††thanks: Presented at 2021 Zakopane school

# Instanton effects in Euclidean vacuum,
real time production and in the light front wave functions

Edward Shuryak  Ismail Zahed  Center for Nuclear Theory, Stony Brook
University
Stony Brook NY 11794 USA

###### Abstract

Nontrivial topological structures of non-Abelian gauge fields were discovered
in the 1970’s. Instanton solutions, describing vacuum tunneling through
topological barriers, have fermionic zero modes which are at the origin of ′t
Hooft effective Lagrangian. In the 1980’s instanton ensembles have been used
to explain chiral symmetry breaking. In the 1990’s a large set of numerical
simulations were performed deriving Euclidean correlation functions. The
special role of scalar diquarks in nucleons, and color superconductivity in
dense quark matter have been elucidated.

In these lectures, we discuss further developments of physics related to gauge
topology. We show that the instanton-antiinstanton “streamline” configurations
describe “sphaleron transitions” in high energy collisions, which result in
production of hadronic clusters with nontrivial topological/chiral charges.
(They are not yet observed, but discussions of dedicated experiments at LHC
and RHIC are ongoing.)

Another new direction is instanton effects in hadronic spectroscopy, both in
the rest frame and on the light front. We will discuss their role in central
and spin-dependent potentials, formfactors and antiquark nuclear “sea”.
Finally, we summarize the advances in the semiclassical theory of
deconfinement, and chiral phase transitions at finite temperature, in QCD and
in some of its “deformed” versions.

PACS numbers come here

##  1 Brief overview of gauge topology

We start these lectures with a map, outlining the role and interdependence of
various topological structures in the QCD vacuum. The phenomenon of color
confinement (left blue region in Fig.1) was first related with center
vortices, associated with a phase $\pi$ for a quark looping around, changing
the sign of the linking Wilson line. Two such vortices combined together, lead
to a singularity with a phase $2\pi$, shown by white an arrow, known as the
Dirac string. Their ends are identified as magnetic monopoles (blue disks).
Confinement is their Bose-Einstein condensation, perhaps the most physical
signature of this phenomenon.

The right yellow region of Fig.1 is associated with another major
nonperturbative phenomenon, chiral symmetry breaking. The pink disk refers to
instantons, 4D solitons in Euclidean space-time. Fermions in this field have
zero modes, elevating each instanton into ′t Hooft multi-fermion operator. The
four black arrows correspond to the case of two quark flavors, $u,d$. The
resulting 4-fermion vertex is similar to Nambu-Jona-Lasinio hypothetical
interaction, and also breaks spontanously $SU(N_{f})_{a}$ chiral symmetry
provided the instanton density is sufficiently large.

Figure 1: Map of gauge topology, explanations are in the text.

At finite temperature the Polyakov line has a number of eigenvalues or
$holonomies$ $\mu_{i}(T)$ (see below). The instanton solution amended by such
asymptotics for the $A_{0}$ field, splits into instanton constituents, known
as instanton-dyons or instanton-monopoles111Both names were criticized: they
are neither the dyons of Schwinger, nor monopoles of Dirac, but Euclidean
self-dual objects, with equal $E$ and $B$ up to a sign. Perhaps a new name,
emphasizing their Euclidean nature, is needed: can it e.g. be $instantino$?  .
Like instantons, they have topological charges. Unlike instantons, the charges
are not quantized to integers $Q$. This is possible since they are connected
by Dirac strings due to their magnetic charges. Ensembles of these topological
objects will be shown to generate both confinement, and chiral breaking at the
end of these lectures.

Last but not least is the phenomenon of Poisson duality, started as a
technical observation relating partition functions derived in terms of
monopoles and in terms of instanton-dyons. As we will discuss below, it has
wider meaning: their equality brings to mind equivalence of two other great
approaches to classical dynamics, by Hamilton and Lagrange, respectively.

The detailed discussion of these configurations, in addition to discussions
about the QCD flux tubes and holographic QCD etc, can be found in [1].

##  2 Brief history of instantons

###  2.1 Semiclassical theory based on path integrals: instantons and fluctons

The quantum mechanics courses include semiclassical methods based on certain
representation of the wave function, starting with the celebrated Bohr-
Sommerfeld quantization condition, applied to the oscillator and hydrogen
atom, and followed by Wentzel-Kramers-Brillouin (WKB) approximation, developed
in 1926. Unfortunately, subsequent study has shown that generalization of
those to physical systems with more than one degree of freedom, as well as to
systematic order-by-order account for quantum fluctuations, are not possible.

By definition the Feynman path integral gives the density matrix in coordinate
representation (see e.g. a very pedagogical Feynman-Hibbs book)

$\rho(x_{i},x_{f},t_{tot})\ =\
\int_{x(0)=x_{i}}^{x(t_{tot})=x_{f}}Dx(t)e^{i\,S[x(t)]/\hbar}\ .$ (1)

Note that it is a function of the initial and final coordinates, as well as
the time needed for the transition. Here $S$ is the classical action of the
system, e.g. for a particle of mass $m$ in a static potential $V(x)$ it is

$S\ =\
\int_{0}^{t_{tot}}dt\,\bigg{[}\frac{m}{2}{\bigg{(}\frac{dx}{dt}\bigg{)}}^{2}-V(x)\bigg{]}\
,$

Feynman has shown that the oscillating exponent along the path, provides the
correct weight of the paths integral (1). For reference, the same object can
also be written in a form closer to that used in quantum mechanics courses.
Heisenberg wrote it as a matrix element of the time evolution operator, the
exponential of the Hamiltonian 222Here, we assume that the motion happens in a
time-independent potential, for otherwise the exponential is time ordered.

$\rho(x_{i},x_{f},t_{tot})\ =\langle x_{f}|e^{i\hat{H}t_{tot}}|x_{i}\rangle$
(2)

between states in which a particle is localized in-out.

Schroedinger set of stationary states $\hat{H}|n\rangle=E_{n}|n\rangle$ can
also be used as a basis set. Because the Hamiltonian is diagonal in this
basis, there is a single (not double) sum

$\rho(x_{i},x_{f},t)\ =\sum_{n}\psi_{n}^{*}(x_{f})\psi_{n}(x_{i})e^{iE_{n}t}$
(3)

with $\psi_{n}(x)=\langle n|x\rangle$. Oscillating weights for different
states are often hard to calculate, and one may wonder if it is possible to
analytically continue in time, to the Euclidean version with $i$ absorbed into
it. For reasons which will soon become clear, we will also define this
imaginary time on a circle with circumference $\beta$

$\tau=i\,t\in[0,\beta]$

In this way we will be able to describe $quantum+statistical$ mechanics of a
particle in a heat bath with temperature $T$ related to the circle
circumference

$\beta={\hbar\over T}$

Such periodic time is known as the Matsubara time. The expression (3) is now

$\rho(x,x,t)\ =\sum_{n}|\psi_{n}(x)|^{2}e^{-E_{n}/T}$ (4)

which sums the quantum-mechanical probabilities to find a particle at point
$x$, times its thermal weight. Performing the integral over all $x$, and using
the normalization of the weight functions, one finds the expression for the
thermal partition function

$Z=\sum_{n}e^{-E_{n}/T}$ (5)

We will use this expression below in a Feynman path integral form, with (i)
taken over all $periodic$ paths with the same endpoints, and with (ii)
Euclidean or rotated time. The probability to find a particle at a certain
point is then

$P(x_{0},t_{tot})=\int_{x(0)=x_{0}}^{x(\beta)=x_{0}}Dx(\tau)e^{-S_{E}[x(\tau)]/\hbar}\
.$ (6)

Note that here, the exponent is not oscillating and equals the Euclidean
action

$S_{E}\ =\ \int_{0}^{\beta}d\tau\,[\frac{m}{2}{(\frac{dx}{d\tau})}^{2}+V(x)]$
(7)

in which the sign of the potential is reversed, and the time derivative is
understood to be over $\tau$.

