# How to observe the QCD instanton/sphaleron processes at hadron colliders?

Edward Shuryak and Ismail Zahed  Center of Nuclear Theory, Department of
Physics and Astronomy,
Stony Brook University, Stony Brook, NY 11794, USA

###### Abstract

The instanton/sphaleron processes involve gauge fields with changing topology,
including a nonzero variation of the Chern-Simons number $\Delta N_{CS}=\pm
1$. In QCD such processes lead to the production of $2N_{f}\Delta N_{CS}$
units of the axial charge, proportional to the number of light quark flavors
$N_{f}=3$, for $u,d,s$. In the electroweak theory such processes lead to the
production of 12 fermions, with $\Delta B=\Delta L=\pm 3$ units of baryon and
lepton number. While this is all known for a long time, and is one of the
pillars of the nonperturbative theory of the QCD vacuum, in this paper we
discuss what we call a “reclined tunneling”, in which external forces are
added to the tunneling processes and create certain gluonic objects with
positive invariant mass. The idea to observe these objects experimentally at
hadronic colliders have been proposed before, but insofar without success.
Motivated by the recent CERN workshop on the topic, we review these ideas. We
put forward our own suggestions, in particular to utilize a double-diffractive
(Pomeron-Pomeron) collisions to this goal, which we believe maximizes the
entrance factor and minimizes the backgrounds. We consider clusters of small
($M=3-10\,{\rm GeV}$), medium ($M=10-30\,{\rm GeV}$) and high $M\sim 100\,{\rm
GeV}$ invariant masses, subsequently. Among the proposed signals are specific
flavor combination of channels, originating from well-defined 6-, 8- and
10-quark-antiquark operators, as well as correlation of quark chiralities to
be potentially detected via $\Lambda$ hyperon decays.

††journal: Physics Reports

##  1 Introduction

###  1.1  The instanton/sphaleron processes

In going to the mountains, one needs a reliable map and preferably from a
variety of sources. Building a tunnel is expensive, so one needs to think a
lot. Should it be horizontal or reclined? Where should the entrance be, and
where should the exit be? As we will see shortly, all such questions also
appear when we think of optimal paths producing topologically nontrivial final
states in a collision.

Some of these questions have already been answered in theory decades ago, but
some still require further analyses and calculations. There is a vast number
of applications of instanton-induced effects in vacuum and hadronic structure.
The instanton-sphaleron processes, producing certain topologically nontrivial
gluonic clusters, are only seen in low-mass domain, like $\eta_{c}$ decays.
This paper is about possible ways to see such objects at hadron colliders,
with variable masses.

One can split the expected cross section into three distinct parts: (i)
tunneling action; (ii) the semiclassical prefactor; (iii) the entrance factor;
and, last but not least, (iv) branching ratios into particular exclusive
channels, or sequentially

$\sigma\sim\bigg{[}{{\rm entrance\over factor}}\bigg{]}\bigg{[}{{\rm
semiclassical\over
prefactor}}\bigg{]}\bigg{[}e^{-S_{cl}}\bigg{]}\bigg{[}B({\rm
final\,state})\bigg{]}$ (1)

As we show below, the classical part is well under control. The semiclassical
prefactor is not yet calculated for gauge theory beyond one loop, but it was
done in relevant toy models, so it can be evaluated if needed. The entrance
factor is really the difficult part still in deliberation.

The produced unstable magnetic objects, generically known as “sphalerons”,
explode into outgoing gluon and quark waves, which eventually “hadronize” into
mesons and baryons going to the detectors. This initial part of this process
is described by a solution of the classical YM equation, see section 4. Quark
pair production is described by the solution of the Dirac equation in the
exploding background. For all light quark flavors one has the same solution,
which results in an effective 6-quark operator of the flavor composition
$(\bar{u}u)(\bar{d}d)(\bar{s}s)$ and with specific chiral properties. Its
projection onto the distribution amplitudes of pertinent mesons, provides the
decay probabilities to exclusive final states, e.g. for three mesons those are
$KK\pi,\pi\pi\eta,\pi\pi\eta^{\prime}$.

A further discussion of heavy ($c,b$) quark production is done in section 7.3.
We will estimate the sphaleron scales at which such production becomes
possible, and then discuss 8- and 10-fermion operators and their final states
such as $KKDD$ and $BBD_{s}DK$.

###  1.2 Double diffraction and Pomerons, glueballs and clusters

In multiple papers the instanton/sphaleron process was calculated with the
assumption that it is initiated at a partonic level, gluon-gluon or quark-
antiquark collisions. In this case the secondaries originated from sphalerons
are imbedded in complicated $pp$ collisions, together with many other jets and
secondaries produced. The separation of the signal from backgrounds is in this
case very difficult.

Perhaps the main idea first expressed in [1], is that one may avoid large
backgrounds associated with min.bias $pp$ collisions, by using double
diffractive events (also known as Pomeron-Pomeron abbreviated as PP, distinct
from $pp$). We start with a brief summary of the experimental situation,
before elaborating on various theoretical models.

The two early experiments of interest are WA102 [2] and WA8 [3], both of which
were carried decades ago at CERN .

WA102 was driven by an idea that the PP processes provide a “glueball filter”,
allowing to separate glueball-type hadrons from other mesons. It provided
important data on the distribution over the azimuthal angle $\phi$, between
the transverse momentum transfers to both surviving protons. The collaboration
identified a certain number of scalar, pseudoscalar and tensor hadronic
resonances in the invariant mass distribution. As a subsequent discussion has
shown, the collaboration has indeed seen not only a scalar glueball candidate,
but also ta tensor $2^{+}$ glueball at a mass of $2.2\,{\rm GeV}$. Both have
angular distributions in $\phi$ completely different from those for ordinary
mesons.

WA8 was focused on larger invariant masses. Unfortunately, its central
detector was just a calorimeter, so no exclusive channels were studied. This
collaboration reported observation of some isotropically decaying “clusters”,
with a mass peaked at $M\approx 3\,{\rm GeV}$, which we will discuss further
in sections 6.2 and 11. They also reported non-isotropic background events
with masses $M=5-20\,GeV$.

The revival of the diffractive field at the LHC has began relatively recently.
The Atlas Forward Proton (AFP) detector was installed and ran, but it is
focused on producing systems with very large mass in the range of hundreds of
GeVs. In our understanding, ATLAS ALFA forward detectors were designed to
measure forward elastic scatterings. On the other hand, the CMS-TOTEM
collaborations, working together, have addressed soft PP reactions, and
recently reported soft exclusive $\pi^{+}\pi^{-}$ data [4]. They focused on
small invariant mass region $M(\pi^{+}\pi^{-})<2\,{\rm GeV}$ and have
confirmed the production of (at least) four scalar and tensor mesons, all seen
previously by WA102. From a wider perspective, apart from focusing on glueball
spectroscopy, these experiments have provided important insights into the
structure of the Pomeron-Pomeron-hadron vertices, and thus the structure of
the Pomeron itself.

From the theory side, during the last decade there were basically two major
developments:

(i) The Pomeron is defined as a virtual state, described by continuation of a
glueball Regge trajectory from positive to small negative $t$ domain, where it
is observed in scattering. Note that the nearest physical state to it is the
tensor $2^{++}$ glueball. In holographic models of QCD this glueball is
identified with the excitation of bulk gravitons [5, 6]. This, and other
consideration, relate Pomeron exchanges with Reggeized graviton exchanges.
These ideas have explained well the WA102 data production of pseudoscalar [7]
and tensor [8] glueballs. In this latter paper the PP-$2^{+}$ vertex used was
a 3-tensor coupling deduced from the Einstein-Hilbert action of general
relativity. Earlier suggestions that the Pomeron phenomenology needs some
polarization tensor were made by the Heidelberg group, see review in [9].

(ii) From the 1970’s the perturbative theory of the Pomeron was based first on
two-gluon exchanges, and then developed into inclusion of ladder diagrams with
gluonic rungs. This theory culminated in the famed papers [10] and is known as
the BFKL Pomeron. It is successful in describing diffraction with large
$|t|>few\,GeV$. Yet at small $|t|$ (or large impact parameter $b\sim
1/\sqrt{|t|}$) there is no reason to use perturbative QCD. Alternative theory
for this case, based on “stringy instanton” solution to Nambu-Goto string
exchange was developed in [11], and is known as the BKYZ Pomeron. Although the
two theories start from very different descriptions, with different
interpretations of the parameters, the scattering amplitude is nearly the
same. The latter BKYZ version, unlike BFKL, tells when the transition in
impact parameter should happen, as strings undergo Hagedorn transition.

##  2 The topological landscape

The Hilbert space of all possible gauge potentials $A^{\mu}_{a}(x)$ is the
manifold over which we need to integrate the Feynman path integral. Of course,
we would not discuss infinite-dimensional maps, and focus on two main
variables, for all static ( 3-dimensional and purely magnetic) configurations
of the lowest energy, consistent with the walue of those coordinates. One of
the coordinates is the topological Chern-Simons number

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

which is integer for pure gauge configurations, at which the energy is zero.
Those points will be referred to as ”valley minima”. The other coordinate is
the mean square radius of the magnetic field strength squared

$\rho^{2}\equiv{\int d^{3}x\vec{x}^{2}\vec{B}^{2}\over\int d^{3}x\vec{B}^{2}}$
(3)

Without fixing it, there is no minimal energy, as simple rescaling can change
it.

By the “topological landscape” we mean the 2-dimensional profile of the
minimal energy $U_{\rm min}(N_{CS},\rho)$ of gauge field configurations with
those two coordinates fixed. For pure gauge theory, such minimal energy
configurations themselves, are known as the “sphaleron path”: as changing
$N_{CS}$ from one integer to the next we are leading from one topological
valley to another, keeping minimal energy at any point. Those were derived by
Carter, Ostrovsky and one of us [12] by two different methods. The one related
with the instanton/sphaleron process will be discussed in the next section.
The second method is minimization with two Lagrange multipliers times two
conditions, (2) and (3). The minimal energy along the path was obtained in a
parametric form

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

The result shows a profile of the “topological mountain” of the gauge theory,
see Fig. 1, also known as “the sphaleron path”. Its maximum, at $\kappa=0$,
has $N_{CS}=\frac{1}{2}$ and its energy is known as the sphaleron mass

$\displaystyle M_{sph}=U_{\rm
min}\bigg{(}\frac{1}{2},\rho\bigg{)}={3\pi^{2}\over g^{2}\rho}$ (5)

Figure 1:  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 $N_{CS}=1$.

When the momentum scale $1/\rho$ is high, the gauge fields are very strong and
one can neglect the “vacuum structure” effects and keep only the classical
Yang-Mills equation. Of course, this equation is scale-invariant, and
therefore $U_{\rm min}\sim 1/\rho$. However, if $1/\rho$ is small enough, one
can no longer neglect vacuum effects.

In the electroweak theory these “vacuum effects” are described by the Higgs
field $\phi$. The scale is set by its VEV $v=\langle\phi\rangle\approx
246\,{\rm GeV}$. Ignoring first the Weinberg angle and using variational
methods, Klinkhamer and Manton [13] have included the Higgs field and solved
the YM+scalar equations. They observed that the Higgs field must vanish at the
origin $r=0$ to keep the topology of the gauge field intact, and therefore the
sphaleron is a “semi-empty bag” inside the Higgs vacuum. This adds some
energy: e.g. the leading term at large $\rho$ proportional to the bag volume
and the highest power of the Higgs VEV

$U_{\rm Higgs}\sim\rho^{3}v^{4}$

The $1/\rho$ gauge term plus Higgs terms with positive powers of the size
leads to a minimum at a certain $\rho^{*}$, which fixes the configuration
completely. The mass of the electroweak sphaleron is found to be about 9 TeV.
Thus, going from one topological valley of the electroweak theory to another,
one needs at least that much energy. All of that was clear since 1984.

Proceeding to our main object of interest, the sphaleron path in QCD, we note
that the QCD vacuum also has a rather complicated vacuum structure, with
nontrivial VEVs of various operators (known as “vacuum condensates”). There
are lattice data and models of the QCD vacuum structure, that provides some
understanding of the topological landscape at large $\rho$. It will be
discussed below, and here we just state that they also force the minimal
energy $U_{\rm min}(1/2,\rho)$ to start growing at large $\rho$, also
producing some optimal size $\rho^{*}$.

##  3 The tunneling paths

The first “tunneling path” was the original BPST instanton discovered in 1975.
It is the solution of the Euclidean YM equation. In terms of the landscape
illustrated in Fig. 1, it is a horizontal path, at energy zero, leading
directly from the bottom of one valley to the bottom of the next one. The
tunneling amplitude along it is $\sim{\rm exp}(-8\pi^{2}/g^{2})$, given by the
action of the instanton.

Many of the discussions on how to improve the tunneling path, including the
final state gluon radiation, were the subject of complicated technical papers
in the late 1980’s. We would not describe them, but just state that Khoze and
Ringwald [14], as well as Verbaarschot and Shuryak [15], have suggested that
4-dimensional configurations describing the $optimal$ instanton-antiinstanton
$I\bar{I}$ also provide the optimal path for sphaleron production. Since then,
most calculations follow this idea.

Before going to the details, let us explain why it is the case, following
important insights from [12, 16, 17]. The $I\bar{I}$ configuration is
schematically shown in the upper plot of Fig. 2. The shaded circles indicate
regions where the instanton and antiinstanton fields are strong. In one of
them, the fields are (close to) self-dual $\vec{E}=\vec{B}$, and in the other
antiself-dual $\vec{E}=-\vec{B}$. If both have the same sizes and color
orientations, symmetry of the plot suggest that the electric field $\vec{E}$
changes sign at $t\rightarrow-t$, and is therefore zero at $t=0$. The
3-dimensional plane in the middle of the plot must contain a pure magnetic
configuration, corresponding to semiclassical “turning points” where the
momentum vanishes. At such points the paths go from a tunneling process in
Euclidean time to a “real life” process in Minkowski space.

So, the $t=0$ plane is nothing else but the “unitarity cut”, familiar from
perturbative Feynman diagrams. The object on it $is$ the object produced. the
two halfs of the $I\bar{I}$ 4-dimensional configurations are the amplitude and
the conjugated amplitude. They describe the probability to produce this
object. Any discussion of the multi-gluon production, with the complicated
interferences between them, are not needed at all: what is produced is this
classical magnetic object.

Now, why is it that this tunneling path is better than the original instanton
(or $R\rightarrow\infty$ limit)? It may be confusing at first glance, since
the $I\bar{I}$ configurations are $not$ solutions of the YM equations. They
are tunnels $inclided$ $upward$ as indicated by the arrows in the lower plot
of Fig. 2. Indeed, to go uphill an external force is needed: but on those
paths the action is reduced! (As we will see, roughly by a factor 2, or many
decades in the rates.)

Instead of ending at the bottom of the valley, these paths end at some points
on the slopes (indicated by red and green balls). After the corresponding
magnetic objects are born, they roll down (explode) classically. Their action
is real, $|exp(iS)|=1$, and therefore their decays have probablity one.
Whatever decays they have, it does not affect the cross section.

As it is clear from the lower plot of Fig. 2, for small $R$ the instanton and
antiinstanton nearly annihilate each other and their combined action can be
reduced all the way to zero. If the product has $N_{CS}<1/2$, it will not roll
to the second valley, but return to the original one. The anomaly relation
then would indicate no fermionic transitions. In summary: there exist
topologically trivial and topologically nontrivial tunnelling paths! One
cannot economize more than (roughly) $half$ of the instanton+antiinstanton
action.

The issue of instanton-antiinstanton ($I\bar{I}$) interactions has its own
long history. When instanton and aniinstanton are well separated, the simple
“sum ansatz” gives twice the action, but what if the distance between them is
comparable to their size? One can invent many arbitrary interpolations.

The idea in [18] was to “follow the force” using $\partial S/\partial
x(\tau)$. In this way one gets a set of configurations, which are action
minima for infinitely many perturbations, except along one direction – the set
itself. One of us has independently generated this set numerically [19] for
the double well 111We did not know then that in mathematics our “sreamlines”
were known in complex analysis as “Lefschitz thimbles”, special lines
connecting saddles in the complex plane. . Yung [20] proposed the “Yung
ansatz” solving the streamline equation at large $R$.

For gauge theory instantons the problem looked more complicated. First of all,
even a sum ansatz could not be used, as special cancellations near the centers
(in singular gauge) were spoiled as the field strength gets singular. For this
purpose a “ratio ansatz” was developed to cure this. Furthermore, it looked
that the interaction should depend on at least 3 variables,
$\rho_{I},\rho_{A},R$, even for identical color orientations. Verbaarschot
[21] however, noticed that since the classical YM theory has conformal
symmetry, the answer should depend on their single conformal-invariant
combination 222Many years later, it was realized that this is a geodesic
distance between two points in AdS5 space, if $\rho$ is the extra coordinate.

${R^{2}+(\rho_{1}^{2}-\rho_{2}^{2})^{2}\over\rho_{1}\rho_{2}}$ (6)

Using an appropriate conformal map, the antiinstanton was set inside the
instanton, and the problem was mapped onto the double-well potential. As a
general surprise, Verbaarschot’s configurations happen to be described rather
accurately by Yung ansatz, not only at large $R$ as was originally claimed,
but in fact all the way to $R\rightarrow 0$ 333The formula itself remained
complicated, and nobody – Yung included – suspected it to be a pure gauge,
with zero field strength!.

Figure 2: The upper plot shows a schematic picture of the instanton-
antiinstanton configuration. The horizontal axis is (Euclidean) time $t$, and
$R$ is the distance between their centers. The blue triangle indicates $t=0$,
a 3-dimensional hyper-surface in which the produced magnetic object resides.
The lower plot with the red arrow, refers to the “reclined tunnel”
corresponding to large $R$. The green arrow on the left indicates tunneling
for small $R$, with a Chern-Simons number of the produced object $N_{CS}<1/2$.
For this last case, the classical explosion returns the system to the original
valley.

To complete the story, let us mention that in [12] it was shown that the
“streamline” $I\bar{I}$ configurations, well approximated by Yung ansatz, do
indeed describe $U_{\rm min}(N_{CS},\rho)$ in complete agreement with the
constrained minimization already shown in Fig 1. The parameter $\kappa$ is
played by the relative separation $R$.

Figure 3: The inversion of the 3-sphere $S_{\rho}^{3}$ of radius $\rho$ onto
the upper part of Minkowski space with $t\geq 0$ or $R_{+}^{4}$, through the
3-sphere $S^{3}_{2\rho}$ of radius $2\rho$.

##  4 Classically exploding sphalerons

Both static and time-dependent exploding solutions for the pure-gauge
sphaleron have been originally discussed by Carter,Ostrovsky and Shuryak (COS)
[12]. A simpler derivation, to be used below, has been subsequently given in
[22]. The technique relies on an off-center conformal transformation of the
$O(4)$ symmetric Euclidean instanton solution, which is analytically continued
to Minkowski space-time. However, the chief focus of that work [22] was the
description of the fermion production.

