# Hadronic structure on the light-front I
Instanton effects and quark-antiquark effective potentials

Edward Shuryak<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794–3800, USA  Ismail Zahed<EMAIL_ADDRESS>Center for
Nuclear Theory, Department of Physics and Astronomy, Stony Brook University,
Stony Brook, New York 11794–3800, USA

###### Abstract

This is the first of a sequence of papers that address the non-perturbative
origin of the central, and spin-dependent forces between quarks. Its main
thrust is a focus on meson spectroscopy in the center of mass frame. We
suggest a novel “dense instanton ensemble” model for the QCD vacuum, to
explain the interquark forces in mesons, from quarkonia to heavy-light and
light-light ones. The sequels will show how to export these interactions to
the light front, and derive the corresponding Hamiltonians, mesonic and
baryonic light front wavefunctions. The ultimate aim of the series is to
bridge the gap between hadronic spectroscopy and partonic observables.

##  I Introduction

The physics of hadrons is firmly based in Quantum Chromodynamics, a theory
over half a century old. One might think that by now this subject has reached
a solid degree of maturity with most issues settled. Unfortunately this is not
yet the case. The field of non-perturbative QCD consists of several subfields,
which can be defined as follows:
1\. hadronic spectroscopy (masses and wave functions of mesons, baryons,
pentaquarks, etc);
2\. QCD vacuum structure (vacuum condensates, Euclidean correlation functions,
etc);
3\. light-front observables (distributions amplitudes, parton distribution
functions or PDFs, etc).
They are relatively weakly connected, although in the past decade considerable
efforts to bridge them has taken place. Our set of papers is another effort in
this direction. In this introductory section we will give a brief description
of the two aforementioned subfields. The third one will be discussed in the
papers to follow.

###  I.1 Hadronic structure

This field originated in the early 1960’s, and experienced a rapid expansion
in the last decade, due to the discoveries of multi-quark hadrons in the
heavy-light sector. We will not discuss these “exotic hadrons” in this work
but focus on the underlying physics at the origin of the inter-quark forces.
Parameterizing and understanding them is a needed step, setting the stage for
derivation of the effective Hamiltonians on the light front.

Early nonrelativistic quark models achieved resounding success, in part
through simple formulae for the masses and magnetic moments of baryons made of
light quarks. The revolutionary discoveries of 1970’s of “quarkonia”, the
bound states made of heavy $c,b$ quarks, revealed families of narrow bound
states well described by the celebrated Cornell potential

$V_{\rm Cornell}(r)=-{4\alpha_{s}\over 3}{1\over r}+\sigma_{T}r$ (1)

Further theory effort developed effective non-relativistic theories, such as
non-relativistic QCD (NRQCD) and perturbative non-relativistic QCD (pNRQCD).
They put spectroscopy of quarkonia close to that of atoms and nuclei in
accuracy. A number of $universal$ (flavor independent) central and spin-
dependent potentials were defined, expressed $via$ certain nonlocal
correlators of vacuum gauge fields. For short but concise summary see e.g.
Pineda and Vairo (2001).

The Cornell potential attributes the short-distance potential to perturbative
one-gluon exchange, and its large distance ${\cal O}(r)$ contribution to the
tension of the confining flux tube (QCD string). The issues to be discussed in
this paper are the nonperturbative origins of the inter-quark interaction at
$intermediate$ distances $r\sim 0.2-0.5\,$fm. Those are especially important
for small size-hadrons, such as bottomonia or pions (see below).

A static quark potential $V_{C}(r)$ is defined through a vacuum average of a
pair of parallell Wilson lines,

$e^{-V_{C}(r)T}=\langle W(\vec{x}_{1})W^{\dagger}(\vec{x}_{2})\rangle$ (2)

defined by

$W(\vec{x})={\rm
Pexp}\bigg{(}ig\int_{W}dx_{4}A_{\mu}^{a}(x_{4},\vec{x})\hat{T}^{a}\bigg{)}$
(3)

where color generator $\hat{T}^{a}=t^{a}/2$, with $t^{a}$ Gell-Mann matrices.
$tPexp$ is path-ordered matrix product. The parallel lines are running in the
Euclidean time direction, separated by the spatial distance
$r=|\vec{x}_{1}-\vec{x}_{2}|$. The lines are connected at times $\pm T/2$ ,
but eventually the limit of large time $T\rightarrow\infty$ is assumed, this
is mostly unimportant. Historically, the derivation of this potential has been
at the forefront of lattice studies of the QCD vacuum. In a way, it is similar
in spirit to the Born-Oppenheimer approximation in molecular physics, where
the probe quarks back-reaction is ignored. For recent developments and
references, we refer to the work by Brambilla et al Brambilla  _et al._
(2018).

Traditionally, the central potential was used as linear $V_{c}(r)=\sigma_{T}r$
with $\sigma_{T}$ be the tension of the confining flux tube (known also as QCD
string). So, one might think the issue of the central potential is very
simple. Unfortunately, it is far from being so, and there are several
impediments for usage of the flux tubes.

The first is that the simple linear term is expected to be correct at large
distances only, while at smaller $r$ there are other nonperturbative effects
as well. The simplest corrections are due to quantum vibrations of the flux
tube. The first of them is the famous “Lusher term”, attractive $\sim 1/r$. If
the string is described by a Nambu-Goto geometrical action, these vibrations
are summed up in the so called Arvis form Arvis (1983)

$E(r)=\sigma_{T}r\sqrt{1-{\pi\over 6}{1\over\sigma_{T}r^{2}}}$ (4)

This expression predicts that potential vanishes at $r\approx 1/3$ fm.
Furthermore, the QCD string may have much more complicated action, so the
Arvis potential cannot be really trusted when the corrections are large. Yet
the two first expansion terms are universal, see more in Aharony and
Klinghoffer (2010). Lattice studies of these corrections (e.g. Athenodorou and
Teper (2021)) in the setting with two static charges, or a flux tube wrapped
around the lattice, do confirm these two terms.

Important issue to be discussed is the derivation of the relativistic
corrections $\sim(v/c)^{2}\sim(\Lambda_{QCD}/m_{Q})^{2}$ where $m_{Q}$ is
heavy quark mass and $\Lambda_{QCD}$ stands for a “QCD scale”. Such
perturbative terms are well known in atomic and nuclear physics, and changing
from one-photon to one-gluons exchange is simple. So it is clear how spin-
spin, spin-orbit and tensor force $V_{SS},V_{SL},V_{T}$ arise from
perturbative contributions, being all just certain derivatives of the Coulomb
potential.

The question is what is the nonperturbative contributions to these potentials.
Those can be related to Wilson lines decorated by two extra field strengths
Callan  _et al._ (1978); Eichten and Feinberg (1981). Schematically, they have
the form

$\langle WG_{\mu\nu}(x)WW^{\dagger}G_{\mu\nu}(y)W^{\dagger}\rangle$

More details of the invariant spin potentials given in section B.

In a nonrelativistic setting, spin-dependent forces originate from quark
(gluo)-magnetic moments interacting with vacuum fluctuations of (mostly) gluo-
magnetic fields. Unfortunately, the QCD flux tubes carry electric flux: models
of their structure and lattice studies specify mostly distribution of the
gluo-electric fields in them, not the gluo-magnetic ones needed for the spin
forces. So, one has to think about other origins of those in the vacuum.

In principle, the pertinent correlation functions can be calculated on the
lattice, so one might think that all spin-dependent forces are by now well
documented and their origins explained. Unfortunately, this is not the case as
quantitative lattice studies have only started recently. Below we will
investigate to what extent such correlators, evaluated using certain models of
vacuum fields, can reproduce the observed spin-dependent potentials. We will
find that in heavy quarkonia instanton-induced forces can provide
contributions to perturbative ones, leading together to a good description of
splittings for a number of the low-lying states.

However, for spin splittings in light quark states these forces coming from
Wilson lines are not sufficient. The resolution comes from additional terms in
the quark propagator, related to their disappearance/creation in/from the
Dirac sea technically described by parts of the propagators related to
fermionic zero modes. In other words, there are spin forces induced by ’t
Hooft effective Lagrangian for light quarks. In Fig.1 we show schematically
how these vertices contribute to the pion and rho meson.

Figure 1:  Schematic representation of the role of instanton-induced ’t Hooft
Lagrangian in the structure of the pion (upper) and rho meson (lower plot).

###  I.2 QCD vacuum and Instanton Liquid Model

Nonperturbative physics of strong interactions started before the development
of QCD in 1970’s. Nambu and Jona-Lasinio (NJL) Nambu and Jona-Lasinio (1961),
inspired by BCS theory of superconductivity, have qualitatively explained that
strong enough attraction of quarks can break $SU(N_{f})_{A}$ chiral symmetry
spontaneously and, among many other effects, create near-massless pions.
Chiral effective Lagrangians and related theory have lead to one important
input, a nonzero quark condensate $\langle\bar{q}q\rangle\neq 0$.

The discussion of the “QCD vacuum structure” started with the QCD sum rules
Shifman  _et al._ (1979), where the short-distance description of Euclidean
point-to-point correlation functions via the Operator Product Expansion (OPE)
was related to their long-distance description in terms of the lightest
hadrons. The (Euclidean) correlators of QCD operators were calculated in terms
of the leading vacuum condensates, $\langle
G^{2}\rangle,\langle\bar{q}q\rangle,$ etc. Yet it was soon realized that the
strongest nonperturbative effects (in the scalar and pseudoscalar channels)
are $not$ described by the “mean fied” OPE predictions Novikov  _et al._
(1981).

Extensive studies of point-to-point (Euclidean) correlation functions in
multiple mesonic channels, based on phenomenology Shuryak (1993), have indeed
found striking differences between correlators of operators with different
quantum numbers. While for vector currents $(\bar{q}\gamma_{\mu}q)$ made of
$light$ quarks $q=u,d,s$ only small deviations from free quark propagation is
observed, correlators of the scalar and pseudoscalar operators
($\gamma_{\mu}\rightarrow 1,\gamma_{5}$) show strong splittings from them at
surprisingly small distances. As shown in detail in Shuryak (1993), these
splittings found their explanation in terms of topological fluctuations of
gluonic fields, described semiclassically by instantons Belavin  _et al._
(1975).

The semiclassical model of the QCD vacuum structure based on instantons
Shuryak (1982a) is known as the ”instanton liquid model” (ILM) . It introduced
an important scale parameter of the nonperturbative vacuum – the typical
instanton size

$\rho\sim\frac{1}{3}\,{\rm fm}$ (5)

Because it is rather small compared to sizes of most hadrons (except for the
lowest $\Upsilon$s and the pion, see below), where possible we will use a
quasi-local approximation in which it is going to zero.

In the ILM the vacuum fields are very inhomogeneous: blobs of strong gauge
fields (at the instanton centers) surrounded by “empty” space-time, free from
nonperturbative fields. Since mid-1990s such field distributions were obtained
from lattice configurations by means of various “cooling” methods. A picture
is better than many words: so we reproduce in Fig. 2 one visualization of the
topological charge distribution from Leinweber (1999); Biddle  _et al._
(2020a, b) which reveals a number of instantons and anti-instantons.

Note that the topological clusters are also threaded by thin center vortices
or Z3-fluxed strings, which span world-sheet surfaces in 4-dimensions. While
the center vortices are important for enforcing confinement at long distances,
Fig. 2 shows that they are on average decoupled from the inhomogeneous and
strong topological fields. Moreover, their field strength in the vicinity of
these topological fields is about $\sigma_{T}\bar{\rho}\approx 0.3\,{\rm
GeV}$, which is weaker than the typical chromo-electric or chromo-magnetic
field in the instanton center $\sqrt{E}=\sqrt{B}\approx 2.5/\bar{\rho}\approx
1.5\,{\rm GeV}$, but crucial for long-distance color correlations.

Figure 2: Instantons (yellow) and anti-instantons (blue) configurations in the
“deep-cooled” Yang-Mills vacuum. After center projection, they are threaded by
center P-vortices Biddle  _et al._ (2020a, b). They constitute the primordial
gluon epoxy (hard glue) at the origin of the light hadron masses Zahed (2021).
The center P-vortices are responsible for confinement. See text.

Instantons are not just semiclassical solitons made of glue. As discovered by
’t Hooft ’t Hooft (1976), they generate 4-dimensional fermionic zero modes
which lead to multifermion effective Lagrangian. The instanton-induced 4-quark
not only solves the famous “$U_{A}(1)$ problem” – by making the
$\eta^{\prime}$ non-Goldstone and heavy – but it also produces a strong
attraction in the $\sigma$ and $\pi$ channels. Including those in the
framework of the ILM Shuryak (1982b) one gets a microscopic understanding of
chiral symmetry breaking, chiral perturbation theory, the pion properties etc.
’T Hooft instanton-induced effective Lagrangian is a QCD substitute to the
hypothetical Nambu-Jona-Lasinio Nambu and Jona-Lasinio (1961) four-quark
interaction.

Another version of this theory look at quarks “hopping” from one instanton to
another. The resulting loops in the quark determinant can be either short or
long (infinite in thermodynamic limit $V_{4}\rightarrow\infty$). The latter
component is responsible for the localization of the Dirac eigenvalues close
to zero $\lambda\sim 1/V_{4}$,

Two parameters of the original ILM are the size and the density of instantons

$\rho=\frac{1}{3}\,{\rm fm},\qquad n_{I+\bar{I}}=\frac{1}{R^{4}}=1\,{\rm
fm}^{-4}$ (6)

which in turn defines the so called diluteness parameter

$\kappa\equiv\pi^{2}\rho^{4}n_{I+\bar{I}}$ (7)

So, the ILM predicts it to be small $\kappa_{\rm ILM}\sim{\cal O}(1/10)\ll 1$.
These parameters have withstood the scrutiny of time, and describe rather well
the chiral dynamics related to pions, the Euclidean correlation functions in
the few femtometers range, and much more, see Schäfer and Shuryak (1998) for a
review.

Another formulation of chiral symmetry breaking relates it to the
collectivization of the instanton zero modes, into the so-called “zero mode
zone” (ZMZ). Due to nonzero matrix elements of the Dirac operator, near-zero
Dirac eigenvalues are residing within a strip of small width

$|\lambda|\sim{\rm width\,(ZMZ)}\sim{\rho^{2}\over R^{3}}\sim 20\,{\rm MeV}$
(8)

as predicted by the original ILM with the parameters given above (6). In
Glozman  _et al._ (2012) the meson and baryon spectroscopy was studied, with
all Dirac states inside a certain strip of eigenvalues $|\lambda|<\Delta$
eliminated. A strong restructuring of the light hadronic spectra was indeed
observed if $\Delta>{\rm width\,(ZMZ)}$. In particular, the Nambu-Goldstone
modes (pions) totally disappear from the spectra, as expected. It would be
interesting to extend the same analysis to the heavy-light sectors, and to
address more generally what happens with all forms of spin-dependent forces.

Further statistical description of interacting instanton ensembles was
developed by mean field methods and statistical simulations, for review see
Schäfer and Shuryak (1998). Some related lattice studies are Chu  _et al._
(1994); Faccioli and DeGrand (2003).

###  I.3 Topological landscape and $I\bar{I}$ molecules

The “landscape” refers to the minimal energy gauge field configurations, as a
function of two main variables. The first is the topological Chern-Simons
number

$N_{CS}\equiv{\epsilon^{\alpha\beta\gamma}\over 16\pi^{2}}\int
d^{3}x\left(A^{a}_{\alpha}\partial_{\beta}A^{a}_{\gamma}+{1\over
3}\epsilon^{abc}A^{a}_{\alpha}A^{b}_{\beta}A^{c}_{\gamma}\right)$ (9)

For the second we will use the r.m.s. size of the configuration, related to
the field strength

$\rho_{r.m.s.}={\int r^{2}G^{2}d^{3}r\over\int G^{2}d^{3}r}$ (10)

For fixed $\rho$, the landscape is shown in Fig.3. It has been defined in Ref
Ostrovsky  _et al._ (2002) and is given in a parametric form

$\displaystyle U_{\rm min}(k,\rho)$ $\displaystyle=$
$\displaystyle(1-k^{2})^{2}{3\pi^{2}\over g^{2}\rho}$ (11)
$\displaystyle{N}_{CS}(\kappa)$ $\displaystyle=$ $\displaystyle\frac{1}{4}{\rm
sign}(k)(1-|k|)^{2}(2+|k|)$

with parameter $\kappa$. At $\kappa=\pm 1$ the minimal energy is zero, while
$k=0$ corresponds to a maximum of the energy. This point was called ”the
sphaleron” in electroweak theory Klinkhamer and Manton (1984). The set of
configurations at arbitrary $\kappa$ is called the sphaleron path. Those
consist of static 3d magnetic field configurations, known also as the turning
points (by analogy to points in quantum mechanics where the semiclassical
momentum vanishes).

Figure 3: The topological landscape: minimal energy (times r.m.s. size) versus
Chern-Simons number $N_{CS}$ (left); thimble path (center); tunneling at
nonzero energy (right).

The (anti)instanton is a tunneling path connecting the $bottoms$ of two
subsequent valleys, at energy zero. Since they have topological charge $Q=\pm
1$, they result in change of Chern-Simons number $\Delta N_{CS}=\pm 1$.