Looking for the periodic paths with the minimal action, one may start with the
simplest, i.e. a particle at rest with $x(\tau)=const$! Such path is dominant
for small time (Matsubara circle), with $\beta\rightarrow 0$ (or high
temperature $T$)333 Note that it is opposite to the limit discussed above in
the Hamiltonian approach. If one ignores the time dependence and velocity on
the paths, there is no kinetic term and only the potential one in the action
contributes. So

$P(x_{0},\beta)\sim e^{-\frac{V(x_{0})}{T}}\ ,$ (8)

which corresponds to classical444Note that if we would keep $\hbar\neq 1$, the
one in $\beta$ and in the exponent $exp(-S/\hbar)$ would cancel out,
confirming the classical nature of this limit.  thermal distribution for a
particle in a potential $V$.

Since the weight is $exp\big{(}-S[x(\tau)]\big{)}$, the paths with the
smallest action should give the largest contribution. These paths satisfy the
classical (Euclidean) equation of motion, as we will carry below. The
semiclassical approximation – the $dominance$ of these paths – would be
justified, as soon as the corresponding action is large

$S_{cl}\equiv S[x_{cl}(\tau)]\gg\hbar$ (9)

Such classical paths were called fluctons in [2]. These paths should have
Euclidean time period $\beta=\hbar/T$. For simplicity, let us start with
“cold” QM, or vanishingly small $T$, $\beta\rightarrow\infty$. Little thinking
of how to arrange a classical path with a very long period leads to the
following solution: the particle should roll to the top of the (flipped)
potential with exactly such energy as to sit there for very long time, before
it will roll back to the (arbitrary) point $x_{0}$ from which the path
started. The classical paths corresponding to relaxation toward the potential
bottom take the form of a path “climbing up” from an arbitrary point $x_{0}$
to the maximum, see Fig. 2

Figure 2: Sketch of the flucton path climbing toward the (flipped) minimum of
the potential.

Let us start with the harmonic oscillator, as the unavoidable first example.
For simplicity, let us use the canonical units, in which the particle mass is
$m=1$, and the oscillator frequency is $\Omega=1$, so that our (Euclidean)
Lagrangian 555Note again the flipped sign of the potential term. In Minkowski
time the potential has sign minus. is simplified to

$L_{E}=\frac{\dot{x}(\tau)^{2}}{2}+\frac{x(\tau)^{2}}{2}\ .$ (10)

Because of this sign, in Euclidean time $\tau$ the oscillator does not
oscillate $e^{it}$ but relaxes $e^{-\tau}$. For a harmonic oscillator, the
classical equations of motion (EOM) are of course not difficult to solve.
However, it is always easier to get solutions using energy conservation (first
integral of motion for static potentials). Since we are interested in
solutions with zero energy $E=0$, they correspond to $\dot{x}^{2}=2V(x)$. The
boundary conditions are $x_{0}$, at $\tau=0$ plus periodicity on a circle with
circumference $\beta$. This solution is

$x_{flucton}(\tau)\ =\ x_{0}\ \frac{\left(e^{\beta-\tau}\ +\
e^{\tau}\right)}{e^{\beta}+1}\ .$ (11)

defined for $\tau\in[0,\beta]$. The particle moves toward $x=0$ and reaches
some minimal value, at $\tau=\beta/2$, and then returns to the initial point
$x_{0}$ again at $\tau=\beta$. Due to the periodicity in $\tau$, one may shift
its range to $\tau\in[-\beta/2,\beta/2]$. The minimal value at $\tau=\beta/2$

$x_{min}={x_{0}\over cosh(\beta/2)}\rightarrow_{\beta\rightarrow\infty}0$

is exponentially small at low temperature. The climbing to the potential top
at $x=0$ is nearly accomplished, if the period is large.

The solution in the zero temperature or $\beta\rightarrow\infty$ limit
simplifies to $x_{0}e^{-|\tau|}$. In the opposite limit of small $\beta$ or
high $T$, there is no time to move far from $x_{0}$, so in this case the
particle does not move at all. The classical action of the flucton path is

$S_{flucton}\ =\ x_{0}^{2}\ \tanh\bigg{(}\frac{\beta}{2}\bigg{)}\ ,$ (12)

The result means that the particle distribution

$P(x_{0})\sim\exp\left(-\frac{x_{0}^{2}}{\coth(\frac{\beta}{2})}\right)\ ,$
(13)

is Gaussian at any temperature. Note that the width of the distribution

$<x^{2}>\ =\ \frac{1}{2}\coth\bigg{(}\frac{\beta}{2}\bigg{)}\ =\
\frac{1}{2}+\frac{1}{e^{\beta}-1}\ ,$ (14)

is the ground state energy plus the contribution due to the thermal
excitations, which we already mentioned at the beginning of the chapter. So,
the flucton reproduces the well known results for the harmonic oscillator, see
e.g. Feynman’s Statistical Mechanics.

Flucton solutions for several well used QM problems (anharmonic oscillator,
the quartic double well, sin-like potential) were discussed in detail in [12].
Unlike WKB, one can define and calculate semiclassical series in $\hbar$ by
well-defined procedure, using Feynman diagrams. Finite temperature fluctons
were used for anharmonic oscillator and 4-nucleon clustering in [11].

Quantum mechanical instantons were introduced666It is interesting that the
first application to the QFT problem – pairs production in constant electric
field – were done already in 1931, by Sauter [3] .  by [4] in the context of
tunneling in the double-well potential. In the inverted potential it is a path
going from one maximum to another.

For an early pedagogical review, including the one-loop (determinant)
calculation by the same tedious scattering-phase method see ABC’s of
instantons [5]. Two-loop corrections were first calculated in [6], some
technical errors in it were corrected by Shuryak and Wohler in [7]. Three loop
corrections have been calculated by Escobar-Ruiz, Turbiner and Shuryak in [8].

The calculation goes in the same standard way, the path is written as a
classical plus quantum fluctuation $\delta x(\tau)$, and then the action is
expanded in powers of $\delta x(\tau)$. One technical (but important)
difference between the $instantons$ and $fluctons$, is that in the former case
the fluctuation operator ($O(\delta x(\tau)^{2})$) has a zero mode, related to
time-shift symmetry. Therefore the Green function for the fluctuations, needs
to be defined in a nonzero mode subspace of Hilbert space. This complication
leads to new features, such as additional diagrams not following directly from
the Lagrangian.

The Gauge theory instanton was found in the famous work by Belavin, Polyakov,
Schwartz and Tyupkin, and since known as the BPST instanton [9]. Let us recall
how it was obtained, as some of the expressions will be useful for later use.
To find the classical solution corresponding to tunneling, BPST used the
following 4-dimensional spherical ansatz depending on the $radial$ trial
function $f$

$gA_{\mu}^{a}=\eta_{a\mu\nu}\partial_{\nu}F(y),\,\,\,\,F(y)=2\int_{0}^{\xi(y)}d\xi^{\prime}f(\xi^{\prime})$
(15)

with $\xi=ln(x^{2}/\rho^{2})$ and $\eta$ the ’t Hooft symbol defined by

$\displaystyle\eta_{a\mu\nu}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rcl}\epsilon_{a\mu\nu}&&\mu,\,\nu=1,\,2,\,3,\\\
\delta_{a\mu}&&\nu=4,\\\ -\delta_{a\nu}&&\mu=4.\end{array}\right.$ (19)

We also define $\overline{\eta}_{a\mu\nu}$ by changing the sign of the last
two equations. Upon substitution of the gauge fields in the gauge Lagrangian
$(G_{\mu\nu})^{2}$ one finds that the effective Lagrangian has the form