The original $O(4)$-symmetric solution is given by the following ansatz

$gA_{\mu}^{a}=\eta_{a\mu\nu}\partial_{\nu}F(y),\,\,\,\,F(y)=2\int_{0}^{\xi(y)}d\xi^{\prime}f(\xi^{\prime})$
(7)

with $\xi={\rm log}(y^{2}/\rho^{2})$ and $\eta$ the ’t Hooft symbol. Upon
substitution of the gauge fields in the gauge Lagrangian one finds the
effective action for $f(\xi)$

$S_{\rm eff}=\int d\xi\left[{\dot{f}^{2}\over 2}+2f^{2}(1-f)^{2}\right]$ (8)

corresponding to the motion of a particle in a double-well potential. In the
Euclidean formulation, as written, the effective potential is inverted

$V_{E}=-2f^{2}(1-f)^{2}$ (9)

and the corresponding solution is the well known BPST instanton, a path
connecting the two maxima of $V_{E}$, at $f=0,f=1$. Any other solution of the
equation of motion following from $S_{\rm eff}$ obviously generalizes to a
solution of the Yang-Mills equations for $A_{\mu}^{a}(x)$ as well. The
sphaleron itself is the static solution at the top of the potential between
the minima $f=-1/2$.

The next step is to perform an off-center conformal transformation as
illustrated in Fig. 3

$(x+a)_{\mu}={2\rho^{2}\over(y+a)^{2}}(y+a)_{\mu}$ (10)

with $a_{\mu}=(0,0,0,\rho)$. It changes the original spherically symmetric
solution to a solution of Yang-Mills equation depending on new coordinates
$x_{\mu}$, with a separate dependences on the time $x_{4}$ and the
3-dimensional radius $r=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$.

The last step is the analytic continuation to Minkowski time $t$, via
$x_{4}\rightarrow it$. The original parameter $\xi$ in terms of these
Minkowskian coordinates, which we still call $x_{\mu}$, has the form

$\xi={1\over 2}{\rm log}\bigg{(}{y^{2}\over\rho^{2}}\bigg{)}={1\over 2}{\rm
log}\left({(t+i\rho)^{2}-r^{2}\over(t-i\rho)^{2}-r^{2}}\right)$ (11)

which is purely imaginary. To avoid carrying the extra $i$, we use the real
combination

$\xi_{E}\rightarrow-i\xi_{M}={\rm arctan}\left({2\rho t\over
t^{2}-r^{2}-\rho^{2}}\right)$ (12)

and in what follows we will drop the suffix $E$. Switching from imaginary to
real $\xi$, corresponds to switching from the Euclidean to Minkowski spacetime
solution. It changes the sign of the acceleration, or the sign of the
effective potential $V_{M}=-V_{E}$, to that of the normal double-well problem.

The solution of the equation of motion is given in [22] 444There was a
misprint in the index of this expression in the original paper.

$f(\xi)={1\over 2}\left[1-\sqrt{1+\sqrt{2\epsilon}}\,{\rm
dn}\left(\sqrt{1+\sqrt{2\epsilon}}(\xi-K),{1\over\sqrt{m}}\right)\right]$ (13)

where ${\rm dn}(z,k)$ is one of the elliptic Jacobi functions,
$2\epsilon=E/E_{s},2m=1+1/\sqrt{2\epsilon}$, $E=V(f_{\rm in})$ is the
conserved energy of the mechanical system normalized to that of the sphaleron
$E_{s}=V(f=1/2)=1/8$. (13)reduces to the SO(4) solution derived by Luscher
[23], and the hypertorus solution obtained by Schechter [24]. Since starting
from exactly the maximum takes a divergent rolling time, we will start from
the nearby turning point with

$f(0)=f_{\rm in}={1\over 2}-\tau,\,\,\,\,\,\,f^{\prime}(0)=0$ (14)

where a small displacement $\tau$ ensures that the “rolling downhill” from the
maximum takes a finite time and that the half-period $K$ – given by an
elliptic integral – in the expression is not divergent. In the plots below we
will use $\kappa=0.01$, but the results dependent on its value very weakly.

The solution above describes a particle tumbling periodically between two
turning points, and so the expression above defines a periodic function for
all $\xi$. However, as it is clear from (12), for our particular application
the only relevant domain is $\xi\in[-\pi/2,\pi/2]$. Using the first 3 nonzero
terms of its Taylor expansion

$f\approx 0.4929-0.00706\,\xi^{2}-0.0011\,\xi^{4}-0.000078\,\xi^{6}$ (15)

we find a parametrization with an accuracy of $10^{-5}$, obviously invisible
in the plot and more than enough for our considerations.

The gauge potential has the form [22]

$gA_{4}^{a}=-f(\xi){8t\rho
x_{a}\over[(t-i\rho)^{2}-r^{2}][(t+i\rho)^{2}-r^{2}]}$ (16) $gA^{a}_{i}=4\rho
f(\xi){\delta_{ai}(t^{2}-r^{2}+\rho^{2})+2\rho\epsilon_{aij}x_{j}+2x_{i}x_{a}\over[(t-i\rho)^{2}-r^{2}][(t+i\rho)^{2}-r^{2}]}$

which is manifestly real. From those potentials we generate rather lengthy
expressions for the electric and magnetic fields, and eventually for CP-
violating operators, using Mathematica.

Note that the sphaleron solution corresponds to $t=0$ or static, which is a
pure magnetic solution with $gA_{4}^{a}=0$. The magnetic field squared is
spherically symmetric and simple

$\vec{B}^{2}={96\rho^{4}\over(\rho^{2}+r^{2})^{4}}$ (17)

Figure 4: Upper plot: kinematics for two emerging quarks with momenta
$\vec{k}_{1,2}$ from an exploding sphaleron, fusing into a meson of momentum
$\vec{p}$; Lower plot: squared projection of outgoing quark waves into a
mesonic distribution amplitude with momentum $p\,({\rm GeV})$.

The ensuing explosion starting from the sphaleron at $t=0$, we describe using
the YM equations. This is justified only if the sphaleron size is sufficiently
small $\rho\ll\rho_{\rm max}$ compared to the maximal size defined by the
vacuum effects. If $\rho v\sim 1$ (where $v$ is the Higgs VEV in the
electroweak theory, or the dual Higgs VEV in QCD), the equations get modified
by additional terms with scalar fields.

It is known for a long time, that the Adler-Bell-Jackiw chiral anomaly relates
the divergence of the axial current to the topological charge $Q$. Also
divergence of the Chern-Simons topological current is proportional to it. As a
result, their combination is conserved, and changes in the axial charge and
Chern-Simons number are locked

$\Delta Q_{5}=(2N_{f})\Delta N_{CS}$ (18)

where $N_{f}=3$ is the number of light quark flavors.

In agreement with these general arguments, we have constructed in [22] the
analytical solutions of the Dirac equation in the field of exploding
sphalerons. As a function of time, one can see how quarks are accelerated from
the lower Dirac sea to the physical continuum of positive energies. The
spectrum of outgoing quarks has a simple form

${dN\over dk}=4\pi k^{2}|\psi(k)|^{2}=\rho(2k\rho)^{2}e^{-2k\rho}$ (19)

which is similar to a Planckian spectrum with an effective temperature $T_{\rm
eff}=1/(2\rho)$. For the maximum value set by the vacuum instanton size
distribution $\rho_{\rm max}\approx 0.3\,{\rm fm}\approx 1/(600\,{\rm MeV})$,
the effective temperature is $300\,{\rm MeV}$. The mean momentum is

$\langle k\rangle={3\over 2\rho}\sim 900\,{\rm MeV}$

A simple quantum estimate of the production amplitude, follows from the
projection of the emerging quarks onto the distribution amplitudes (DAs) of
the outgoing mesons at any momenta as illustrated in Fig.4 (upper). More
specifically, we have

$\langle\vec{k}_{1},\vec{k}_{2}|\vec{p}\rangle\sim\int
dk_{1\perp}^{2}dk_{1l}dk_{1\perp}^{2}dk_{1l}\psi(\vec{k}_{1})\psi(\vec{k}_{2})f(p_{\perp}^{2})\varphi(x=k_{1l}/P)\delta(k_{1\perp}^{2}-p_{\perp}^{2})\delta(k_{2\perp}^{2}-p_{\perp}^{2})\delta(k_{1l}+k_{2}l-P)$
(20)

where $\psi(k)$ is the outgoing quark wave (19), and the functions
$\varphi(x)f(p_{\perp}^{2})$ are the DAs of the corresponding mesons with
longitudinal fraction ($x$) and transverse ($p_{\perp}$) momenta. For
simplicity, we take a Gaussian $f(p_{\perp})\sim{\rm exp}(-{\rm
const}*p_{\perp}^{2})$ and a flat $\varphi(x)=1$ which approximate well say a
pion. The squared projection for a sphaleron of size $\rho=0.3\,{\rm
fm}=1/(0.6\,{\rm GeV})$ is shown in the lower plot, as a function of the
outgoing meson momentum $p_{\perp}$.

(Note that in this estimate we ignored all the other particles produced. In
reality the total mass of the cluster puts its own kinematical restrictions.
For example, for three-meson decay modes to be discussed, the tail at large
momenta is cutoff above $M/3$.)

##  5  The instanton size distribution in the QCD vacuum

By now, the subject of instantons in the QCD vacuum is well established and
broad, and clearly goes beyond the scope of this review. For us, the only
relevant issue is the instanton size distribution $dn/d\rho$ in the vacuum. It
has been evaluated in various models and on the lattice. For definiteness we
use the lattice results from [25]. The average size was found to be
$\langle\rho\rangle\approx 0.30\pm 0.01\,{\rm fm}$, a bit smaller than in the
ILM. The mean distance was found instead to be $0.61\pm 0.02\,fm$. The data on
the instanton size distribution are shown in Fig.5. (The figure is taken from
[26] and the lattice data from Hasenfratz et al [25]). The left plot shows the
size distribution itself. Recall that the semiclassical theory predicts it to
be $dn/d\rho\sim\rho^{b-5}$ at small sizes, with $b=11N_{c}/3=11$ for pure
gauge $N_{c}=3$ theory. The right plot – in which this power is taken out –is
constant at small $\rho$, which agrees with the semiclassical prediction.

The other feature is a peak at $\rho\approx 0.3\,{\rm fm}$ – the value first
proposed phenomenologically in [27], decades before the lattice data. The peak
is due to a suppression at large sizes. Trying to understand its origin, one
may factor out all known effects. The right plot shows that after this is
done, a rather simple suppression factor $\sim{\rm exp}(-{\rm
const}*\rho^{2})$ describes it well, for about 3 decades. What is the physical
origin of this suppression?

There are two answers to that question, which are perhaps “Poisson dual” to
each other [28]. The first is that it is due to the mutual repulsion between
an instanton and the rest of the instanton-antiinstanton ensemble. (It is
described in the mean field approximation and numerically, see the review
[29]).

Another one, proposed in [26], is that the coefficient is proportional to the
dual magnetic condensate, that of Bose-condensed monopoles. It has been
further argued there that it can be related to the string tension $\sigma$, so
that the suppression factor should be

${dn\over d\rho}=\bigg{(}{dn\over d\rho}\bigg{)}_{\rm
semiclassical}exp[-2\pi\sigma\rho^{2}]$ (21)

where the Higgs VEV is traded for the string tension $\sigma$ via the dual
Higgs model of confinement. If this idea is correct, this suppression factor
should be missing at $T>T_{c}$, in which the dual magnetic condensate is
absent. However, in this regime, quantum/thermal fluctuations generate at high
$T$ a similar factor [30]

${dn\over d\rho}=\bigg{(}{dn\over d\rho}\bigg{)}_{T=0}exp[-(2N_{c}+{N_{f}\over
3})(\pi\rho T)^{2}]$ (22)

related to the scattering of the quarks and gluons of the quark-gluon-plasma
(QGP) on the instanton [31]. Empirically, the suppression factor at all
temperatures looks Gaussian in $\rho$, interpolating between those limiting
regimes.

Figure 5: (left) The instanton size $\rho$ [fm] distribution $dn/(d\rho
d^{4}z)$. (right) The combination $\rho^{-6}dn/d\rho d^{4}z$, in which the
main one-loop behavior drops out for $N_{c}=3,N_{f}=0$. The points are from
the lattice work for this theory, with $\beta$=5.85 (diamonds), 6.0 (squares)
and 6.1 (circles). Their comparison should demonstrate that results are
lattice-independent. The line corresponds to the proposed expression, see
text.

##  6 Quark flavor signatures of instanton/sphaleron process at small
$M=3-10\,{\rm GeV}$

###  6.1 The instanton-induced glueball and $\eta_{c}$ decays

As emphasized earlier, the instantons are virtual paths at zero energy,
playing a significant role in the theory of the QCD vacuum, for a review see
e.g. [29]. Their existence has been unambiguously revealed by the cooled
lattice simulations carried by many groups, e.g. Leinweber [32].

The subject of this work, the instanton/sphaleron process, requires external
forces able to provide the amount of energy needed for the sphaleron
production. Furthermore, one should look for reactions involving gluons in
combinations to which instantons are naturally coupled. Those are in scalar
combinations or $G_{\mu\nu}^{2}$ and pseudoscalar combinations or
$G_{\alpha\beta}G_{\gamma\delta}\epsilon^{\alpha\beta\gamma\delta}$. In order
of increasing mass, they are:
(i) decays of a scalar glueball ($M\approx 1.6\,{\rm GeV}$);
(ii) decays of a pseudoscalar glueball ($M\approx 2.5\,{\rm GeV}$);
(iii) decays of pseudoscalar quarkonia555We are not aware of any study of
$\eta_{b}$ decays in this resepect. ($M_{\eta_{c}}\approx 3\,{\rm GeV}$)

The main idea of this work is that gluon-gluon or (especially) Pomeron-Pomeron
collisions also couple to such operators by fusion, and through them one can
possibly investigate instanton/sphaleron processes of larger masses $M_{\rm
sph}$ at hadron colliders.

Historically, the issue of instanton-induced decays was first noticed by
Bjorken [33]. He pointed out that $\eta_{c}$ decays have 3 large 3-body modes,
several percents each of the total width:

$\eta_{c}\rightarrow KK\pi;\,\,\,\pi\pi\eta;\,\,\,\pi\pi\eta^{\prime}$

Note that there is no $\pi\pi\pi$ decay mode, or other decay modes we may
think of without strangeness: the ’t Hooft vertex $must$ have all light quark
flavors in it, including $\bar{s}s$.

More generally, charmonium decays are especially nice since one can use the
well known $J/\psi$ decays as a ′′control group′′, since its three-gluon
annihilation mode does not go to operators that couple naturally to
instantons. Indeed, the branchings of the 2- and 3-body decays of $J/\psi$ are
much smaller, and the average multiplicity of its decays is much larger than
3. That is why the few-percent branching of these 3-meson $\eta_{c}$ decays is
remarkable by itself.

The actual calculations of $\pi\pi,KK,\eta\eta$ decays of scalar and
$KK\pi,\eta\pi\pi,\eta^{\prime}\pi\pi$ decays of $\eta_{c}$ were made by
Zetocha and Schafer [34]. Their results contain rather high power of the
instanton radius and therefore strongly depend on its value. So the authors
used the inverted logic, evaluating from each data point the corresponding
value of the mean instanton size $\bar{\rho}$. The results reproduced the
decay widths reasonably well. Furthermore, these calculations provided about
the most accurate evaluation of the average instanton size available, in the
range of $\bar{\rho}=0.29-0.30\,fm$, common to all decay channels.

Let us start with the flavor composition. The specific form of the effective
operator between light quarks induced by instantons has been derived by ’t
Hooft

$(\bar{u}_{R}u_{L})(\bar{d}_{R}d_{L})(\bar{s}_{R}s_{L})+(L\leftrightarrow R)$

with the coefficients following from the LSZ reduction of the zero modes. For
the the lightest quark clusters, of mass $M\sim 3\,{\rm GeV}$, we expect the
production of 6 quarks in combination $uuddss$. Before discussing the decay
modes, we need to look closer at $N_{f}=3$.

###  6.2 The simplified forms of the effective Lagrangian

One issue is that in [34] the emitted mesons are treated in the “local
approximation”, with the vertex Lagrangian directly projected to meson “decay
constants” (which are the values of the wave functions at the origin). In
reality, mesons fly with a momentum of about $1\,{\rm GeV}$, for which the
projections from the initial quasi-local state to the final state (the form
factors) are not exactly one. This issue of “projection” (which we have
already addressed above) gets more important as one considers sphalerons with
larger masses.

Another (more technical) issue is the correct inclusion of all diagrams.
Whatever form chosen for the 6-q Lagrangian, there are $3!=6$ channels
(various connections of quarks and antiquarks), and they all needs to be
included. While it is possible to go from one form to to another with Fierz
transformation of operators, we were not sure all 6 connections were included
on equal footing, and decided to repeat the calculation.

Another reason for redoing it is that the 3-flavor version of the Lagrangian
used in [34] (their (26)) is very complicated, containing expressions with
three color matrices and structure constants of the $SU(3)$ group

$f^{abc}\lambda^{a}\lambda^{b}\lambda^{c},\,\,\,\,d^{abc}\lambda^{a}\lambda^{b}\lambda^{c}$

As we will show, these can be eliminated in favor of a simpler form, which we
use. (In B.1 we will also show one more generic form of this Lagrangian based
on the so called Weingarten coefficients of the product of unitary matrices
averaged over the groups.)

We start by explaining the two main technical complications in this problem. A
single instanton (or sphaleron) is constructed in a color $SU(2)$ setting,
with the gauge fields $A^{a}_{\mu}T^{a}_{ij}$ with color indices $i,j=1,2$.
The quark zero modes $\psi_{\alpha}^{j}$ in the chiral representation
($\gamma_{5}$ is diagonal) carry a matching spin index $\alpha=1,2$ and color
index $j=1,2$, coupled to each other in a “hedgehog” way.

However, the gauge group in QCD is $SU(3)$ and in it, the instanton appears
rotated from its standard form (just described) to some arbitrary 2d plane
$\in SU(3)$ by a unitary $3\times 3$ matrix, e.g.
$\psi^{a}_{\alpha}\rightarrow U^{a}_{i}\psi^{i}_{\alpha}$. The ’t Hooft
effective Lagrangian contains $2N_{f}=6$ (or 8 or 10) quark zero modes, and
therefore the 6-th (or 8-th or 10-th) power of this matrix. Although the
explicit expressions for “isotropic” averaging of these powers of $U$ have
been known for some time (see e.g. appendix of [35]) these formulae are rather
complicated, and contain convolutions with the structure constants
$f^{abc},d^{abc}$ of $SU(3)$. Schafer and Zetocha [34] used such form, see
eqn(26) of their paper.