However instanton paths are not the only form of topological fluctuations
which may occur in this landscape. Indeed, we will discuss $two$ additional
sets of paths: (i) those that travel the landscape along the sphaleron path
(or ′streamline′); (ii) those that travel the landscape at fixed energy,
including the tunneling; all of which are illustrated in Fig.3

The first set is described by a constrained Yang-Mills equation with a nonzero
r.h.s., or external current, which drags them up (or down) the potential along
the gradient. In the mathematical literature this construction is known as
“Lefschetz thimbles”, a special path connecting two extrema of a function
following its gradient. In QCD-related literature they are known as
“streamline” configurations described by overlapping instanton-antiinstanton
pairs or molecules.

The second set is described by Yang-Mills equation with a $zero$ r.h.s., and
thus is occurring at fixed energy. The path’s history passes the “turning
points ” $twice$, with Euclidean time solution in between them, known as
tunneling at nonzero energy. In general, those should be complemented by
Minkowskian time solutions, before and after turning points. Technical name
for those is “zigzag path” indicating a turn from real to imaginary time and
then back to real.

Both sets of solutions, with proper references, we detail in the Appendices.
What we would like to emphasize here is the fact that they describe gauge
field fluctuations different from well separated instantons. They are neither
self-dual nor possess (near) zero Dirac eigenvalues – and therefore they do
not contribute to chiral symmetry breaking and were not included in the
original ILM. And still, they do describe certain fluctuations of vacuum gauge
fields, and therefore can contribute to certain observables: especially the
Wilson lines. Therefore, we will investigate their contributions to central
and spin-dependent forces between quarks. The inclusion of “molecules” or
”zigzag paths” is the novel element of novel vacuum model we develop here.

Close instanton-antiinstanton pairs are of course known, and in particular
there were observed on the lattice. They are however not seen in Fig. 2
obtained by so called “deep cooling” of configurations, during which close
instanton-antiinstanton pairs are already annihilated. The application of the
“molecular component” of the vacuum was made previously in connection to phase
transitions in hot/dense matter. Indeed, (if quark masses are neglected) this
component is the only one which survives at temperatures $T>T_{c}$, where
chiral symmetry is restored. Account for “atomic” and “molecular” components
together started with Ilgenfritz and Shuryak (1989). The “molecular component”
was also shown to be important at high baryonic densities, where it
contributes to quark pairing and color superconductivity Rapp  _et al._
(1998). More recently, we have explored it in the context through non-
perturbative contributions to the mesonic form factors Shuryak and Zahed
(2021a) and matching kernels Liu and Zahed (2021).

The theory of sphaleron processes ( sketched in the middle of Fig.3. is
related to he issue of $I\bar{I}$ interaction. Numerical calculation of the
“streamline” along the action gradient was first done for quantum-mechanical
instantons in Shuryak (1988), for gauge theory the “streamline equation was
derived in Balitsky and Yung (1986), solved approximately by Yung (1988) and
numerically by Verbaarschot (1991). The surprising finding of the latter paper
was that “Yung ansatz” was rather accurate not only at large distances
$R\gg\rho$ between instanton and antiinstanton, where it was derived, but in
fact for any distance till zero. Note that the last two papers used conformal
inversion at the center of the instanton, making $I$ and $\bar{I}$ co-central.

Consider $I$ and $\bar{I}$ of the same size $\rho$ and same color orientation,
with 4-distance between their centers $R$ serving as the parameter of the set,
with $x1_{\mu}=(0,0,0,R/2),x2_{\mu}=(0,0,0,-R/2)$. In the mathematical
literature such set is known as Lefschets thimble, it connects one extremum at
large $R$, the independent $I$ and $\bar{I}$, with another, the zero field at
$R=0$. Note that at the locations $y_{4}<0$ the gauge fields are approximately
antiselfdual $\vec{E}\approx-\vec{B}$, and at $y_{4}>0$ they are approximately
selfdual $\vec{E}\approx\vec{B}$. At $y_{4}=0$ the electric field vanishes. As
shown in Ostrovsky  _et al._ (2002), these 3d magnetic objects obtained using
Yung ansatz are very close to the sphaleron path configurations obtained by
constrained energy minimization.

The instanton-antiinstanton streamline therefore provides a semiclassical
description of the sphaleron production. In the context of electroweak theory
it was first used in Khoze and Ringwald (1991) and Shuryak and Verbaarschot
(1992) three decades ago. Recent interest in the production of QCD sphalerons
at LHC and RHIC colliders is discussed in our recent paper Shuryak and Zahed
(2021b).

Let us now move to lattice observables, to which the “molecular component” of
the instanton ensemble contributes. The simplest of which is the so called
gluon condensate $\langle G_{\mu\nu}^{2}\rangle$ introduced in the context of
the QCD sum rules framework Shifman  _et al._ (1979). The accuracy of this
number was later questioned and it was revised to a larger value. Further
discrepancies were shown in lattice studies, that were attempting to extract
local (or nonlocal) observables with powers of the gauge field strength
$G_{\mu\nu}$.

Nowadays, when “cooling” by the gradient flow method can be consistently
related to the renormalization group (RG) flow Luscher and Weisz (2011), one
can put the appropriate scale dependence of the “molecular component” on a
firm basis. Skipping several decades, let us take as an example the recent
work Athenodorou  _et al._ (2018) which studies the 3- and 4-point gluon field
correlators and related their evolution to topology during cooling. Their
original motivation was to extract the gluon coupling $\alpha_{s}(k)$, so the
observable was chosen to be the ratio of the 3-point to 2-point Green function
leading to effective coupling

$\alpha_{MOM}(k)={k^{6}\over 4\pi}{\langle
G^{(3)}(k^{2})\rangle^{2}\over\langle G^{(2)}(k^{2})\rangle^{3}}$ (12)

In the “uncooled” quantum vacuum (with gluons) the effective coupling starts
running downward at large $k>1\,{\rm GeV}$, as rexpected by asymptotic
freedom. However at low $k\rightarrow 0$, one finds another persisting
positive power of $k$, with a slope that matches exactly the one following
from an instanton ensemble Boucaud  _et al._ (2003)

$\alpha_{MOM}(k)\rightarrow\frac{k^{4}}{18\pi n_{I+\bar{I}}}$ (13)

Furthermore, it was observed that with increasing cooling time $\tau$, the
same power spreads to higher momenta, $k>1\,{\rm GeV}$. It was observed that
“cooling” eliminates not only perturbative gluons, but close instanton-
antiinstanton pairs as well.

Figure 4: The dependence of the mean instanton sizes (upper plot) and density
(lower plot) on the gradient flow cooling time $\tau$ (arbitrary units). The
quantum vacuum of course corresponds to an extrapolation to $\tau\rightarrow
0$.

The dependence of the mean instanton sizes and density as a function of the
(gradient flow) cooling time is shown in Fig.4, from Athenodorou  _et al._
(2018) . Their main conclusion from this analysis was that the size and
density – extrapolated to $zero$ cooling time ($\tau\rightarrow 0$) – are

$\,\,\rho\rightarrow{1\over 3}\,fm,\,\,\,n\rightarrow 10\,fm^{-4}$ (14)

Therefor one can see that total instanton density is significantly larger than
known before from “deep cooling” (large $\tau$) of $1/{\rm fm}^{4}$, as in the
original ILM. In fact, the density may even be larger than that, because the
definition of instantons used in this paper exclude pairs (“molecules”) which
are too close. We will use these numbers, as indicative, for our “dense
instanton liquid model” below.

###  I.4 Content of this paper

In section II we will argue that the $upper$ limit on the total instanton
density can be deduced from known magnitude of the central potentials, e.g.
from quarkonium spectroscopy.

The main purpose of the present paper is to quantify nonperturbative spin-
dependent forces. We begin by reviewing their phenomenology in section III.1
and then the theory in III.4 which is based on the standard ”Wilson line plus
two field strengths” correlators. The contribution of an instanton to a pair
of Wilson lines can be readily calculated analytically. This approach has been
first used to calculate the static $\bar{Q}Q$ potential by the original
Princeton group Callan  _et al._ (1978). Two decades later we generalized it
to high energy $\bar{Q}Q$ and dipole-dipole scatterings Shuryak and Zahed
(2000). In the scattering approach, the Wilson lines are first assumed to
cross at an angle $\theta_{12}$ in Euclidean geometry. The final result is
analytically transformed to a scattering amplitude (near) the light front by
the substitution of the angle $\theta_{12}$ between two Wilson lines, to the
hyperbolic angle $iy$ with relative rapidity $y$ between two colliding
particles in the ultra-relativistic limit. Unfortunately, neither of these
works were widely noticed or actually used by phenomenologists (except for
Giordano and Meggiolaro (2008, 2010)).

An extensive study of the instanton-induced effects on heavy quarkonia has
been revived recently by Musakhanov et al (see e.g. Yakhshiev  _et al._
(2017a)), who calculated the magnitude of the effect for the central as well
as the spin-spin, spin-orbit and tensor forces. Using the original ILM
parameters of Shuryak (1982c), it was found that the central potential is of
magnitude $\sim 150\,{\rm MeV}$ at large $r$, a relatively small correction to
the phenomenological potential. The spin-spin potential was also calculated,
and was found to be short ranged and of order of about $30\,{\rm MeV}$, also a
small correction. The spin-orbit and tensor forces are even smaller.

We will modify these results by using our new model for the instanton
ensemble, and compare the results to the lattice results for spin-spin forces
in section III.3, and to the phenomenological vector-pseudoscalar mass
splittings in section III.4. We will see that in this setting the splittings
in heavy-heavy quarkonia are reproduced.

The next challenge using the novel model, is to extend the same analysis to
heavy-light mesons such as $B,D...$ and eventually light-light mesons. The
goal is to find out wether these lighter systems can also be described
analogously using variants of the “constituent quark” models.

Not all instanton-induced effects are included using temporal Wilson loops
with straight fermion lines. The part of the quark propagation, related to the
chiral anomaly, instanton zero modes and the ’t Hooft Lagrangian has to be
treated separately as we detail in section IV. Only the inclusion of this
effective interaction can describe pions.

Yet to get the correct mass and wave function of the vector ($\rho$) mesons,
we aldo need the effective forces originating from $I\bar{I}$ molecules, see
IV.5. The discussion of the heavy-light systems is in section IV.6. The
conclusions reached in this study are summarized in section V.

##  II Central potentials

###  II.1 Flux tubes vs the instantons

The static quark potential $V_{C}(r)$ is defined via the vacuum average of the
Wilson line

$W=Pexp\big{[}ig\int dx^{\mu}A_{\mu}^{a}\hat{T}^{a}\big{]}$ (15)

over a closed rectangle $r\times T$. In the limit when the time extent is much
larger than the spacial extent $T\gg r$, one can ignore the small integrals
over the spatial direction $r=|\vec{x}_{1}-\vec{x}_{2}|$, and keep only two
over the Euclidean time direction, defining the potential by

$e^{-V_{C}(r)T}=\langle W(\vec{x}_{1})W^{+}(\vec{x}_{2})\rangle$ (16)

For decades this Wilson’s definition is used in lattice studies.

We already noted that due to quantum vibrations of the QCD string, the flux
tube contribution to $V_{C}(r)$ is in fact rather uncertain at the distance
scale $\sim 1/3\,fm$. Yet precisely this distance range happens to be the most
important one for spin-dependent forces.

Another important point is that the quark spins, due to the magnetic moments,
interact with $magnetic$ fields. The spin-dependent potentials can be defined
via the average of two Wilson lines to which either two magnetic field
strengths or product of electric and magnetic fields are added. The QCD flux
tubes description, however, provides a description of the $electric$ fields
only. Therefore, one might think that they cannot contribute significantly to
the spin forces.

The instantons (antiinstantons) are selfdual (antiselfdual) vacuum
fluctuations, in which the modulus of the magnetic and electric fields are
equal. Furthermore, the BPTS $instanton$ fields have “hedgehog” structure
$A^{a}_{\mu}\sim\eta^{a}_{\mu\nu}x^{\nu}$, and since along the straight line
$dx^{\mu}$ is the same vector, the colors are rotated around the same
direction. Therefore, the accumulated rotation angle is given by a an integral
along the line Callan  _et al._ (1978); Eichten and Feinberg (1981). The
instanton-induced central potential has the form

$V_{\rm instanton}(r)={4\pi n_{\bar{I}+I}\rho^{3}\over
N_{c}\rho}I\bigg{(}{r\over\rho}\bigg{)}$ (17)

Here $n_{\bar{I}+I}$ is the instanton plus antiinstanton 4-dimensional
density, $\rho$ is the typical instanton size, and the function $I(x)$ is
defined by an integral over the location of the instanton center $y_{\mu}$.

$\displaystyle
I(x)=\int_{0}^{\infty}dyy^{2}\int_{-1}^{1}dc\bigg{[}1-cos(\alpha_{1})cos(\alpha_{2})-{y+xc\over\sqrt{y^{2}+x^{2}+2xyc}}sin(\alpha_{1})sin(\alpha_{2})\bigg{]}$
(18)

in which $c$ is the cosine of the angle between $\vec{r}$ and $\vec{y}$, going
through the instanton center. The two color rotation angles are

$\alpha_{1}=\pi{y\over\sqrt{y^{2}+\rho^{2}}},\,\,\,\alpha_{2}=\pi\sqrt{y^{2}+x^{2}+2xyc\over
y^{2}+x^{2}+2xyc+\rho^{2}}$ (19)

Note that they vanish for zero impact parameter of the line $y=0$, and become
$\pi$ if their impact parameter gets large. In the former case the integrand
above vanishes as $cos(\alpha)=1$, in the latter $cos(\alpha)=-1$ and the
integrand is maximal.

Using the instanton parameters of the ILM (6) one finds the large distance
value of about 150 MeV Yakhshiev  _et al._ (2017a). We however extend the
original ILM including also a “molecular $I\bar{I}$ component”. As a first
approximation (to be improved later) we will do so by just enhancing the
density, keeping the function with the same instanton size $\rho=1/3\,{\rm
fm}$ (as justified by lattice data of Fig.4(a) ).

The “molecular density” $n_{\rm mol}$ should be limited from above by the
phenomenological value of the central potential. The instanton contribution to
$V_{C}(r)$ is shown in Fig.5 (solid line) for

$n\equiv n_{\rm mol}+n_{\rm ILM}=7.\,{\rm fm}^{-4}$ (20)

Note that this density is about twice lower than indicated by a triangle and
extrapolation lines in Fig.4(b).

If so, the instanton-induced central potential gets quite close to the
phenomenologically known one, for intermediate distances $r<3\,{\rm
GeV}^{-1}$. Assuming it is indeed the case, we proceed to calculate the spin-
dependent effects below, and will show that the results are quite reasonable.

Note further that with this choice, the diluteness parameter $\kappa\approx
1$. In other words, the ensemble is very dense, with the mean interparticle
distance as low as

$R_{\rm dense}\equiv n^{-1/4}=0.61\,{\rm fm}\approx 2\rho$

(It may appear too dense: but remember that the $I\bar{I}$ molecules are not
in fact two independent instantons, and their fields and action are partially
cancelling each other.)

Figure 5: Comparison of the central potentials $V_{C}(r)$ versus the inter-
quark distance $r$ in ${\rm GeV}^{-1}$ (so $1\,{\rm fm}\approx 5{\rm
GeV}^{-1}$ on this plot). The solid line is the one derived from the “dense”
instanton ensemble. The potential produced by the confining flux tube is
illustrated by two blue dashed lines, the upper linear line is the classical
string, and the lower is the Arvis potential (4) acounting for quantum string
vibrations Arvis (1983) .

But before we do so, let us take a well studied example – the bottonium states
– and check to what extent the difference between the linear and instanton-
induced potential at $r>1\,{\rm fm}$ is reflected in the mass spectra. We have
calculated the levels of bottomonium $\bar{b}b$ states using both potentials.
In Fig.6 we show their non-relativistic energies for four radial excitations,
with $S$-shell, orbital momentum $L=0$. (As the spin-dependent forces were not
included – we return to those in the next subsections– therefore the
calculation corresponds to the spin averaged combination, of $J=1$ Upsilons
and $J=0$ $\eta_{b}$ states.) The difference between these two potentials at
large distances $r>5\,{\rm GeV}^{-1}\approx 1\,{\rm fm}$ does indeed
translates into different predictions for radially excited states. At large
$n$, the instanton-induced version does not keep with the expected Regge
behavior $m_{n}^{2}\sim n$. However, for the select bottomia considered there
is little practical difference between the potentials used. (In the matrix
elements used for lighter systems below, we will still use the wave functions
obtained with the usual Cornell potential.)

Figure 6: Nonrelativisitc energies $M_{i}-2M_{b}$ for spin averaged $\bar{b}b$
states as a function of the principal quantum number $n$. Five red squares
show the experimentally determined masses of
$\Upsilon[1S],\eta_{b}[1S],\Upsilon[2S],\Upsilon[3S],\Upsilon[4S]$, the blue
circles correspond to the standard Cornell potential, while the black closed
circles are based on the instanton-induced potential shown in Fig.5.

###  II.2 Contributions to $V_{C}$ of $I\bar{I}$ molecules

We start with $I\bar{I}$ pairs, with a fixed distance $R_{\mu}$ between the
centers. The contribution to the central potential strongly depends on the
orientation of this vector. When it is lined in time direction,
$R_{\mu}=(0,0,0,R)$, as shown in the left of Fig.7, the electric field and
potential $\vec{E}(x),A_{4}(x)$ are time-odd, with the opposite sign between
lower and upper sub-volumes, the instanton and antiinstanton. Therefore, the
color rotation angles in the Wilson lines – integrated over time along the
lines with arrows – get contributions of opposite sign which $cancel$. We thus
come to the somewhat surprising conclusion that this configuration would $not$
contribute to the central potential.