$L_{eff}=\int d\xi\left[{\dot{f}^{2}\over 2}+2f^{2}(1-f)^{2}\right]$ (20)

corresponding to the motion of a particle in a double-well potential. The
Euclidean solution is that of a quantum mechanical instanton, connecting the
$maxima$ of the flipped potential. The corresponding field is

$A^{a}_{\mu}(x)={2\over g}{\eta_{a\mu\nu}x_{\nu}\over x^{2}+\rho^{2}}$ (21)

Here $\rho$ is an arbitrary parameter characterizing the size of the
instanton. Like in the potential we discussed in the preceding section, its
appearance is dictated by the scale invariance of the classical Yang-Mills
equation. The ansatz itself perhaps needs some explanation. The ’t Hooft
symbol projects to sefl-dual fields, which best captured by the identity

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{4g^{2}}\int
d^{4}x\,G^{a}_{\mu\nu}G^{a}_{\mu\nu}\;=\;\frac{1}{4g^{2}}\int
d^{4}x\,\left[\pm
G^{a}_{\mu\nu}\tilde{G}^{a}_{\mu\nu}+\frac{1}{2}\left(G^{a}_{\mu\nu}\mp\tilde{G}^{a}_{\mu\nu}\right)^{2}\right],$
(22)

where $\tilde{G}_{\mu\nu}=1/2\epsilon_{\mu\nu\rho\sigma}G_{\rho\sigma}$ is the
dual field strength tensor (the field tensor in which the roles of electric
and magnetic fields are interchanged). Since the first term is a topological
invariant (see below) and the last term is always positive, it is clear that
the action is minimal if the field is (anti)self-dual777This condition is
written in Euclidean space. In Minkowski space there is an extra $i$ in the
electric field.

$G^{a}_{\mu\nu}=\pm\tilde{G}^{a}_{\mu\nu},$ (23)

In a simpler language, it means that the Euclidean electric and magnetic
fields are the same888In the BPST paper the self-duality condition (1-st order
differential equation) was solved, rather than (2-nd order) EOM for the
quartic oscillator. . The action density is given by

$(G^{a}_{\mu\nu})^{2}=\frac{192\rho^{4}}{(x^{2}+\rho^{2})^{4}}.$ (24)

It is spherically symmetric, and very well localized, at large distances it is
$\sim x^{-8}$. The action depends on the scale only via the running coupling

$S={8\pi^{2}\over g^{2}(\rho)}$ (25)

The one-loop (determinant) quantum corrections were calculated in the classic
paper by ′t Hooft [10]. The two-loop and higher order quantum corrections have
not been calculated to this day, due to the difficulties with the part of the
Green functions related to the nonzero modes.

###  2.2 Chiral symmetry breaking in QCD

Chiral symmetries are additional symmetries that appear when quarks are
massless. Since the mass term is the only one in the QCD Lagrangian connecting
spinors with right (R) and left (L) chiralities, the $SU(N_{f})$ symmetry of
flavor quark rotations, gets doubled by acting on the L- and R-quarks
separately. Also, it can be viewed as $vector$ and $axial-vector$
transformations, in which rotations on the L- and R-spinors in the same or
opposite directions. The latters are further split into $U(1)_{A}$ (acting on
all quarks by $exp(i\gamma_{5}\theta)$ and $SU(N_{f})_{a}$.

The physics of the nonperturbative vacuum of strong interaction started even
before QCD. Nambu and Jona-Lasinio (NJL) [13], 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.

Instantons, the basis for a semiclassical theory of the QCD vacuum and
hadrons, has been discovered in 1970’s [9], and soon ′t Hooft has found their
fermionic zero modes and formulated his famous effective Lagrangian [10]. Not
only it solved 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. In the framework of the so called instanton
liquid model (ILM) [14], it provided a microscopic (QCD-based) basis for
chiral symmetry breaking, chiral perturbation theory and the pion properties.
Its two parameters

$\rho=\frac{1}{3}\,{\rm fm},\qquad n_{I+\bar{I}}=\frac{1}{R^{4}}=1\,{\rm
fm}^{-4}$ (26)

play the same role as the cutoff and coupling in the NJL model. 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, interacting ensemble of instantons, and much more, see
[15] for a review.

The subject of these lectures is many other uses of instantons discussed in
recent works. We will consider quark pair production, and its role in the
isospin asymmetry of the nucleon “sea”, the inclusion of $\bar{I}I$ molecules
in mesonic formfactors, and forces between quarks in hadrons.

###  2.3 Euclidean correlation functions

This analysis originally started from the small-distance OPE and the QCD sum
rule method, it moved to intermediate distances (see review e.g. in [16]),

First, the experimentally known correlation functions were reproduced by the
instanton liquid model at small distances at a quantitative level. Then, for
many mesonic channels [18, 19], significant numerical efforts were made,
allowing to calculate the relevant correlation functions till larger distances
(of about 1.5 fm), where they decay by few decades. As a result, the
predictive power of the model has been explored in substantial depth. Many of
the coupling constants and even hadronic masses were calculated, with good
agreement with experiment and lattice.

###  2.4 Diquarks and color superconductivity

Subsequent calculations of baryonic correlators [20] has revealed further
surprising facts. In the instanton vacuum the nucleon was shown to be made of
a “constituent quark” plus a deeply bound $diquark$, with a mass nearly the
same as that of constituent quarks. On the other hand, decuplet baryons (like
$\Delta^{++}$) had shown no such diquarks, remaining weakly bound set of three
constituent quarks. Deepling bound scalar diquarks are a direct consequence of
the ’t Hooft Lagrangian, a mechanism that is also shared by the Nambu-Jona-
Lasinio interaction [21], but ignored for a long time.

It also leads to the realization that diquarks can become Cooper pairs in
dense quark matter, see [22] for a review on “color superconductivity”.

##  3 The topological landscape and sphaleron production processes

There has been significant development in the understanding of the
“topological landscape” in relation between instantons and sphaleron
production processes, see [23]. Recently there have been renewed interest, due
to the possible experimental searches for sphaleron production at RHIC/LHC
[24]. The sphalerons are 3-dimensional, static and purely magnetic
configurations, that minimize the energy functional. To quantify them, we will
focus on two main variables. Their topological Chern-Simons number, and their
squared field size

$N_{CS}\equiv{\epsilon^{\alpha\beta\gamma}\over 16\pi^{2}}\int
d^{3}x\left(A^{a}_{\alpha}\partial_{\beta}A^{a}_{\gamma}+{1\over
3}\epsilon^{abc}A^{a}_{\alpha}A^{b}_{\beta}A^{c}_{\gamma}\right),\,\,\,\,\rho^{2}\equiv{\int
d^{3}x\vec{x}^{2}\vec{B}^{2}\over\int d^{3}x\vec{B}^{2}}$ (27)

If those are kept constant, by adding pertinent Lagrange multipliers to the
action, we can find the energy and Chern-Simons number in parametric form

$\displaystyle U_{\rm min}(\kappa,\rho)=(1-\kappa^{2})^{2}{3\pi^{2}\over
g^{2}\rho},\,\,\,{N}_{CS}(\kappa)=\frac{1}{4}{\rm
sign}(\kappa)(1-|\kappa|)^{2}(2+|\kappa|)$ (28)

where $\kappa=0$ corresponds to top of the barrier, known as the “sphaleron”.

Figure 3:  The potential energy $U_{\rm min}\bigg{(}N_{CS},\rho\bigg{)}$ (in
units of $1/g^{2}\rho$) versus the Chern-Simons number ${N}_{CS}$ for the
“sphaleron path” between $N_{CS}=0$ and $N_{CS}=1$.