Another technical issue is as follows. Multifermion operators can be
identically represented in many different forms, since one can color-couple
different $\bar{q}q$ pairs. (Or even $qq$ pairs as is convenient for color
superconductivity and baryon production.) Those “Fierzing” transformations are
jus identities, but with flavor, color and Dirac indices involved, they can
create a multitude of operators. Yet, whatever form of the vertex operator is
used, one still need to include it in all possible channels. For example, for
the operator structure $\bar{q}\bar{q}\bar{q}qqq$ projected onto 3 mesons,
there are obviously $3!=6$ ways to relate it to three mesonic $\bar{q}q$ wave
functions. For 8-q operators there are $4!$, etc.

Here and in Appendix B we show how one can simplify the operator structure, by
keeping its flavor determinantal form intact, which has no structure constants
as quoted in [36], and used for heavy-light multi-quark systems [37]. We then
explicitly do all possible convolutions with all mesonic wave functions in
Mathematica, avoiding Fierzing altogether. More specifically [36] (See eq.
2.56)

$\displaystyle{\cal V}^{L+R}_{qqq}=$
$\displaystyle\frac{\kappa}{N_{c}(N_{c}^{2}-1)}\bigg{(}\frac{2N_{c}+1}{2(N_{c}+2)}\,{\rm
det}(UDS)$ $\displaystyle+\frac{1}{8(N_{c}+1)}\,\bigg{(}{\rm
det}(U_{\mu\nu}D_{\mu\nu}S)+{\rm det}(U_{\mu\nu}DS_{\mu\nu})+{\rm
det}(UD_{\mu\nu}S_{\mu\nu})\bigg{)}\bigg{)}+(L\leftrightarrow R)$ (23)

with a strength

$\kappa=\frac{{n_{I+\bar{I}}}}{2}\bigg{(}\frac{4\pi^{2}\rho^{3}}{M\rho}\bigg{)}^{3}$
(24)

and the short hand notations ($Q\equiv U,D,S$)

$Q=\overline{q}_{R}q_{L}\qquad
Q_{\mu\nu}=\overline{q}_{R}\sigma_{\mu\nu}q_{L}\qquad
Q^{a}=\overline{q}_{R}\sigma^{a}q_{L}$ (25)

with $\sigma_{\mu\nu}=\frac{1}{2}[\gamma_{\mu},\gamma_{\nu}]$. Note that the
spin contribution is sizably suppressed by $1/8N_{c}$ in the large $N_{c}$
limit, when compared to the scalar contribution. It is clear from our Fierzing
arguments in Appendix B and symmetry, that only two determinantal invariants
will survive after Fierzing as in (6.2), with only the structures $UDS$ and
$U_{\mu\nu}D_{\mu\nu}S$ and their permutations allowed. The only non-trivial
results are their weight coefficients. This observation holds for 4, 8 and 10
quark vertices as well, assuming they allow for zero modes. For instance, for
4-quark vertices the general structure is

$\displaystyle{\cal V}^{L+R}_{qq}=\,{\kappa}_{2}\,A_{2N}\,\bigg{(}{\rm
det}(UD)+B_{2N}\,{\rm det}(U_{\mu\nu}D_{\mu\nu})\bigg{)}+(L\leftrightarrow R)$
(26)

which is readily checked to hold with

$\displaystyle A_{2N}=\frac{(2N_{c}-1)}{2N_{c}(N_{c}^{2}-1)}\qquad
B_{2N}=\frac{1}{4(2N_{c}-1)}$ (27)

Note that at large $N_{c}$, the suppression of the spin contribution is still
exactly $1/8N_{c}$, with $A_{\\#q}\approx 1/N_{c}^{\\#}$. Remarkably, both the
value of the $A,B$ coefficients and their determinantal structures are fixed
uniquely in this limit by symmetry and scaling. For completeness, the 8-quark
vertices are of the form

$\displaystyle{\cal V}^{L+R}_{qqqq}={\kappa}_{4}\,A_{4N}$
$\displaystyle\bigg{(}{\rm det}(UDSC)+B_{4N}\,\bigg{(}{\rm
det}(U_{\mu\nu}D_{\mu\nu}SC)+\,{\rm perm.}\bigg{)}$ (28)
$\displaystyle+C_{4N}\bigg{(}{\rm
det}(U_{\mu\nu}D_{\mu\nu}S_{\alpha\beta}C_{\alpha\beta})+\,{\rm
perm.}\bigg{)}\bigg{)}+(L\leftrightarrow R)$

although for the heavy charm the use of the $L,R$ zero modes may not be
justified.

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 (6.2) can be simplified

$\displaystyle{\cal V}^{L+R}_{qqq}=$
$\displaystyle\frac{\kappa}{N_{c}(N_{c}^{2}-1)}\bigg{(}\frac{2N_{c}+1}{2(N_{c}+2)}\,{\rm
det}(UDS)$ $\displaystyle-\frac{1}{2(N_{c}+1)}\,\bigg{(}{\rm
det}(U^{a}D^{a}S)+{\rm det}(U^{a}DS^{a})+{\rm
det}(UD^{a}S^{a})\bigg{)}\bigg{)}+(L\leftrightarrow R)$ (29)

${\cal V}_{qqq}$ is only active in flavor singlet 6-quark states. The flavor
determinantal interactions can be made more explicit by using the permutation
operators in flavor space as the symmetric group $S_{3}$ of permutations is
composed of $3!$ terms only, 3 cyclic-permutations with positive signature and
3 pair-permutations with negative signature. Clearly, the 3-body instanton
induced interaction (6.2) does not vanish only in flavor singlet $uds$ states
(repulsive). Its 2-body reduction is attractive in states with a pair of
antisymmetric flavor diquarks $ud$, $us$, $ds$ (attractive). A more explicit
writing of (6.2) suitable for numerical analysis in terms of explicit $3\times
3$ flavor determinants is

$\displaystyle{\cal
V}^{L+R}_{qqq}=\frac{\kappa}{N_{c}(N_{c}^{2}-1)}\bigg{[}\bigg{(}\frac{2N_{c}+1}{2(N_{c}+2)}\bigg{)}\begin{Vmatrix}\overline{u}_{R}u_{L}&\overline{u}_{R}d_{L}&\overline{u}_{R}s_{L}\\\
\overline{d}_{R}u_{L}&\overline{d}_{R}d_{L}&\overline{d}_{R}s_{L}\\\
\overline{s}_{R}u_{L}&\overline{s}_{R}d_{L}&\overline{s}_{R}s_{L}\\\
\end{Vmatrix}$
$\displaystyle-\frac{1}{2(N_{c}+1)}\sum_{a=1}^{3}\bigg{(}\begin{Vmatrix}\overline{u}_{R}\sigma^{a}u_{L}&\overline{u}_{R}\sigma^{a}d_{L}&\overline{u}_{R}s_{L}\\\
\overline{d}_{R}\sigma^{a}u_{L}&\overline{d}_{R}\sigma^{a}d_{L}&\overline{d}_{R}s_{L}\\\
\overline{s}_{R}u_{L}&\overline{s}_{R}d_{L}&\overline{s}_{R}s_{L}\\\
\end{Vmatrix}+\begin{Vmatrix}\overline{u}_{R}\sigma^{a}u_{L}&\overline{u}_{R}d_{L}&\overline{u}_{R}\sigma^{a}s_{L}\\\
\overline{d}_{R}u_{L}&\overline{d}_{R}d_{L}&\overline{d}_{R}s_{L}\\\
\overline{s}_{R}\sigma^{a}u_{L}&\overline{s}_{R}d_{L}&\overline{s}_{R}\sigma^{a}s_{L}\\\
\end{Vmatrix}+\begin{Vmatrix}\overline{u}_{R}u_{L}&\overline{u}_{R}d_{L}&\overline{u}_{R}s_{L}\\\
\overline{d}_{R}u_{L}&\overline{d}_{R}\sigma^{a}d_{L}&\overline{d}_{R}\sigma^{a}s_{L}\\\
\overline{s}_{R}u_{L}&\overline{s}_{R}\sigma^{a}d_{L}&\overline{s}_{R}\sigma^{a}s_{L}\\\
\end{Vmatrix}\bigg{)}\bigg{]}$ $\displaystyle+(L\leftrightarrow R)$ (30)

###  6.3 Mesonic decays of sphalerons

Convoluting the vertex function in the form (6.2) with various mesons wave
functions in all possible $3!=6$ ways we obtain the matrix elements for a
number of 3-meson decay channels, listed in the Table. The meson definitions
and couplings are defined in Appendix A.

|  | PDG2020 | input [34] | —M—
---|---|---|---|---
$K\bar{K}\pi$ | $K^{+}K^{-}\pi^{0}$ |  $7.3\pm 0.4$ (all 4) |  $5.5\pm 1.7$ (all 4) | 5.07 $K_{K}^{2}K_{\pi}$
| $K^{+}\bar{K}^{0}\pi^{-}$ |  |  | 7.27 $K_{K}^{2}K_{\pi}$
| $K^{0}\bar{K}^{0}\pi^{0}$ |  |  | 5.07$K_{K}^{2}K_{\pi}$
| $K^{-}K^{0}\pi^{+}$ |  |  | 7.27$K_{K}^{2}K_{\pi}$
$\pi\pi\eta$ | $\pi^{+}\pi^{-}\eta$ | $1.7\pm 0.5$ |  $4.9\pm 1.8$ (both) | 4.92 $K_{\pi}^{2}K_{\eta}^{s}$
| $\pi^{0}\pi^{0}\eta$ |  |  | 2.46$K_{\pi}^{2}K_{\eta^{\prime}}^{s}$
$\pi\pi\eta^{\prime}$ | $\pi^{+}\pi^{-}\eta^{\prime}$ |  $4.1\pm 1.7$ (both) |  $4.1\pm 1.7$ (both) | 5.20 $K_{\pi}^{2}K_{\eta^{\prime}}^{s}$
| $\pi^{0}\pi^{0}\eta$ |  |  | 2.60$K_{\pi}^{2}K_{\eta^{\prime}}^{s}$
$\bar{K}K\eta$ | $K^{+}K^{-}\eta$ |  $1.36\pm 0.15$ (both) |  | 3.68 $K_{K}^{2}F_{\eta}^{q}$
| $K^{0}\bar{K}^{0}\eta$ |  |  | 3.68 $K_{K}^{2}F_{\eta}^{q}$
$\bar{K}K\eta^{\prime}$ | $K^{+}K^{-}\eta^{\prime}$ |  |  | 3.53 $K_{K}^{2}F_{\eta^{\prime}}^{q}$
| $K^{0}\bar{K}^{0}\eta^{\prime}$ |  |  | 3.53 $K_{K}^{2}F_{\eta^{\prime}}^{q}$
$\eta\eta\eta$ |  |  |  | 1.32$(K_{\eta}^{q})^{2}K_{\eta}^{s}$
Table 1: The first column gives the generic names of the decay channels of
$\eta_{c}$, while the second column records the specific channels. The third
column contains the corresponding branching ratio (percents) according to the
Particle Data Table 2020. For comparison, we show in the fourth column the
corresponding numbers used in [34]. The last column gives the decay matrix
elements. The meson-specific constants (wave function at the origin) are
defined in Appendix A.

Our first comment on the table is that in that two decades from the work in
[34], some of the experimental branching ratios have improved their accuracy,
while some are substantially modified. This needs to be kept in mind when
comparing the predictions to experiment.

Of course, we can construct many ratios out of the Table. Here, we will
mention two in particular

$\displaystyle{\Gamma(K\bar{K}\pi)\over\Gamma(\eta\pi\pi)}$ $\displaystyle=$
$\displaystyle{\sum|M|^{2}\over\sum|M|^{2}}{0.111\over 0.135}\approx
10\,\,\,\,\,\,\,\bigg{(}{\rm exp}:{7.3\pm 0.4\over 2.55\pm 0.75}\bigg{)}$ (31)
$\displaystyle{\Gamma(\eta\pi\pi)\over\Gamma(\eta^{\prime}\pi\pi)}$
$\displaystyle=$ $\displaystyle{\sum|M|^{2}\over\sum|M|^{2}}{0.135\over
0.0893}\approx 0.9\,\,\,\bigg{(}{\rm exp}:{2.55\pm 0.75\over 4.1\pm
1.7}\bigg{)}$

where the last ratio corresponds to the three-body phase space. The ratios are
only in qualitative agreement with the reported measurements. Clearly more
studies are needed.

###  6.4 Baryonic decays of sphalerons

Another fascinating issue, not so far discussed in the literature, is whether
the 6-fermion effective Lagrangian can be used to produce a baryon-antibaryon
pair, rather than 3 mesons.

At first sight one finds a problem which looks quite severe. The operator
involves the quark set $u,d,s$, and the natural baryons to look at are
$\Lambda$ or $\Sigma^{0}$ baryons. As for any baryon, their color wave
function is antisymmetric in color, $\epsilon_{c_{1}c_{2}c_{3}}$. The flavor
determinant is also antisymmetric in flavor $\epsilon_{f_{1}f_{2}f_{3}}$.
Fermi statistics then require the remaining part of the wave function to be
antisymmetric. In the lowest shell of the quark model, this remaining part is
made of three quark spins. It is not possible to create
$\epsilon_{s_{1}s_{2}s_{3}}$ spin wave functions, since the spin indices are
$1,2$ only. Indeed it is well known, that the lowest baryonic octet does not
have the ninth state, a singlet, unlike the mesons.

This notwithstanding, let us also approach the issue phenomenologically. we
assume, following Bjorken, that $\eta_{c}$ decays possess topological paths,
while $J/\psi$ decays do not. So, what do we see in the baryonic sector?

Both $\eta_{c}$ and $J/\psi$ have observed decays into $\bar{\Lambda}\Lambda$,
with similar branching $\sim 10^{-3}$. Yet their absolute widths differ by
roughly a factor of 200, in favor of the former case. What speaks against the
topological mechanism is the fact that in both cases the channels $\bar{p}p$
and $\Sigma^{+}\Sigma^{-}$ have similar yields to $\bar{\Lambda}\Lambda$,
although they cannot follow from our 6-fermion effective vertex.

In summary, topological production of $\bar{\Lambda}\Lambda$ does not appear
to take place in $\eta_{c}$ decays.

Yet, if the sphaleron mass is larger than 3 GeV, one may think of production
of two higher mass baryon resonances. Among those there are known $SU(3)$
singlets, which are anti-symmetric in flavor. The lowest of them are

$\Lambda(1405)\,\,:\,\,J^{P}=1/2^{-},\qquad\Lambda(1520)\,\,:\,\,J^{P}=3/2^{-}$

Their negative parity is explained in the quark model by a unit of orbital
momentum $L=1$. Unfortunately, this negative parity also protect them from any
mixing with the usual $\Lambda$. As they are well known resonances, their
back-to-back production should be quite noticeable. Their widths are
$\Gamma=50$ and $157\,{\rm MeV}$, respectively. They both have very
characteristic decays,

$\Lambda(1405)\rightarrow\Lambda\gamma,\,\,\,\Lambda(1520)\rightarrow\Sigma\pi$

with branching ratios of the order of 50% for both.

###  6.5 Chirality correlation in baryonic decays

The $\Lambda\bar{\Lambda}$ channel is the most interesting for two reasons:

(i) Zero isospin means that $(ud)$ diquark has spin zero, and therefore the
whole spin of $\Lambda$ is carried by its strange quark;
(ii) weak decays of $\Lambda$ hyperon allows to observe its polarization.
Indeed, in the decay $\Lambda\rightarrow p\pi^{-}$ of polarized hyperon, the
direction of the proton is mostly along the initial polarization direction.

As a measure of $s$ quark chirality one can use

$\xi_{\Lambda}\equiv{\rm
cos}\big{(}\theta(\vec{p}_{\Lambda}\vec{p}_{p})\big{)}$ (32)

and calculate the distributions $P(\chi_{\Lambda\bar{\Lambda}})$ over the
relative chiralities, the product of

$\chi_{\Lambda\bar{\Lambda}}\equiv\xi_{\Lambda}\times\xi_{\bar{\Lambda}}$ (33)

Ordinary perturbative diagrams with one or two gluons (or photons) leading to
the production of a strange quark pair are “vector-like”, meaning that the
chiralities are the same,

$(\bar{s}_{L}s_{L})+(\bar{s}_{R}s_{R})$

This means either both $\xi$ are positive, or both negative, leading to
$\chi_{\Lambda\bar{\Lambda}}$ positive.

On the other hand, the instanton/sphaleron-induced vertex is, $non-diagonal$
in chirality

$(\bar{s}_{L}s_{R})+(\bar{s}_{R}s_{L})$

The produced $\Lambda$ and $\bar{\Lambda}$ should therefore have the
$opposite$ chiralities, and $\chi_{\Lambda\bar{\Lambda}}$ is negative. We are
not aware of such study even in any inclusive reactions, in which a pair of
Lambda hyperon decays are identified with some reasonable statistics.

As we discussed above, the exclusive production of $\bar{\Lambda}\Lambda$ from
the t’ Hooft-like Lagrangian is not possible. Yet strong chirality
correlations in question would perhaps persist in channels with other
associate hadrons. For example, in the production of
$\bar{\Lambda}(1405)\Lambda(1405)$ with their subsequent radiative decays into
$\Lambda+\gamma$, there should remain rather significant correlation of
polarizations.

If observed, it would be an excellent indication of the topological origin of
the vertex, pointing to an explicit violation of the $U_{A}(1)$ symmetry.

##  7 Sphaleron decays at medium masses:
$M=10-20\,{\rm GeV}$ and 10-fermion operators

###  7.1 Charm pairs and decays with 8-fermion operators

The field magnitude at the center of the instantons is comparable to
$m_{c}^{2}$. Although charm is not usually treated as a light flavor, it must
have a certain coupling to instantons. The fact of large instanton-induced
decays of $\eta_{c}$ confirms this idea.

The 8-flavor operators have also a flavor-asymmetric structure

$V\sim(\bar{u}u)(\bar{d}d)(\bar{s}s)(\bar{c}c)$

with a typical determinantal structure . There are $4!=24$ 4-meson channels
one should think of: for example those can be the same 3-meson channels
discussed above, $\pi\pi\eta,\pi\pi\eta^{\prime},KK\pi$ with an $added$
$\eta_{c}$. The other quite distinct channels are e.g. $KKDD$ $without$ any
pions. Note also that 4 pseudoscalar mesons without orbital momentum
correspond to operator $G^{2}$, not the pseudoscalar one.

###  7.2 Multi-glueball production

We already mentioned several times, that the total sphaleron mass is ${\cal
O}(3\pi^{2}/g^{2}\rho)$ while the energy of produced gluons are only
$O(1/\rho$) . The gauge quanta multiplicity is therefore $\sim
1/g^{2}\sim{\cal O}(100)$ in the electroweak theory and ${\cal O}(10)$ in QCD.