Figure 7: Instanton-antiinstanton molecule oriented along the time axes (left)
and space axes (right).

This is not so if $R_{\mu}$ has other orientations, e.g. in one of the space
direction. Then the time-oriented Wilson lines would go through either the
self-dual or anti-selfdual parts of the “molecule”. At small enough distance
between both of them $r<\rho$ they are likely to go through the same duality,
and therefore the result would be qualitatively the same as for a single
instanton (discussed in the preceding subsection). At large $r\sim 2\rho$ the
opposite would be dominant, with one of the angles in expression (18) changing
sign.

###  II.3 Fixed energy tunneling events and their contribution to $V_{C}$

Tunneling in vacuum in topological landscape proceed at energy zero. But for
quarks moving inside hadrons, or during some hadronic reactions, a certain
amount of kinetic energy is available. Therefore tunneling through topological
barriers may occur not via BPST instanton but with modified tunneling
solution, at some nonzero energy. These paths were sketched in the right side
of Fig.3: they include two Minkowskian sections of the path, and an Euclidean
one. The analytic solutions for a family of such paths, of Yang-Mills
equations, can be obtained.

Their Euclidean part interpolates between two “turning point” magnetic
configurations. As a result, the change in Chern-Simons number is fractional.
Depending on the energy, one can obtain a single-parameter family of
solutions, which interpolate between the bottom of the topological barrier
(instanton) and its top (the sphaleron configuration). When analytically
continued to Minkowski signature they describe implosion and explosion to and
from the magnetic turning points.

Step 1 to obtain such solutions use the O(4) static and symmetric ansatz for
the SU(2) configuration in regular gauge

$A^{a}_{M}(y)=2\,\bar{\eta}_{aMN}\,\frac{y_{N}}{y^{2}}f(\xi)$ (21)

with the conformal variable $\xi(y)=\frac{1}{2}{\rm ln}(y^{2}/\rho^{2})$. The
passage to singular gauge follows from the substitution $f(\xi)\rightarrow
f(\xi)-1$. The corresponding gauge invariant action is

$S_{S}=-\frac{1}{4}\int d^{4}y\,{F}^{a}_{MN}{F}^{a}_{MN}$ (22)

or in terms of the conformal variable

$\displaystyle S_{S}={24\pi^{2}}\int d\xi\left(\frac{f^{\prime
2}(\xi)}{2}+V(f(\xi))\right)$ (23)

with the inverted double well potential

$-V(f)=-2f^{2}(1-f)^{2}$

in Euclidean signature. The O(4) profile $f(\xi)$ extremizes (22) and solves
the Jacobi equation

$\frac{d^{2}f}{d\xi^{2}}=4(f^{2}-f)(2f-1)\,\,.$ (24)

The solution to (24) with a sphaleron-like turning point at $\xi=0$ with zero
momentum $f^{\prime}(\xi=0)=0$, is

$f_{k}(\xi)=\frac{1}{2}\bigg{(}1+\left(\frac{2k^{2}}{1+k^{2}}\right)^{\frac{1}{2}}{\rm
sn}\bigg{(}\xi\bigg{(}\frac{2}{1+k^{2}}\bigg{)}^{\frac{1}{2}}-K(k),k\bigg{)}$
(25)

with $sn$ the Jacobi sine function. (25) forms a k-family of $\xi$-periodic
functions with the period

$T_{k}=4K(k)\left(\frac{1+k^{2}}{2}\right)^{\frac{1}{2}}$ (26)

with $K(k)$ the elliptic function.

The solutions (25) carries energy (conjugate to $\xi$-coordinate) at the
turning point

$E_{k}=\frac{24\pi^{2}}{\rho}V(f_{k}(\xi=0))=E_{0}\bigg{(}\frac{1-k^{2}}{1+k^{2}}\bigg{)}^{2}$
(27)

with $E_{0}=3\pi^{2}/\rho$. (27) interpolates continuously between the
sphaleron and the instanton for $0\leq k\leq 1$. At $k=0$ the energy $E_{0}$
is maximal and corresponds to sphaleron mass, the period is
$T_{0}=\sqrt{2}\pi$ and the profile $f_{0}(\xi)=\frac{1}{2}$ is constant. At
$k=1$ the instanton energy is $E_{1}=0$, the period $T_{1}=\infty$ and the
profile is

$f_{1}(\xi)=\frac{1}{2}+\frac{1}{2}{\rm
sn}(\xi-K(1),1)=\frac{e^{2(\xi-K(1))}}{1+e^{2(\xi-K(1))}}$ (28)

Note that the argument shift with $K(1)$ in (28), amounts to a rescaling of
the instanton size in the conformal coordinate
$\rho\rightarrow\rho\,e^{K(1)}$.

In general, the solution (25) carries Chern-Simons number $N_{k}$ as well,
that is tied to the energy $E_{k}$ through the profile of the potential

$\left(\frac{E_{k}}{E_{0}}\right)=16N^{2}_{k}(1-N_{k})^{2}$ (29)

with $N_{k=1}=1$ the instanton topological charge, and $N_{k=0}=\frac{1}{2}$
the sphaleron Chern-Simons number. Only the solution with $N_{1}=1$ is self-
dual and stable. All the other sphaleron-like configurations with $N_{k}<1$
are unstable extrema. The flavor analogue of these configurations were
recently used to construct stable holographic tetraquark states Liu  _et al._
(2019a, b).

Step 2 is applying the conformal (stereographic) mapping of the previous
solutions that projects the O(4) turning points on the sphere at
$\xi^{2}=\rho^{2}$ onto the planar turning point at $t_{E}=0$,

$\displaystyle(x+a)_{\mu}={2\rho^{2}\over(y+a)^{2}}(y+a)_{\mu}$ (30)

with $a_{\mu}=(0,0,0,\rho)$. As a result, the gauge fields depend separately
on the Euclidean coordinates $t_{E}=x_{4}-Z_{4}$ and $r=|\vec{x}-\vec{Z}|$
(and parametrically on $k$).

$\displaystyle
A^{a}_{4}(t_{E},r;k)=\bigg{(}{8\rho\,t_{E}x_{a}\over[(t_{E}+\rho)^{2}+r^{2}][(t_{E}-\rho)^{2}+r^{2}]}\bigg{)}f_{k}(\xi_{E})$
$\displaystyle
A^{a}_{i}(t_{E},r;k)=\bigg{(}{\delta_{ai}(-t_{E}^{2}-r^{2}+\rho^{2})+2\rho\epsilon_{aij}x_{j}+2x_{i}x_{a}\over[(t_{E}+\rho)^{2}+r^{2}][(t_{E}-\rho)^{2}+r^{2}]}\bigg{)}4\rho
f_{k}(\xi_{E})$

with $Z_{\mu}=(\vec{Z},Z_{4})$ the collective position, and

$\displaystyle\xi_{E}=\frac{1}{2}{\rm
ln}\bigg{(}\frac{(t_{E}+\rho)^{2}+r^{2}}{(t_{E}-\rho)^{2}+r^{2}}\bigg{)}$ (32)

In Euclidean signature, $A_{4}$ and the electric field are real and vanishing
at $t_{E}=0$ as they should, while $A_{i}$ and the magnetic field are finite
and real. Similar configurations were originally discussed in Luscher (1977);
Schechter (1977). The half periodicity (26) maps onto the tunneling time
($r=0$)

Figure 8: Zig-zag path for a temporal Wilson line describing a heavy-quark
traveling from one vacuum to another with a different topological charge, and
riding a fixed energy tunneling configuration with tunneling time ${\cal
T}_{k}$.

${\cal
T}_{k}=\rho\,\bigg{(}\frac{e^{\frac{1}{2}T_{k}}-1}{e^{\frac{1}{2}T_{k}}+1}\bigg{)}=\rho\,{\rm
tanh}\bigg{(}\frac{1}{4}T_{k}\bigg{)}\leq\rho$ (33)

The conformal transforms of the electric and magnetic fields in (D) are
lengthy. The sphaleron configuration with $k=0$ and $f_{0}=\frac{1}{2}$, has
$A_{4}=0$ at the turning point $t_{E}=0$, with an O(3) symmetric and well
localized squared magnetic field

$\displaystyle\vec{B}^{2}(0,r;0)={96\rho^{4}\over(\rho^{2}+r^{2})^{4}}$ (34)

When analytically continued to Minkowski signature $t_{E}\rightarrow-it$, the
gauge fields (II.3) describe a spherically outgoing (incoming) luminal thin
shell, as the exiting sphaleron explodes (implodes) on its way downhill
(uphill). The luminal shell supports only light-like fields which are purely
transverse, and fall off as $1/t$ at large times. It was previously used to
describe “sphaleron explosion” in Ostrovsky  _et al._ (2002).

The fixed energy topological solutions contribute to Wilson lines, as a heavy
quark travels from one vacuum to another vacuum with one added (subtracted)
topological charge. In Minkowski signature, it corresponds to the zig-zag path
shown in Fig. 8,

$\displaystyle{\cal C}=$ $\displaystyle]-\infty+i{\cal T}_{k},i{\cal
T}_{k}]\cup[i{\cal T}_{k},0]\cup[0,+\infty[$

where on the purely imaginary $[i{\cal T}_{k},0]$ path, the heavy quark is
riding the tunneling process. A similar path choice was also advocated for
scattering processes through instantons in Rubakov  _et al._ (1992).

The chief contribution to the time-like Wilson line follows from this path,
since the gauge field on the paths with real time support is luminal and
transverse asymptotically,

$\displaystyle W(r;k)={\rm exp}\bigg{(}i\bigg{[}\int_{-\infty+i{\cal
T}_{k}}^{i{\cal T}_{k}}dt\,\frac{1}{2}A_{4}^{a}(-it,r;k)\tau^{a}+\int_{i{\cal
T}_{k}}^{0}dt\,\frac{1}{2}A_{4}^{a}(-it,r;k)\tau^{a}+\int_{0}^{\infty}dt\,\frac{1}{2}A_{4}^{a}(-it,r;k)\tau^{a}\bigg{]}\bigg{)}$
(36)

At the sphaleron point with $f_{k=0}=1/2$, we can check that $W(r;k=0)=1$ and
no self-energy is generated in going through a finite energy tunneling
configuration. This result follows by deforming the contour on the real axis,
without encountering poles for ${\cal T}_{k=0}<\rho$, and noting that $A_{4}$
is time-odd. Away from the sphaleron point, a fraction of the instanton self-
energy can be picked, through the emergence of complex singularities solution
to $f_{k}(\xi)=\infty$.

##  III Spin-dependent potentials

###  III.1 Phenomenology, from heavy to light quarks

To start, let us consider the first $1S$-shell, with zero orbital momentum
$L=0$, to recall the magnitude of the spin-spin forces. Starting from heavy
quarkonia and proceeding to lighter systems, we focus on the following
selection of mass splittings (all in MeV)

$\displaystyle M(\Upsilon)-M(\eta_{b})$ $\displaystyle=$ $\displaystyle
61.;\,\,\ $ (37) $\displaystyle M(J/\psi)-M(\eta_{c})$ $\displaystyle=$
$\displaystyle 116.;\,\,\,M(\psi^{2S})-M(\eta_{c}^{2S})=51.$ $\displaystyle
M(B^{*})-M(B)$ $\displaystyle=$ $\displaystyle
35.5;\,\,\,\,\,\,M(D^{*})-M(D)=137.\,\,\,$ $\displaystyle M(K^{*})-M(K)$
$\displaystyle=$ $\displaystyle 398.;\,\,\,M(\rho)-M(\pi)=636.\,\,\,$

Note that these splittings grow for lighter quarks, eventually getting
comparable to the scale of the mesonic masses: therefore in those case spin
forces cannot be treated as a perturbation.

Naively, the flavor dependence of the spin-spin forces should just follow from
the product of the quark magnetic moments, which is $\sim
1/m_{Q_{1}}m_{Q_{2}}$, times some universal magnetic fields in the QCD vacuum
as defined in the correlators of section B. Yet the select splittings above
show that it is not the case even for heavy quarks: e.g. charm and bottom
effects are different by a factor of two, not 10 as the mass ratio would
suggest. Yet universal vacuum correlators /potentials do not imply universal
(mass-independent) matrix elements, since all these mesons have vastly
different wave functions. Especially interesting is the case of charmonium
(second line in (37) ) in which one has both the splittings of 1S and 2S
levels. Not only the overall volume and $\psi(r=0)$ are different, but also
the wave function of 2S state has a node. The ratio of wave functions squared
is shown in Fig.9, and one can see that it changes from $0.538$ at $r=0$ to
zero at $r\approx 2\,GeV^{-1}$. The ratio of the observed splittings is
$51./113.\approx 0.45$ is comfortably in between, but much closer to the
former number than to the latter one. This observation alone tells us that
$V_{SS}$ must be concentrated at very small distances.

Going further in phenomenology, we now proceed to the 1-P shell, with $L=1$.
For any flavor combination, there are 4 states we will be interested in as
above, starting from heavy quarkonia, to heavy-light and all the way to mesons
made of light quarks. For the light mesons we select two channels, the
strange-light $K$ mesons and charge $I=1$ sector of the light-light. (The
$I=0$ sector has complicated mixing between quark states and glueballs, which
we prefer not to include.) The results are listed in Table 1. For example, the
$\chi_{b2}$ state is the $\bar{b}b$ state with $S=1,L=1,J=2$, and the $h_{b}$
state has $S=0,L=1,J=1$. All the names and masses are from the 2020 PDG tables

Figure 9: The ratio of wave functions squared
$\psi_{2S}(r)^{2}/\psi_{1S}(r)^{2}$ as a function of distance $r(GeV^{-1})$.

With $L=1$ all the three spin-dependent terms come into play, and with four
masses one has three differences which allow us to solve for each individual
contribution. We define the differences from the $\chi$ to $h$ states as

$a\langle(\vec{S}\cdot\vec{L})\rangle+b\langle
S_{12}\rangle+c\langle(\vec{S}_{1}\cdot\vec{S}_{2})\rangle$

We can solve for $a,b,c$ by using the corresponding quantum numbers for these
states, namely

$\displaystyle M(\chi_{b2})-M(h_{b})$ $\displaystyle=$ $\displaystyle
a-(2/5)b+c,$ $\displaystyle M(\chi_{b1})-M(h_{b})$ $\displaystyle=$
$\displaystyle-a+2b+c,$ $\displaystyle M(\chi_{b0})-M(h_{b})$ $\displaystyle=$
$\displaystyle-2a-4b+c$ (38)

The results are listed in the last three columns of Table 1. The ensuing
comments are:

(i) The spin-orbit matrix element $a$ for heavy quarks, is dominant;
(ii) The spin-orbit gets smaller for light quarks. Why?
(iii) The tensor force grows for heavy-light but then flips sign for light
quarks. Why?
(iv) The spin-spin term grows and is dominant for light quarks.

We now proceed to see whether these observations can be explained by the
theory we discuss below.

Table 1: The first column is the flavor composition. The next four columns refer to four states of the $n=1,P$-shell with the quantum numbers explained in the text. The last three columns are matrix elements of spin-orbit, tensor and spin-spin terms, in MeV. | S=1, J=2 | S=1,J=1 | S=1,J=0 | S=0,J=1 | a | b | c
---|---|---|---|---|---|---|---
$\bar{b}b$ | $\chi_{b2}(9912)$ | $\chi_{b1}(9893)$ | $\chi_{b0}(9859)$ | $h_{b}(9899)$ | 13.7 | 3.3 | 0.5
$\bar{c}c$ | $\chi_{c2}(3556)$ | $\chi_{c1}(3511)$ | $\chi_{c0}(3415)$ | $h_{c}(3525)$ | 35. | 10. | 0.1
$\bar{c}q$ | $D_{2}^{*}(2461)$ | $D_{1}(2430)$ | $D^{*}_{0}(2300)$ | $D_{1}(2421)$ | 34. | 16. | 12.
$\bar{s}q$ | $K_{2}(1430)$ | $K_{1}(1403)$ | $K_{0}(1430)$ | $h_{s}(1270)$ | 6.7 | -5.6 | 151.
$\bar{q}q$ | $a_{2}(1317)$ | $a_{1}(1230)$ | $a_{0}(1450)$ | $b_{1}(1235)$ | 0.5 | -36. | 68.