Production of sphaleron-path states can be described semi-classically using
instanton-antiinstanton “streamline” [25, 26, 17], and their explosion in
Minkowski space-time by [27]. The new results consist in estimates of how one
can produce QCD sphalerons, as topologically-charged clusters in double-
diffractive events, with a cluster of few GeV mass at the center. Its decay
modes into 3 mesons were calculated using ′t Hooft Lagrangian [24].

##  4 Mesons and baryon light-front wave functions and isospin asymmetry of
the nucleon “sea”

High energy processes, as in the famed electron-nucleon deep inelastic
scattering, produced a rich “parton phenomenology”, in the form of parton
distribution functions (PDFs), distribution amplitudes (DAs), transverse
momentum distribution (TMDs) etc. They are pertinent matrix elements of light
front wave functions (LFWFs), which, are not well understood from first
principles. In [28, 29] light front DAs and PDFs were explored in the
instanton liquid model using the large momentum effective theory (LaMET)
formulation [30], although the LaMET matching kernels may well be contaminated
by non-perturbative small size instanton and anti-instanton contributions
(also $\bar{I}I$ molecules as we discuss below) [31].

Furthermore, the understanding of the light front Hamiltonians is only in its
initial stage. In [32] LFWFs for several mesons and baryons were calculated,
including for the first time the 5-quark component of the baryon, in relation
to the isospin asymmetry puzzle. The canonical process of quark pair
production via gluons is “flavor blind”, and yet the antiquark sea of the
nucleon is very asymmetric, as shown in Fig.4 (right). As noted iin [33], the
’t Hooft Lagrangian is flavor asymmetric, say a $d$ quark can produce a
$\bar{u}u$ pair (Fig.4 left) but $not$ a $\bar{d}d$. If it would be the only
process, the $\bar{d}/\bar{u}$ ratio would be 2, as there are two $u$ quarks
and only one $d$ in the nucleon. This observation is not very far from the
currently reported data for a given range of $x$.

A calculation of the admixture of the 5-q states to the nucleon LFWF, leads
the right magnitude for the asymmetry and shape, of the first-generation
antiquarks in agreement with the data.

Figure 4: Sea generation via ’t Hooft Lagrangian (left) and the data on the
$\bar{d}/\bar{u}$ ratio versus $x$ (right).

##  5 “Dense instanton ensemble”

The original instanton liquid model focused on the quark condensate and chiral
symmetry breaking, and therefore on the stand-alone instantons, whose zero
modes get collectivized. But the instanton ensemble also includes close
instanton-antiinstanton pairs, or molecules. In so far, the application of the
“molecular component” of the semiclassical ensemble was made only in
connection to the phase transitions in hot/dense matter. Indeed, this
component is the only one which survives at temperatures $T>T_{c}$, where
chiral symmetry is restored. Account for both components together started with
[35]. The “molecular component” was also shown to be important at high
baryonic densities, where it contributes to quark pairing and color
superconductivity [36].

The $\bar{I}I$ molecules were also observed on the lattice. The stand-alone
instantons are seen via “deep cooling” , during which the instanton-
antiinstanton molecules get annihilated. The close $\bar{I}I$ pairs have been
qualitatively studied in the recent work in [37], which studied their
evolution during cooling, see Fig.5. When extrapolated to zero cooling (left
side of the plots), one sees that while the instanton size fits previous
expectations (26), the density seems to actually be significantly larger.

Figure 5: The dependence of the mean instanton sizes (left plot) and density
(right plot) on the gradient flow cooling time $\tau=t/t_{0}$ (arbitrary
units). The quantum vacuum corresponds to an extrapolation to $\tau\rightarrow
0$.

Figure 6: Left: two diagrams with quark propagators in the instanton fields.
Right: vector formfactor of the pion $Q^{2}F_{\pi}(Q^{2})$. Open points
correspond to some experimental data points, thin curve on the left is
standard dipole fit. The dotted line on the top is the sum of the perturbative
and instanton contributions

##  6 Formfactors: including instantons in the hard block

More recently, we have explored the non-perturbative contributions to the
“hard block” of mesonic form factors [38]. We calculated the photon, scalar,
graviton and dilaton FFs for the pion, rho and scalar $a_{0}$ (brother of
$\eta^{\prime}$). The field at the instanton center is rather strong

$G_{rms}\equiv\sqrt{\big{(}G^{a}_{\mu\nu}(0)\big{)}^{2}}=\sqrt{192}/\rho^{2}\approx
5\,{\rm GeV}^{2},$

for a typical size of $\rho=1/3\,fm$. This scale is comparable to $Q^{2}$ in
the so called semi-hard region.

One example, which was studied a lot experimentally, is the electromagnetic
pion formfactor, shown in Fig.6 right. It is dominated by the diagram with
three propagators made of non-zero Dirac modes (the left-most), not the one
with Dirac zero modes (in the middle). The contributions of the “dense
instanton liquid” , through $\kappa=\pi^{2}n_{I+\bar{I}}\rho^{4}\rightarrow
1$, is shown by the middle curve in Fig. 6 right. The sum of the perturbative
and instanton-induced formfactor reproduce the data.

The expressions for the perturbative, non-zero-mode and zero mode parts are

$\displaystyle
V^{\pi}_{a}(Q^{2})=\epsilon_{\mu}(q)(p^{\mu}+p^{\prime\mu})\,(e_{u}+e_{\overline{d}})\,\bigg{[}\bigg{(}\frac{2C_{F}\pi\alpha_{s}f_{\pi}^{2}}{N_{c}Q^{2}}\bigg{)}$
$\displaystyle\times\int
dx_{1}dx_{2}\bigg{(}{1\over\bar{x}_{1}\bar{x}_{2}+m_{\rm
gluon}^{2}/Q^{2}}\bigg{)}\bigg{(}\varphi_{\pi}(x_{1})\varphi_{\pi}(x_{2})$
$\displaystyle+2{\chi_{\pi}^{2}\over
Q^{2}}\bigg{(}\varphi_{\pi}^{P}(x_{1})\varphi^{P}_{\pi}(x_{2})\bigg{(}{1\over\bar{x}_{2}+E_{\perp}^{2}/Q^{2}}-1\bigg{)}+\frac{1}{6}\varphi_{\pi}^{P}(x_{1})\varphi^{\prime\,T}_{\pi}(x_{2})\bigg{(}{1\over\bar{x}_{2}+E_{\perp}^{2}/Q^{2}}+1\bigg{)}\bigg{)}\bigg{)}\bigg{]}$
$\displaystyle V_{c}^{\pi}=$
$\displaystyle\epsilon_{\mu}(q)(p^{\mu}+p^{\prime\mu})\,(e_{u}+e_{\overline{d}})\,\bigg{[}{\kappa\pi^{2}f_{\pi}^{2}\chi_{\pi}^{2}\over
N_{c}M_{Q}^{2}}\langle\rho^{2}{\mathbb{G}}_{V}(Q\rho)\rangle$ (30)
$\displaystyle\times\int
dx_{1}dx_{2}\overline{x}_{1}\bigg{(}\varphi_{\pi}^{P}(x_{1})\varphi_{\pi}^{P}(x_{2})-\frac{1}{36}\varphi_{\pi}^{{}^{\prime}T}(x_{1})\varphi_{\pi}^{{}^{\prime}T}(x_{2})\bigg{)}\bigg{]}$
$\displaystyle
V_{d}^{\pi}=-\epsilon_{\mu}(q)(p^{\mu}+p^{\prime\mu})\,(e_{u}+e_{\overline{d}})\,$
$\displaystyle\times\bigg{[}\bigg{(}\frac{1}{N_{c}^{2}(N_{c}+1)}\bigg{)}\frac{4\kappa\pi^{2}f_{\pi}^{2}\chi_{\pi}^{2}}{3M_{Q}^{2}}\left<\rho^{2}\frac{K_{1}(Q\rho)}{Q\rho}\right>\int
dx_{1}dx_{2}\varphi_{\pi}^{P}(x_{1})\varphi_{\pi}^{{}^{\prime}T}(x_{2})\bigg{]}$

Here $\varphi_{\pi}^{P},\varphi_{\pi}^{T}$ are the pion DAs associated with
the pseudoscalar $\gamma_{5}$ and tensor $\sigma^{\mu\nu}$ channels. The
leading twist $\varphi_{\pi}$ is the standard matrix element of the axial
current $\gamma_{\mu}\gamma_{5}$. In the figure, all three DAs are taken to be
just constant, independent of $x$. (Therefore all tensor DAs with an
x-derivative vanish.)