Therefore, in the “medium sphaleron mass” range indicated, the energy per
gluon is in the range $E_{g}=1-2\,{\rm GeV}$. We know, from lattice studies
and models, that under such conditions gluons are not yet “partons” with
independent life. Instead, they are paired into the lowest glueball states
with masses

$M_{0^{+}}\approx 1.6\,{\rm GeV}\,\qquad M_{2^{+}}\approx 2.2\,{\rm GeV}$ (34)

So, we propose that in this mass range a signficant part of the energy will go
into several lowest glueballs. All of them are now reasonably well identified
with scalar and tensor resonances.

###  7.3 Semiclassical production in constant electric field and in sphaleron
explosion

Another new interesting option is the production of $\bar{b}b$ pair by
sphaleron decays.

Before we provide some estimates, let us recall how the related classic
problem of the pair production of $e^{+}e^{-}$ in a constant electric field
can be described semiclassically. It is widely known as the Schwinger process,
as he solved it in detail in 1950. However, we will discuss neither the
Schwinger paper, nor even the earlier Heisenberg-Euler paper, but the much
earlier semiclassical work [38] from 1931 (well before anyone else).

The EOM of a charge relativistically moving in constant electric field is a
classic problem which everybody had encounter in E/M classes. Writing it in
the usual Minkowski form

${dp\over dt}={d\over dt}\bigg{(}{v\over\sqrt{1-v^{2}}}\bigg{)}={eE\over
m}\equiv a$ (35)

yields the hyperbolic solution

$v_{M}(t)={at\over\sqrt{1+a^{2}t^{2}}},\,\,\,\,x_{M}(t)={1\over
a}\big{(}\sqrt{1-a^{2}t^{2}}-1\big{)}$ (36)

for a particle that starts at rest with a nonrelativistic acceleration,
$x_{m}=at^{2}/2$. At large times, the motion turns ultrarelativistic with
$v_{M}\rightarrow 1$.

The analytical continuation of the the trajectory to Euclidean time $\tau=it$
yields

$v_{E}={a\tau\over\sqrt{1-a^{2}\tau^{2}}},\,\,\,x_{E}(\tau)={1\over
a}\big{(}\sqrt{1+a^{2}\tau^{2}}-1\big{)}$ (37)

At time $\tau=0$ the particle is at the origin with zero velocity,
$x_{E}=0,v_{E}=0$. For $-1/a<\tau<1/a$ it describes an Euclidean path in shape
of a semicircle. In Euclidean space, the world-time is no different from the
other coordinates, and the electric field $G_{01}$ is not different from the
magnetic field, so in the 0-1 plane the paths are circles, like they are in
all other planes.

To understand the physical meaning of the semicircle (blue dashed line) in the
Dirac sea interpretation one needs to split it into two halves. The path
describes tunneling through the “mass gap”, between energies $-M$ and $M$ in
the spectrum of states. In Minkowski space there are no states between
$E=-\sqrt{p^{2}+m^{2}}$ and $E=\sqrt{p^{2}+m^{2}}$ with real momentum $p$, In
the Euclidean world however, the momentum is imaginary!

Let us note that the rapidity $y_{M}={\rm arctanh}(v_{M})$ in Minkowski space,
becomes in Euclidean space a rotation angle

$y_{E}={\rm arctan}(v_{E})={\rm
arctan}\bigg{(}{a\tau\over\sqrt{1-a^{2}\tau^{2}}}\bigg{)}$ (38)

and at $\tau=\pm 1/a$ it is $\pm\pi/2$. The action $S=\int(-mds-eExdt)$ when
evaluated with the Euclidean path gives $S_{E}=\pi m^{2}/2eE$. The
semiclassical probability (square of the amplitude) of the pair production is
then

$P\sim e^{-2S_{E}}\sim e^{-{\pi m^{2}}/{eE}}$ (39)

When discussing the heavy quark production we will assume that the back
reaction of the produced quark on the gauge field can be neglected. This
implies that the total sphaleron mass is much larger than that of the produced
pair

${3\pi^{2}\over g^{2}(\rho)\rho}\gg 2M$ (40)

For $\rho\sim 1/3\,{\rm fm}$, corresponding to the maximum in the instanton
size distribution, the l.h.s. is about $M_{\rm sph}\approx 3\,{\rm GeV}$.
Therefore, the condition is satisfied for the strange quark pair $\bar{s}s$
but not for the charm quark pair $\bar{c}c$.

The electric field during the sphaleron explosion $E^{m}_{a}(\vec{x},t)$
follows from the expressions for the field potential (16). The component with
$a=3$ corresponds to the diagonal Pauli generator $\tau^{3}$, so that quark
color remains unchanged. The maximal magnitude of the field corresponds to the
$x^{3}$ or $z$ direction, $E^{3}_{3}$. The formula is a bit long to give here,
but its behavior is shown in Fig. 6

Figure 6: The snapshots of the electric field component $E^{3}_{3}(r,t)$ in
units of $1/g\rho^{2}$, as a function of $x^{3}/\rho$, for times $t/\rho=0.5$
(dashed), $t/\rho=1$ (solid) and $t/\rho=2$ (dotted) curves.

From our discussion of the semiclassical quark pair production in a
$homogeneus$ electric field it follows that the field must produce an
impulsion

$\Delta p=\int Edt\sim M$

to reach, in Euclidean notations, the rotation angle $y_{E}=\pi/2$ needed to
make it real. For an estimate, one can take $E\approx 1.5/\rho^{2}$ and
$\delta t\approx\rho$ ( following Fig. 6), from which the estimated impulsion
is

$\Delta p=E\delta t\approx{1.5/\rho}$

This gives an estimated limit for the mass $M$ of a quark which is $likely$ to
be produced

$M<1.5/\rho$ (41)

Using $1/\rho=0.6\,GeV$, one finds the r.h.s. to be $\approx 1\,GeV$. This
implies that strange quarks, with $M_{s}\sim 0.14\,GeV$ can be produced, but
not the charmed ones, with $M_{c}\sim 1.5\,GeV$. To satisfy this estimate, one
would need to decrease $\rho$ by about a factor 2. To produce a $b$ quark,
with $M_{b}\approx 5\,GeV$, one would need to decrease $\rho$ by a factor 6 or
so.

In order to get quantitative semiclassical description on has to do the
following: (i) convert the expressions for the field to Euclidean time; and
(ii) solve the relativistic classical EOM

$M{du^{\mu}\over ds}=F^{\mu\nu}u_{\nu}$

where $u^{\mu}=dx^{\mu}/ds$ and $s$ is the proper time,
$ds^{2}=dt^{2}-d\vec{x}^{2}$. Comparable and rather complex electric and
magnetic fields make the paths quite complicated. On top of that, the result
depends on the starting location of the particle. So, at this time, we have no
results on such a calculation to report.

Figure 7: Schematic picture illustrating the configuration of QCD strings in
four-jet events. The left corresponds to the usual case, when jets originate
from collisions at the origin in the transverse plane. The right corresponds
to an exploding sphaleron in which the strings are not connected to the
origin, but are close to the expanding shell (dotted circle).

##  8 Sphaleron decays at large masses, $M\sim 100\,GeV$

These masses fall in the range discussed in the theoretical literature, and
mostly searched at HERA and LHC, as they promise to have ${\cal O}(10)$ well
recognized gluonic and quark jets. Background evaluation for such events were
attempted, using current event generators tuned to multi-jet events.

Two comments are in order:

(i) In this mass range, it is possible to use double diffractive events at
LHC, which are expected to reduce backgrounds substantially in comparison to
ambient $pp$ collisions.

(ii) In the usual parton-parton collisions, the fragmentation function of
${\cal O}(10\,GeV)$ gluons is essentially the product of string breaking. One
end of these strings is on the leading gluon, the other ends at throughgoing
partons, or the origin in the transverse plane, see Fig. 7 left. But sphaleron
decay leaves out the interior of the exploding shell “empty” (more precisely,
with a pure gauge configuration). As these gluons become physical and
separated from each other, the strings would go $between$ them close to the
shell, rather than extending to he origin, see Fig. 7 right. This suggests the
appearance of some unusual gluon jet fragmentation functions, with a
significant depletion at small $p_{t}$.

##  9 Magnetic field and Chiral Magnetic Effect

In the presence of an electromagnetic field $\vec{b}_{EM}$ and chiral
disbalance of fermions, there exists the so called Chiral Magnetic Effect
(CME), an electric current along $\vec{b}_{EM}$ is created, for a review see
[39]. Experiments aimed at observing it in heavy ion collisions at RHIC have
been made, lately utilizing beams of nuclei with the same atomic weight but
different charges, but the analysis is not conclusive. The CME was observed in
semi-metals [39].

An interesting manifestation of the phenomenon must also be present in the
instanton-sphaleron production processes, which we propose in this paper.

The first part of it, related with instantons, has already been studied. While
calculating nucleon form factors using instantons [40], Faccioli observed that
the instanton zero mode acquires an electric dipole along the magnetic field.
(Of course, adding the antiinstanton contribution cancels it, unless the
topological $\theta$ angle is nonzero). Lattice simulations in a magnetic
field carried in [41] have confirmed reality of this effect.

The extensive analytic study of the electric dipole moment of a zero mode is
due to Basar, Dunne and Kharzeev [42]. Let us just recall a qualitative
explanation of the mechanism from their work. The Dirac equation in the
instanton background conserves total

$\vec{J}=\vec{L}+\vec{S}+\vec{T}$

the sum of orbital momentum, spin and (SU(2)) isospin. As noticed by t’Hooft,
the isospin-orbit $(\vec{T}\cdot\vec{L})$ term forces orbital momentum $not$
to be a good quantum number. Therefore, it is possible to mix opposite parity
states666This discussion is in the instanton background, which is CP non-
invariant. The QCD vacuum and other states are CP conserving. . The external
electromagnetic field $\vec{b}$ couples spin to angular momentum, and induces
such mixing. The zero fermion and antifermion modes are deformed in different
directions, creating a nonzero electric dipole along the $\vec{b}$ field.
(Antiinstantons of course would do the opposite: we discuss local, not global
CP violation.) No observable consequences of this effect have so far (to our
knowledge) been proposed.

Now, consider e.g. very peripheral heavy ion collisions, in which Pomeron-
Pomeron (or $\gamma$-Pomeron or $\gamma\gamma$) collision initiate instanton-
sphaleron production process. As usual, the large electric charges of the ions
generate a strong magnetic field at the production point. The exploding
instanton and sphaleron gauge field remains spherical, but, since the quark
zero mode is deformed, the exploding electric field will carry them
differently in direction of the magnetic field, as compared to two other
directions, producing anisotropy of the quark momenta.

The effect of a magnetic field can be seen by comparing $pp$ and $PbPb$
peripheral collisions. The orientation of the collision plane (to which
$\vec{b}$ is normal) can be deduced from the direction of two forward-moving
protons (or ion remnants).

The sign of the effect for a quark depends on its electric charge, and the
light quarks in question, $u,d,s$ have charges $+2/3,-1/3,-1/3$. So, for the
mesons $\pi^{+}$ versus $\pi^{-}$ (kaons, etc) the effects are added,
resulting in a charge-dependent deformation of the distribution. The instanton
and antiinstanton events still produce opposite signs of the effect. So one
may construct a meson $correlation$ observable inside an event ($all$
$\pi^{+}$ have one preferred direction, opposite to $\pi^{-}$).

Another (statistically challenging) option is to select events with $\Lambda$
(or $\bar{\Lambda}$) decays, which tells us about the strange quark chirality.
Statistically selecting “more-likely-instanton” or “more-likely-antiinstanton”
events, one can perhaps see an electric dipole in the event directly.

Concluding this discussion, let us again emphasize the following. The
instanton-sphaleron process is the “hydrogen atom” of topological effects. The
chiral correlations produced in it is maximally possible, as all quarks and
antiquarks have fixed chiralities. (For comparison, in heavy ion collisions we
think the axial charge of the fireball is between $\pm 20$ or so, on top of
thousands of unpolarized ones. While the proposed measurements may appear
quite exotic, the cross sections are relatively large, allowing LHC with the
right trigger to produce billions of diffractive events.

##  10 The “entrance factor”

So far our discussion was mostly semiclassical, we considered (Euclidean)
tunneling paths and (Minkowski) sphaleron explosions. Then we projected the
outgoing quanta to some final hadronic states.

Now we focus on the very initial stage of the collision. Here we first have to
make hard choices, selecting theoretical tools for its description. Here are
these choices: colliding protons can be seen as colliding (i) partons (gluons
or quarks); (ii) color dipoles; (iii) glueballs; or (iv) Pomerons. Let us
briefly discuss them subsequently.

At high mass scale, e.g. the sphaleon mass squared

$s^{\prime}=M^{2}_{\rm sph}\gg 100\,{\rm GeV}^{2}$

the parton-parton collision is the natural perturbative description, developed
since the 1970’s in the context of jets or heavy quanta production. It was
applied by Ringwald and Schrempp for DIS at HERA [43], and now by Khoze et al.
[44] to LHC setting, incorporating Mueller perturbative correction [45]. For
comparison of these predictions with HERA data see [46], and with LHC [47]. At
high mass regime the theoretical predictions have reasonable uncertainties,
such as in the semiclassical prefactor, but the main difficulty in it is large
background induced by ambient events. The main observable is a spherical
multi-jet events.

To illustrate the situation, we show the gluon-gluon cross section
$\sigma(\sqrt{s^{\prime}})$ of the instanton-sphaleron process in Fig.8. The
points are from Table 1 by Khoze et al. [44]. The instanton size in the range
of the plot changes from $1/\rho=1\,{\rm GeV}$ to $75\,{\rm GeV}$, and the
number of outgoing gluons at the high end reaches about a dozen.

The line, shown for comparison, is $1/M^{9}$. Recall that the one-loop
coefficient of the QCD beta function is $b=(11/3)N_{c}-(2/3)N_{f}$, or 9 if
the number of light flavors $N_{f}=3$. The effective action is twice that of
the instanton $2S_{inst}$ minus its depletion for sphaleron production
$-\Delta S$. The cross section should be, by dimension,

$\sigma\sim{1\over M^{2}}\bigg{(}{\Lambda_{QCD}\over M}\bigg{)}^{b(2-\Delta
S/S_{0})}$

One can see that the original estimates

${\Delta S\over S_{0}}\approx 1$ (42)

are supported by actual multidimensional integration.

The calculation along a path including both the Euclidean and Minkowskian
times has been preformed in [48], in the electroweak setting. Their result
(solid line in Fig.6 of that paper) shows that the action is reduced from
$\approx 12.5$ at the zero sphaleron mass to about $\approx 7$ at the large
sphaleron mass. It also supports the estimate (42).

The backgrounds come from multiple QCD reactions, which have cross sections

$\sigma_{background}\sim{\alpha_{s}^{2}\over M^{2}}$

It is therefore clear that the task of separating the signal from the
background becomes much harder as the cluster mass $M$ grows.

Figure 8: Cross section $\sigma(M)$ in pb of the process gluon-gluon-
instanton/sphaleron, versus the sphaleron mass $M(GeV)$.The points are
explained in the text, the line is the $1/M^{9}$ background dependence shown
for comparison.

As the momentum transfer scale decreases, one may naturally think of coherent
contribution of two (or more) partons. Color neutrality can be implemented
starting from color dipoles. Unlike partons, the colliding color dipoles have
a natural scale given by their sizes $a_{1},a_{2}$. Correlating dipoles with
instantons via Wilson loops has been done by us [16]. The cross sections
obtained can be directly tested in double-inelastic electron-positron (or
$\gamma^{*}\gamma^{*}$) collisions. Unfortunately, a description of a proton
in terms of color dipoles is not yet (to our knowledge) developed.

At small momentum transfer scale – in diffractive processes we will focus on –
the colliding objects are described in terms of (reggeized) hadron exchanges,
especially by tensor glueballs or (their extension to the whole Regge
trajectory), the Pomerons. In a number of relatively recent works it has been
shown that Pomeron exchanges require effective description in terms of
symmetric tensor. These facts, together with overall development of
holographic QCD models, had strengthen the Pomeron-graviton connection.

##  11 Sphaleron production in Pomeron-Pomeron collisions

We start with some general remarks related to experimental setting.

Double-Pomeron production processes at LHC are only at their initial stage. A
general but brief discussion of the existing detectors was made in section
1.2. Let us only add that apparently in the standard high luminocity LHC runs,
one cannot access masses less than say hundreds of ${\rm GeV}$, but can go to
any masses provided the dedicated low-luminocity runs are performed.

As we will detail below, Pomeron-Pomeron vertices are coupled to two
operators, scalar $G^{2}$ and pseudoscalar $G\tilde{G}$. The former is maximal
when electric field directions are parallel, in the second orthogonal.
Momentum kicks to two protons are directed with electric field. Therefore,
locaing roman pots in different azimuth, one can in principle “tag” and
separate these two contributions.

Returning now to current experiments at LHC, we note that exclusive channel
$\pi^{+}\pi^{-}$ has been already studied by CMS [49], for invariant masses
till $M<2\,{\rm GeV}$. It shows a $\rho$-meson peak, and contains hints at
some other resonances.

Heavier clusters unfortunately were only studied by the old UA8 collaboration
[3] with $p\bar{p}$ collisions at the SPS. The production cross section in
Pomeron-Pomeron collisions at the peak $M\sim 3\,{\rm GeV}$ is rather large
$\sigma_{PP}\sim 4\,{\rm mb}$.

But, as usual, what really matters is the Signal/Background (S/B) ratio. In
this respect our only normalization process is the $\eta_{c}$ decay. Assuming
standard $\bar{c}c\rightarrow gg$ annihilation, into
$G_{\mu\nu}\tilde{G}_{\mu\nu}$ pseudoscalar operator, one realizes that the
gluon can either enter the instanton/sphaleron tunneling, or produce quark
pairs perturbatively. Most likely, at fixed $M_{gg}$ it does not matter
whether $gg$ came from $\bar{c}c$ annihilation or Pomeron-Pomeron collisions.

In total, the three prominent 3-meson channels ascribed to the former have
certain branching ratio, from which one can conclude that

$\bigg{(}{Signal\over Background}\bigg{)}(M\approx 3\,{\rm GeV})>0.15$

One can think of many other modes, e.g. with glueballs, scalar or
pseudoscalar, but at a mass of only $M_{\eta_{c}}\approx 3\,{\rm GeV}$ those
would be suppressed by phase space 777 For a more accurate estimate, one needs
to figure out which other modes come from the signal. Unfortunately there are
many modes and their branching ratios are at the level less than a percent, so
it would be hard if not impossible to get an accurate number.. Perhaps it
increases the signal by a factor of 2 or something like that. Alternatively,
PP collisions can be approximated as $gg+gg$ collisions, or operators with
$two$ stress tensors. This perhaps increases the background, say also by a
factor of 2, so that these effects cancel out in the S/B ratio.