###  III.2 Wilson lines and the five potentials

Eichten and Feinberg Eichten and Feinberg (1981) defined general spin-
dependent interactions of heavy quarks in terms of a priori five potentials
(see their definitions and further discussion in appendix A) which contribute
as follows

$\displaystyle V_{SD}=$ $\displaystyle\bigg{(}\frac{S_{Q}\cdot
L_{Q}}{2m^{2}_{Q}}-\frac{S_{\bar{Q}}\cdot
L_{\bar{Q}}}{2m_{\bar{Q}}^{2}}\bigg{)}\bigg{(}\frac{1}{r}\frac{d}{dr}\big{(}V_{C}(r)+2V_{1}(r)\big{)}\bigg{)}$
$\displaystyle+$ $\displaystyle\bigg{(}\frac{S_{\bar{Q}}\cdot
L_{Q}}{m_{Q}m_{\bar{Q}}}-\frac{S_{Q}\cdot
L_{\bar{Q}}}{m_{\bar{Q}}m_{\bar{Q}}}\bigg{)}\bigg{(}\frac{1}{r}\frac{d}{dr}V_{2}(r)\bigg{)}+\frac{(3S_{Q}\cdot\hat{r}S_{\bar{Q}}\cdot\hat{r}-S_{Q}\cdot
S_{\bar{Q}})}{3m_{Q}m_{\bar{Q}}}V_{3}(r)+\frac{1}{3}\frac{S_{Q}\cdot
S_{\bar{Q}}}{m_{Q}m_{\bar{Q}}}V_{4}(r)$ (39)

$\vec{S}_{Q,\bar{Q}}$ and $\vec{L}_{Q,\bar{Q}}$ are the spin and orbital
angular momenta of the $\bar{Q}Q$ pair. $V(r)$ is the central static
potential, $V_{1}(r)$ and $V_{2}(r)$ are obtained by inserting a chromo-
electric or chromo-magnetic field on the temporal Wilson loop. The spin-spin
and tensor contributions $V_{3,4}(r)$ follow from the insertion of two chromo-
magnetic fields on the Wilson loop. (III.2) is exact to order $1/m_{Q}^{2}$.
The last spin-spin part is fixed as $V_{4}(r)=2\nabla^{2}V_{2}(r)$ Eichten and
Feinberg (1981). Lorentz invariance ties the central potential $V_{C}(r)$ to
the spin-orbit potentials $V_{1,2}(r)$ through the so called Gromes relation
Gromes (1984)

$V_{C}(r)=V_{2}(r)-V_{1}(r)$ (40)

The spin-dependent contributions emerging from the string were discussed by
Buchmuller Buchmuller (1982) and others Pisarski and Stack (1987); Gromes
(1984). Since the spin-spin interactions are short ranged, only the spin-orbit
contributions survive at large separation $b_{\perp}$. This is manifest from
(40) with $V_{2}(r)\rightarrow 0$ and $V_{C}(r)\rightarrow-V_{1}(r)$
asymptotically, hence Buchmuller (1982); Pisarski and Stack (1987); Gromes
(1984)

$\displaystyle V_{SL,{\rm string}}(b_{\perp})$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{S_{Q}\cdot
L_{Q}}{2m_{Q}^{2}}-\frac{S_{\bar{Q}}\cdot
L_{\bar{Q}}}{2m_{\bar{Q}}^{2}}\bigg{)}\frac{\sigma_{T}}{b_{\perp}}$ (41)
$\displaystyle\rightarrow$
$\displaystyle-\frac{\sigma_{T}}{2m_{Q}^{2}b_{\perp}}\,S\cdot L$

with $V_{C}(r)=\sigma_{T}r$, $\vec{S}=\vec{S}_{Q}+\vec{S}_{\bar{Q}}$ and
$\vec{L}=\vec{L}_{Q}=-\vec{L}_{\bar{Q}}$. Since the electric flux tube is
confined to the string, the spin-orbit contribution is only due to Thomas
precession which is of opposite sign and half the spin-orbit contribution from
the cental potential. This can be understood from (III.2-40) if we assume that
$V_{2}(r)$ is short range, so that $V_{C}(r)=-V_{1}(r)$. This string-induced
spin-orbit effect is dubbed scalar-like in contrast to the Coulomb-induced
spin-orbit effect which is vector-like.

Figure 10: Perturbative (upper plot) and instanton-induced (lower plot) spin-
dependent potentials for charmonium. The black solid, blue dashed and red
dash-dotted lines are for $r^{2}V_{SS},r^{2}V_{SL},r^{2}V_{T}$, in ${\rm
GeV}^{-1}$, versus $r$ in ${\rm GeV}^{-1}$.

###  III.3 Lattice studies

The static spin-spin potentials have been evaluated on the lattice, using
correlators of Wilson lines with explicit field strengths. In particular, Koma
and Koma Koma and Koma (2007) find that while $V_{SS}$ is indeed rather short
range, it does not fit to the vector+scalar exchange paradigm: a pseudo-scalar
glueball exchange has been proposed. Skipping a decade, Kawanai and Sasaki
Kawanai and Sasaki (2015) accurately derived the central and spin-spin
potentials for the $\bar{c}c$ and $\bar{c}s$ families, and used them in
effective Schrodinger equations for a spectroscopic analysis with nice
agreement with the spectroscopic data. In Fig. 13 we show (solid line) their
exponential fit (for $\bar{c}c$).

$V_{SS}^{\bar{c}c}\equiv\alpha e^{-\beta r},\,\,\,\alpha=2.15\,{\rm
GeV},\,\,\,\beta=2.93\,{\rm GeV}$ (42)

Let us try to interpret this result: Naively, the coefficient in the exponent
should be the mass of the object exchanged: for a gluon it should be about
half of glueball masses, of the order of $1\,GeV$ or so. It is in fact larger
by a factor of 3. Why is it so large?

Fortunately, one can check it phenomenologically and find that it does
describe well the $2S$ to $1S$ splitting in charmonium. Using this $V_{SS}$
parameterization from Kawanai and Sasaki (2015) we calculate 1S triplet-
singlet splitting to be $113.7\,MeV$ (exp: $113\,MeV$) and 2S to be $47.3,\
MeV$ (exp: $51\,MeV$). While not perfect, this version of $V_{SS}$ clearly
does the job, at least in charmonium.

Figure 11: Notations of vectors in evaluation of Wilson lines with two field
strengths.

###  III.4 Spin-dependent forces from instantons

We decided first not to rely on general relations, sometimes leading to
singular functions, and evaluated the correlators of Wilson lines with fields
directly.

The notations we use for the evaluation of the spin-dependent potentials are
shown in Fig. 11. The Wilson line $W_{1}$ is located at the origin of the
spatial coordinates $\vec{0}$, and $W_{2}$ at $\vec{r}=(r,0,0)$. Two magnetic
fields are inserted on them, and act at times $t_{1}$ and $t_{2}$,
respectively. The SU(2) color matrix associated with the field $B_{m}$ is
$\tau^{m}$, and the overall color trace defines the following tensor

$C^{m\bar{m}}={\rm
Tr}\big{[}W_{1}^{upper}(W_{2}^{upper})^{+}\tau^{\bar{m}}(W_{2}^{lower})^{+}W_{1}^{lower}\tau^{m}\big{]}$

This construction is implicitly included in what we denote by the “double
average”, for example the spin-spin potential is related to correlator of two
magnetic fields integrated over their time difference

$V_{4}(r)=\int dt\langle\langle
B^{ma}(\vec{0},0)B^{ma}(\vec{r},t)\rangle\rangle$ (43)

The center of the instanton is put at time zero and spatial location
$\vec{y}$. On dimensional grounds, the instanton of size $\rho$ should
contribute to those potentials $\sim\kappa{1/\rho^{3}}F(r/\rho)$.

Figure 12: The dependence of the integrated correlator of two magnetic fields
(43) on the distance, in units of $r/\rho$, is shown by points. The two thin
lines are drawn for comparison, the upper is $\sim 1/r$ and the lower $\sim
1/r^{2}$.

The field strength for an instanton (in regular gauge), located at point $y$,
is

$G^{a}_{\mu\nu}(x)=-4\eta^{a}_{\mu\nu}{\rho^{2}\over\big{(}(x-y)^{2}+\rho^{2}\big{)}^{2}}$
(44)

Four Wilson lines are written in their canonical form
$W=cos(A)+i(\vec{n}\vec{\tau})sin(A)$ with certain angles, which in regular
gauge are

$\displaystyle
A_{1}^{lower}=-\big{[}\pi+2arctan({t_{1}\over\sqrt{y^{2}+\rho^{2}}})\big{]}{1\over\sqrt{1+\rho^{2}/y^{2}}}$
$\displaystyle
A_{2}^{lower}=-\big{[}\pi+2arctan({t_{2}\over\sqrt{r^{2}+y^{2}-2rycos(\theta)+\rho^{2}}})\big{]}{1\over\sqrt{1+\rho^{2}/(r^{2}+y^{2}-2rycos(\theta))}}$
(45)

The “upper” lines are given by the same expressions with opposite signs of
arctan. If there is no field insertions, these terms cancel as they should,
and their sum become the angles we already met in the expression for central
potentials.

In total, the correlator includes integration over $d^{3}y,t_{1},t_{2}$ . With
the azimuthal angle $\phi_{y}$ being irrelevant, it is 4-dimensional integral
performed numerically. The results are shown in Fig.12. In this log plot one
can see that the resulting spin-spin force rapidly decreases with distance.
For comparison, we show two lines with $1/r$ and $1/r^{2}$, which
approximately reproduces the behavior. One should not however conclude that
they give correct analytic asymptotic: in fact, expanding the integrand in
inverse powers of $r$ leads to divergences of the remaining integrals,
indicating that the true asymptotic behavior cannot be just powers.

In the instanton vacuum, with (anti) selfdual fields
$\vec{E}^{a}=\pm\vec{B}^{a}$ in Euclidean signature, all potentials in (III.2)
can be tied to the central potential $V_{C}(r)$ Callan  _et al._ (1978);
Eichten and Feinberg (1981); Yakhshiev  _et al._ (2017a)

$\displaystyle V_{C}(r)+2V_{1}(r)\rightarrow 0\qquad
V_{2}(r)\rightarrow\frac{1}{2}V_{C}(r)\qquad
V_{3}(r)\rightarrow-\bigg{(}\frac{1}{r}V_{C}^{\prime}-V_{C}^{\prime\prime}\bigg{)}$
(46)

and the simplified spin-orbit contribution

$\bigg{[}\frac{1}{2m_{Q}^{2}}\frac{1}{r}\frac{d}{dr}V_{C}(r)\bigg{]}(S_{Q}+S_{\bar{Q}})\cdot
L_{Q}\equiv\bigg{[}\frac{1}{2m_{Q}^{2}}\frac{1}{r}\frac{d}{dr}V_{C}(r)\bigg{]}S\cdot
L$ (47)

Schematically, the central electric interaction $\langle
E_{i}^{a}(x)[x,0]^{ab}E_{i}^{b}(0)\rangle$ is amenable to the spin-orbit
interaction $\langle E_{i}^{a}(x)[x,0]^{ab}B_{i}^{b}(0)\rangle$ and also the
spin-spin and tensor interaction $\langle
B_{i}^{a}(x)[x,0]^{ab}B_{i}^{b}(0)\rangle$.

Figure 13: Spin-spin potential for $\bar{c}c$ system multiplied by distance
squared $r^{2}V_{SS}\,(GeV^{-1})$ versus $r\,(GeV)^{-1}$.The solid line is the
exponential fit (42) to lattice measurements Kawanai and Sasaki (2015). The
dash-dotted line shows the regulated laplacian of the Coulomb potential (48)
with $\delta=0.6\,GeV^{-1}$. The blue dashed line show the instanton
contribution. Note that the area under the last two curves is roughly equal to
that under solid one, the lattice fit.

In Fig.13 we compare the perturbative (dashed-dotted-black) and
nonperturbative (dashed-blue) spin-spin potentials with the lattice result
(solid-black). Recall that the Laplacian of the Coulomb force is a delta
function, well known in atomic and positronium physics. However, one should
use Coulomb potential $1/r$ in some regulated form, for several reasons.
First, when the potential gets as deep as (minus) mass, the nonrelativistic
approximation itself should break down, so the spread cannot be smaller than
$O(1/m_{Q})$. Second, lattice studies include non-locality in the form of the
lattice spacing $a$. Third, a Gaussian-smeared delta function is actually used
in many spectroscopic calculations, see e.g. Barnes  _et al._ (2005). For the
perturbative spin-spin interaction, we use a simpler version, as the Laplacian
of a Coulomb contribution regulated by a fixed parameter $\delta=0.6\,{\rm
GeV}^{-1}$

$V_{SS}^{C}(r)=-{A\over
3m_{Q}m_{\bar{Q}}}\vec{\nabla}^{2}{1\over\sqrt{r^{2}+\delta^{2}}}$ (48)

For the nonperturbative spin-spin part, we use the instanton contribution,
which is given by the Laplacian acting on the central potential

$V_{SS}^{\rm inst}(r)={1\over 3m_{Q}m_{\bar{Q}}}\vec{\nabla}^{2}V_{C}^{\rm
inst}(r)$ (49)

All the potentials shown are multiplied by $r^{2}$, as they appear in the
volume integrals. So, if the wave functions are roughly constant at this small
scale, the area under the curves in Fig. 13 gives an estimate of the relative
contributions. So, one finds that the perturbative and non-perturbative parts
contribute $comparably$ to the total spin-spin potential, in sum roughly
reproducing the lattice-based result. This observation is among the main
findings of this work.

Of course, the actual magnitude of the spin splittings depends on the quark
masses involved, explicitly via $1/m_{Q}^{2}$ factors and implicitly, via the
wave functions. Lighter quarks lead generally to hadrons of larger size, or
smaller wave functions at small distances.

The matrix elements of all three potentials shown in Fig. 13 were evaluated
using wavefunctions from the Cornell potential irrespective of flavor. The
results for the spin-spin contributions are in Table 2. The results show that
perturbative and non-perturbative contributions for heavy quarkonia do
reproduce the splittings if taken together. Already for heavy-light mesons one
can see that some additional effect seems to be missing: it will indeed so as
will be explained below. Furthermore, for the mesons containing only light
quarks both contributions together definitely fail to describe large
splittings seen experimentally. Clearly some other effects are in play here,
to be discussed below.

We continue our phenomenological description of the spin-dependent
interactions, by considering the next four states in the 1P shell, with the
onset of the spin-orbit and tensor potentials The empirical information to be
used on several mesonic families is listed in 3. The theoretical estimates
follow from 1P wave functions derived with standard Cornell potential. For
both components of the central potential – perturbative and instanton-induced
– we calculated the matrix elements of the three spin-dependent potentials
using the formulae given above, and then calculated their matrix elements in
the 1P shell.

One observation from this table is that the “theory” (the sum of the last two
rows) and the lattice-induced one (the second row) are basically in agreement.
The other is that their disagreement with experiment (the upper raw) is
dramatically growing for light quark systems. This implies that for light
quarks something important is missed, in the description as developed so far.

Table 2: “Hyperfine” splittings of certain $L=0$ mesons with $J=1$ and $J=0$. The first row of numbers shows the experimental values (MeV) (rounded to 1 MeV). The second row gives the matrix elements of the lattice-based spin-spin potential (42), the next two rows are the (regulated) Coulomb and instanton-induced spin-spin contributions. flavors | $M_{\Upsilon}-M_{\eta_{b}}$ | $M_{J/\psi}-M_{\eta_{c}}$ | $M(D^{*})-M(D)$ | $M(K^{*})-M(K)$ | $M(\rho)-M(\pi)$
---|---|---|---|---|---
Exp | 61. | 116. | 137. | 398. | 636.
$\langle V_{SS}^{\rm lat}/3m_{Q}m_{\bar{Q}}\rangle$ | 46. | 108. | 98. | 170. |
$\langle\vec{\nabla}^{2}V_{C}/3m_{Q}m_{\bar{Q}}\rangle$ | 28. | 58. | 48. | 82. |
$\langle\vec{\nabla}^{2}V_{\rm inst}/3m_{Q}m_{\bar{Q}}\rangle$ | 7. | 30. | 48. | 90. |
Table 3: Groups of three matrix elements of spin-spin, spin-orbit and tensor potentials $V_{SS},V_{SL},V_{T}$, respectively. The first one in each group is the observed “exp” value from Table 1, the second and the third values are the perturbative and instanton-induced contributions, corresponding to the Coulomb and instanton-induced parts of the central potential.  | SS “exp” | SS pert | SS inst | SL “exp” | SL pert | SL inst | T “exp” | T pert | T inst
---|---|---|---|---|---|---|---|---|---
$\bar{c}c$ | 0.1 | 0.56 | 1.9 | 35. | 3.2 | 3.8 | 10. | 5.8 | -5.7
$\bar{s}q$ | 151. | 5. | 29. | 6.7 | 31. | 38. | -5.6 | 58. | -46.
$\bar{q}q$ | 68. | 8. | 48. | 0.5 | 52. | 64. | -36. | 96. | -78.

###  III.5 Contributions of $I\bar{I}$ molecules to spin forces

Like in subsection II.2, let us start with the $I\bar{I}$ molecules oriented
in the time direction, for which $\vec{E}\sim G_{4m}$ and $A_{4}$ are time-
odd. Therefore, as we noted before, there is no contribution to the central
potential.

Yet the spin-spin and tensor forces are not at all zero, as the main
contributors to the correlator come from the $magnetic$ fields $\vec{B}$,
which have the same sign in the $I$ and $\bar{I}$ parts of the molecule!

A detailed description of the $I\bar{I}$ configuration can be achieved by the
Yung ansatz, which is quite accurate. The corresponding expressions for field
strengths were obtained in Mathematica, which are way too complicated to be
given here. Yet the field strengths are approximately additive, and this
approximation will be used in what follows.

Let us start with the correlator of two magnetic fields, measured at
integrated times $t_{1}$ and $t_{2}$ on two temporal lines separated by a
spatial distance $r$

$\displaystyle\int dt\,\langle B^{ma}(0,\vec{0})B^{ma}(t,\vec{0})\rangle=$
$\displaystyle\int dt_{1}dt_{2}d^{3}y(48n_{mol}\rho^{4})$ (50)
$\displaystyle\times\bigg{(}{1\over((t_{1}-R/2)^{2}+\rho
2+y^{2})^{2}}+{1\over((t_{1}+R/2)^{2}+\rho 2+y^{2})^{2}}\bigg{)}$
$\displaystyle\times\bigg{(}{1\over((t_{2}-R/2)^{2}+\rho
2+y_{2}^{2})^{2}}+{1\over((t_{2}+R/2)^{2}+\rho 2+y_{2}^{2})^{2}}\bigg{)}$

Here $y^{2}$ and $y_{2}^{2}=y^{2}+r^{2}-2rycos(\theta_{ry})$ are the squared
3d distances to the first and second Wilson lines. We assumed additivity of
the magnetic fields from $I$ and $\bar{I}$, and ignored the color rotation
angles on the Wilson lines (which vanish at the instanton center, as we
already know).