##  7 Instanton-induced inter-quark forces

Historically, hadronic spectroscopy got a solid foundation in 1970’s, with the
discovery of nonrelativistic quarkonia made of heavy $c,b$ quarks. In the
first approximation, those are well described by the simple Cornell potential

$V_{\rm Cornell}(r)=-{4\alpha_{s}\over 3}{1\over r}+\sigma_{T}r$ (32)

which correctly attributes the short-distance potential to the perturbative
gluon exchange, and its large distance ${\cal O}(r)$ contribution to the
tension of a confining flux tube (the QCD string). The issues to be discussed
deal with the nonperturbative origins of the inter-quark interactions at
$intermediate$ distances $r\sim 0.2-0.5\,$fm.

Later developments in [39, 40] connected the static interquark potential, to
the correlator of static Wilson lines (central)

$e^{-V_{C}(r)T}=\langle W(\vec{x}_{1})W^{\dagger}(\vec{x}_{2})\rangle$ (33)

The spin-dependent forces were related to such correlators with two magnetic
fields ($V_{SS},V_{tensor}$), or a magnetic and electric field for the spin-
orbit one. To evaluate such nonlocal quantities, one needs to use lattice
simulations, or rely on certain model of the vacuum fields. In Fig.7 (left) we
show the predictions from a “dense instanton model” to the central potential,
compared to the linear potential, and its version from [41], including the
string quantum vibrations (resummed “Lusher terms”). One can see that
instantons can complement smoothly, the flux tube at intermediate distances.

Figure 7: The central potential $V_{c}$ (left) and spin-spin $r^{2}V_{SS}(r)$
(right) versus distance $r,(GeV^{-1})$. The lattice result is one of the
parameterization given in [42], the perturbative is Laplacian of (regulated)
Coulomb term.

One problem with the electric flux tube model is that it does not provide
magnetic fields, while instantons are self-dual and have $B$ (as large as $E$)
badly needed to generate the spin forces. We calculated the instanton
contributions to spin forces, see $V_{SS}$, in comparison to the lattice
potential, perturbative and instanton-induced in Fig.7 (right). Note that the
area below the perturbative and instanton-induced terms is comparable. The
corresponding matrix elements (using the Cornell wave functions with proper
quark masses) is shown in the table. One can see that e.g. for charmonium, the
magntitude of the spin-spin term is in agreement with the lattice estimate,
and the level splitting. We also considered the $L=1$ families of mesons, from
heavy to light, and considered other spin-dependent potentials. We also
discussed $\bar{I}I$ molecules, which provide somewhat different potentials
due to their different field content.

However for heavy-light and light-light cases this $V_{SS}$ is not enough. The
missing part is attributed to the part of the quark propagators containing
zero modes (′t Hooft Lagrangian). It works well for heavy-light and the pions,
as expected.

##  8 Hadrons on the light front

In the rest frame, the non-relativistic description of hadrons and their
constituents is most natural for heavy quarks, with the use of non-
relativistic kinectic energy and potentials. However, this approach is not
justified for light quarks, as masses and transverse momenta are comparable.
Their quantum motion is complicated as illustrated in Fig. 8 (left). The
situation however is much more “democratic” on the light front, where all
constituents motion are frozen anyway in Fig. 8 (right). All interactions can
be deduced from the pertinent Wilson loops, modulo tunneling due to zero
modes.

Figure 8: $\bar{Q}Q$ meson in the rest frame (a) and in the light-front frame
(b).

###  8.1 Light front Hamiltonian from the QCD vacuum

Indeed, let $Q\bar{Q}\equiv Q_{1}Q_{2}$ be the world-lines assigned to the
Wilson lines composing a meson in Fig. 8 (right). In Euclidean signature, the
Wilson lines are sloped at an angle $\theta$, and their light-like limit
follows by analytical continuation $\theta\rightarrow-i\chi$ [43, 44, 45, 46].
For the squared meson mass operator, or light front Hamiltonian $H_{LF}$, the
result [46]

$\displaystyle H_{LF}\approx$
$\displaystyle\frac{k_{\perp}^{2}+m_{Q}^{2}}{{x\bar{x}}}+2P^{+}P^{-}$
$\displaystyle\approx$
$\displaystyle\frac{k_{\perp}^{2}+m_{Q}^{2}}{{x\bar{x}}}+2M(\mathbb{V}_{Cg}(\xi_{x})+\mathbb{V}_{C}(\xi_{x})+\mathbb{V}_{SD}(\xi_{x},b_{\perp})+\mathbb{V}_{TH}(\xi_{x},b_{\perp}))$

On the light front, the invariant distance $\xi_{x}$ is

$\displaystyle
M\xi_{x}=\big{(}\big{|}{id/dx}\big{|}^{2}+M^{2}{b_{\perp}}^{2}\big{)}^{\frac{1}{2}}$
(35)

The longitudinal distance $\gamma b_{3}=id/dx/M$, is the conjugate of
Bjorken-x or $x=k^{3}/P^{3}$.

The one-gluon exchange potential $V_{Cg}$ in the instanton vacuum is

$\displaystyle\mathbb{V}_{Cg}(\xi_{x})=-\frac{g^{2}T_{1}^{A}T_{2}^{A}}{2\pi^{2}}\frac{1}{\xi_{x}}\int_{0}^{\infty}\frac{dx\,x{\rm
sin}x}{x^{2}+(\xi_{x}m_{G}(x\rho/\xi_{x}))^{2}}\rightarrow-\frac{g^{2}T_{1}^{A}T_{2}^{A}}{4\pi}\frac{e^{-m_{G}\xi_{x}}}{\xi_{x}}$

with the running gluon mass [47]

$\displaystyle m_{G}(k\rho)$ $\displaystyle=$ $\displaystyle
m_{G}\,\bigg{(}k\rho\,K_{1}(k\rho)\bigg{)}$ $\displaystyle m_{G}\rho$
$\displaystyle\approx$ $\displaystyle
2\bigg{(}\frac{6\kappa}{N_{c}^{2}-1}\bigg{)}^{\frac{1}{2}}\approx 0.55$ (37)

using the estimate $\kappa=\pi^{2}\rho^{4}n_{I+\bar{I}}$ in the right-most
result.