What is however even more important, if one looks in the “instanton-induced”
channels there would hardly be any background. It follows from comparison of
the $\eta_{c}$ decays to $KK\pi,\pi\pi\eta$ yields with pure pion final
states.

Now, we address the sphaleron mass distribution in PP collisions. We already
discussed in 5 the instanton size distribution

${dn\over d\rho}\sim\bigg{(}{1\over\rho^{5}}\bigg{)}(\rho\Lambda)^{b}{\rm
exp}\big{(}-2\pi\sigma\rho^{2}\big{)}$ (43)

with $b=(11/3)N_{c}-(2/3)N_{f}$ is the first coefficient of the beta function,
and $\sigma\approx(0.42\,{\rm GeV})^{2}$ the string tension. When combined
with the (pure gauge) expression of the sphaleron mass
$M_{sph}=3\pi^{2}/g^{2}\rho$, it yields the distribution over this mass. In
Fig. 9 we compare it with the data points from the UA8 experiment [3] (left
plot), and with the background (right plot) expected to scale as
$d\sigma/dM\sim 1/M^{3}$ and normalized to the high-mass UA8 data.

Figure 9: Left plot: Semiclassical distribution over the cluster mass
$M\,({\rm GeV})$, compared to the data points from the UA8 experiment; Right
plot is the logarithmic representation of the same curve (solid), now compared
with the dashed line representing the perturbative background.

As one can see, the vacuum distribution in instanton sizes and the
experimental cluster mass distribution are in qualitative agreement in shape,
and in predicted position of a peak. This supports our suggestion that those
clusters may in fact be QCD sphalerons.

Let us make few other points:
(i) the instanton-sphaleron production seems to be dominant at $M\sim{\rm few-
GeV}$;
(ii) the cross section to $M\sim 20\,{\rm GeV}$ decreases by about three
orders of magnitude. With the efficiency of the forward trigger of UA8, about
1300 events were detected at $theSp\bar{p}S$. The luminocity of LHC is more
than enough to cover these extra three orders of magnitude; (iii) where the
signal-to-background ratio is about $1:20$. The background – mostly two-jet
events – is very anisotropic, so perhaps the identification of an isotropic
signal can still be possible, even at $M\sim 20\,{\rm GeV}$.;
(iv) we recall that the clusters with $M\sim 20\,{\rm GeV}$ are the ones in
which 10 quark operators (with pairs of $\bar{b}b$ quarks) can be produced.

##  12 Sphaleron-gluon-quark vertex

To make explicit the sphaleron-quark vertex, we note that the instanton
density in (24) follows from the averaging of the gluon field in the instanton
vacuum

$n_{I+\bar{I}}=n_{I}+n_{\bar{I}}=\frac{\alpha_{s}}{8\pi}\langle{G^{2}}\rangle\qquad
n_{I-\bar{I}}=n_{I}-n_{\bar{I}}=\frac{\alpha_{s}}{8\pi}\langle{G\tilde{G}}\rangle\rightarrow
0$ (44)

where the second averaging is zero for $\theta=0$. The normalizations in (44)
are fixed by the QCD scale and axial anomalies in the instanton vacuum,
respectively. Note that in (44) $G^{2}$ and $G\tilde{G}$ count the number of
instantons and anti-instantons.

The single instanton-six-quark vertex and anti-instanton-six-quark vertex
follow by omitting the vacuum averaging in (44), and recalling that the left-
vertex is induced by an instanton, and the right vertex by an anti-instanton.
More specifically, from (6.2) and (44) we obtain

$\displaystyle{\cal
L}_{Gqqq}=\bigg{[}\frac{\alpha_{s}}{16\pi}\bigg{(}G^{2}+G\tilde{G}\bigg{)}\bigg{]}\bigg{(}\frac{4\pi^{2}\rho^{3}}{M\rho}\bigg{)}^{3}\bigg{[}\frac{{\cal
V}^{L}_{qqq}}{\kappa}\bigg{]}+\bigg{[}\frac{\alpha_{s}}{16\pi}\bigg{(}G^{2}-G\tilde{G}\bigg{)}\bigg{]}\bigg{(}\frac{4\pi^{2}\rho^{3}}{M\rho}\bigg{)}^{3}\bigg{[}\frac{{\cal
V}^{R}_{qqq}}{\kappa}\bigg{]}$ (45)

The sphaleron produced in the diffractive process is half a tunneling process
with half the topological charge, and not self-dual. At the turning point, the
sphaleron drags six quark zero modes out of the QCD vacuum with at $t=0$ the
vertex mediated by $G_{S}^{2}=2B^{2}$ since $G\tilde{G}_{S}=4E\cdot B=0$

$\displaystyle{\cal L}_{Sqqq}(t=0)=$
$\displaystyle\bigg{(}\bigg{[}\frac{\alpha_{s}}{16\pi}\,G_{S}^{2}\bigg{]}\bigg{(}\frac{4\pi^{2}\rho^{3}}{M\rho}\bigg{)}^{3}\bigg{[}\frac{{\cal
V}^{L}_{qqq}+{\cal V}^{R}_{qqq}}{\kappa}\bigg{]}\bigg{)}_{t=0}$
$\displaystyle+$
$\displaystyle\bigg{(}\bigg{[}\frac{\alpha_{s}}{16\pi}\,\dot{K}_{0S}\bigg{]}\bigg{(}\frac{4\pi^{2}\rho^{3}}{M\rho}\bigg{)}^{3}\bigg{[}\frac{{\cal
V}^{L}_{qqq}-{\cal V}^{R}_{qqq}}{\kappa}\bigg{]}\bigg{)}_{t=0}$ (46)

and by the rate of the Chern-Simons charge density $\dot{K}_{0S}$ following
from $\partial^{\mu}K_{\mu}=G\tilde{G}$.

##  13 Witten amplitude for diffractive production $pp\rightarrow ppX$

QCD is difficult to track in the infrared since its fundamental quark and
gluon constituents repackage in confined hadrons. The ensuing hadronic
dynamics is strongly coupled. Holographic QCD is a proposal guided by the
AdS/CFT or gauge/gravity duality discovered in string theory. The holographic
duality or principle states that boundary operators in the gauge theory can be
mapped to a higher dimensional string theory in bulk in a curved anti-deSitter
space. The original correspondence holds for type IIB superstring theory in
$AdS_{5}\times S_{5}$, but is commonly assumed to hold for a string theory in
a general background. The srting theory is in general difficult to solve, but
in the double limit of a large number of colors and strong gauge coupling
$\lambda=g^{2}N_{c}$, it reduces to a weakly coupled supergravity in the
classical limit, with a weak string coupling $g_{s}=g^{2}/4\pi$. The gauge
invariant operators at the boundary are mapped onto supergravity fields in
bulk with pertinent anomalous dimensions.

The n-point functions at the boundary of $AdS_{5}$ follow from variation of
the on-shell supergravity action in bulk with respect to the boundary values.
The results are tree-level Feynman graphs with fixed end-points on the
boundary also known as Witten diagrams. The Witten diagram for the diffractive
process of interest in this work $pp\rightarrow ppX$ is illustrated in Fig.
10. The insertions on the $AdS_{5}$ boundary refer to 4 nucleon operator
insertions ${\cal O}_{pi=1,...,4}$ and a glueball insertion ${\cal
O}_{X=G^{2},G\tilde{G}}$. ${\cal O}_{p}$ sources a Dirac fermion in bulk with
anomalous dimension $M=\tau-3/2$ ($\tau$ refers to the twist, with typically
$\tau=3$ to reproduce the hard scattering rules), and the ${\cal O}_{X}$
sources a dilaton or axion in bulk with anomalous dimension $\Delta_{X}=4$. In
the Regge kinematics, the exchange in bulk is mediated by two closed string
producing a dilaton $h$ or an axion $a$ which is equivalent to Pomeron-Pomeron
fusion into $h,a$ (${\bf P}{\bf P}\rightarrow h,a$).

The Holographic construction provides a first principle description of the
Pomeron (${\bf P}$) as a dual to a close string exchange [50, 11] or Reggeized
graviton in bulk [5, 6]. In this section we will use this framework to extract
the differential cross section for the reaction $pp\rightarrow ppX$ through
${\bf PP}$ fusion as discussed in [7] for pseudoscalar emission, and in [51]
for Higgs production. After a brief description for the pertinent kinematics
for this process in the Reggeized limit, we will summarize the main formula
for the production of $G^{2}$ and $G\tilde{G}$ glueballs with most of the
details given in D-E-F.

###  13.1 Kinematics for the Reggeized limit

For the process $p(p_{1})+p(p_{2})\rightarrow p(p_{3})+p(p_{4})+X(p_{5})$ we
will set the incoming protons back-to-back

$p_{1}=(E,0,0,p)\qquad p_{2}=(E,0,0,-p)\qquad p_{i=3,4,5}=(E_{i},q_{\perp
i},p_{iz})$ (47)

with longitudinal fractions $p_{3z}=x_{1}p$ and $p_{4z}=-x_{2}p$ for the
outgoing protons. The transverse momenta in the azimuthal plane are 2-vectors

$q_{\perp 3}=(q_{3}\,{\rm cos}\,\theta_{3},q_{3}\,{\rm sin}\,\theta_{3})\qquad
q_{\perp 4}=(q_{4}\,{\rm cos}\,\theta_{4},q_{4}\,{\rm sin}\,\theta_{4})$ (48)

The Mandelstam kinematics suggests five invariants

$\displaystyle s=(p_{1}+p_{2})^{2}\qquad t_{1}=(p_{1}-p_{3})^{2}\qquad
t_{2}=(p_{2}-p_{4})^{2}\qquad s_{1}=(p_{3}+p_{5})^{2}\qquad
s_{2}=(p_{4}+p_{5})^{2}$

which are reduced to four by azimuthal symmetry. We will choose the four
invariants as $t_{1},t_{2}$, the relative azimuthal angle
$\theta_{34}=\theta_{4}-\theta_{3}$, and the relative momentum fraction
$x_{F}=x_{1}-x_{2}$. Following [7], the Reggeized limit is characterized by
large $\sqrt{s}$ and small scattering or azimuthal angles,

$s\gg s_{1},s_{2}\gg-t_{1},-t_{2},m_{N}^{2}\qquad{\rm
and\,\,fixed\,}\qquad\mu=\frac{s_{1}s_{2}}{s}$ (50)

with the simplified kinematics

$s_{1,2}\approx\sqrt{\mu s}\qquad{t_{1,2}}\approx-q^{2}_{\perp
3,4}\qquad\mu\approx p_{5}^{2}+t_{1}+t_{2}+2\sqrt{t_{1}t_{2}}\,{\rm
cos}\,\theta_{34}$ (51)

In this limit and using (51), the differential cross section for
$pp\rightarrow pp0\mp$ is dominated by $x_{F}\approx 0$ [7], and reads

$\frac{d\sigma_{\mp}}{d\theta_{34}dt_{1}dt_{2}}\approx\frac{1}{16\pi^{4}s^{2}}\,{\rm
ln}\bigg{(}\frac{s}{\mu}\bigg{)}\,\frac{1}{4}\sum_{\rm spin}\bigg{|}{\cal
A}_{pp\rightarrow pp0\mp}\bigg{|}^{2}$ (52)

for the production amplitudes (G.4) and (G.4) respectively. We now make
explicit the production amplitude for scalar and pseudoscalar production.

Figure 10: Witten diagram for diffractive $pp\rightarrow ppX$ with
$X=(a,f)\equiv(G\tilde{G},G^{2})$ for the sphaleron at the boundary

###  13.2 Differential cross section: $G\tilde{G}$

The main elements entering the Witten amplitude in Fig 10 for $pp\rightarrow
pp0-$ through ${\bf PP}$ fusion are detailed in the Appendices D-E-F, and
specialized to the soft wall model in a slab of AdS5. Other models in AdS5
yield similar results. The ensuing squared spin-averaged amplitude entering
(52) is

$\displaystyle\frac{1}{4}\sum_{\rm spins}\bigg{|}{\cal A}_{pp\rightarrow
pp0-}\bigg{|}^{2}=\frac{16}{4}\,A^{2}(t_{1})\,A^{2}(t_{2})\,$
$\displaystyle\times\bigg{[}\bigg{(}\frac{\lambda}{\pi^{2}}+1\bigg{)}^{2}\bigg{(}\frac{\sqrt{\lambda}}{2\pi}\bigg{(}\gamma+\frac{\pi}{2}\bigg{)}\bigg{)}^{4}\frac{e^{-\sqrt{\lambda}(\gamma+\pi/2)^{2}(1/{\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2})+1/{\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))}}{({\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2}){\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))^{3}}\,\bigg{(}\frac{s_{1}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{1})-2)}\bigg{(}\frac{s_{2}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{2})-2)}\bigg{]}$
$\displaystyle\times\bigg{[}\bigg{(}B_{1}^{-}(t_{1},t_{2})\eta_{\beta\bar{\beta}}+B_{2}^{-}(t_{1},t_{2})k_{2\beta}k_{1\bar{\beta}}\bigg{)}\,\epsilon_{\alpha\bar{\alpha}\gamma\delta}k_{1}^{\gamma}k_{2}^{\delta}\bigg{]}$
$\displaystyle\times\bigg{[}\bigg{(}B_{1}^{-}(t_{1},t_{2})\eta_{\underline{\beta}\bar{\underline{\beta}}}+B_{2}^{-}(t_{1},t_{2})k_{2\underline{\beta}}k_{1\bar{\underline{\beta}}}\bigg{)}\,\epsilon_{\underline{\alpha}\bar{\underline{\alpha}}\underline{\gamma}\underline{\delta}}k_{1}^{\underline{\gamma}}k_{2}^{\underline{\delta}}\bigg{]}$
$\displaystyle\times\bigg{[}p^{\alpha}p^{\beta}p^{\underline{\alpha}}p^{\underline{\beta}}+\frac{1}{16}\bigg{(}t_{1}\eta_{\alpha\underline{\alpha}}p_{\beta}p_{\underline{\beta}}+t_{1}\eta_{\alpha\underline{\beta}}p_{\beta}p_{\underline{\alpha}}+t_{1}\eta_{\beta\underline{\alpha}}p_{\alpha}p_{\underline{\beta}}+(t_{1}\eta_{\beta\underline{\beta}}-k_{1\beta}k_{1\underline{\beta}})p_{\alpha}p_{\underline{\alpha}}\bigg{)}\bigg{]}$
$\displaystyle\times\bigg{[}\underline{p}^{\bar{\alpha}}\underline{p}^{\bar{\beta}}\underline{p}^{\underline{\bar{\alpha}}}\underline{p}^{\bar{\underline{\beta}}}+\frac{1}{16}\bigg{(}t_{2}\eta_{\bar{\alpha}\bar{\underline{\alpha}}}p_{\bar{\beta}}p_{\bar{\underline{\beta}}}+t_{2}\eta_{\bar{\alpha}\bar{\underline{\beta}}}p_{\bar{\beta}}p_{\bar{\underline{\alpha}}}+t_{2}\eta_{\bar{\beta}\bar{\underline{\alpha}}}p_{\bar{\alpha}}p_{\bar{\underline{\beta}}}+(t_{2}\eta_{\bar{\beta}\bar{\underline{\beta}}}-k_{1\bar{\beta}}k_{1\bar{\underline{\beta}}})p_{\bar{\alpha}}p_{\bar{\underline{\alpha}}}\bigg{)}\bigg{]}$
(53)

with the vertex

$\displaystyle 2B_{1}^{-}(t_{1},t_{2})-\mu
B_{2}^{-}(t_{1},t_{2})=4\bigg{(}{\bf C}^{-}_{1}-{\bf
C}^{-}_{2}\sqrt{t_{1}t_{2}}\,{\rm cos}\,\theta_{34}\bigg{)}$ (54)

and ${\bf C}^{-}_{1,2}$ given in (F.2-135) for the soft-wall model. Here, the
Pomeron trajectory is the Reggeized graviton trajectory

$\displaystyle\alpha_{\mathbb{P}}(t)=\alpha_{\mathbb{P}}(0)+\frac{\alpha^{\prime}}{2}t\equiv
2-\frac{2}{\sqrt{\lambda}}+\frac{\alpha^{\prime}}{2}t$ (55)

after restoring the t-dependence, with $\alpha^{\prime}=l_{s}^{2}$ (the
squared string length). For simplicity we have set $\lambda\rightarrow\infty$
in the A-form factors in (13.2), i.e. $A(j_{0}\rightarrow 2,k)\equiv A(t)$,
although this can be relaxed. A similar observation was made originally in [1]
(instantons) and more recently in [7] (holography) for the double diffractive
production of $\eta,\eta^{\prime}$ .

We recall that for the nucleon as a Dirac fermion, the energy momentum tensor
is characterized by three invariant form factors

$\left<p_{2}|T^{\mu\nu}(0)|p_{1}\right>=\overline{u}(p_{2})\left(A(k)\gamma^{(\mu}p^{\nu)}+B(k)\frac{ip^{(\mu}\sigma^{\nu)\alpha}k_{\alpha}}{2m_{N}}+C(k)\frac{k^{\mu}k^{\nu}-\eta^{\mu\nu}k^{2}}{m_{N}}\right)u(p_{1})\,,$
(56)

with $p=(p_{1}+p_{2})/2$ and $k=p_{2}-p_{1}$, and the normalization
$\left<p|T^{\mu}_{\mu}|p\right>=2A(0)m_{N}^{2}$. In Appendix D, $A(t)$ is
given explicitly for the soft wall model, which is found to be well
parametrized by the dipole form [52]

$A(k)=\frac{A(0)}{\bigg{(}1+\frac{K^{2}}{m_{A}^{2}}\bigg{)}^{2}}=\frac{A(0)}{\bigg{(}1-\frac{t}{m_{A}^{2}}\bigg{)}^{2}}\equiv
A(t)$ (57)

with $m_{A}=1.124$ GeV. It compares well with the reported lattice value
$m_{A,\,\rm lattice}=1.13$ GeV by the MIT group [53], for $A(0)=0.58$. The
gravitational form factor $A(k)$ is saturated by the $2^{++}$ glueball
trajectory without any quark mixing, essentially a quenched result. The
holographic C-term $C(k)=-4A(k)$ is also in good agreement with the lattice
results. The holographic B-term is found to vanish which is consistent with
the lattice findings. Note that (57) corresponds to a squared gravitational
radius of the nucleon as a bulk Dirac fermion

$\displaystyle\left<r^{2}_{G}\right>=6\bigg{(}\frac{d{\rm
log}A^{(}t)}{dt}\bigg{)}_{t=0}=(0.61\,{\rm fm})^{2}$ (58)

which is smaller than the nucleon squared charge radius
$\left<r^{2}_{C}\right>=(0.831\,{\rm fm})^{2}$ [54].