At zero distance between the line $r=0$, all integrations can be done
analytically, with the result

$\int dt\,\langle
B^{ma}_{1}(0,\vec{0})B^{ma}_{2}(t,\vec{0})\rangle_{r=0}=12\pi^{4}\rho n_{mol}$
(51)

For qualitative orientation, one can compare this expression to (the
extrapolation of) the lattice spin-spin potential (42) to $r=0$: they are
equal if $n_{mol}\approx 7\,fm^{-4}$.

Note that in this approximation there is no dependence on the (time)
separation $R$ of $I$ and $\bar{I}$ centers. For nonzero $r$ the volume
integral over $y,cos(\theta_{ry})$ is done numerically, see the results in
Fig. 14.

Now let us calculate the same correlator, but for a molecule rotated so that
its vector $R_{\mu}$ joining the centers, lines in the spatial direction, say
the same as the vector $\vec{r}$ between the Wilson lines. The electric and
certain magnetic fields exchange places and the expression changes into

$\displaystyle\int dt\,\langle
B^{am}(0,\vec{0})\vec{B}^{am}(t,\vec{0})\rangle$ $\displaystyle=$
$\displaystyle\int dt_{1}dt_{2}dy_{1}2\pi
y_{\perp}dy_{\perp}(48n_{mol}\rho^{4})$
$\displaystyle\times\bigg{(}{1\over(t_{1}^{2}+\rho^{2}+(y_{1}-R/2)^{2}+y_{\perp}^{2})^{2}}-{1\over(t_{1}^{2}+(y_{1}+R/2)^{2}+\rho^{2}+y_{\perp}^{2})^{2}}\bigg{)}$
$\displaystyle\times\bigg{(}{1\over(t_{2}^{2}+\rho^{2}+(y_{1}-R/2+r)^{2}+y_{\perp}^{2})^{2}}-{1\over(t_{2}^{2}+(y_{1}+R/2+r)^{2}+\rho^{2}+y_{\perp}^{2})^{2}}\bigg{)}$

In Fig. 14 we compare the correlators for two molecule orientations with
averaging factors, $(1/4)(\ref{eqn_molec_t})$ and $(3/4)(\ref{eqn_molec_x})$
by upper and lower points, as well as their sum (the line). So, while the two
contributions have different dependence on $r$, the orientation-average
$V_{SS}(r)$ decreases monotonously, and has the effective width (at half
maximum) close to the average instanton size $\rho$. Note that this range is
shorter than $V_{SS}(r)$ from the instantons alone (calculated from the
Laplacian of the contribution to the central potential in the previous
subsection), and it does not have a negative part. It is therefore closer in
shape to what was found on the lattice.

Figure 14: The normalized correlators $(1/4)(\ref{eqn_molec_t})$ (upper
points) and $(3/4)(\ref{eqn_molec_x})$ for $R=\rho$ (lower points) in units of
$n_{mol}\rho$, versus the distance between the Wilson lines $r/\rho$. The
solid line shows their sum.

The spin-orbit force is qualitatively different from spin-spin ones, being
related to a correlator $\int dt\vec{E}\vec{B}$. For a time-oriented molecule
$R_{\mu}=(0,0,0,R)$ $\vec{E}(t)$ is time-odd and $\vec{B}(t)$ is time-even, so
for this molecule orientation this correlator vanishes. The situation is
different for a space-oriented molecule, say along $x_{1}$. Now
$G_{1m},m=2,3,4$ fields are odd in $x_{1}$ inversion, but everything is even
along the time axes. The expression for the correlator can easily be
constructed and calculated in a way similar to what was done above for
$V_{SS}(r)$

We conclude that the “molecule-induced” spin forces are quite similar to the
instanton induced ones (apart of course normalization to quite different
densities) at small distances, but are significantly reduced at $r>2\rho$ due
to the partial cancellations by different orientations.

In the previous section we have seen that, in the approximation of molecules
represented as two independent pseudoparticles, the non-perturbative
$V_{SS(r)}$ have about the right magnitude, but somewhat too large range. We
now see that a better description of the field structure has the potential to
remedy this problem.

###  III.6 Contributions of the fixed energy tunneling configurations to spin
forces

The finite energy tunneling configurations contribute to the spin forces,
through close Wilson loops dressed by $E,B$ fields. However, since these
configurations cease to be self-dual for $0\leq k<1$, the induced spin
potentials are in general independent. Moreover, these configurations carry
t-odd electric fields, and are purely magnetic at the exit points ($t_{E}=0$).
They may only contribute to the spin-spin and tensor potentials.

The explicit expression of the magnetic field in Euclidean time is

$\displaystyle B^{ma}(t_{E},\vec{x})$ $\displaystyle=$
$\displaystyle\delta^{ma}\bigg{(}-\frac{16\rho^{2}}{\mathbb{D}}f+\frac{8\rho^{2}r^{2}}{\mathbb{D}^{2}}fF+\frac{32\rho^{3}t_{E}r^{2}}{\mathbb{D}^{2}}f^{\prime}+\frac{32r^{2}\rho^{2}}{\mathbb{D}^{2}}f^{2}\bar{F}\bigg{)}$
(53) $\displaystyle+$ $\displaystyle
x^{m}x^{a}\bigg{(}-\frac{8\rho^{2}}{\mathbb{D}^{2}}fF-\frac{32t_{E}\rho^{3}}{\mathbb{D}^{2}}f^{\prime}+\frac{32\rho^{2}}{\mathbb{D}^{2}}f^{2}F\bigg{)}$
$\displaystyle+$
$\displaystyle\epsilon^{mai}x^{i}\bigg{(}\frac{16\rho}{\mathbb{D}}f+\frac{4\rho}{\mathbb{D}^{2}}fF\bar{F}+\frac{16t_{E}\rho^{2}}{\mathbb{D}^{2}}f^{\prime}\bar{F}-\frac{32\rho^{3}}{\mathbb{D}^{2}}(2r^{2}+\bar{F})f^{2}\bigg{)}$

with $f^{\prime}=\partial_{\xi}f_{k}(\xi)$,

$\displaystyle\mathbb{D}=(t_{E}^{2}+r^{2}+\rho^{2})^{2}-4t_{E}^{2}\rho^{2}$
(54)

and

$\displaystyle
F=t_{E}^{2}+r^{2}+\rho^{2}\qquad\bar{F}=-t_{E}^{2}-r^{2}+\rho^{2}$ (55)

The magnetic field in Minkowski signature follows from the substitution
$t_{E}\rightarrow-it$.

The induced spin-spin interaction follows from the complexified path in the
t-plane shown in Fig. 8. An estimate of the magnitude of the spin-spin
interaction, follows from zero separation between a path ${\cal C}$ and
$\bar{\cal C}$

$\displaystyle\int_{\cal C}dt\int_{\bar{\cal
C}}d\bar{t}\,\left<B^{ma}(-it,\vec{0})B^{ma}(-i\bar{t},\vec{0})\right>=[3\,2^{8}\rho^{4}n_{k}]\int_{\cal
C}dt\int_{\bar{\cal C}}d\bar{t}\,\int
d^{4}Z\bigg{(}\frac{f\bar{f}}{\mathbb{D}\overline{\mathbb{D}}}\bigg{)}$ (56)

with now

$\displaystyle\mathbb{D}\rightarrow((-it-
Z_{4})^{2}+{\vec{Z}}^{2}+\rho^{2})^{2}-4(-it-Z_{4})^{2}\rho^{2}$
$\displaystyle\overline{\mathbb{D}}\rightarrow((-i\bar{t}-Z_{4})^{2}+{\vec{Z}}^{2}+\rho^{2})^{2}-4(-i\bar{t}-Z_{4})^{2}\rho^{2}$

and $F=f_{k}(\xi)$ and $\bar{f}=f_{k}(\bar{\xi})$ with

$\displaystyle\xi\rightarrow\frac{1}{2}{\rm ln}\bigg{(}\frac{(-it-
Z_{4}+\rho)^{2}+{\vec{Z}}^{2}}{(-it-Z_{4}-\rho)^{2}+{\vec{Z}}^{2}}\bigg{)}$
$\displaystyle\bar{\xi}\rightarrow\frac{1}{2}{\rm
ln}\bigg{(}\frac{(-i\bar{t}-Z_{4}+\rho)^{2}+{\vec{Z}}^{2}}{(-i\bar{t}-Z_{4}-\rho)^{2}+{\vec{Z}}^{2}}\bigg{)}$

with the Euclidean-Minkowski time assignments in (II.3).

At the sphaleron point $f_{k=0}=1/2$, and (56) reduces to

$\displaystyle\int_{\cal C}dt\int_{\bar{\cal
C}}d\bar{t}\,\left<B^{ma}(-it,\vec{0})B^{ma}(-i\bar{t},\vec{0})\right>\rightarrow[192\rho^{4}n_{k=0}]\int_{\cal
C}dt\int_{\bar{\cal C}}d\bar{t}\,\int d^{4}Z$
$\displaystyle\times\frac{1}{(((-it-
Z_{4})^{2}+{\vec{Z}}^{2}+\rho^{2})^{2}-4(-it-
Z_{4})^{2}\rho^{2})(((-i\bar{t}-Z_{4})^{2}+{\vec{Z}}^{2}+\rho^{2})^{2}-4(-i\bar{t}-Z_{4})^{2}\rho^{2})}$

Overall time translational invariance allows us to set $Z_{4}=0$ in the
integrand, with the final result

$\displaystyle\frac{\int_{\cal C}dt\int_{\bar{\cal
C}}d\bar{t}\,\left<B^{ma}(t,\vec{0})B^{ma}(\bar{t},\vec{0})\right>}{\int
dZ_{4}}=\frac{\pi^{4}}{4\rho^{3}}$ (60)

This result follows by deforming the complexified path to the real axis,
without encountering poles for ${\cal T}_{k=0}<\rho$. The spin-spin
interaction (60) stemming from the finite energy tunneling configurations, is
comparable to the spin-spin interaction (51) from the thimbles. At the
sphaleron point and for zero separation, they are comparable when

$\displaystyle\frac{n_{k=0}}{n_{\rm
mol}}=\frac{12\pi^{4}}{192\pi^{4}/4}=\frac{1}{4}$ (61)

##  IV Interquark forces induced by instanton zero modes
(’t Hooft effective Lagrangian)

###  IV.1 Pseudoscalar mesons which are and are not the Nambu-Goldstone modes

It is known since 1970’s that in the chiral limit light-quark pseudoscalar
mesons consist of octet $\pi,K,\eta$ Nambu-Goldstone modes and a singlet
$\eta^{\prime}$ which does $not$ belong to that set, and is instead very
heavy. It happens because the t’ Hooft effective Lagrangian violates
$U_{A}(1)$ and quite effectively generates flavor-changing annihilation
processes

$\bar{u}u\leftrightarrow\bar{d}d\leftrightarrow\bar{s}s$

When a strange quark mass is added, one finds $\eta-\eta^{\prime}$ mixing
which describes well the phenomenology of their decays. It sharply contrasts
with the vector counterparts, $\rho,\omega,\phi$ mesons, for which the mixing
is very small.

To avoid these annihilation processes, one can discuss “flavored” channels,
like $\pi^{+}=\bar{d}u$, in which case the t’ Hooft effective Lagrangian acts
as a force between the quark and antiquark. Yet it is still different from the
conventional forces we so far discussed, in that it exists for quarks of
different flavors only. In Fig.1 we illustrate how the t’ Hooft induced vertex
operates in the pion and rho charged meson channels.

Before we turn to specifics, let us comment on some lattice implementations of
theories with different quarks. The simplest case is a single quark flavor
theory, $N_{f}=1$. In it the ’t Hooft action is a 2-fermion operator: it gives
the quark an effective mass, but does not generate annihilations or
interactions as we noted. The only pseudoscalar mesons is the analog of
$\eta^{\prime}$, no $\pi,K,\eta$ modes exist.

In lattice implementations nowadays one uses gauge ensemble for physical QCD,
but still use certain diagram and mass selections aiming to study certain
unphysical particles. For example, in Koponen  _et al._ (2017), an artificial
particle called $\eta_{s}$ made of $\bar{s}s$ was intoduced, with no mixing to
other flavors. Its properties are deduced from the $connected$ diagram,
without the disconnected one in which the flavor is changed. One may explain
its absence by introducing an $extra$ valence quark species $s^{\prime}$, and
view source/sink operators as flavor-nondiagonal $\bar{s}s^{\prime}$. Our
comment is that such setting, while eliminating the annihilations, still keeps
the ’t Hooft effective Lagrangian producing a force between the quark and
antiquark, as they are not of the same flavor. Apart from the mass values,
this channel is no different from say $\pi^{+}=\bar{d}u$ we consider.

In our first take on spin-spin forces in (37) we already observed anomalously
large splitting between the $\rho,S=1$ and $\pi,S=0$ states in the lowest
light quark shell, and have seen that the magnitude of $V_{SS}(r)$ discussed
so far, is not sufficient to explain it.

(In particular, adding $V_{SS}(r)$ as detailed above to the basic quark model,
with the Cornell potential and a light quark constituent mass of $0.35\,{\rm
GeV}$, reduces the ground state 1S mass substantially, from $1.4\,{\rm GeV}$
to about $0.85\,{\rm GeV}$. Yet it is still far from the observed pion mass of
$0.138\,{\rm GeV}$ and from zero, which it should be in the chiral limit.) Of
course, a correct pion mass can only be explained in an approach, consistently
explaining chiral symmetry breaking, as well as other parameters of chiral
perturbation theory. Two famous models – NJL and ILM – are well known examples
of that.

Before we focus on the theory, let us return for a brief moment to the
phenomenological constituent quark model, to see what kind of interaction one
needs to add in order to obtain the experimentally observed pions. Let us
assume that there is an additional interaction, that operates only between
quarks of different flavors, on top of the flavor-blind forces so far
discussed. Note that the usual discussion of $\pi-\eta^{\prime}$ splitting is
based on “annihilation channels” (like $\bar{u}u\leftrightarrow\bar{d}d$) as
we continue to consider only charged mesons, like $\bar{d}u$.

The ’t Hooft Lagrangian is usually written as a local operator a la NJL one,
corresponding to a potential of the type $G_{\rm Hooft}\delta^{3}(\vec{r})$
potential. This form assumes that the instanton size $\rho$ is much smaller
than the hadronic sizes, as it appears in the standard LSZ “amputation” of
external propagator lines. Unfortunately, in the Euclidean setting of the
instanton calculus, taking these lines on-shell amounts to the limit
$p^{\mu}\rightarrow 0$ for all components. And yet, it is clear that the
process is nonlocal, and that the scale of its size is $\rho$. Thus in model
applications one include form-factors, usually the Fourier transform of the
zero modes. We will use another “regulated delta function” form described in
section C.

(Comment: When the nonlocality induced by the parameter $\rho$ is small
compared to the hadron size, it can be well approximated by a delta function.
However, this is not true for pions. This can be seen from the pion wave
function (with or without the effective ’t Hooft term): its
$r^{2}\psi^{2}_{\pi}(r)$ peaks at $r\approx\rho$. In this case. Therefore, in
the pion case there a remains certain dependence on a choice of the non-local
approximation.)

Finally, the coupling strength for the t’ Hooft effective interaction to add
to the discussed potentials above needed to reproduce the physical pion mass
is found to be

$G_{\rm Hooft}=17\,{\rm GeV}^{-2}$ (62)

###  IV.2 Chiral anomaly and instanton zero modes

It was well understood already in 1970’s that instantons provide an example of
explicit violation of the $U_{A}(1)$ chiral symmetry. The divergence of the
corresponding axial current is proportional to the topological charge density
$G\tilde{G}$, which in turn is a divergence of the topological Chern-Simons
current. Thus the axial charge of fermions, or the number of right minus left-
polarized quarks, is intimatly related to gauge topology.

How this happens in the instanton case was explained by t’ Hooft ’t Hooft
(1976), who found the existence of 4-dimensional bound state, or the fermion
zero mode of the Dirac operator in the instanton background. Formally it
amounts to an additional term in the 4-dimensional quark propagator

$S(x,y)=\sum_{\lambda}{\psi_{\lambda}(x)\psi_{\lambda}(y)^{+}\over\lambda+im}$
(63)

with the zero mode $\lambda=0$. The physical meaning is that the negative
energy Dirac sea creates a new state, into which the original quark is pushed
into. At the same time, a quark of opposite chirality emerges from the Dirac
sea as a physical (positive energy) state. The infinite hotel story.