The instanton-induced central potential $V_{C}$ is

$\mathbb{V}_{C}(\xi_{x})=\bigg{(}\frac{4\kappa}{N_{c}\rho}\bigg{)}{\bf
H}(\tilde{\xi}_{x})$ (38)

with the integral operator

$\displaystyle{\bf H}(\xi_{x})=$
$\displaystyle\int_{0}^{\infty}y^{2}dy\int_{-1}^{+1}dt$
$\displaystyle\times\bigg{[}1-{\rm cos}\bigg{(}\frac{\pi
y}{\sqrt{y^{2}+1}}\bigg{)}{\rm
cos}\bigg{(}\pi\bigg{(}\frac{y^{2}+\tilde{\xi}_{x}^{2}+2\xi_{x}yt}{y^{2}+\tilde{\xi}_{x}^{2}+2\tilde{\xi}_{x}yt+1}\bigg{)}^{\frac{1}{2}}\bigg{)}$
$\displaystyle-\frac{y+\xi_{x}t}{(y^{2}+\xi_{x}^{2}+2\tilde{\xi}_{x}yt)^{\frac{1}{2}}}{\rm
sin}\bigg{(}\frac{\pi y}{\sqrt{y^{2}+1}}\bigg{)}{\rm
sin}\bigg{(}\pi\bigg{(}\frac{y^{2}+\tilde{\xi}_{x}^{2}+2\tilde{\xi}_{x}yt}{y^{2}+\xi_{x}^{2}+2\tilde{\xi}_{x}yt+1}\bigg{)}^{\frac{1}{2}}\bigg{)}\bigg{]}$

with the dimensionless invariant distance on the light front
$\tilde{\xi}_{x}=\xi_{x}/\rho$. ${\bf H}(\tilde{\xi}_{x})$ admits the short
distance limit

$\displaystyle{\bf
H}(\tilde{\xi}_{x})\approx+\bigg{(}\frac{\pi^{3}}{48}-\frac{\pi^{3}}{3}J_{1}(2\pi)\bigg{)}\tilde{\xi}_{x}^{2}$
$\displaystyle+\bigg{(}-\frac{\pi^{3}(438+7\pi^{2})}{30720}+\frac{J_{2}(2\pi)}{80}\bigg{)}\tilde{\xi}_{x}^{4}$
(40)

and large distance limit

$\displaystyle{\bf H}(\xi_{x})\approx-\frac{2\pi^{2}}{3}\bigg{(}\pi
J_{0}(\pi)+J_{1}(\pi)\bigg{)}+\frac{C}{\tilde{\xi}^{p}_{x}}$ (41)

with $p\ll 1$ and $C>0$. The large asymptotic is to be subtracted in the
definition of the potential. In the dense instanton vacuum [46], the central
potential (38) is almost linear at intermediate distances $0.2-0.5$ fm, and is
expected to be taken over by the linearity of the confining potential at
larger distances.

The spin potentials receive contribution from both the perturbative and non-
perturbative parts of the underlying gluonic fields. In particular, the non-
perturbative instanton-induced spin potentials can be derived in closed form

$\displaystyle\mathbb{V}_{SD}(\xi_{x},b_{\perp})=$ $\displaystyle
2M\bigg{(}\bigg{(}\frac{l_{1\perp}\cdot
S_{2\perp}}{m_{Q1}m_{Q2}}-\frac{l_{2\perp}\cdot
S_{1\perp}}{m_{Q1}m_{Q2}}\bigg{)}\frac{1}{\xi_{x}}\mathbb{V}_{C}^{\prime}(\xi_{x})$
$\displaystyle+\frac{1}{m_{Q1}m_{Q2}}(S_{1\perp}\cdot\hat{b}_{\perp}S_{2\perp}\hat{b}_{\perp}-\frac{1}{2}S_{1\perp}\cdot
S_{2\perp})\frac{2b_{\perp}^{2}}{\xi_{x}}\mathbb{V}_{C}^{\prime\prime}(\xi_{x})\bigg{)}$

with the respective spins $\vec{S}_{1,2}=\vec{\sigma}_{1,2}/2$, and transverse
orbital momenta

$\displaystyle l_{1,2\perp}=\pm(b_{\perp}\times
m_{Q1,2}s_{1,2}\hat{3})_{\perp}\qquad s_{1,2}={\rm
sgn}(v_{1,2})\rightarrow\frac{Mx_{1,2}}{m_{Q1,2}}$ (43)

The contributions stemming from the zero modes
$\mathbb{V}_{TH}(\xi_{x},b_{\perp})$ are not included. They follow from the ′t
Hooft determinantal interaction properly continued to the light front.

###  8.2 Light front spectra and wavefunctions

We have used the light front Hamiltonian (8.1) to analyze heavy and light
mesons and baryons. One strategy consists at linearizing the confining part of
the potential (instanton induced at small distances, and string induced at
large distances), and treating the Coulomb and spin contributions
perturbatively. More specifically, the central part of the potential can be
linearized using the “einbein trick”

$\displaystyle
2M\mathbb{V}_{C}(x,b_{\perp})=2\sigma_{T}\big{(}|id/dx|^{2}+M^{2}b_{\perp}^{2}\big{)}^{\frac{1}{2}}\rightarrow\sigma_{T}\bigg{(}\frac{|id/dx|^{2}+b\,b_{\perp}^{2}}{a}+a\bigg{)}$
(44)

with $a,b$ variational parameters. The minimization with respect to $a$ would
lead back to a previous expression, but the trick is to do it $after$ the
Hamiltonian is diagonalized. It is also followed by the substitution
$b\rightarrow M^{2}\approx(2m_{Q})^{2}$ for heavy mesons, and most light ones.
The light front Hamiltonian is then

$H_{LF}=H_{0}+\tilde{V}+V_{perp}+V_{spin}$ (45)

with

$H_{0}={\sigma_{T}\over a}\bigg{(}-{\partial^{2}\over\partial
x^{2}}-b{\partial^{2}\over\partial\vec{k}_{\perp}^{2}}\bigg{)}+\sigma_{T}a+4(m_{Q}^{2}+k_{\perp}^{2})$
(46)

and $\tilde{V}$

$\tilde{V}(x,\vec{k}_{\perp})\equiv(m_{Q}^{2}+k_{\perp}^{2})\bigg{(}{1\over
x\bar{x}}-4\bigg{)}$ (47)

where $\bar{x}=1-x$. $H_{0}$ is diagonal in the functional basis used [46]. We
use the momentum representation, with $\vec{k}_{\perp}$ as variable. The
residual interaction has non-zero matrix elements $\langle
n_{1}|V(x,\vec{k}_{\perp})|n_{2}\rangle$ for all $n_{1},n_{2}$ pairs. The
perturbative part $V_{perp}$ for heavy quarks is the Coulomb term, with
running coupling and other radiative corrections. Finally, the last term
$V_{spin}$ contains matrices in spin variables and in orbital momenta.

To solve the problem we have two numerical strategies. First we used a
(truncated) basis set and write the Hamiltonian in the form of a 12$\times$ 12
matrix, and diagonalize $H_{0}+\tilde{V}$, to find its eigenvalues as a
function of the remaining parameter $a$. The results for the three lowest
states $n=1,2,3$ are shown in Fig. 9 (left). The near flateness in $a$ around
the minima, suggest $a=25$ as a common value. This procedure preserves the
orthonormality of the eigen-spectrum.

Another approach worked out later is direct numerical solution in 3-d space of
transverse momenta and $x$. To our delight, the wave functions found in both
methods agree within the width of the lines.

Figure 9: Top: Squared masses $M^{2}_{n}$ for $\bar{b}b$ mesons for $n=1,2,3$
versus the variational parameter $a$. Bottom: Squared masses for $n=0..5$
(left to right) and orbital momentum $m=0,1,2$ (down to up), calculated from
the light front Hamiltonian $H_{LF}$ (red triangles), and shifted by a
constant, $M^{2}_{n+1}-5\,GeV^{2}$. For comparison, the blue-circles show the
squared masses $M^{2}_{n+1}$ calculated from Schroedinger equation in the CM
frame, with only linear plus centrifugal potentials.

The calculated masses (shifted by a constant “mass renormalization”, to make
$n=0,m=0$ states the same) are shown in Fig.9 (right). The bottom part shows
good agreement between the masses obtained solving the Schroedinger equation
in the rest frame (blue circles), and the masses following from the light-
front frame (red-triangles). The slope is correct, and is determined by the
same string tension $\sigma_{T}$. The splittings in orbital momentum are of
the same scale, but not identical. This is expected, as we compare the
2-dimensional $m$-states on the light front, with the 3-dimensional $L$-states
in the center of mass frame. The irregularity between the third and fourth set
of states, is due to our use of a modest basis set, with only three radial
functions (altogether 12 functions if one counts them with 4 longitudinal
harmonics). This can be eliminated using a larger set. The corresponding
wavefunction for bottomium is shown in Fig. 10 (left), and for a typivcal
light meson $\bar{q}q$ is shown Fig. 10 (right). Further details can be found
in our subsequent papers.