In the Reggeized limit, the dominant contributions in the last two brackets
stem from the first two terms with $p^{2},\underline{p}^{2}\sim s$ as the lead
gravitational coupling to the energy-momentum tensor, with the result

$\displaystyle\frac{1}{4}\sum_{\rm spins}\bigg{|}{\cal A}_{pp\rightarrow
pp0-}\bigg{|}^{2}\approx\frac{1}{4}\,A^{2}(t_{1})\,A^{2}(t_{2})\,$
$\displaystyle\times\bigg{[}\bigg{(}\frac{\lambda}{\pi^{2}}+1\bigg{)}^{2}\bigg{(}\frac{\sqrt{\lambda}}{2\pi}\bigg{(}\gamma+\frac{\pi}{2}\bigg{)}\bigg{)}^{4}\frac{e^{-\sqrt{\lambda}(\gamma+\pi/2)^{2}(1/{\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2})+1/{\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))}}{({\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2}){\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))^{3}}\,\bigg{(}\frac{s_{1}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{1})-2)}\bigg{(}\frac{s_{2}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{2})-2)}\bigg{]}$
$\displaystyle\times s^{4}t_{1}t_{2}\bigg{(}2B_{1}^{-}(t_{1},t_{2})-\mu
B_{2}^{-}(t_{1},t_{2})\bigg{)}^{2}\,{\rm sin}^{2}\,\theta_{34}$ (59)

The corresponding differential cross section is

$\displaystyle\frac{d\sigma_{-}}{d\theta_{34}dt_{1}dt_{2}}\approx\frac{1}{(8\pi)^{4}}\,A^{2}(t_{1})\,A^{2}(t_{2})\,t_{1}t_{2}\bigg{(}2B_{1}^{-}(t_{1},t_{2})-\mu
B_{2}^{-}(t_{1},t_{2})\bigg{)}^{2}\,{\rm sin}^{2}\,\theta_{34}$
$\displaystyle\times\bigg{[}\bigg{(}\frac{\lambda}{\pi^{2}}+1\bigg{)}^{2}\bigg{(}\frac{\sqrt{\lambda}}{2\pi}\bigg{(}\gamma+\frac{\pi}{2}\bigg{)}\bigg{)}^{4}\frac{e^{-\sqrt{\lambda}(\gamma+\pi/2)^{2}(1/{\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2})+1/{\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))}}{({\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2}){\rm ln}(s_{2}/\tilde{\kappa}_{N}^{2}))^{3}}$
$\displaystyle\times\bigg{[}\,\bigg{(}\frac{s_{1}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{1})-2)}\bigg{(}\frac{s_{2}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{2})-2)}\bigg{]}\bigg{[}s^{2}{\rm
ln}\bigg{(}\frac{s}{\mu}\bigg{)}\bigg{]}$ (60)

For $s_{1,2}\sim\sqrt{\mu s}$ and fixed $\mu$ in the Reggeized limit, (13.2)
asymptotes

${\rm ln}\,s\,\bigg{[}\bigg{(}\frac{e^{-\\#\sqrt{\lambda}/{\rm ln}\,s}}{({\rm
ln}\,s)^{\frac{3}{2}}}\bigg{)}^{2}\,s^{1+\frac{1}{2}(\alpha_{\mathbb{P}}(t_{1})+\alpha_{\mathbb{P}}(t_{2})-4)}\bigg{]}^{2}$
(61)

with the first logarithm accounting for the $x_{F}\sim 0$ phase space
enhancement [7], and the remainder for the 2-Pomeron fusion exchange with its
holographic diffusive signature in $5-2=3$ transverse directions. The rise in
$s$ can be tamed through the Eikonalization of the exchange.

Finally, we note the transverse plane dependence on ${\rm sin}^{2}\theta_{34}$
of the differential cross section (also through $\mu$ in (51)) as inherited
from the abnormal parity nature of the Chern-Simons vertex (129) at the origin
of the pseudo-scalar emission in bulk. The diffractive emission of the pseudo-
scalar glueball is strongly anisotropic in the transverse plane.

###  13.3 Differential cross section: $G^{2}$

A similar reasoning applies for the spin-averaged squared amplitude for
$pp\rightarrow pp0+$ which reads

$\displaystyle\frac{1}{4}\sum_{\rm spins}\bigg{|}{\cal A}_{pp\rightarrow
pp0+}\bigg{|}^{2}=\frac{16}{4}\,A^{2}(t_{1})\,A^{2}(t_{2})\,$
$\displaystyle\times\bigg{[}\bigg{(}\frac{\lambda}{\pi^{2}}+1\bigg{)}^{2}\bigg{(}\frac{\sqrt{\lambda}}{2\pi}\bigg{(}\gamma+\frac{\pi}{2}\bigg{)}\bigg{)}^{4}\frac{e^{-\sqrt{\lambda}(\gamma+\pi/2)^{2}(1/{\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2})+1/{\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))}}{({\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2}){\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))^{3}}\,\bigg{(}\frac{s_{1}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{1})-2)}\bigg{(}\frac{s_{2}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{2})-2)}\bigg{]}$
$\displaystyle\times\bigg{[}B^{+}(t_{1},t_{2},p_{5}^{2})\eta_{\alpha\bar{\alpha}}\eta_{\beta\bar{\beta}}\bigg{]}\times\bigg{[}B^{+}(t_{1},t_{2},p_{5}^{2})\eta_{\underline{\alpha}\underline{\bar{\alpha}}}\eta_{\underline{\beta}\bar{\underline{\beta}}}\bigg{]}$
$\displaystyle\times\bigg{[}p^{\alpha}p^{\beta}p^{\underline{\alpha}}p^{\underline{\beta}}+\frac{1}{16}\bigg{(}t_{1}\eta_{\alpha\underline{\alpha}}p_{\beta}p_{\underline{\beta}}+t_{1}\eta_{\alpha\underline{\beta}}p_{\beta}p_{\underline{\alpha}}+t_{1}\eta_{\beta\underline{\alpha}}p_{\alpha}p_{\underline{\beta}}+(t_{1}\eta_{\beta\underline{\beta}}-k_{1\beta}k_{1\underline{\beta}})p_{\alpha}p_{\underline{\alpha}}\bigg{)}\bigg{]}$
$\displaystyle\times\bigg{[}\underline{p}^{\bar{\alpha}}\underline{p}^{\bar{\beta}}\underline{p}^{\underline{\bar{\alpha}}}\underline{p}^{\bar{\underline{\beta}}}+\frac{1}{16}\bigg{(}t_{2}\eta_{\bar{\alpha}\bar{\underline{\alpha}}}p_{\bar{\beta}}p_{\bar{\underline{\beta}}}+t_{2}\eta_{\bar{\alpha}\bar{\underline{\beta}}}p_{\bar{\beta}}p_{\bar{\underline{\alpha}}}+t_{2}\eta_{\bar{\beta}\bar{\underline{\alpha}}}p_{\bar{\alpha}}p_{\bar{\underline{\beta}}}+(t_{2}\eta_{\bar{\beta}\bar{\underline{\beta}}}-k_{1\bar{\beta}}k_{1\bar{\underline{\beta}}})p_{\bar{\alpha}}p_{\bar{\underline{\alpha}}}\bigg{)}\bigg{]}$
(62)

Again, in the Reggeized limit, the dominant contributions in the last two
brackets stem from the first two terms with $p^{2},\underline{p}^{2}\sim s$,
with the result

$\displaystyle\frac{d\sigma_{+}}{d\theta_{34}dt_{1}dt_{2}}\approx\frac{1}{(8\pi)^{4}}\,A^{2}(t_{1})\,A^{2}(t_{2})\,B^{+2}(t_{1},t_{2},p_{5}^{2})$
$\displaystyle\times\bigg{[}\bigg{(}\frac{\lambda}{\pi^{2}}+1\bigg{)}^{2}\bigg{(}\frac{\sqrt{\lambda}}{2\pi}\bigg{(}\gamma+\frac{\pi}{2}\bigg{)}\bigg{)}^{4}\frac{e^{-\sqrt{\lambda}(\gamma+\pi/2)^{2}(1/{\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2})+1/{\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))}}{({\rm
ln}(s_{1}/\tilde{\kappa}_{N}^{2}){\rm
ln}(s_{2}/\tilde{\kappa}_{N}^{2}))^{3}}\bigg{]}$
$\displaystyle\times\bigg{[}\bigg{(}\frac{s_{1}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{1})-2)}\bigg{(}\frac{s_{2}}{{\tilde{\kappa}_{N}}^{2}}\bigg{)}^{2(\alpha_{\mathbb{P}}(t_{2})-2)}\bigg{]}\bigg{[}s^{4}{\rm
ln}\bigg{(}\frac{s}{\mu}\bigg{)}\bigg{]}$

with the scalar vertex $B^{+}(t_{1},t_{2},p_{5}^{2})$

$B^{+}(t_{1},t_{2},p_{5}^{2})=\bigg{(}{\bf
C}_{1}^{+}(p_{5}^{2}+(t_{1}+t_{2}))-2{\bf C}_{2}^{+}-{\bf
C}_{3}^{+}p_{5}^{2}\bigg{)}$ (64)

and $p_{5}^{2}=m^{2}_{0^{+}}$ on mass-shell. ${\bf C}_{1,2,3}^{+}$ are given
in (F.1-128) for the soft-wall model. Modulo $\mu$, the emission of the scalar
glueball is isotropic in the transverse plane, in contrast to the pseudo-
scalar glueball emission.

##  14 Summary

The subject of this paper – instanton-sphaleron processes, both in QCD and
electroweak setting – were at the forefront of the theoretical discussions in
the late 1980s. The optimal tunneling path was suggested by Yung [20] and
Verbaarschot [21]. The idea to use it to evaluate cross sections was suggested
by Ringwald and Khoze [14], as well as Verbaarschot and Shuryak [15]. A more
detailed description of the topological landscape and the realization that
this path leads to the production of sphalerons came in 2002, from Ref. [12].
The analytic form of a (pure gauge) sphaleron classical explosion was also
obtained in that work, with a complementary derivation by us in [22],
clarifying the production of fermions. So, from the theoretical perspective,
one might think the issue was completely clarified many years ago.

Ringwald and Schrempp [43] put a lot of efforts to identify the instanton-
induced reactions in deep-inelastic scattering at HERA. The large $Q^{2}$
scale involved, pushes the instantons to be very small, causing the
corresponding cross sections to be also small. However, the real problem
turned out to be the separation of the “instanton events” from the multi-jet
background. The more recent version for the LHC, by Khoze et al [44], relates
the instanton size to the fraction of the collision energy, again with a cross
section that is strongly decreasing with the invariant mass
$M=\sqrt{s^{\prime}}$ of the produced cluster. This mechanism was recently
revisited using single-diffractive medium cluster production and a more
favorable choice of kinematics, but with stringent cuts [55].

While we do not disagree with these approaches, we still think one needs a
different experimental strategy. One better starts with smaller clusters, with
invariant masses of several ${\rm GeV}$, which offer clearly identifiable
mesonic and perhaps baryonic final states, which can be related to existing
theoretical predictions. The background issue would be helped if instead of
min.bias $pp$ collisions one would use the double diffraction, or PP
collisions. Last but not least, using gluon-rich but colorless fusing
Pomerons, rather than fusing gluonic partons, eliminates the need for color
flux tubes and cleans up the stage.

Can it all be done in practice, and if so, why now? It looks like LHC has
reached certain stage of multiple searches at which all which remains to do is
to increase statistics. The search for instanton/sphaleron clusters may not be
“beyond the standard model” but is still worth trying.

The double diffractive direction we advocate is in especially nice position.
The first roman pot (RP) detectors are installed and running. ATLAS AFP is
tuned to a huge mass $M\sim 1\,{\rm TeV}$, while CMS-TOTEM has carried studies
of exclusive reactions at $M\sim 1\,{\rm GeV}$. The in between region we
suggest to explore is still basically “terra incognita”. It remained untouched
for decades, since the pioneering $UA8$ experiment.

Last but not least, the QCD sphalerons and their explosions are close
relatives of the electroweak sphalerons. We do not know how one may produce
those in the lab, but we do expect them to be very important at the
cosmological electroweak transition [56].

Acknowledgements This work has been triggered by the CERN workshop ”
Topological Effects in the Standard Model: Instantons, Sphalerons and Beyond
at LHC”, Dec-2020. We thank its organizers, and especially Matthias Schott,
for organizing it and for reviving this field in general. This work is
supported by the Office of Science, U.S. Department of Energy under Contract
No. DE-FG-88ER40388.

##  Appendix A Some meson couplings used

For completeness, we give here some standard PCAC formulae and their numerical
values,

$K_{\pi}\equiv\langle
0|\bar{d}\gamma^{5}u|\pi^{+}\rangle=i{m_{\pi}^{2}f_{\pi}\over
m_{u}+m_{d}}\approx i(500\,MeV)^{2}$ (65) $K_{K}\equiv\langle
0|\bar{s}\gamma^{5}u|K^{+}\rangle=i{m_{K}^{2}f_{K}\over m_{u}+m_{s}}\approx
i(523\,MeV)^{2}$ (66)

The $\eta,\eta^{\prime}$ couplings depend on the mixing angle. We will use the
same values adopted in [34]

$\displaystyle K^{q}_{\eta}$ $\displaystyle\equiv$ $\displaystyle\langle
0|\bar{u}\gamma^{5}u|\eta\rangle=\langle
0|\bar{d}\gamma^{5}d|\eta\rangle=-i(358\,MeV)^{2}$ (67) $\displaystyle
K^{q}_{\eta^{\prime}}$ $\displaystyle\equiv$ $\displaystyle\langle
0|\bar{u}\gamma^{5}u|\eta^{\prime}\rangle=\langle
0|\bar{d}\gamma^{5}d|\eta^{\prime}\rangle=-i(320\,MeV)^{2}$ $\displaystyle
K^{s}_{\eta}$ $\displaystyle\equiv$ $\displaystyle\langle
0|\bar{s}\gamma^{5}s|\eta\rangle=i(435\,MeV)^{2}$ $\displaystyle
K^{s}_{\eta^{\prime}}$ $\displaystyle\equiv$ $\displaystyle\langle
0|\bar{s}\gamma^{5}s|\eta^{\prime}\rangle=i(481\,MeV)^{2}$

The other issue is to define the $\bar{s}s$ part of $\eta,\eta^{\prime}$
responsible for the observed $\eta\pi\pi$ and $\eta^{\prime}\pi\pi$ decay
rates. The $\eta-\eta^{\prime}$ mixing has been well researched. In terms of
singlet and octet states

$\displaystyle\eta_{8}=$
$\displaystyle{1\over\sqrt{6}}(\bar{u}u+\bar{d}d-2\bar{s}s)$ (68)
$\displaystyle\eta_{1}=$
$\displaystyle{1\over\sqrt{3}}(\bar{u}u+\bar{d}d+\bar{s}s)$

the standard definition of the mixing is

$\displaystyle\eta^{\prime}$ $\displaystyle=$ $\displaystyle{\rm
sin}(\theta_{p})\eta_{8}+{\rm cos}(\theta_{p})\eta_{1}$ (69)
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle{\rm
cos}(\theta_{p})\eta_{8}-{\rm sin}(\theta_{p})\eta_{1}$

Using the mixing angle $\theta_{p}\approx-14.6^{o}$, one finds that the ratio
of the $\bar{s}s$ probabilities (squares of amplitudes) is

${ss_{\eta^{\prime}}\over{ss}_{\eta}}=\bigg{(}{0.764\over
0.644}\bigg{)}^{2}=1.40$ (70)

##  Appendix B Instanton-induced $uuddss$ interactions

In this Appendix we outline the general structure of the induced ′t Hoot
vertex for multi-quark states. In particular, we show the explicit steps that
lead to the 6-quark vertex both in the color bleached and bleached form. its
generalization to 8- and 10-quark states follows. An alternative but tedious
formula involving color averaging over the color moduli with also be discussed
which can be generalized to even higher multi-quark states.