This process can be described by a “chirality flip” of a quark. However, the
phenomenon is not reduced just to this: $all$ light quark flavors must
experience this together and flip their chiralities $simultaneously$. Because
of the Pauli principle, only the zero modes with different flavors can undergo
simultaneous tunneling, the t’ Hooft effective Lagrangian is therefore a six-
quark operator, with $\bar{u}u\bar{d}d\bar{s}s$ quarks. As emphasized by t ’
Hooft, it is repulsive in the $\eta^{\prime}$ channel, and solves the famous
the famous $U_{A}(1)$ problem. But the fact that it is $attractive$ in the
scalar $\sigma$ and pseudoscalar $\pi,K,\eta$ octet channels leads to even
more important consequences, breaking the $SU(N_{f})$ chiral symmetry
spontaneously Shuryak (1982b). The account for these effects led to the
statistical studies of instanton ensembles in the 1990’s, for a review see
Schäfer and Shuryak (1998). Many more studies based on these observations,
were carried in the last two decades as well: let us just mention our recent
study of the so called instanton-sphaleron production process, which can be
studied experimentally in colliders Shuryak and Zahed (2021b).

For clarity, let us now discuss some technical issues already discussed in the
literature. The first is the explicit analytic form of the ’t Hooft operator,
which by Fierz transformations, can take several different forms. A
straightforward reduction of the product of six unitary color rotation
matrices, averaged over the $SU(3)$ group, leads to color structures
containing $f^{abc}\lambda^{a}\lambda^{b}\lambda^{c}$ and
$d^{abc}\lambda^{a}\lambda^{b}\lambda^{c}$ terms, which is difficult to use in
practice. However they (and in fact all color matrices) can be eliminated
using specific properties of zero modes producing the following result
Chernyshev  _et al._ (1996)

$\displaystyle{\cal
V}^{L+R}_{qqq}=\frac{G_{Hooft}}{N_{c}(N_{c}^{2}-1)}\bigg{[}\bigg{(}\frac{2N_{c}+1}{2(N_{c}+2)}\bigg{)}\begin{Vmatrix}\overline{u}_{R}u_{L}&\overline{u}_{R}d_{L}&\overline{u}_{R}s_{L}\\\
\overline{d}_{R}u_{L}&\overline{d}_{R}d_{L}&\overline{d}_{R}s_{L}\\\
\overline{s}_{R}u_{L}&\overline{s}_{R}d_{L}&\overline{s}_{R}s_{L}\\\
\end{Vmatrix}$
$\displaystyle-\frac{1}{2(N_{c}+1)}\sum_{a=1}^{3}\bigg{(}\begin{Vmatrix}\overline{u}_{R}\sigma^{a}u_{L}&\overline{u}_{R}\sigma^{a}d_{L}&\overline{u}_{R}s_{L}\\\
\overline{d}_{R}\sigma^{a}u_{L}&\overline{d}_{R}\sigma^{a}d_{L}&\overline{d}_{R}s_{L}\\\
\overline{s}_{R}u_{L}&\overline{s}_{R}d_{L}&\overline{s}_{R}s_{L}\\\
\end{Vmatrix}+\begin{Vmatrix}\overline{u}_{R}\sigma^{a}u_{L}&\overline{u}_{R}d_{L}&\overline{u}_{R}\sigma^{a}s_{L}\\\
\overline{d}_{R}u_{L}&\overline{d}_{R}d_{L}&\overline{d}_{R}s_{L}\\\
\overline{s}_{R}\sigma^{a}u_{L}&\overline{s}_{R}d_{L}&\overline{s}_{R}\sigma^{a}s_{L}\\\
\end{Vmatrix}+\begin{Vmatrix}\overline{u}_{R}u_{L}&\overline{u}_{R}d_{L}&\overline{u}_{R}s_{L}\\\
\overline{d}_{R}u_{L}&\overline{d}_{R}\sigma^{a}d_{L}&\overline{d}_{R}\sigma^{a}s_{L}\\\
\overline{s}_{R}u_{L}&\overline{s}_{R}\sigma^{a}d_{L}&\overline{s}_{R}\sigma^{a}s_{L}\\\
\end{Vmatrix}\bigg{)}\bigg{]}$ $\displaystyle+(L\leftrightarrow R)$ (64)

with the strength of the 6-q operator

$G_{\rm Hooft}={n_{I+\bar{I}}\over
2}\bigg{(}4\pi^{2}\rho^{3}\bigg{)}^{3}\bigg{(}{1\over
m^{*}_{u}\rho}\bigg{)}\bigg{(}{1\over m^{*}_{d}\rho}\bigg{)}\bigg{(}{1\over
m^{*}_{s}\rho}\bigg{)}$ (65)

with the effective quark masses $m^{*}_{q}$ to be explained a bit later.

In (IV.2) all quarks and antiquarks have opposite chiralities. In a completely
massless theory we cannot have “loop” quarks by a propagator conserving
chirality. However, the spontaneous breaking of chiral symmetry in the QCD
vacuum leads to nonzero quark condensates $\langle\bar{q}_{f}q_{f}\rangle\neq
0$. This fact leads to “dimensional reduction” of the six-quark operator to
four (and even two-) quark expressions, by “looping” some quark-antiquark
pairs into the corresponding condensates. For example, strange quark loops
generate an effective $ud$-interaction. It also has a structure of two-by-two
determinants, which (with some abuse of notations) we write as

$\displaystyle{\cal V}^{L+R}_{qq}={\kappa}_{2}\,A_{2N}\,\bigg{(}{\rm
det}(UD)+B_{2N}\,{\rm det}(U_{\mu\nu}D_{\mu\nu})\bigg{)}+(L\leftrightarrow R)$
(66)

with

$\displaystyle A_{2N}=\frac{(2N_{c}-1)}{2N_{c}(N_{c}^{2}-1)}\qquad
B_{2N}=\frac{1}{4(2N_{c}-1)}$

The coefficient

$\kappa_{2}=3!\,G_{\rm Hooft}\langle\overline{s}_{R}s_{L}\rangle=3\,G_{\rm
Hooft}\langle\overline{s}s\rangle<0\,\,,$

is negative and thus attractive. In the Weyl basis,
$\sigma_{\mu\nu}\rightarrow i\eta^{a}_{\mu\nu}\sigma^{a}$ with the ′t Hooft
symbol satisfying $\eta^{a}_{\mu\nu}\eta^{b}_{\mu\nu}=4\delta^{ab}$, and (66)
can be simplified further

$\displaystyle{\cal V}^{L+R}_{qq}=\,{\kappa}_{2}\,A_{2N}\,\bigg{(}{\rm
det}(UD)-4B_{2N}\,{\rm det}(U^{a}D^{a})\bigg{)}+(L\leftrightarrow R)$ (68)

Another possible “looping” of quarks is possible if we account for nonzero
quark masses, in particular the largest strange quark mass $m_{s}$. As a
result, the effective $ud$ interaction (containing the $m_{s}$ term) is
$stronger$ than $us,ds$ interactions, as they only contain negligibly small
light quark masses. As noticed by one of us and Rosner Shuryak and Rosner
(1989), the ratio of the coefficients for light-light and light-strange
4-fermion terms is

${G(\bar{u}u\bar{d}d)\over G(\bar{u}u\bar{s}s)}={m_{q}^{*}+m_{s}\over
m_{q}^{*}}\approx 1.5$ (69)

One consequence of this is the relatively large violation of $SU(3)_{f}$
symmetry. Another is that the ’t Hooft-operator-induced spin-dependent forces
have flavor ratios similar to the magnetic moment ratios (provided one ignores
the difference between the onshell “constituent quark mass” and the
determinantal masses $m^{*}$.) Therefore, one cannot trace their origin simply
by the mass dependence.

Now let us define $m^{*}$ in the Lagrangian coefficient, following Faccioli
and Shuryak (2001), see also Forkel and Nielsen (1995); Blotz and Shuryak
(1997); Braguta and Onishchenko (2004). If there is a single instanton in the
“empty” vacuum, as considered in the original ’t Hooft paper, those are just
quark masses $m_{q}$. The product of the quark masses is however also present
in the instanton density (in this setting), and they all cancel out, leaving a
finite Lagrangian even in the chiral limit $m_{q}\rightarrow 0$.

In an ensemble of instantons, the quark propagator in the QCD vacuum and
restricted to the zero modes, is approximated by the form

$S(x,y)=S_{Z}(x,y)+\sum_{I,J}\psi_{0I}^{*}(x)\bigg{(}{1\over
T}\bigg{)}_{I,J}\psi_{0J}(y)$ (70)

where $T_{IJ}$ denotes the so called “instanton hopping” matrix, constructed
out of the Dirac zero modes overlaps between neighboring instantons $I,J$.
Note that it contributes as an $inverse$ matrix, as propagators are inverse to
Dirac operators. So, when one discusses a process in which both points $x,y$
are inside the same instanton $I^{*}$, as per the definition of a hard block,
we can restrict the sum to only the term with the zero mode of this very
instanton. This leads to the following re-definition of the “determinantal
mass”

$\frac{1}{m_{u}}\equiv\bigg{<}\bigg{(}{1\over T}\bigg{)}_{I^{*}I^{*}}\bigg{>}$
(71)

Furthermore, in the diagrams containing $two$ quark propagators of different
flavors one has a different averaging

$\frac{1}{m^{2}_{uudd}}\equiv\bigg{<}\bigg{(}{1\over
T}\bigg{)}^{2}_{I^{*}I^{*}}\bigg{>}$ (72)

These two quantities were evaluated in the random and interacting instanton
liquid model, with

$\frac{1}{m_{u}^{2}}\approx{1\over(177\,{\rm
MeV})^{2}}\ll\frac{1}{m^{2}_{uudd}}\approx{1\over(103\,{\rm MeV})^{2}}$ (73)

The chief consequence of these substantial deviations from mean field can be
captured by a “’t Hooft operator enhancement factor”

$f_{\rm tHooft}\equiv\bigg{(}{m_{u}\over m_{uudd}}\bigg{)}^{2}\approx 3$ (74)

This enhancement of 4-fermion operator relative to 2-fermion squared has also
been observed on the lattice in Faccioli and DeGrand (2003).

###  IV.3 Spin and flavor-dependent interactions from zero-modes

Since the t′ Hooft induced interaction is non-local, there are additional
contributions besides the local terms retained earlier. More specifically, for
two flavors its generic form is

$\displaystyle{\cal
L}=\kappa_{2}A_{2N}\,\int\prod_{i=1}^{4}\frac{d^{4}k_{i}}{(2\pi)^{4}}\bigg{(}\frac{M(k_{i})}{M(0)}\bigg{)}^{\frac{1}{2}}\,(2\pi)^{4}\delta^{4}(k_{1}+k_{3}-k_{2}-k_{4})$
$\displaystyle\times\frac{1}{2}\epsilon^{f_{1}f_{2}}\epsilon^{g_{1}g_{2}}\bigg{(}\overline{\psi}_{Rf_{1}}(k_{1})\psi_{Lg_{1}}(k_{2})\overline{\psi}_{Rf_{2}}(k_{3})\psi_{Lg_{2}}(k_{4})-4B_{2N}\overline{\psi}_{Rf_{1}}(k_{1})\sigma^{a}\psi_{Lg_{1}}(k_{2})\overline{\psi}_{Rf_{2}}(k_{3})\sigma^{a}\psi_{Lg_{2}}(k_{4})\bigg{)}+(L\rightarrow
R)$

with form-factors, from a Fourier transform of the quark zero mode

$\frac{M(k)}{M(0)}=\bigg{(}2z\bigg{[}I_{0}(z)K_{1}(z)-I_{1}(z)K_{0}(z)-\frac{1}{z}I_{1}(z)K_{1}(z)\bigg{]}\bigg{)}^{2}_{z=k\rho/2}$
(76)

We note that the antisymmetric operator for flavor exchange can be rewritten
as

$\epsilon^{f_{1}f_{2}}\epsilon^{g_{1}g_{2}}=\frac{1}{2}\bigg{(}\delta^{f_{1}g_{1}}\delta^{f_{2}g_{2}}-\tau^{a,f_{1}g_{1}}\tau^{a,f_{2}g_{2}}\bigg{)}\rightarrow\frac{1}{2}(1-\tau_{1}\cdot\tau_{2})$
(77)

so that (IV.3) reads

$\displaystyle{\cal
L}=\frac{1}{2}\kappa_{2}A_{2N}\,\int\prod_{i=1}^{4}\frac{d^{4}k_{i}}{(2\pi)^{4}}\bigg{(}\frac{M(k_{i})}{M(0)}\bigg{)}^{\frac{1}{2}}\,(2\pi)^{4}\delta^{4}(k_{1}+k_{3}-k_{2}-k_{4})$
$\displaystyle\bigg{[}\bigg{(}\overline{\psi}(k_{1})\psi(k_{2})\overline{\psi}(k_{3})\psi(k_{4})+\overline{\psi}(k_{1})\gamma_{5}\psi(k_{2})\overline{\psi}(k_{3})\gamma_{5}\psi(k_{4})$
$\displaystyle-\overline{\psi}(k_{1})\tau^{a}\psi(k_{2})\overline{\psi}(k_{3})\tau^{a}\psi(k_{4})-\overline{\psi}(k_{1})\tau^{a}\gamma_{5}\psi(k_{2})\overline{\psi}(k_{3})\tau^{a}\gamma_{5}\psi(k_{4})\bigg{)}$
$\displaystyle-4B_{2N}\bigg{(}\overline{\psi}(k_{1})\sigma^{a}\psi(k_{2})\overline{\psi}(k_{3})\sigma^{a}\psi(k_{4})+\overline{\psi}(k_{1})\sigma^{a}\gamma_{5}\psi(k_{2})\overline{\psi}(k_{3})\sigma^{a}\gamma_{5}\psi(k_{4})$
$\displaystyle-\overline{\psi}(k_{1})\sigma^{a}\tau^{b}\psi(k_{2})\overline{\psi}(k_{3})\sigma^{a}\tau^{b}\psi(k_{4})-\overline{\psi}(k_{1})\sigma^{a}\tau^{b}\gamma_{5}\psi(k_{2})\overline{\psi}(k_{3})\sigma^{a}\tau^{b}\gamma_{5}\psi(k_{4})\bigg{)}\bigg{]}$
(78)

Figure 15: Kinematics in the center of mass frame of the two-flavor instanton
induced interaction with in-out on-shell Dirac fermions.

Going ahead of ourselves, we note that in the sequel on the light-front
Hamiltonian, we will use this operator with light cone kinematics,
$ultrarelativistic$ on-shell spinor, in spin or chiral basis. In this paper,
however, we are in the center of mass frame, in the non-relativistic setting
of a constituent quark model. Therefore, we perform the non-relativistic
reduction of (IV.3) with on-mass-shell normalized Dirac spinors

$U_{s}(p)=\begin{pmatrix}f(p)\chi_{s}\\\ g(p)\sigma\cdot
p\chi_{s}\end{pmatrix}\qquad V_{s}(p)=\begin{pmatrix}g(p)\sigma\cdot
p\chi^{C}_{s}\\\ f(p)\chi^{C}_{s}\end{pmatrix}$ (79)

with standard normalization factors

$f(p)=\bigg{(}\frac{E_{p}+m}{2E_{p}}\bigg{)}^{\frac{1}{2}}=(E_{p}+m)g(p)$ (80)

and $E_{p}=(p^{2}+m^{2})^{\frac{1}{2}}$. For the reduction to central, spin-
orbit and spin-spin interactions, we set the kinematics in the center of mass
defined in Fig. 15 with $k_{1}=(E_{p+q},\vec{p}+\vec{q})$,
$k_{2}=(E_{p},\vec{p})$,
$k_{3}=(E_{p^{\prime}-q},\vec{p}^{\,\prime}-\vec{q})$,
$k_{4}=(E_{p^{\prime}},\vec{p}^{\,\prime})$, insert (79) into (IV.3) and
expand up to order $p^{2}/m^{2}$. Recall that in a meson state $\bar{q}q$, the
labels in Fig. 15 refers also to the flavor and spin $1,2\rightarrow U$ with
mass $m_{Q}$ for the quark, and $3,4\rightarrow V$ with mass $m_{\bar{Q}}$ for
the anti-quark. More specifically after setting $m=m_{Q}=m_{\bar{Q}}$, the
order $q^{0}/m^{0}$ amounts to the two-body central and spin-spin operators

$-\frac{1}{4}|\kappa_{2}|A_{2N}\bigg{(}1-\tau_{1}\cdot\tau_{2}\bigg{)}\bigg{(}1-16B_{2N}\vec{S}_{1}\cdot\vec{S}_{2}\bigg{)}$
(81)

in the U(1) or $\eta^{\prime}$ channel. For clarity, the form factor is
temporarily set to 1 (zero size approximation). Taking into account the
overall negative sign in $\kappa_{2}$, and $16B_{2N}=4/5$, and the values of
$\vec{S}_{1}\cdot\vec{S}_{2}=-3/4$ for $S=0$ states, one finds that the last
bracket simplifies to $-8/5$ for the pseudoscalar (pion) channel. Note that
(IV.3) is only active in the $(\sigma,\pi,\sigma_{5},\pi_{5},\eta^{\prime})$
channels. (IV.3) does not contribute to the vector channels.