Figure 10:  Distribution amplitudes for $\bar{b}b$ (upper) and a “generic”
light $\bar{q}q$ meson (lower) as a function of $x$, for the three lowest
states $n=0,1,2$. For bottomium, the difference between the three curves is
too small to be visible. For the light meson, the differences are visible.
With increasing $n$, the DAs become narrower and higher at $x=\frac{1}{2}$.

##  9 Semiclassical theory of deconfinement and chiral phase transitions based
on instanton-dyons

Instanton-dyons are topological solutions of YM equations at finite
temperatures. Their semiclassical ensembles were studied by a number of
methods, including direct Monte-Carlo simulation, for SU(2) and SU(3)
theories, with and without fermions. We present these results and compare them
with those from lattice studies. We also consider two types of QCD
deformations. One consists in adding operators with powers of the Polyakov
line, affecting deconfinement. Another consists in changing the quark
periodicity condition, affecting the chiral transition. We will also discuss
how lattice configurations (with realistic quark masses) can be used to
unravel the zero and near-zero Dirac modes of the underlying gauge
configurations. The results are in good agreement with the analytic instanton-
dyon theory. In sum, the QCD phase transitions are well described in terms of
such semiclassical configurations.

At finite temperature the Polyakov line has certain nonzero expectation
values, or $holonomies$ $\mu_{i}(T)$ (see below). The instanton solution
deformed by an asymptotics $A_{0}$ field, splits into instanton constituents,
known as instanton-dyons or instanton-monopoles Like instantons, they have
topological charges. Unlike instantons, those are not quantized to integer
$Q$, which is possible because they are wired by Dirac strings due to their
magnetic charges.

Remarkably, the partition function for monopoles follows from that of
instanton-dyons by Poisson duality, which is tightly related to Hamilton and
Jacobi duality when describing dynamical systems. More details regarding this,
and further discussions on the QCD flux tubes are in [1].

###  9.1 Instanton-dyons on the lattice

A thorough discussion of the Kraan-van Baal solution for an instanton,
containing $N_{c}$ instanton-dyons, or the formalism leading to their zero
modes, is not possible in this short review, and we refer to the original
papers for that. For completeness, we recall that the eigenvalues of Polyakov
line, designated as $\mu_{i}(T),i=1..N_{c}$ can be regarded as locations on a
unit circle. Their differences $\nu_{i}=\mu_{I+1}-\mu_{i}$ are lengths of the
corresponding fractions of the circle, $S_{i}=\nu_{i}S$ where

$S=8\pi^{2}/g^{2}=\bigg{(}{11\over 3}N_{c}-{2\over
3}N_{f}\bigg{)}log\bigg{(}{T\over\Lambda_{QCD}}\bigg{)}$

Also, if quarks are given different periodicity phases $z_{f},f=1..N_{f}$ over
the Matsubara circle, the normalizable physical zero mode belongs to the dyon
for which $z_{f}$ belongs to the sector $z_{f}\in[\mu_{i},\mu_{i+1}]$. So,
using different $z_{f}$ one can see all dyon types.

Let us start with the description of the efforts to identify these objects on
the lattice. The key method is “cooling” of vacuum configurations, which was
used efficiently to identify ensembles of instantons in the 1990’s. The first
lattice observation of instanton-dyons was via “constrained minimization” by
Langfeld and Ilgenfritz [48], fixing $\langle P\rangle$, in which selfdual
clusters with non-integer topological charges were seen. Gattringer et al and
Ilgenfritz et al have since introduced and refined the “fermionic filter”
allowing to identify the instanton-dyons via distinct zero modes and variable
periodicity phases.

Figure 11: (Top) Space slice of density of an exact zero mode from QCD
simulaiton at $T=T_{c}$. The three colors refer to dyons of three different
types. (Bottom) Tau dependence of a dyon, perturbed by nearby dyons. See text.

Some of the recent progress in these directions can be found in [49, 50]. QCD
simulations with realistic masses were performed at and near $T_{c}$, using
domain wall fermions with good chiral symmetry. Using overlap fermions with
exact chiral symmetry, the focus was on exact zero modes (and near-zero ones).
The top part in Fig.11 shows a typical landscape of the zero mode densities.
There are three different dyon types for $N_{c}=3$. The shape of isolated
peaks are well described by analytic formulae from van Baal and collaborators,
and derived for a single dyon. They are apparently undisturbed by millions of
perturbative gluons present in the ensemble.

Previous works however have not analyzed the “topological clusters”, the
situations in which two or three dyons overlap strongly. The Kraan-van Baal
solution allows these cases, and good agreement was also found in the
numerical analysis of instanton-dyon ensembles in [49, 50]. The right part in
Fig.11, is an example of a (Euclidean time) $\tau$-dependence of the density.
An isolated dyon should show no such dependence at all, and what is seen is a
result of an interference with overlapping dyons. Locating those and using
analytic expressions for the zero mode density, confirms their identity. The
semiclassical description of zero and near-zero Dirac modes on the lattice is
quite accurate, at least in terms of the zero mode shapes.

###  9.2 Studies of instanton-dyon ensembles

The simplest limiting case is weak coupling or very high temperature
($T\rightarrow\infty$), in which the dyon density is exponentially small, with
their interactions and back reactions negligible. In QCD with $N_{f}$ quarks,
all twist factors are equal with $z_{f}=\pi$, and all the $L$ dyons combine in
a ′t Hooft vertex with $2N_{f}$ legs. For $N_{f}=1$, $U(1)_{A}$ chiral
symmetry is explicitly broken at any $T$, with exponentially small
$\langle\bar{q}q\rangle$. For $N_{f}>1$ and $SU(N_{f})_{A}$ chiral symmetry is
unbroken. So there should be “molecules” $\bar{L}L$, similar to $\bar{I}I$
molecules originally discussed in [35, 51, 52], and called “bions” by Unsal
and collaborators.

The opposite case is a dense ensemble at $T\sim T_{c}$, discussed by mean
field approximation in a number of settings [53, 54, 55]. Here we will only
discuss the numerical simulations for the $SU(2)$ in [56, 57] and $SU(3)$ in
[58, 59] color groups, without ($N_{f}=0$) and with two quark flavors
($N_{f}=2$) .

The first physics issue is the $deconfinement$ phase transition. Recall that
the Gross-Pisarski-Yaffe (GPY) perturbative potential for the Polyakov line,
favors a trivial $\langle P\rangle=1$ case, and disfavors confinement l
$\langle P\rangle=0$. Therefore, to have confinement, the nonperturbative
effects – with a sufficient density of the dyons in the simulations – should
$overcome$ the GPY potential. In Fig.12 we see that this happens differently
for SU(2) and SU(3) pure gauge theories. In the former the minimum gradually
shifts to the confining value of $\nu=1/2$, and stay there at high densities.
In the latter, there is a jump, indicating first order transition. There is no
place here for comparison with lattice data. We just mention that the SU(3)
case $\langle P\rangle$ jumps to a value 0.4, remarkably close to the reported
lattice value. The $deformation$ of the SU(3) gauge theory by an operator
$\sim P^{2}$ in the instanton-dyon ensembles, is also in agreement with the
lattice simulation. In particular, the deconfinement temperature is observed
to increase, with the size of the jump decreasing.

Figure 12: Free energy $f$ versus the holonomy parameter $\nu$, for SU(2)
(top) and SU(3) (bottom) pure gauge theories. Different curves are for
different instanton densities (or temperatures). See text.