###  B.1 Details of the color-spin reduction

The momentum-space kernel for the LSZ reduced zero-mode propagator in the
instanton background reads

$\displaystyle
L\equiv\bigg{[}\frac{(p^{2}\varphi^{\prime}(p))_{0}^{2}}{m\rho}\bigg{]}\bigg{[}\frac{1}{8}\,U(\delta_{\mu\nu}+i{\overline{\eta}}_{\mu\nu}^{a}\tau^{a})U^{\dagger}\times\gamma_{\mu}\gamma_{\nu}\bigg{]}\frac{1-\gamma_{5}}{2}$
(71)

in the zero momentum limit $(p^{2}\varphi^{\prime 2}(p)_{0}\equiv\sqrt{C}$.
The 6-quark $uuddss$ vertex stemming from (71) follows from averaging over the
color matrices $U$ the product

$\displaystyle\int dU\,[L]^{3}$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{1}{8}\,U(\delta_{\alpha\beta}+i{\overline{\eta}}_{\alpha\beta}^{a}\tau^{a})U^{\dagger}\times\gamma_{\alpha}\gamma_{\beta}\bigg{]}$
(72) $\displaystyle\times$
$\displaystyle\bigg{[}\frac{1}{8}\,U(\delta_{\mu\nu}+i{\overline{\eta}}_{\mu\nu}^{b}\tau^{b})U^{\dagger}\times\gamma_{\mu}\gamma_{\nu}\bigg{]}\times\bigg{[}\frac{1}{8}\,U(\delta_{\lambda\tau}+i{\overline{\eta}}_{\lambda\tau}^{c}\tau^{c})U^{\dagger}\times\gamma_{\lambda}\gamma_{\tau}\bigg{]}$

using the identity [37]

$\displaystyle\int dU\,[U^{a}_{i}U^{\dagger
j}_{b}]^{3}=\bigg{[}\frac{1}{N_{c}}\delta^{a_{1}}_{b_{1}}\delta_{i_{1}}^{j_{1}}\,{\bf
1}_{1}\bigg{]}\bigg{[}\frac{1}{N_{c}}\delta^{a_{2}}_{b_{2}}\delta_{i_{2}}^{j_{2}}\,{\bf
1}_{2}\bigg{]}\bigg{[}\frac{1}{N_{c}}\delta^{a_{3}}_{b_{3}}\delta_{i_{3}}^{j_{3}}\,{\bf
1}_{3}\bigg{]}$
$\displaystyle+\bigg{(}\bigg{[}\frac{1}{N_{c}}\delta^{a_{1}}_{b_{1}}\delta^{j_{1}}_{i_{1}}\,{\bf
1}_{1}\bigg{]}\bigg{[}\frac{1}{4(N_{c}^{2}-1)}[\lambda_{2}^{A}]^{a_{2}}_{b_{2}}[\lambda_{3}^{A}]^{a_{3}}_{b_{3}}[\lambda_{2}^{B}]^{j_{2}}_{i_{2}}[\lambda_{3}^{B}]^{j_{3}}_{i_{3}}\bigg{]}+2\,{\rm
perm.}\bigg{)}$
$\displaystyle+\bigg{(}\frac{N_{c}}{8(N_{c}^{2}-1)(N_{c}^{2}-4)}\bigg{[}d^{ABC}([\lambda_{1}^{A}]^{a_{1}}_{b_{1}}[\lambda_{2}^{B}]^{a_{2}}_{b_{2}})[\lambda_{3}^{C}]^{a_{3}}_{b_{3}}\bigg{]}\bigg{[}d^{IJK}([\lambda_{1}^{I}]^{j_{1}}_{i_{1}}[\lambda_{2}^{J}]^{j_{2}}_{i_{2}})[\lambda_{3}^{K}]^{j_{3}}_{i_{3}}\bigg{)}\bigg{]}$
$\displaystyle+\bigg{(}\frac{1}{8N_{c}(N_{c}^{2}-1)}\bigg{[}f^{ABC}([\lambda_{1}^{A}]^{a_{1}}_{b_{1}}[\lambda_{2}^{B}]^{a_{2}}_{b_{2}})[\lambda_{3}^{C}]^{a_{3}}_{b_{3}}\bigg{]}\bigg{[}f^{IJK}([\lambda_{1}^{I}]^{j_{1}}_{i_{1}}[\lambda_{2}^{J}]^{j_{2}}_{i_{2}})[\lambda_{3}^{K}]^{j_{3}}_{i_{3}}\bigg{]}\bigg{)}$
(73)

which follows from the projection onto the color singlet channel. The result
after some algebra is

$\displaystyle\bigg{[}\frac{C}{m\rho N_{c}}\bigg{]}^{3}$
$\displaystyle\bigg{(}$ $\displaystyle
UDS+\frac{N_{c}(N_{c}-2)}{4(N_{c}^{2}-1)}\bigg{[}U^{A}D^{A}S+U^{A}DS^{A}+UD^{A}S^{A}\bigg{]}$
(74)
$\displaystyle-\frac{N_{c}^{2}}{16(N_{c}^{2}-1)}\bigg{[}U_{\mu\nu}^{A}D_{\mu\nu}^{A}S+U_{\mu\nu}^{A}DS_{\mu\nu}^{A}+UD_{\mu\nu}^{A}S_{\mu\nu}^{A}\bigg{]}\bigg{)}$
$\displaystyle-\frac{N_{c}^{3}(N_{c}-2)}{96(N_{c}^{2}-1)(N_{c}^{2}-4)}d^{ABC}\bigg{[}U_{\mu\nu}^{A}D_{\mu\nu}^{B}S^{C}+U_{\mu\nu}^{A}D^{B}S_{\mu\nu}^{C}+U^{A}D_{\mu\nu}^{B}S_{\mu\nu}^{C}\bigg{]}$
$\displaystyle+\frac{N_{c}^{2}(N_{c}-2)}{8(N_{c}^{2}-1)(N_{c}+2)}d^{ABC}U^{A}D^{B}S^{C}-\frac{N_{c}^{2}}{32(N_{c}^{2}-1)}if^{ABC}U_{\mu\nu}^{A}D_{\nu\rho}^{B}S_{\rho\mu}^{C}\bigg{)}$

where we have defined

$\displaystyle U=\overline{u}_{R}u_{L}\qquad
U^{A}=\overline{u}_{R}\lambda^{A}u_{L}\qquad
U_{\mu\nu}^{A}=\overline{u}_{R}\lambda^{A}\gamma_{\mu}\gamma_{\nu}u_{L}$
$\displaystyle D=\overline{d}_{R}d_{L}\qquad
D^{A}=\overline{d}_{R}\lambda^{A}d_{L}\qquad
D_{\mu\nu}^{A}=\overline{d}_{R}\lambda^{A}\gamma_{\mu}\gamma_{\nu}d_{L}$
$\displaystyle S=\overline{s}_{R}s_{L}\qquad
S^{A}=\overline{s}_{R}\lambda^{A}s_{L}\qquad
S_{\mu\nu}^{A}=\overline{s}_{R}\lambda^{A}\gamma_{\mu}\gamma_{\nu}s_{L}$ (75)

Throughout in the notation $\gamma_{\mu}\gamma_{\nu}$ the condition $\mu<\nu$
is subsumed. (B.1) was originally derived in [57] for $N_{c}=3$.

(74) can be considerably reduced by successive Fierzing to bleach the color
and bring it to a determinantal form typical of instanton induced interactions
that are manifestly $SU(N_{f})$ symmetric but $U(1_{f})$ violating. Since the
procedure is considerably lengthy, we will show how the Fierzing works for the
typical blocks in (74).

Consider the typical colored scalar block in (74)

$\displaystyle U^{A}D^{A}\equiv$
$\displaystyle\overline{u}_{R}\lambda^{A}u_{L}\overline{d}_{R}\lambda^{A}d_{L}$
$\displaystyle=$
$\displaystyle\overline{u}_{Ra}u_{Lb}\overline{d}_{Rc}d_{Ld}\,[\lambda^{A}]^{ab}[\lambda^{A}]^{cd}=2\overline{u}_{Ra}u_{Lb}\overline{d}_{Rb}d_{La}-\frac{2}{N_{c}}\overline{u}_{R}u_{L}\overline{d}_{R}d_{L}$
$\displaystyle=$
$\displaystyle-\frac{2}{N_{c}}\overline{u}_{R}u_{L}\overline{d}_{R}d_{L}-\overline{u}_{R}d_{L}\overline{d}_{R}u_{L}+\frac{1}{4}\overline{u}_{R}\gamma_{\mu}\gamma_{\nu}d_{L}\overline{d}_{R}\gamma_{\mu}\gamma_{\nu}u_{L}$
(76)

where we made use of the color identity in the second line

$\displaystyle[\lambda^{A}]^{ab}[\lambda^{A}]^{cd}=2\delta^{ad}\delta^{bc}-\frac{2}{N_{c}}\delta^{ab}\delta^{cd}$
(77)

and the Fierz re-arrangement in the third line

$\displaystyle\overline{u}_{Ra}u_{Lb}\overline{d}_{Rb}d_{La}=-\frac{1}{2}\overline{u}_{R}d_{L}\overline{d}_{R}u_{L}+\frac{1}{8}\overline{u}_{R}\gamma_{\mu}\gamma_{\nu}d_{L}\overline{d}_{R}\gamma_{\mu}\gamma_{\nu}u_{L}$
(78)

The same procedure carries for the colored tensor block with the result

$\displaystyle U_{\mu\nu}^{A}D_{\mu\nu}^{A}\equiv$
$\displaystyle\overline{u}_{R}\lambda^{A}\gamma_{\mu}\gamma_{\nu}u_{L}\overline{d}_{R}\lambda^{A}\gamma_{\mu}\gamma_{\nu}d_{L}$
$\displaystyle=$
$\displaystyle-\frac{2}{N_{c}}\overline{u}_{R}\gamma_{\mu}\gamma_{\nu}u_{L}\overline{d}_{R}\gamma_{\mu}\gamma_{\nu}d_{L}+12\overline{u}_{R}d_{L}\overline{d}_{R}u_{L}+\overline{u}_{R}\gamma_{\mu}\gamma_{\nu}d_{L}\overline{d}_{R}\gamma_{\mu}\gamma_{\nu}u_{L}$
(79)

using the Fierz re-arrangement

$\displaystyle\overline{u}_{Ra}\gamma_{\mu}\gamma_{\nu}u_{Lb}\overline{d}_{Rb}\gamma_{\mu}\gamma_{\nu}d_{La}=6\overline{u}_{R}d_{L}\overline{d}_{R}u_{L}+\frac{1}{2}\overline{u}_{R}\gamma_{\mu}\gamma_{\nu}d_{L}\overline{d}_{R}\gamma_{\mu}\gamma_{\nu}u_{L}$
(80)

The bleaching of color from the contributions involving the structure factors
$f^{ABC}$ and $d^{ABC}$ can be simplified by noting the respective identities

$\displaystyle
if^{ABC}[\lambda^{A}]^{ab}[\lambda^{B}]^{cd}[\lambda^{C}]^{ef}=2\bigg{(}\delta^{ad}\delta^{cf}\delta^{eb}-\delta^{af}\delta^{cb}\delta^{ed}\bigg{)}$
(81)

and

$\displaystyle d^{ABC}[\lambda^{A}]^{ab}[\lambda^{B}]^{cd}[\lambda^{C}]^{ef}=$
$\displaystyle
2\bigg{(}\delta^{ad}\delta^{cf}\delta^{eb}+\delta^{af}\delta^{cb}\delta^{ed}\bigg{)}-\frac{4}{N_{c}^{2}}\delta^{ab}\delta^{cd}\delta^{ef}$
$\displaystyle-$
$\displaystyle\frac{2}{N_{c}}\bigg{(}\delta^{ab}[\lambda^{A}]^{cd}[\lambda^{A}]^{ef}+\delta^{cd}[\lambda^{A}]^{ab}[\lambda^{A}]^{ef}+\delta^{ef}[\lambda^{A}]^{ab}[\lambda^{A}]^{cd}\bigg{)}$
(82)

(81-B.1) can be established by Fierzing in color and using the identities

$\displaystyle\frac{1}{8}d^{ABC}[\lambda^{A}]^{a}_{b}[\lambda^{B}]^{b}_{c}[\lambda^{C}]^{c}_{d}=\frac{N_{c}^{2}-4}{2N_{c}}C_{F}\,\delta^{a}_{d}$
$\displaystyle\frac{1}{8}f^{ABC}[\lambda^{A}]^{a}_{b}[\lambda^{B}]^{b}_{c}[\lambda^{C}]^{c}_{d}=\frac{i}{2}N_{c}C_{F}\,\delta^{a}_{d}$
(83)

with $C_{F}=(N_{c}^{2}-1)/2N_{c}$ the Casimir in the fundamental
representation. (B.1) can be totally color reduced using (77). Inserting
(B.1-B.1-81-B.1) in (74) yield (6.2) after lengthy algebraic re-arrangements.
Note that in (6.2) we are still using the notation $Q_{\mu\nu}$ but with
$\sigma_{\mu\nu}$ instead of $\gamma_{\mu}\gamma_{\nu}$ with $\mu<\nu$
subsumed.

###  B.2 Alternative form

An alternative form for the 6-quark ′t Hooft vertex, can be reached by
reconsidering the full LSZ reduced amplitude in Weyl notations

$\displaystyle\bigg{[}\frac{n_{I}}{2}\bigg{]}_{\rho}\,$
$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{d^{4}p_{2}}{(2\pi)^{4}}\frac{d^{4}q_{1}}{(2\pi)^{4}}\frac{d^{4}q_{2}}{(2\pi)^{4}}\bigg{[}(2\pi)^{4}\delta^{4}(k_{1}+p_{1}+q_{1}-k_{2}-p_{2}-q_{2})\bigg{]}$
(84)
$\displaystyle\times\bigg{[}\bigg{<}u_{R}^{\dagger}(k_{2})k_{2}\bigg{[}\sqrt{2}\varphi^{\prime}(k_{2})\hat{k}_{2}\epsilon
U\bigg{]}\frac{1}{m}\bigg{[}\sqrt{2}\varphi^{\prime}(k_{1})U^{\dagger}\epsilon\hat{k}_{1}\bigg{]}k_{1}u_{L}(k_{1})$
$\displaystyle\qquad\times
d_{R}^{\dagger}(p_{2})p_{2}\bigg{[}\sqrt{2}\varphi^{\prime}(p_{2})\hat{p}_{2}\epsilon
U\bigg{]}\frac{1}{m}\bigg{[}\sqrt{2}\varphi^{\prime}(p_{1})U^{\dagger}\epsilon\hat{p}_{1}\bigg{]}p_{1}d_{L}(p_{1})$
$\displaystyle\qquad\times
s_{R}^{\dagger}(q_{2})q_{2}\bigg{[}\sqrt{2}\varphi^{\prime}(q_{2})\hat{q}_{2}\epsilon
U\bigg{]}\frac{1}{m}\bigg{[}\sqrt{2}\varphi^{\prime}(q_{1})U^{\dagger}\epsilon\hat{q}_{1}\bigg{]}q_{1}s_{L}(q_{1})\bigg{>}_{U}\bigg{]}+{L\leftrightarrow
R}$

after averaging over the instanton Z-position in 4-Euclidean space, without
taking the zero momentum limit as in (71). The left zero mode in the instanton
background in momentum space is

$\displaystyle\psi^{\alpha}_{iI}(p)=\sqrt{2}\varphi^{\prime}(p)(\hat{p}\epsilon
U)^{\alpha}_{i}\qquad\qquad\varphi^{\prime}(p)=\pi\rho^{2}\bigg{(}I_{0}K_{0}-I_{1}K_{1}\bigg{)}^{\prime}(z=\rho
p/2)$ (85)

in terms of which (84) reads

$\displaystyle\bigg{[}\frac{n_{I}}{2}\bigg{]}_{\rho}\,$
$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{d^{4}p_{2}}{(2\pi)^{4}}\frac{d^{4}q_{1}}{(2\pi)^{4}}\frac{d^{4}q_{2}}{(2\pi)^{4}}\bigg{[}(2\pi)^{4}\delta^{4}(k_{1}+p_{1}+q_{1}-k_{2}-p_{2}-q_{2})\bigg{]}$
$\displaystyle\times\bigg{[}\frac{1}{m^{3}}(2k_{1}k_{2}\varphi^{\prime}(k_{1})\varphi^{\prime}(k_{2}))(2p_{1}p_{2}\varphi^{\prime}(p_{1})\varphi^{\prime}(p_{2}))(2q_{1}q_{2}\varphi^{\prime}(q_{1})\varphi^{\prime}(q_{2}))\bigg{]}$
$\displaystyle\times\bigg{<}\bigg{[}u_{R}^{\dagger}(k_{2})\epsilon
U\bigg{]}\bigg{[}U^{\dagger}\epsilon
u_{L}(k_{1})\bigg{]}\,\bigg{[}d_{R}^{\dagger}(p_{2})\epsilon
U\bigg{]}\bigg{[}U^{\dagger}\epsilon
d_{L}(p_{1})\bigg{]}\,\bigg{[}s_{R}^{\dagger}(q_{2})\epsilon
U\bigg{]}\bigg{[}U^{\dagger}\epsilon
s_{L}(q_{1})\bigg{>}_{U}\bigg{]}+{L\leftrightarrow R}$

The color bracket is explicitly

$\displaystyle\bigg{<}U^{a_{1}}_{c_{1}}U^{\dagger
b_{1}}_{d_{1}}U^{a_{2}}_{c_{2}}U^{\dagger
b_{2}}_{d_{2}}U^{a_{3}}_{c_{3}}U^{\dagger
b_{3}}_{d_{3}}\bigg{>}_{U}\bigg{[}\epsilon^{i_{1}c_{1}}\epsilon_{j_{1}b_{1}}\,\epsilon^{i_{2}c_{2}}\epsilon_{j_{2}b_{2}}\,\epsilon^{i_{3}c_{3}}\epsilon_{j_{3}b_{3}}\bigg{]}$
$\displaystyle\times$
$\displaystyle\bigg{[}u^{\dagger}_{Ra_{1}i_{1}}(k_{2})u_{L}^{d_{1}j_{1}}(k_{1})\,d^{\dagger}_{Ra_{2}i_{2}}(p_{2})d_{L}^{d_{2}j_{2}}(p_{1})\,s^{\dagger}_{Ra_{3}i_{3}}(q_{2})s_{L}^{d_{3}j_{3}}(q_{1})\bigg{]}$
(87)

The color averaging over the unitary matrices can be written in terms of the
Weingarten coefficients

$\displaystyle\bigg{<}U^{a_{1}}_{c_{1}}U^{\dagger
b_{1}}_{d_{1}}U^{a_{2}}_{c_{2}}U^{\dagger
b_{2}}_{d_{2}}U^{a_{3}}_{c_{3}}U^{\dagger b_{3}}_{d_{3}}\bigg{>}_{U}=$
$\displaystyle\frac{(N_{c}^{2}-2)}{N_{c}(N_{c}^{2}-1)(N_{c}^{2}-4)}\sum_{n=1}^{3!}\delta^{a_{1}a_{2}a_{3}}_{(d_{1}d_{2}d_{3})_{n}}\delta^{c_{1}c_{2}c_{3}}_{(b_{1}b_{2}b_{3})_{n}}$
$\displaystyle-$
$\displaystyle\frac{1}{(N_{c}^{2}-1)(N_{c}^{2}-4)}\sum_{n=1}^{3!}\delta^{a_{1}a_{2}a_{3}}_{(d_{1}d_{2}d_{3})_{n}}\bigg{(}\delta^{c_{1}c_{2}c_{3}}_{(b_{2}b_{1}b_{3})_{n}}+\delta^{c_{1}c_{2}c_{3}}_{(b_{1}b_{3}b_{2})_{n}}+\delta^{c_{1}c_{2}c_{3}}_{(b_{3}b_{2}b_{1})_{n}}\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{2}{N_{c}(N_{c}^{2}-1)(N_{c}^{2}-4)}\sum_{n=1}^{3!}\delta^{a_{1}a_{2}a_{3}}_{(d_{1}d_{2}d_{3})_{n}}\bigg{(}\delta^{c_{1}c_{2}c_{3}}_{(b_{3}b_{1}b_{2})_{n}}+\delta^{c_{1}c_{2}c_{3}}_{(b_{2}b_{3}b_{1})_{n}}\bigg{)}$
(88)

The short-hand notation

$\displaystyle\delta^{a_{1}a_{2}a_{3}}_{(d_{1}d_{2}d_{3})_{n}}\equiv\sum^{3!}_{n=1}\bigg{(}\delta^{a_{1}}_{d_{1}}\delta^{a_{2}}_{d_{2}}\delta^{a_{3}}_{d_{3}}+{\rm
perm.}\bigg{)}$ (89)

refers to the product of three kroneckers in the sum over the n! permutations
of the permutation group $S_{3}$. For completeness, we note the analogous
relations for lower color averagings for two flavors