The relativisitc correction $\sim q^{2}/m^{2}$ amounts to the two-body
operators (with again the form factor set to 1)

$\displaystyle-$
$\displaystyle\frac{1}{4}|\kappa_{2}|A_{2N}\bigg{(}1-\tau_{1}\cdot\tau_{2}\bigg{)}\bigg{(}\frac{1}{4m^{2}}\bigg{)}\bigg{(}-L_{S}-4S_{T}$
(82) $\displaystyle+$ $\displaystyle 4B_{2N}\,(3L_{S}+4S_{T}+8q^{2}S_{1}\cdot
S_{2})\bigg{)}$

with spin-orbit and tensor operators

$\displaystyle L_{S}$ $\displaystyle=$
$\displaystyle(S_{1}+S_{2})\cdot(iq\times(p-p^{\prime}))$ (83) $\displaystyle
S_{T}$ $\displaystyle=$ $\displaystyle S_{1}\cdot qS_{2}\cdot
q,\,\,\,\,S_{12}=3(S_{1}\hat{q})(S_{2}\hat{q})-(S_{1}S_{2})$

with $L_{S}$ symmetrized using $\vec{p}+\vec{p}^{\,\prime}=0$. The translation
$iq\rightarrow\nabla$ to configuration space will be understood.

###  IV.4 Zero-mode-induced potential for pions

In section IV.1 we have already studied the pion case, and defined the value
of the ’t Hooft effective coupling (62) needed to put the pion at the
empirical mass. The first question we now discuss is whether the ILM
(component of the instanton ensemble) can provide a coupling of this
magnitude. The answer to this question is affirmative.

The “determinantal masses” induced by the quark condensates are given by the
following expressions for light and strange quarks

$m^{*}_{q}={2\pi^{2}\rho^{2}\over
3}|\langle\bar{q}q\rangle|,\,\,\,\,m^{*}_{s}={2\pi^{2}\rho^{2}\over
3}|\langle\bar{s}s\rangle|+m_{s}$ (84)

Using them, for a fixed instanton radius $\rho=0.33\,{\rm fm}\approx 1.4\,{\rm
GeV}^{-1}$, we get the following value of the coupling of $\bar{u}d\bar{d}u$
’t Hooft vertex

$G_{ud}=A_{2N}{3n\over
8}{(4\pi^{2})^{3}\rho^{9}\over(m^{*}_{q}\rho)^{2}(m^{*}_{s}\rho)}|\langle\bar{s}s\rangle|\approx
27\,{\rm GeV}^{-2}$ (85)

It is larger than the value of $17\,{\rm GeV}^{-2}$ we fitted from the pion
mass using the Schroedinger equation above, by about 50%. However, note that
it contains a large (sixth) power of the instanton size, which is not yet
defined with sufficiently high precision. The numbers would match if one
reduces it from $1/3\,fm$ to $0.29\,fm$.

Now, assuming that the ’t Hooft-induced potential puts the pion at the right
mass, we now turn our attention to the $\rho$ meson. In the previous section
we have found that there is an attractive ’t Hooft-induced potential which, at
the leading nonrelativistic order (81), is just a half of that for the pion.
Naively, one might think that this potential will pull the mass of the $\rho$
to half of what it does to the pion, putting it right at the experimentally
observed value. Unfortunately, it is not the case. The effects of the ’t Hooft
operator are too large to be treated as small perturbations. A factor two
difference in a potential, does not imply a factor two in the energy, because
the wave function itself depends strongly on it. What we found is that, due to
the ’t Hooft-induced potential, the pion’s size get reduced, making it
comparable to the instanton potential range $\sim\rho$. Half of it, acting
against the repulsive $V_{SS(r)}$ from the earlier contributions, fail to
shrink the $\rho$ size, which remains roughly twice the size of $\pi$. This
reduces the wave function at small distances roughly by a factor of $2^{3}$,
and effectively reduces the matrix element of the $V^{\rm Hooft}(r)$ in the
$\rho$ case by a comparable factor. The corresponding potentials and wave
functions are shown in Fig.16.

###  IV.5 The $\rho$ meson and the “molecule-induced” potential

This leaves us with the model prediction for the $\rho$ meson mass $m_{\rho}$
in the vicinity of the mass following from a “generic Cornell meson”, with
$m\approx 1.4\,\,GeV$, far from its experimental mass. The theory part is
missing something very important for the vector light quark channel.

Fortunately, there is still one more effect, already discussed in the
literature, the second-order ’t Hooft effective forces due to “molecular”
configurations, see the lower plot in the Introduction, Fig.1 . In particular,
the effective Lagrangian induced by these molecules was used in the theory of
color superconductivity at high density Rapp  _et al._ (2000), from which we
borrow its analytic form

$\displaystyle L_{\rm mol}/G_{\rm mol}=$ $\displaystyle{1\over
N_{c}^{2}}\big{[}(\bar{\psi}\gamma_{\mu}\psi)^{2}+(\bar{\psi}\gamma_{\mu}\gamma_{5}\psi)^{2}\big{]}-{1\over
2N_{c}(N_{c}-1)}\big{[}(\bar{\psi}\gamma_{\mu}\lambda^{a}\psi)^{2}+(\bar{\psi}\gamma_{\mu}\gamma_{5}\lambda^{a}\psi)^{2}\big{]}$
(86) $\displaystyle-{1\over
N_{c}^{2}}\big{[}(\bar{\psi}\gamma_{\mu}\psi)^{2}-(\bar{\psi}\gamma_{\mu}\gamma_{5}\psi)^{2}\big{]}-{2N_{c}-1\over
2N_{c}(N_{c}^{2}-1)}\big{[}(\bar{\psi}\gamma_{\mu}\lambda^{a}\psi)^{2}-(\bar{\psi}\gamma_{\mu}\gamma_{5}\lambda^{a}\psi)^{2}\big{]}$

It follows by using the same color orientation for the instanton and anti-
instanton, which locks them to their most attractive channel. The quark fields
come from zero modes of $I$ and $\bar{I}$, so there is no Pauli principle and
the flavors can be both the same or different. Note that in in this case the
chiralities of the quarks and antiquarks are the same in each term, which is
suitable for the $\rho$ meson channel. Choosing the vector
$\gamma_{\mu}\lambda^{a}$ terms and reducing the color matrices for the meson
case, we can reduce it to

$L_{\rm mol}=G_{\rm
mol}(\bar{u}\gamma_{\mu}u)_{x}(\bar{d}\gamma_{\mu}d)_{y}F_{mol}(x-y)$ (87)

where we introduced another form-factor to account for the nonlocality of the
instanton-antiinstanton molecule. In principle, one can evaluate (87) in a way
that is analogous to our treatment of the ’t Hooft Lagrangian. However, it
would now include $two$ 4-dimensional integrations over positions of both
centers, plus the relativiely complicated $I\bar{I}$ interaction depending on
the relative color orientations. Qualitatively, one expects its non-locality
size to approximately double, so we may just substitute the instanton size
$\rho\rightarrow 2\rho$ in the instanton form factor as an estimate. As a
result, its coefficient is reduced by $1/2^{3}$ in the formfactor (C). Such a
size reduction of the potential range is precisely what is needed to obtain
the correct $\rho$ meson mass.

More specifically, setting the corresponding potential into the Schroedinger
equation, along with the Cornell and spin-spin terms, we find that the correct
$\rho$ mass is indeed obtained for

${G_{\rm mol}\over G_{\rm Hooft}}={120\,{\rm GeV}^{2}\over 17\,{\rm
GeV}^{2}}\sim{n_{\rm mol}+n_{\rm ILM}\over n_{\rm ILM}}\approx 7$ (88)

Note that this ratio of couplings (or densities) agrees well with what was
deduced from the central potential fitting the linear potential discussed
above.

(As a parting comment, we remind the reader that the NJL model – the
pioneering effort to describe chiral symmetry breaking and nonperturbative
interactions in hadrons – involves flavor-diagonal terms like the one we used
here. In later papers based on the NJL model, those contributions were
attributed to the “strong coupling part” of one gluon exchange.)

Figure 16: The total potentials including the ’t Hooft ones (upper plot) and
wave functions as $r^{2}\psi(r)^{2}$ (lower plot) for the $\rho$ channel
(solid curves), the $\pi$ channel (blue dashed curves) and the generic Cornell
potential (black dots). All potentials are shown versus the distance $r$ in
${\rm GeV}^{-1}$.

###  IV.6 Zero mode effects in heavy-light systems

The spin interactions for heavy-light quarks split into the contributions
solely from the non-zero modes which give rise to the same general spin
dependent interactions as in (III.2).

For simplicity, we start with the heavy-light mesons, with a single light
quark. The contributions to the quark propagators come from the zero-mode for
the light quark, and the non-zero-modes for the heavy-quark. The resulting
spin-independent interaction was derived in Chernyshev  _et al._ (1996)

$\displaystyle{\cal L}_{qQ}=-\bigg{(}\frac{\Delta m_{Q}\Delta
m_{q}}{2nN_{c}}\bigg{)}\bigg{(}\overline{\bf Q}\frac{1+\gamma^{0}}{2}{\bf
Q}\,\overline{\bf q}{\bf
q}+\frac{1}{4}\overline{Q}\frac{1+\gamma^{0}}{2}\lambda^{a}{\bf
Q}\,\overline{\bf q}\lambda^{a}{\bf q}\bigg{)}$ (89)

while the spin-dependent interaction is more suppressed

$\displaystyle{\cal L}^{\rm spin}_{qQ}=-\bigg{(}\frac{\Delta m^{\rm
spin}_{Q}\Delta m_{q}}{2nN_{c}}\bigg{)}\times\frac{1}{4}\bigg{(}\overline{\bf
Q}\frac{1+\gamma^{0}}{2}\lambda^{a}\sigma^{\mu\nu}{\bf Q}\,\overline{\bf
q}\lambda^{a}\sigma^{\mu\nu}{\bf q}\bigg{)}$

with $\Delta m^{\rm spin}_{Q}/\Delta m_{q}\sim 1/100$ (charm). The local
potential stemming from (89-IV.6) is

$\displaystyle V_{qQ}(r)=\bigg{(}\frac{\Delta m_{Q}\Delta
m_{q}}{2nN_{c}}\bigg{)}\bigg{(}1+\frac{1}{4}\lambda_{q}^{a}\lambda_{Q}^{a}\bigg{)}\,\delta^{3}(r)$
(90)

In the mesons, the colors on the two lines are opposite and the second term
dominates the first, so this local potential is negative. For the $D$ meson it
amounts to $\langle V_{qQ}\rangle\approx-180\,{\rm MeV}$ Chernyshev  _et al._
(1996), to be compared with the non-relativistic energy due to the standard
Hamiltonian, kinetic plus central potential average

$\langle P^{2}/2m_{q}+V_{c}(r)\rangle\approx+250\,{\rm MeV}.$

The spin-spin potential is

$\displaystyle V^{\rm spin}_{qQ}(r)=-\bigg{(}\frac{\Delta m^{\rm
spin}_{Q}\Delta
m_{q}}{2nN_{C}}\bigg{)}\,{S}_{q}\cdot{S}_{Q}\,\lambda^{a}_{q}\lambda_{Q}^{a}\,\delta^{3}(r)$

For the D meson both the relative spin and color are negative, so the
contribution of this term is negative as well, giving $\langle V^{\rm
spin}_{qQ}\rangle\approx-42\,{\rm MeV}$ Chernyshev  _et al._ (1996). The
contribution of this term to the spin splitting is then

$M(D^{*})-M(D)=(4/3)42\,{\rm MeV}=56\,{\rm MeV}$

to be compared with our calculations above giving together $\langle V^{\rm
pert}_{SS}+V^{\rm inst}_{SS}\rangle\approx 96\,{\rm MeV}$. We thus conclude
that all the three contributions together – the perturbative one and the two
instanton-induced ones – do finally reproduce this mass difference, close to
the experimental value $M(D^{*})-M(D)=137\,{\rm MeV}$.

##  V Summary and Discussion

The purpose of this work was to have a fresh look at the origins of the
central and spin-dependent potentials, used in the non-relativistic
description of mesons. We focused in the range of “intermediate distances”
$r\sim 0.2-0.5\,{\rm fm}$, where the spin-dependent forces are mostly active.

The phenomenological part of this paper focused solely on mesons, from heavy
charmonia to heavy-light and to light-light ones. Using data for the spin
splittings in the 1S shell (2 states) and the 1P shell (4 states), we
evaluated matrix elements of the individual spin-dependent potentials. Among
the important observations is a change of sign of the tensor force for lighter
systems.

The theory part started with a reformulation of the instanton ensemble. Unlike
previous studies using the dilute ILM, and mostly focused on the spontaneous
breaking of chiral symmetry and zero modes, the ensemble we consider now
includes $I\bar{I}$ ”molecules”. It was shown that in this regime the
instanton-induced potential reproduces the magnitude of the central potential,
up to distances $r<1\,{\rm fm}$. Ascribing the central potential $V_{C}(r)$ at
$r<1\,{\rm fm}$ to this extra contribution, we get its normalization fixed. It
turns out to be about twice less dense than the lattice extrapolations of Fig.
4(b), with a diluteness parameter $\kappa\sim 1$.

This leads to evaluations of the nonperturbative part of spin-dependent
potentials. Their application to heavy-heavy systems can be considered
consistent with available lattice data for $V_{SS}(r)$, and also with
phenomenology. One important finding is that the perturbative and instanton-
induced contributions are comparable in magnitude. Together they semi-
quantitavily reproduce lattice-based potential and the spin splitting in heavy
quarkonia.

Going further from heavy to light systems, we analyzed four P-shell states,
attempting to extract spin-orbit and tensor potentials. The reader should
however be aware that for light quarks, we no longer are well below the open
decay channels, whose possible effects on observed masses have been ignored
for now. Experience with some mesons in the past suggests that these effects
are small. An argument that we mostly have it correct is that, as one can see
from Table 3, that for all three potentials the trend, from heavy to light
quarks, seems to be consistent.

Although our straightforward evaluation of perturbative and instanton-induced
potentials have not reproduced the apparently negative tensor force for light
quarks, the negative $sign$ of the instanton-induced term indicates that, with
some play of parameters, it perhaps can be done.

Furthermore, for light quarks there exists “anomalous” processes, in which
quarks do not travel continuously in time, but are instead dumped into the
Dirac sea and substituted by another quark, with different value of the
chirality. The non-relativistic reduction of 6-quark ’t Hooft operator have
produced novel spin-dependent forces. Their application to the pion case is
successful, as well as to the heavy-light systems. The effects related to the
zero modes and the ’t Hooft effective interactions can still be put in the
form of quasi-local flavor-nondiagonal potentials, i.e. central and spin-
dependent. The phenomenological implications of this is a reminder of the two
component instanton model discussed here, meaning that for the ’t Hooft
potential the instanton density is the smaller one, taken from the original
ILM. The magnitude of such ’t Hooft potential is found of the size needed to
put the pion mass at its small physical value. It is in agreement with the
long-known view of the pion as depicted in the upper Fig. 1.

The case of the $\rho$ meson is more complicated. Even with the ’t Hooft
potential its mass remains well above the observed value. However, there is
the so called “molecular” component, schematically shown in the lower part of
Fig. 1. It generates attractive forces diagonal in chirality. Assuming that
with the molecular component the density is 7 times larger than that of the
original ILM, we get an estimate of its coupling which we find to put the
$\rho$ meson in the correct mass range.

The focus of this work has not been on numerical accuracy, but rather on the
microscopic origin of the central and spin-dependent forces. It should be
viewed as an introduction to the use of these concepts for the light front
Hamiltonians and wave functions, which we will address next.

In sum, the paper proposes novel interpretations of interquark potentials,
that should and will be tested. One way to do this is to calculate all spin-
dependent potentials on the lattice during a “cooling” process (e.g. via
gradient flow), and assess how the results depend on the cooling time
(resolution). This will allow for a quantitative discrimination between the
perturbative (gluon-induced), and topological (instanton-induced) effects, at
low resolution.

##  VI Acknowledgements

This work was supported by the U.S. Department of Energy under Contract No.DE-
FG-88ER40388.

##  Appendix A Asymptotics of $V_{C}(r)$

The instanton induced central potental $V_{C}(r)$ in Fig. 5 asymptotes twice
the self-energy $2\Delta m_{Q}$. This asymptotics is reached from above in
singular gauge contrary to expectations. The simplest way to see this, is to
note that

$2\Delta M_{Q}\rho={8\pi n\rho^{4}\over
N_{c}}\int_{0}^{\infty}dyy^{2}\bigg{(}1+cos\bigg{(}{\pi
y\over\rho^{2}+y^{2}}\bigg{)}\bigg{)}\approx{8\pi n\rho^{4}\over N_{c}}2.2$
(92)

is substantially smaller than the potential in Fig. 5. A numerical evaluation
of $I(x)$ as in (18) at large $x$, is shown in Fig. 17. It confirms that the
asymptotics is slowly reached from above.

Figure 17: $I(x)$ versus $x$ as given in (5)

We note that a Taylor expansion of the integrand in $I(x)$ suggests $1/r^{3}$
as the leading contribution, but the remaining integration is divergent,
thereby invalidating the expansion. In Yakhshiev  _et al._ (2017b) it was
suggested that the asymptotics is reached from below through $-\pi^{2}/2r$.
This result is correct in a perturbative regular gauge with no additional
color rotations, but not in a non-perturbative singular gauge with additional
color rotations. Also, the $1/r$ central potential yields a spin-spin
potential $V_{SS}(r)$ as a delta-function which is also not compatible with
the $1/r^{2}$ behavior noted above. Fig. 17 can be fitted numerically by
$2\Delta M_{Q}+C/r^{p}$ with $p\ll 1$ and $C>0$. Its ensuing Laplacian then
gives $p(1-p)/r^{2+p}$, which is consistent with $V_{SS}(r)$ at large $r$.