The second nonperturbative issue is chiral symmetry breaking at low $T$.
Again, it requires a sufficient density of the dyons, so that their zero modes
can get $collectivized$999Note that already in 1961, the NJL model has shown
that one needs a large enough 4-fermion coupling to break chiral symmetry..

Figure 13: Quark condensates (left) and Polyakov line (right) versus the
action parameter $S(T)$. The higher temperatures are recorded on the right
vertical axis. The blue circles in the right plot are for QCD with
$N_{c}=N_{f}=2$, and the red squares are for $Z_{2}QC_{2}D$. A $crossover$ in
the former, changes to a first order transition in the latter. The black
triangles on the left are for QCD with $N_{c}=N_{f}=2$, with restored chiral
$SU(2)_{A}$ symmetry above $S>8$. The blue discs and red squares show two
chiral condensates for $Z_{2}QC_{2}D$. We clearly see a transition from a
symmetric to an asymmetric phase, with no chiral symmetry restoration.

###  9.3 QCD deformation by modified quark periodicity phases

This deformation came first under the name of ”imaginary chemical potentials”
[60]. Its usage is different for $small$ and $large$ deformations. In the
former case, the motivation was due to the fact that imaginary chemical
potentials can be simulated by usual Monte Carlo algorithms, while those with
real $\mu$ cannot. A plot of the lattice results for negative $\mu^{2}<0$,
allows the extrapolation to real $\mu$, e.g. by Taylor series. This strategy
has been used in many lattice studies.

For a large phase, Roberge and Weiss predicted a first order transitions close
to $z=(2k+1)\pi/Nc$, due to different $N_{c}$ branches of the gluonic GPY
potential. Of course, it is a perturbative argument expected to be true at
large $T$ only. And indeed, when dyons are numerous this transition ends, and
according to [61] it happens at $T_{RW}=1.34(7)T_{c}=208(5)\,\,\,MeV$.

Another form of deformed QCD is to select imaginary chemical potentials
proportional to $T$, so that the quark fugacities are
$exp(iz_{f}),f=1..N_{f}$, with certain $T$-independent periodic phases. Moving
from conventional $z_{f}=\pi$ (quarks are fermions) to other values, one
should see multiple phase transitions, each time when one of the $z_{f}$
crosses the Polyakov phases $\mu_{i}(T)$, as the fermion zero modes jump from
one instanton-dyon to another. The ‘ultimate” selection for $N_{c}=N_{f}$
theories was proposed in [62], with $z_{f}$ suggested to be located
“democratically”, with one phase located inside each of the holonomy sectors
$[\mu_{i+1},\mu_{i}]$ (for $T<T_{deconfinement}$).

In Fig.13 we show ensembles of instanton-dyons simulations for $N_{c}=N_{f}=2$
QCD and $Z_{2}QC_{2}D$, with one quark being a fermion and one being a boson.
The plots show drastic changes in both phase transitions. Deconfinement
changes from a crossover to a strong first order, and chiral restoration is
not seen at all 101010Note that SU(2) is somewhat of a special case. If the
group is SU(3), and two sectors of $M_{1},M_{2}$ dyons would have
$z_{1},z_{2}$ located in them, ultimately at high $T$ these sectors shrink as
$\mu_{i}(T)$ move towards zero. When they cross phases $z_{1},z_{2}$ , the
zero modes would return to $L$ dyons, and thus at infinite $T$ would become
similar to undeformed QCD, with chiral symmetry is restored.. We note that the
multiply deformed worlds (such as $Z_{N_{c}}QCD$) are unphysical, and
qualitatively different from ours. We expect them to be separated by
singularities. Indeed, we recall that $Z_{N_{c}}QCD$ has completely different
flavor and chiral symmetries (except at high $T$ when all $\mu_{i}$ are near
zero). Its chiral symmetry is split diagonally for all flavors
$(U(1)_{A})^{N_{f}-1}$, each broken $explicitly$ by diquark ′t Hooft operators
for each dyon type. Obviously, this chiral symmetry breaking has no relation
to the spontaneously broken $SU(N_{f})$ chiral symmetry occuring in our world,
and due to the multi-quark operators with at finite instanton-dyon density.

Another take on this unusual phases with modified quark periodicities,
historically came from supersymmetry. Davies, Hollowood and Khose [63]
considered $\cal N$=1 SYM theory on $R^{3}\times S^{1}$ with a small circle
and $bosonic$ gluinos, and evaluated the quark condensate using instanton-
dyons. In this case the gluino term in the GPY potential changes sign,
canceling the gluon contribution.

Unsal [64] considered theories with more than one flavor of gluinos (adjoint
fermions), $N_{a}>1$. The GPY potential was found to be multiplied by
$(1-N_{a})$, and for $N_{a}>1$ was $inverted$, favoring confinement at weak
coupling (small circle or high $T$). It was suggested that with confinement
present both at small and large $L=1/T$, there would be $continuity$ (no phase
transitions) at any $L$. However, lattice studies [65] have found $two$
deconfined phases in between those two limits, preventing in general such
continuity. The considerations based on the GPY potentials at weak coupling
(high temperature), do not necessarily carry to strong coupling.

##  10 Summary

The physics of instantons is now nearly 50 years old, yet we are still finding
new applications in the vast land of hadronic physics. Indeed, only recently,
the large contributions due to the instanton-antiinstanton molecules came to
focus. Recently, we used these contributions to solve two old puzzles.
Specifically, the nonperturbative nature of the quark-antiquark potentials,
and the large observed mesonic formfactors in the semi-hard regime with
$Q^{2}\sim$ few $GeV^{2}$, can be explained by these fields in the QCD vacuum.

The physics of diquarks, first appearing in connection to color
superconductivity, again came to the front of spectroscopy, with recent
discovery of the double charmed tetraquark state $T_{cc}$ made of
$cc\bar{u}\bar{d}$.

The development of light front Hamiltonians and wave functions entered a stage
at which properties of $all$ hadrons, from bottomonium to pions and with all
kind of baryons and exotica be systematically calculated. Traditional
distinction between heavy and light physics (absence of nonrelativism in the
former case) is no longer present on the light front, so all of them are
studied in the same setting.

Methods for solving the corresponding equations include variational ones,
diagonalization of Hamiltonian written as matrices in appropriate basis, and
even direct numerical solutions. This new ”spectroscopy on the light front”
program is going to bridge the gap between the usual spectroscopy, the
Euclidean version (lattice and instantons) and the light front observables.

The semiclassical theory based on ensembles of instanton-dyons, reproduces
semi-quantitatively the main lattice findings about deconfinement in pure
gauge theories, and chiral phase transitions, in QCD with light quarks.
Deforming QCD by extra action with powers of the Polyakov loop shift/modify
the deconfinement transition. Deforming it via quark periodicity phases, leads
to phases with drastically different deconfinement and chiral transitions.
While all these phases have multiple unphysical properties, their existence
shed light on the mechanisms driving the QCD phase transitions. Their key
feature are “jumps” of the fermion zero modes, from one dyon type to another.

The dyon zero modes from lattice QCD simulations preserve remarkably well
their shapes on the lattice, even in the case of their strong overlaps.
Millions of thermal gluons do not seem to affect them.

Finally, we note that instanton-dyon simulations are simple multiple integrals
over the instanton-dyon collective variables. So it is relatively easy to get
to say $N_{d}\sim 300$ of instanton-dyons. In contrast, the lattice
simulations which are of course based on first principles, their current cost
limit them to about $N_{d}\sim O(10)$ instanton-dyons, as e.g. seen from our
Fig.2. If the instanton-dyon ensembles are important for QCD phase transitions
as we suggest, then their current values for $N_{d}$ are perhaps too small to
get an accurate description of their phases.

There seems to be enough new land to cover for the next 50 years.

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