efimov trimer
and Feshbach dimer. After a hold time $T$, the second short RF pulse turns the
states back into the initial three-atom states. The final atom number count
shows an oscillation $\cos[(E_{t}-E_{d})T]$ from which the binding energy of
the Efimov trimer relative to the dimer energy can be obtained
PhysRevLett.122.200402 . The oscillation is coherent and insensitive to the
loss of Efimov states as long as the hold time $T$ is smaller than the
lifetime of the Efimov states. With this technique, the binding energy of an
Efimov trimer can be measured as a function of the $s$-wave scattering length
yudkin2023reshape . While this technique has the advantage of enabling
coherent control of the Efimov trimer, currently it can only access the $a>0$
region in the vicinity of the dimer+atom breakup threshold of the Efimov
state, because the width of the RF pulse must cover both the Efimov state and
the dimer+atom state to create their superposition.

##### Fermionic Efimov states.

While the Efimov states for bosonic atoms have been successfully observed for
various atomic species, there have been ongoing efforts to observe the Efimov
states with fermions. For the fermionic 6Li atoms, the Efimov states have been
observed for atoms in three different hyperfine states PhysRevLett.103.130404
; PhysRevA.80.040702 ; PhysRevLett.106.143201 ; PhysRevLett.108.210406 . Three
atoms in different internal states can be regarded as distinguishable
particles, so that the Pauli exclusion principle does not play any role and
the properties of the Efimov states are essentially the same as those for
bosons. In order to realize Efimov states affected by the Pauli exclusion
principle, one needs to prepare fermionic atoms in the same hyperfine state.
Since a resonantly large $s$-wave interaction is necessary, the simplest setup
is a system of two fermionic atoms in the same hyperfine state interacting
with another distinguishable atom via a large $s$-wave scattering length.
According to the theoretical arguments efimov1973energy ; PhysRevA.67.010703 ;
kartavtsev2007low ; PhysRevLett.103.153202 ; PhysRevA.86.062703 ;
endo2011universal , such a system can form Efimov trimers if the mass ratio is
above the critical value of $13.6$. Notably, the predicted Efimov trimers of
fermions are characterized by a total orbital angular momentum of $\ell=1$,
which is a new feature compared with the bosonic Efimov states with $\ell=0$.
Therefore, they are expected to manifest different universal properties than
the bosonic ones zhao2023effects ; SciPostPhys.12.6.185 ; OiEndo2024

To realize such a cold-atom system with a large mass ratio, we need a
fermionic atomic species with a large enough mass. A mixture of fermionic Yb
and Li atoms has been successfully cooled down to their Fermi degeneracy
PhysRevLett.106.205304 . While the mass ratio is large enough to support the
Efimov states, Yb-Li mixture is found to have no useful magnetic Feshbach
resonances in an easily accessible range of the experiments
PhysRevLett.108.043201 ; chen2015anisotropy ; PhysRevX.10.031037 ;
PhysRevA.96.032711 , owing to closed-shell electronic structure of Yb atom.

More recently, an ultracold mixture of Er and Li atoms has been realized
PhysRevA.107.L031306 . The Er and Li atoms are cooled down by a sympathetic
cooling method utilizing the Yb atoms as a coolant, which is so powerful that
a new kind of quantum degenerate gases of dual BECs of highly-magnetic atoms
of Er and totally-nonmagnetic atoms of Yb can also successfully be created.
The Er-Li system (mass ratio $\approx 28$) is deemed as a major candidate to
realize the fermionic Efimov states. As both Er and Li atoms have open-shell
atomic structures, they may have magnetic Feshbach resonances broad enough to
accurately control the scattering length PhysRevA.92.022708 . Recently,
systematic measurements of Er-Li Feshbach resonances for various isotope
mixtures have been performed in a magnetic field range below 1 kG, and
succeeded in observing many Feshbach resonances schafer_feshbach_2022 ;
167Er6Li2023 . While the observed Feshbach resonances are mostly narrow,
possibly induced by the anisotropic electrostatic interactions or dipolar
interactions chen2015anisotropy , several broad Feshbach resonances were also
found