##  Appendix B Definitions of the spin-dependent potentials

Following Ref. Eichten and Feinberg (1981), we give for completeness the
correlation functions generating the five spin-dependent potentials. The
integrals over points $z,z^{\prime}$ are taken along two Wilson lines located
at $\vec{x}_{1},\vec{x}_{2}$ and running in the Euclidean time direction, from
$-T/2$ to $T/2$. The limit of large $T\rightarrow\infty$ is not written
explicitly but assumed. The normalization to $\langle 1\rangle$ means
subtraction of the central potential coming from the correlator of Wilson
lines without the extra field strengths

$\displaystyle{r^{k}\over r}{dV_{1}(r)\over dr}=$
$\displaystyle\epsilon_{ijk}\int dzdz^{\prime}({z-z^{\prime}\over
T}){g^{2}\over 2}\langle
B^{i}(\vec{x}_{1},z)E^{j}(\vec{x}_{1},z^{\prime})\rangle/\langle 1\rangle$
$\displaystyle{r^{k}\over r}{dV_{2}(r)\over dr}=$
$\displaystyle\epsilon_{ijk}\int dzdz^{\prime}({z-z^{\prime}\over
T}){g^{2}\over 2}\langle
B^{i}(\vec{x}_{2},z)E^{j}(\vec{x}_{1},z^{\prime})\rangle/\langle 1\rangle$
(93)
$\displaystyle\times\big{[}(\hat{r}^{i}\hat{r}^{j}-\delta_{ij}/3)V_{3}(r)+\delta_{ij}/3)V_{4}(r)\big{]}=\int
dzdz^{\prime}{g^{2}\over T}\langle
B^{i}(\vec{x}_{2},z)E^{j}(\vec{x}_{1},z^{\prime})\rangle/\langle 1\rangle$

Although it involves only the vector product of electric and magnetic fields,
using the Bianchi identity one can express it via a correlator of two $B$
fields, as expected for non-relativistic spin interactions. It is however of
little importance for instantons, which are (anti) self-dual
$\vec{E}=\pm\vec{B}$.

Note that the reduction of the dressed Wilson line “WEW” line with an electric
field insertion can be made into a derivative of the Wilson line over its
location, via the following steps

$\displaystyle D_{m}(y){\bf W}(x_{0},y_{0})-{\bf W}(x_{0},y_{0})D_{m}(y)$
$\displaystyle=\langle
x_{0}|D_{0}\frac{1}{iD_{0}}-\frac{1}{iD_{0}}D_{m}|y_{0}\rangle$
$\displaystyle=\langle
x_{0}|\frac{1}{iD_{0}}[iD_{0},D_{m}]\frac{1}{iD_{0}}|y_{0}\rangle$
$\displaystyle=\langle
x_{0}|\frac{1}{iD_{0}}(-F_{0m})\frac{1}{iD_{0}}|y_{0}\rangle=\int_{-\frac{1}{2}T}^{+\frac{1}{2}T}dz_{0}\,{\bf
W}(-T,z_{0})(-F_{0m})(z_{0}){\bf W}(z_{0},T)$ (94)

and in the large $|T|\rightarrow\infty$ asymptotic the feld is assumed to
vanish, so the covariant derivative can be changed to ordinary one.

Subsequent discussions of the ${\cal O}(1/m^{2})$ of the potentials have been
made, starting from complete effective actions of NRQCD and pNRQCD to this
order, and some modifications of the expressions in Eichten and Feinberg
(1981) were found (See Eqs. 49-51 in Pineda and Vairo (2001)). Modulo
matching, the only difference was noted in the spin-orbit coefficient with no
$\frac{1}{2}$ in (B). However, for the purposes of our work, it is enough to
note that the spin-spin and spin-orbit potentials are related to the
correlators of two magnetic fields, and the spin-orbit to the correlator of
electric and magnetic fields. (Of course, the correlators include the
appropriate Wilson lines, and are integrated over the time difference between
the field strengths.)

Figure 18: Numerically integrated and normalized integral (C) (points),
compared to a fit (solid line). Also for comparison we show (the dashed line),
the normalized “smoothed delta function” as used in (48) for $\delta=\rho$. It
was our initial choice, but as shown here, it does not provide an accurate
description of the form-factor and therefore is not used.

##  Appendix C Form-factor in the ’t Hooft Lagrangian

In many applications, the instanton induced form-factor on the tunneling
quarks, is used in the momentum representation. More specifically, as the
Fourier transform of the zero modes in (76). However, in the effective
potential we use, there are four zero modes, thus the fourth power of $M(k)$.
Its inverse Fourier transform to coordinate space is complicated. We use
instead directly the coordinate expression, with two $densities$ of zero modes
separated by a distance $r$ , with the explicit integration over the location
of the instanton center $z_{\mu}$

$\displaystyle F(r)\sim$ $\displaystyle\int
d^{4}z|\psi_{0}(z^{\mu}-r^{\mu}/2)|^{2}|\psi_{0}(z^{\mu}+r^{\mu}/2)|^{2}$
$\displaystyle\sim$
$\displaystyle\int{z^{3}dzsin^{2}(\theta)d\theta\over(z^{2}+r^{2}/4+zrcos(\theta)+\rho^{2})^{3}(z^{2}+r^{2}/4-zrcos(\theta)+\rho^{2})^{3}}\approx
0.18\,{\rm exp}\big{(}-0.89\,r^{2}\big{)}$ (95)

The last expression is a fit, normalized by $\int d^{3}F(r)=1$, with our
standard size $\rho=1.4\,{\rm GeV}^{-1}$. The comparison between the numerical
integral and the fit is shown in Fig. 18.

##  Appendix D Field strengths through conformal mapping

The field strengths $F_{MN}$ and its dual $\star{F}_{MN}$ for the tunneling at
fixed energy in O(4) symmetric form (Step 1) are given by

$\displaystyle y^{2}\vec{E}=$
$\displaystyle\vec{\sigma}\left(f^{\prime}-\frac{(\vec{y})^{2}}{y^{2}}(f^{\prime}-2f\bar{f})\right)$
$\displaystyle+\frac{(\vec{y}\cdot\vec{\sigma}\vec{y}-y_{4}\vec{\sigma}\times\vec{y})}{y^{2}}(f^{\prime}-2f\bar{f})$
$\displaystyle y^{2}\vec{B}=$
$\displaystyle\vec{\sigma}\left(2f\bar{f}+\frac{(\vec{y})^{2}}{y^{2}}(f^{\prime}-2f\bar{f})\right)$
(96)
$\displaystyle-\frac{(\vec{y}\cdot\vec{\sigma}\vec{y}-y_{4}\vec{\sigma}\times\vec{y})}{y^{2}}(f^{\prime}-2f\bar{f})$

with $E^{i}=F^{i4}$ and $B^{i}=\frac{1}{2}{\epsilon^{ijk}}F^{jk}$. For self-
dual fields $f^{\prime}=2f\bar{f}$ and $\vec{E}=\vec{B}$ are hedgehogs in
color-spin as expected for the instanton path. For the sphaleron path, we have
$f=\frac{1}{2}$ and $(\vec{E}+\vec{B})=\vec{\sigma}/2y^{2}$, as only the
electric plus magnetic field sum is hedgehog in color-spin. We then use the
conformal (stereographic) map (Step 2), as explained in the text.

## References

  * Pineda and Vairo (2001) A. Pineda and A. Vairo, Phys. Rev. D 63, 054007 (2001), [Erratum: Phys.Rev.D 64, 039902 (2001)], arXiv:hep-ph/0009145 .
  * Brambilla  _et al._ (2018) N. Brambilla, G. a. Krein, J. Tarrús Castellà, and A. Vairo, Phys. Rev. D 97, 016016 (2018), arXiv:1707.09647 [hep-ph] .
  * Arvis (1983) J. F. Arvis, Phys. Lett. B 127, 106 (1983).
  * Aharony and Klinghoffer (2010) O. Aharony and N. Klinghoffer, JHEP 12, 058 (2010), arXiv:1008.2648 [hep-th] .
  * Athenodorou and Teper (2021) A. Athenodorou and M. Teper, (2021), arXiv:2106.00364 [hep-lat] .
  * Callan  _et al._ (1978) C. G. Callan, Jr., R. F. Dashen, D. J. Gross, F. Wilczek, and A. Zee, Phys. Rev. D 18, 4684 (1978).
  * Eichten and Feinberg (1981) E. Eichten and F. Feinberg, Phys. Rev. D 23, 2724 (1981).
  * Nambu and Jona-Lasinio (1961) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).
  * Shifman  _et al._ (1979) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979).
  * Novikov  _et al._ (1981) V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 191, 301 (1981).
  * Shuryak (1993) E. V. Shuryak, Rev. Mod. Phys. 65, 1 (1993).
  * Belavin  _et al._ (1975) A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, Phys. Lett. B 59, 85 (1975).
  * Shuryak (1982a) E. V. Shuryak, Nucl. Phys. B 198, 83 (1982a).
  * Leinweber (1999) D. B. Leinweber, in  _Workshop on Light-Cone QCD and Nonperturbative Hadron Physics_ (1999) arXiv:hep-lat/0004025 .
  * Biddle  _et al._ (2020a) J. C. Biddle, W. Kamleh, and D. B. Leinweber, Phys. Rev. D 102, 034504 (2020a), arXiv:1912.09531 [hep-lat] .
  * Biddle  _et al._ (2020b) J. C. Biddle, W. Kamleh, and D. B. Leinweber, EPJ Web Conf. 245, 06010 (2020b), arXiv:2009.12047 [hep-lat] .
  * Zahed (2021) I. Zahed, Phys. Rev. D 104, 054031 (2021), arXiv:2102.08191 [hep-ph] .
  * ’t Hooft (1976) G. ’t Hooft, Phys. Rev. D 14, 3432 (1976), [Erratum: Phys.Rev.D 18, 2199 (1978)].
  * Shuryak (1982b) E. V. Shuryak, Nucl. Phys. B 203, 140 (1982b).
  * Schäfer and Shuryak (1998) T. Schäfer and E. V. Shuryak, Rev. Mod. Phys. 70, 323 (1998), arXiv:hep-ph/9610451 .
  * Glozman  _et al._ (2012) L. Y. Glozman, C. B. Lang, and M. Schrock, Phys. Rev. D 86, 014507 (2012), arXiv:1205.4887 [hep-lat] .
  * Chu  _et al._ (1994) M. C. Chu, J. M. Grandy, S. Huang, and J. W. Negele, Phys. Rev. D 49, 6039 (1994), arXiv:hep-lat/9312071 .
  * Faccioli and DeGrand (2003) P. Faccioli and T. A. DeGrand, Phys. Rev. Lett. 91, 182001 (2003), arXiv:hep-ph/0304219 .
  * Ostrovsky  _et al._ (2002) D. M. Ostrovsky, G. W. Carter, and E. V. Shuryak, Phys. Rev. D 66, 036004 (2002), arXiv:hep-ph/0204224 .
  * Klinkhamer and Manton (1984) F. R. Klinkhamer and N. S. Manton, Phys. Rev. D 30, 2212 (1984).
  * Ilgenfritz and Shuryak (1989) E.-M. Ilgenfritz and E. V. Shuryak, Nucl. Phys. B 319, 511 (1989).
  * Rapp  _et al._ (1998) R. Rapp, T. Schäfer, E. V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998), arXiv:hep-ph/9711396 .
  * Shuryak and Zahed (2021a) E. Shuryak and I. Zahed, Phys. Rev. D 103, 054028 (2021a), arXiv:2008.06169 [hep-ph] .
  * Liu and Zahed (2021) Y. Liu and I. Zahed, (2021), arXiv:2102.07248 [hep-ph] .
  * Shuryak (1988) E. V. Shuryak, Nucl. Phys. B 302, 621 (1988).
  * Balitsky and Yung (1986) I. I. Balitsky and A. V. Yung, Phys. Lett. B 168, 113 (1986).
  * Yung (1988) A. V. Yung, Nucl. Phys. B 297, 47 (1988).
  * Verbaarschot (1991) J. J. M. Verbaarschot, Nucl. Phys. B 362, 33 (1991), [Erratum: Nucl.Phys.B 386, 236–236 (1992)].
  * Khoze and Ringwald (1991) V. V. Khoze and A. Ringwald, (1991).
  * Shuryak and Verbaarschot (1992) E. V. Shuryak and J. J. M. Verbaarschot, Phys. Rev. Lett. 68, 2576 (1992).
  * Shuryak and Zahed (2021b) E. Shuryak and I. Zahed, (2021b), arXiv:2102.00256 [hep-ph] .
  * Luscher and Weisz (2011) M. Luscher and P. Weisz, JHEP 02, 051 (2011), arXiv:1101.0963 [hep-th] .
  * Athenodorou  _et al._ (2018) A. Athenodorou, P. Boucaud, F. De Soto, J. Rodríguez-Quintero, and S. Zafeiropoulos, JHEP 02, 140 (2018), arXiv:1801.10155 [hep-lat] .
  * Boucaud  _et al._ (2003) P. Boucaud, F. De Soto, A. Le Yaouanc, J. P. Leroy, J. Micheli, H. Moutarde, O. Pene, and J. Rodriguez-Quintero, JHEP 04, 005 (2003), arXiv:hep-ph/0212192 .
  * Shuryak and Zahed (2000) E. V. Shuryak and I. Zahed, Phys. Rev. D 62, 085014 (2000), arXiv:hep-ph/0005152 .
  * Giordano and Meggiolaro (2008) M. Giordano and E. Meggiolaro, Phys. Rev. D 78, 074510 (2008), arXiv:0808.1022 [hep-lat] .
  * Giordano and Meggiolaro (2010) M. Giordano and E. Meggiolaro, Phys. Rev. D 81, 074022 (2010), arXiv:0910.4505 [hep-ph] .
  * Yakhshiev  _et al._ (2017a) U. T. Yakhshiev, H.-C. Kim, M. M. Musakhanov, E. Hiyama, and B. Turimov, Chin. Phys. C 41, 083102 (2017a), arXiv:1602.06074 [hep-ph] .
  * Shuryak (1982c) E. V. Shuryak, Nucl. Phys. B 203, 93 (1982c).
  * Liu  _et al._ (2019a) Y. Liu, M. A. Nowak, and I. Zahed, Phys. Rev. D 100, 126023 (2019a), arXiv:1904.05189 [hep-ph] .
  * Liu  _et al._ (2019b) Y. Liu, M. A. Nowak, and I. Zahed, (2019b), arXiv:1909.02497 [hep-ph] .
  * Luscher (1977) M. Luscher, Phys. Lett. B 70, 321 (1977).
  * Schechter (1977) B. M. Schechter, Phys. Rev. D 16, 3015 (1977).
  * Rubakov  _et al._ (1992) V. A. Rubakov, D. T. Son, and P. G. Tinyakov, Phys. Lett. B 287, 342 (1992).
  * Gromes (1984) D. Gromes, Z. Phys. C 26, 401 (1984).
  * Buchmuller (1982) W. Buchmuller, Phys. Lett. B 112, 479 (1982).
  * Pisarski and Stack (1987) R. D. Pisarski and J. D. Stack, Nucl. Phys. B 286, 657 (1987).
  * Koma and Koma (2007) Y. Koma and M. Koma, Nucl. Phys. B 769, 79 (2007), arXiv:hep-lat/0609078 .
  * Kawanai and Sasaki (2015) T. Kawanai and S. Sasaki, Phys. Rev. D 92, 094503 (2015), arXiv:1508.02178 [hep-lat] .
  * Koma  _et al._ (2006) M. Koma, Y. Koma, and H. Wittig, PoS LAT2005, 216 (2006), arXiv:hep-lat/0510059 .
  * Barnes  _et al._ (2005) T. Barnes, S. Godfrey, and E. S. Swanson, Phys. Rev. D 72, 054026 (2005), arXiv:hep-ph/0505002 .
  * Koponen  _et al._ (2017) J. Koponen, A. C. Zimermmane-Santos, C. T. H. Davies, G. P. Lepage, and A. T. Lytle, Phys. Rev. D 96, 054501 (2017), arXiv:1701.04250 [hep-lat] .
  * Chernyshev  _et al._ (1996) S. Chernyshev, M. A. Nowak, and I. Zahed, Phys. Rev. D 53, 5176 (1996), arXiv:hep-ph/9510326 .
  * Shuryak and Rosner (1989) E. V. Shuryak and J. L. Rosner, Phys. Lett. B 218, 72 (1989).
  * Faccioli and Shuryak (2001) P. Faccioli and E. V. Shuryak, Phys. Rev. D 64, 114020 (2001), arXiv:hep-ph/0106019 .
  * Forkel and Nielsen (1995) H. Forkel and M. Nielsen, Phys. Lett. B 345, 55 (1995), arXiv:hep-ph/9408396 .
  * Blotz and Shuryak (1997) A. Blotz and E. V. Shuryak, Phys. Rev. D 55, 4055 (1997), arXiv:hep-ph/9606355 .
  * Braguta and Onishchenko (2004) V. V. Braguta and A. I. Onishchenko, Phys. Lett. B 591, 255 (2004), arXiv:hep-ph/0311146 .
  * Rapp  _et al._ (2000) R. Rapp, T. Schäfer, E. V. Shuryak, and M. Velkovsky, Annals Phys. 280, 35 (2000), arXiv:hep-ph/9904353 .
  * Yakhshiev  _et al._ (2017b) U. T. Yakhshiev, H.-C. Kim, M. M. Musakhanov, E. Hiyama, and B. Turimov, Chin. Phys. C 41, 083102 (2017b), arXiv:1602.06074 [hep-ph] .