The observed broad Feshbach resonances in Er-Li is an important step toward
observing the fermionic Efimov trimers. The Efimov trimers are expected to be
observed from the loss measurement. In particular, the loss process involving
the Er-Er-Li channel, namely two Er accompanied by one Li atoms should be
predominantly enhanced. In Ref 167Er6Li2023 , the atom loss behavior around
the Feshbach resonance at $455$ G is observed in detail for the 167Er and 6Li
mixture. The atom loss is found to be predominantly induced through the Er-Er-
Li channel, namely two Er accompanied by one Li, which is consistent with the
formation of the fermionic Efimov trimers. It is also shown that the shape of
the observed resonance is asymmetric, which is also consistent with the
previous observations of multiple losses originated from the formation of
several Efimov trimers Pires2014 ; Tung2014 . More detailed measurements,
including the determination of the inter-species scattering lengths, will
reveal the behaviors of these resonant losses induced by the fermionic Efimov
trimers in this mass-imbalanced Er-Li mixture. Due to the strong magnetic
dipole interaction between Er atoms, an interplay of the van der Waals and
dipole interaction is predicted to lead to novel universal behaviors of the
Efimov states OiEndo2024 . We also note that the Er-Li mixture is also a
fascinating candidate for realizing a novel $p$-wave superfluid of fermions of
Er through the induced attraction via a BEC of light Li atoms
PhysRevA.61.053601 ; PhysRevLett.117.245302 ; gurarie2007resonantly ;
yamaguchi2017novel , which is also interesting from the viewpoint of the
effect of the many-body background of the novel Efimov trimers.

##  5 Summary

We have reviewed three-body forces and related phenomena from an
interdisciplinary perspective across nuclei and atoms. The three-body forces
have been a key element in accurate accounts of nuclear properties, and we
have reviewed recent theoretical and experimental progress on the nuclear
three-body forces. We have also reviewed some recent attempts to observe,
control, and quantum-simulate three-body and higher-body forces in cold atoms.
We have also shown how the Efimov states appear universally across nuclear and
atomic systems, and overviewed the recent developments in their observation
and understanding. The three-body forces and three-body phenomena have thus
been an active area of research, not only as a specific field of research or
methodology, but involving various disciplines of physics and scales. With
further developments in experimental technologies and theoretical methods, it
is expected that three-body forces will be determined with ever more accuracy,
and new quantum phenomena will be uncovered by further exploring the
implications of three-body physics.

###### Acknowledgements.

This work is supported by Grant-In-Aid for Scientific Research on Innovative
Areas ‘Clustering as a window on the hierarchical structure of quantum
systems’. S.E. is supported by JSPS KAKENHI Grant Nos. JP21H00116, 22K03492,
and 23H01174. S.E. acknowledges support from Institute for Advanced Science,
University of Electro-Communications. E.E. is supported by ERC under the EU
Horizon 2020 research and innovation programme (ERC AdG Nuclear Theory, No.
885150), by DFG and NSFC through funds provided to the Sino-German CRC 110
“Symmetries and the Emergence of Structure in QCD” (DFG Project ID 196253076 -
TRR 110, NSFC Grant No. 11621131001), by the MKW NRW (funding code
NW21-024-A), and by the EU Horizon 2020 research and innovation programme
(STRONG-2020, No. 824093). E.E. acknowledges the work of his collaborators
Ashot Gasparyan, Hermann Krebs, Ulf-G. Meißner, and the members of the LENPIC
Collaboration. P.N. is supported by JSPS KAKENHI Grant No. JP23K03292. Y.N. is
supported by JSPS KAKENHI Grant Nos. JP18H05405 and JP21K03384. K.S. is
supported by JSPS KAKENHI Grant Nos. JP25105502, JP16H02171, JP18H05404,
JP20H05636, and JST ERATO Grant No. JPMJER2304, Japan. Y.T. is supported by
the Grant-in-Aid for Scientific Research of JSPS (Nos. JP17H06138, JP18H05405,
JP18H05228, JP21H01014, JP21K03384, and JP22K20356), JST CREST (Nos.
JPMJCR1673 and JPMJCR23I3), MEXT Quantum Leap Flagship Program (MEXT Q-LEAP)
Grant No. JPMXS0118069021, and JST Moonshot R&D Grant No. JPMJMS2269.

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