$\displaystyle\left<U^{a_{1}}_{c_{1}}U^{\dagger
b_{1}}_{d_{1}}U^{a_{2}}_{c_{2}}U^{\dagger b_{2}}_{d_{2}}\right>_{U}=$
$\displaystyle\frac{1}{N_{c}^{2}-1}\sum_{n=1}^{2!}\delta^{a_{1}a_{2}}_{(d_{1}d_{2})_{n}}\delta^{c_{1}c_{2}}_{(b_{1}b_{2})_{n}}-\frac{1}{N_{c}(N_{c}^{2}-1)}\sum_{n=1}^{2!}\delta^{a_{1}a_{2}}_{(d_{1}d_{2})_{n}}\delta^{c_{1}c_{2}}_{(b_{2}b_{1})_{n}}$
(90)

and the well known one for one flavor

$\displaystyle\left<U^{a}_{c}U^{\dagger
b}_{d}\right>_{U}=\frac{1}{N_{c}}\delta^{a}_{d}\delta_{b}^{c}$ (91)

##  Appendix C Exploding quark modes in Sphaleron background

In so far, we have extracted the sphaleron-six-quark vertex using the $O(4)$
exact zero modes in a self-dual instanton (anti-instanton) background. While
at $t=0$ this is the case, as $t>0$ the zero modes become real and turn to
$O(3)$ exploding quark modes as we discussed in [22]. We now briefly review
how these modes are constructed and then substitute them in (12) to generate
${\cal L}(t>0)$ to describe the final state explosive vertex in the
diffractive $pp\rightarrow ppX$ process,

###  C.1 O(4) and O(3) fermionic zero mode

A Dirac fermion in a general gauge field solves the equation

$\left(\partial_{\mu}-iA_{\mu}\right)\gamma_{\mu}\,{\psi}=0$ (92)

We will use the chiral basis with spin matrices
$\bar{\sigma}_{Ms}=(1,-i\vec{\sigma}_{s})$ and

$\displaystyle\gamma_{5}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\qquad\gamma_{\mu}=\left(\begin{array}[]{cc}0&\sigma_{Ms}\\\
\bar{\sigma}_{Ms}&0\end{array}\right)$ (97)

If we recall that the t′ Hooft symbol satisfies the color identity
$\bar{\sigma}_{Mc}\sigma_{Nc}=\sigma_{ac}\overline{\eta}_{aMN}$ with
$\sigma_{Mc}=(1,-i\vec{\sigma}_{c})$, then the positive chirality mode
$\psi_{+}$ solves (92)

$\left(\bar{\sigma}_{Ms}\partial_{M}+\frac{1}{2}\bar{\sigma}_{Ns}\bar{\sigma}_{Nc}\,\sigma_{Mc}\partial_{M}\,F\right)\,{\psi}_{+}=0$
(98)

with the spin and color matrices commuting. $F(\xi(y))$ for an $O(4)$
symmetric gauge field is given in (7). Note that while writing (98) we have
added a U(1) part to the gauge field for notational simplicity, and will
remove it in the final answer by inspection. A formal solution to (98) is
$(\psi_{+})^{a}_{\mu}=\varphi\epsilon_{\mu}^{a}$ wich is a singlet (hedgehog)
in color-spin space

$\sigma_{Ms}\chi_{Q}=\bar{\sigma}_{Mc}\epsilon\qquad\bar{\sigma}_{Ms}\epsilon=\sigma_{Mc}\epsilon$
(99)

Inserting (99) in (98) yields

$\left(\bar{\sigma}_{Ms}\partial_{M}+\frac{1}{2}\bar{\sigma}_{Ns}\sigma_{Ns}\,\bar{\sigma}_{Ms}\partial_{M}\,F\right)\,\varphi\epsilon=0$
(100)

To remove the redundant U(1) contribution noted above we use

$\bar{\sigma}_{Ns}\sigma_{Ns}\,\epsilon=\left(1+(\vec{\sigma}_{s})^{2}\right)\epsilon\rightarrow(\vec{\sigma}_{s})^{2}\epsilon=3\epsilon$
(101)

after which $\varphi$ is seen to solve
$\varphi^{\prime}+\frac{3}{2}F^{\prime}\varphi=0$, hence

$({\psi}_{+})^{a}_{\mu}(y)={\bf
C}\,e^{-\frac{3}{2}F(\xi(y))}\epsilon^{a}_{\mu}$ (102)

with

$\xi(y)=\frac{1}{2}{\rm
ln}\left(\frac{(t-\rho)^{2}+r^{2}}{(t+\rho)^{2}+r^{2}}\right)$ (103)

The overall normalization ${\bf C}$ is fixed by

${\bf
C}=\left|\int_{T_{\frac{1}{2}}}d^{4}y\,e^{-3F(\xi(y))}\right|^{-\frac{1}{2}}$
(104)

with $T_{\frac{1}{2}}$ the sphaleron tunneling time. (104) is agreement with
the result in [22] (see Eq. 22 with the exponent $2\rightarrow\frac{3}{2}$
when (101) is enforced).

The chiral O(3) symmetric zero mode follows by applying the off-center
inversion (10) in Section 4, onto the O(4) symmetric zero mode in (102). The
corresponding tansform is

${\tilde{\psi}}_{+}(x)=\frac{\sigma^{\dagger}_{\mu}\,(y+a)_{\mu}}{1/(y+a)^{2}}\,{\psi}_{+}(y)$
(105)

or more explicitly ($r=|\vec{x}|$)

$\displaystyle({\tilde{\psi}}_{+})^{a}_{\mu}(t,r)=\frac{{8\bf\,C}\rho^{6}}{((t+\rho)^{2}+r^{2})^{2}}\,e^{-\frac{3}{2}F(\xi(y))}\,\big{[}\left((t+\rho)+i\vec{\sigma}\cdot\vec{x}\right)\,\epsilon\big{]}^{a}_{\mu}$
(106)

This result is in agreement with the one derived in [22] prior to the
analytical continuation to Minkowski space.

##  Appendix D Graviton-nucleon coupling

In bulk, the graviton couples to the Dirac fermion through its energy momentum
tensor [58, 52]

$-\frac{\sqrt{2\kappa^{2}}}{2}\int
d^{5}x\,\sqrt{g}\,h_{\mu\nu}T_{F}^{\mu\nu}=-\frac{\sqrt{2\kappa^{2}}}{2}\int
d^{5}x\,\sqrt{g}\,\bigg{(}\frac{e^{-\phi}}{2g_{5}^{2}}\frac{i}{2}\,z\,\overline{\Psi}\gamma^{\mu}\overset{\leftrightarrow}{\partial^{\nu}}\Psi-\eta^{\mu\nu}\mathcal{L}_{F}\bigg{)}$
(107)

with typically for the Dirac fermions in the soft-wall model

$\displaystyle\mathcal{L}_{F}=$
$\displaystyle\frac{e^{-\phi(z)}}{2g_{5}^{2}}\,\bigg{(}\frac{i}{2}\bar{\Psi}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{D}_{N}-\overleftarrow{D}_{N}\big{)}\Psi-(M+V(z))\bar{\Psi}\Psi\bigg{)}\,,$
(108)

with $M=\tau-3$ the anomalous dimension. The potential
$V(z)=\tilde{\kappa}_{N}^{2}z^{2}$ is added to make the Dirac fermions massive
in bulk. Here $e^{N}_{A}=z\delta^{N}_{A}$ denotes the inverse vielbein, and
the covariant derivatives are defined as

$\displaystyle\overrightarrow{D}_{N}=$
$\displaystyle\overrightarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]$
$\displaystyle\overleftarrow{D}_{N}=$
$\displaystyle\overleftarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]$
(109)

The components of the spin connection are $\omega_{\mu z\nu}=-\omega_{\mu\nu
z}=\frac{1}{z}\eta_{\mu\nu}$, the Dirac gamma matrices satisfy anti-
commutation relation $\\{\Gamma^{A},\Gamma^{B}\\}=2\eta^{AB}$ with the
explicit choice $\Gamma^{A}=(\gamma^{\mu},-i\gamma^{5})$.

In the nucleon as a bulk Dirac fermion, the energy momentum tensor is
characterized by the three invariant form factors in (56). For the soft wall
model, we have explicitly [58, 52]

$A(k)=\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z\,\big{(}\psi_{R}^{2}(z)+\psi_{L}^{2}(z)\big{)}\,\mathcal{H}(k,z)=-\frac{C(k)}{(\alpha
z_{0}m_{N}/2)^{2}}$ (110)

with ${\cal H}(K,z^{\prime})$ the non-normalizable bulk-to-boundary graviton
propagator [59, 60, 61]

$\displaystyle\mathcal{H}(K,z)=$ $\displaystyle 4z^{\prime
4}\Gamma(a_{K}+2)U\Big{(}a_{K}+2,3;2\tilde{\kappa}_{N}^{2}z^{2}\Big{)}=\Gamma(a_{K}+2)U\Big{(}a_{K},-1;2\tilde{\kappa}_{N}^{2}z^{2}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{\Gamma(a_{K}+2)}{\Gamma(a_{K})}\int_{0}^{1}dx\,x^{a_{K}-1}(1-x){\rm
exp}\Big{(}-\frac{x}{1-x}(2\tilde{\kappa}_{N}^{2}z^{2})\Big{)}$ (111)

and $a_{K}={K^{2}}/{8\tilde{\kappa}_{N}^{2}}\geq 0$. We have used the
transformation $U(m,n;y)=y^{1-n}U(1+m-n,2-n,y)$. (D) satisfies the
normalization condition ${\cal H}(0,z^{\prime})={\cal H}(K,0)=1$. The bulk
Dirac fermions $\Psi(x,z)=\psi_{L,R}(z)e^{-ip\cdot x}u_{L,R}(p)$ are

$\displaystyle\psi_{R}(z)=\frac{\tilde{n}_{R}}{\tilde{\kappa}_{N}^{\tau-2}}z^{\frac{5}{2}}\xi^{\frac{\tau-2}{2}}L_{0}^{(\tau-2)}(\xi)\qquad\psi_{L}(z)=\frac{\tilde{n}_{L}}{\tilde{\kappa}_{N}^{\tau-1}}z^{\frac{5}{2}}\xi^{\frac{\tau-1}{2}}L_{0}^{(\tau-1)}(\xi)\,$
(112)

Here $\xi=\tilde{\kappa}_{N}^{2}z^{2}$, $\tau=7/2-1/2=3$ is the twist
parameter, and $L_{n}^{(\alpha)}(\xi)$ are the generalized Laguerre
polynomials, with

$\tilde{n}_{R}=\tilde{n}_{L}\tilde{\kappa}_{N}^{-1}\sqrt{\tau-1}\qquad\tilde{n}_{L}=\tilde{\kappa}_{N}^{\tau}\sqrt{{2}/{\Gamma(\tau)}}\\\
$

More specifically [58, 52] ($k^{2}=-K^{2}=t\leq 0$)

$\displaystyle A(k)$ $\displaystyle=$ $\displaystyle
A(0)\,(a_{k}+1)\bigg{(}-\left(1+a_{k}+2a_{k}^{2}\right)+2\left({a_{k}}+2{a^{3}_{k}}\right)\Phi(-1,1,a_{k})\bigg{)}$
(113) $\displaystyle=$ $\displaystyle
A(0)\bigg{(}(1-2a_{k})(1+a_{k}^{2})+a_{k}(1+a_{k})(1+2a_{k}^{2})\bigg{(}\psi\bigg{(}\frac{1+a_{k}}{2}\bigg{)}-\psi\bigg{(}\frac{a_{k}}{2}\bigg{)}\bigg{)}\bigg{)}$

with $a_{k}={k^{2}}/{8\tilde{\kappa}_{N}^{2}}$, and $B(0)=0$.
$\Phi(-1,1,a^{\prime})$ refers to the LerchPhi function, and $\psi(x)$ refers
to the digamma function or harmonic number $H_{x}=\psi(x)+\gamma$. The
gravitational form factor $C(k)$ is proportional to $A(k)$ modulo a negative
overall constant $-(\alpha z_{0}m_{N}/2)^{2}<0$ which is left undetermined
since $\alpha$ is arbitrary in the tensor decomposition of the graviton [52].
Note that ${\cal H}(k,0)={\cal H}(0,z)$ is arbitrary (1-point function), so
that $A(0)$ is not fixed in holography.

##  Appendix E Spin-j-nucleon coupling

For diffractive scattering at large $\sqrt{s}$, the spin-2 graviton coupling
reggeizes to spin-j coupling. The resummed spin-j couplings transmutes to the
Pomeron coupling. The generic form of the spin-j coupling is [52]

$\displaystyle\mathcal{V}^{\alpha\beta(TT)}_{h\bar{\Psi}\Psi}(j,p_{1},p_{2},k_{z})=-\frac{\sqrt{2\kappa^{2}}}{2}\int
dz\,\sqrt{g}\,e^{-\phi}\,\mathcal{H}(j,K,z)\,\bigg{[}z^{1+2(j-2)}\frac{1}{2}\bar{\Psi}(p_{2},z)\gamma^{[\alpha}p^{\beta]}\,\Psi(p_{1},z)z^{-(j-2)}\bigg{]}$

where ${\cal H}(j,K,z)$ is the spin-j bulk to boundary space-like propagator.
For the pure spin-2 graviton $h_{\mu}^{\mu}=0$ is traceless, so the coupling
to ${\cal L}_{F}$ in (107) drops out. The z-pre-factors in the internal braket
in (E) reflect on the spin-j generalization of the energy-momentum tensor
(higher twist vertices). For the soft-wall model, (E) takes the explicit form

$\mathcal{V}^{\alpha\beta(TT)}_{h\bar{\Psi}\Psi}(j,p_{1},p_{2},k_{z})=-\sqrt{2\kappa^{2}}\times
g_{5}^{2}A(j,K)\times\frac{1}{2}\bar{u}(p_{2})\gamma^{[\alpha}p^{\beta]}u(p_{1})$
(115)

with

$\displaystyle A(j,K)=$ $\displaystyle\frac{1}{2g_{5}^{2}}\int
dz\,\sqrt{g}\,e^{-\phi}\,\mathcal{H}(j,K,z)\bigg{[}z^{1+2(j-2)}\,\big{(}\psi_{R}^{2}(z)+\psi_{L}^{2}(z)\big{)}\,z^{-(j-2)}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2g_{5}^{2}}\frac{2^{2-\Delta(j)}\tilde{\kappa}_{N}^{j-2-\Delta(j)}}{\Gamma(\tilde{a}(j))}\int_{0}^{1}dx\,{x^{\tilde{a}(j)-1}}{(1-x)^{-\tilde{b}(j)}}\big{(}I_{z}^{R}(x)+I_{z}^{L}(x)\big{)}\,,$
(116)

where the fermionic integrals are defined as
($\xi=\tilde{\kappa}_{N}^{2}z^{2}$)

$I_{z}^{R/L}(x)=\int
dz\,\sqrt{g}\,e^{-\phi}\,z^{1+2(j-2)}\,\psi_{R/L}^{2}(z)\,\xi^{\frac{-(j-2)}{2}}\xi^{2-\frac{\Delta(j)}{2}}{\rm
exp}\Big{(}-\frac{2x\xi}{1-x}\Big{)}\,,$ (117)

and j-dependent parameters are

$\displaystyle\tilde{a}(j)=a_{k}+2-\frac{1}{2}\Delta(j)\qquad$
$\displaystyle{\rm and}$ $\displaystyle\qquad\tilde{b}(j)=3-\Delta(j)$
$\displaystyle\Delta(j)=2+\sqrt{2\sqrt{\lambda}(j-j_{0})}\qquad$
$\displaystyle{\rm and}$ $\displaystyle\qquad
j_{0}=2-\frac{2}{\sqrt{\lambda}}$ (118)

Using the bulk Dirac wavefunctions (112) for the soft-wall model, we can
further reduce the fermionic integrals (117) to have [52]

$\displaystyle
I_{z}^{R}(x)=\frac{1}{2}\times\tilde{\kappa}_{N}^{-2(j-2)}\times\bigg{(}\frac{\tilde{n}_{R}}{\tilde{\kappa}_{N}^{\tau-1}}\bigg{)}^{2}\times\Gamma\bigg{(}\frac{j-2}{2}+\tau-\frac{\Delta(j)}{2}+1\bigg{)}\times\bigg{(}\frac{1+x}{1-x}\bigg{)}^{-\frac{j-2}{2}-\tau+\frac{\Delta(j)}{2}-1}\,,$
$\displaystyle
I_{z}^{L}(x)=\frac{1}{2}\times\tilde{\kappa}_{N}^{-2(j-2)}\times\bigg{(}\frac{\tilde{n}_{L}}{\tilde{\kappa}_{N}^{\tau}}\bigg{)}^{2}\times\Gamma\bigg{(}\frac{j-2}{2}+\tau-\frac{\Delta(j)}{2}+2\bigg{)}\times\bigg{(}\frac{1+x}{1-x}\bigg{)}^{-\frac{j-2}{2}-\tau+\frac{\Delta(j)}{2}-2}\,.$

Using (E) in (E), the spin-j form factor of the Dirac fermion reduces to

$\displaystyle A(j,K)=$
$\displaystyle\frac{1}{4g_{5}^{2}}\frac{\tilde{\kappa}_{N}^{-(j-2)-\Delta(j)}}{\Gamma(\tilde{a}(j))}\int_{0}^{1}dx\,x^{\tilde{a}(j)-1}(1-x)^{-\tilde{b}(j)}$
$\displaystyle\times\bigg{(}\bigg{(}\frac{\tilde{n}_{R}}{\tilde{\kappa}_{N}^{\tau-1}}\bigg{)}^{2}\times\Gamma(c(j))\bigg{(}\frac{1+x}{1-x}\bigg{)}^{-c(j)}+\bigg{(}\frac{\tilde{n}_{L}}{\tilde{\kappa}_{N}^{\tau}}\bigg{)}^{2}\times\Gamma(c(j)+1)\bigg{(}\frac{1+x}{1-x}\bigg{)}^{-(c(j)+1)}\bigg{)}\,,$

with $\Delta(j)$, $\tilde{a}(j),\tilde{b}(j)$ given in (E) and

$\displaystyle c(j)=(\tau+1)+\frac{j-2}{2}-\frac{\Delta(j)}{2}$ (121)

$A(j,K)$ is the spin-j generalization of the spin-2 graviton form factor
(113). The x-integral form (E) can be interpreted as the holographic partonic
content of the Dirac fermion at strong coupling for spin-j, with $x$ playing
the role of the Bjorken momentum fraction.

##  Appendix F Pomeron-Pomeron-glueball bulk coupling

The sphaleron at the boundary of the Witten vertex sources either $G^{2}$ or
$G\tilde{G}$. $G^{2}$ is dual to the dilaton in bulk, while $G\tilde{G}$ is
dual to the axion, both of which are described by massive glueballs of even
and odd parity respectively. $G^{2}$ yields a direct scalar Pomeron-Pomeron-
dilaton coupling times the pertinent bulk-to-boundary propagato. $G\tilde{G}$
yields an indirect pseudoscalar Pomeron-Pomeron-pseudoscalar coupling by
mixing to the bulk singlet and Chern-Simons term, times pertinent bulk-to-bulk
and bulk-to-boundary propagators

###  F.1 Bulk dilaton-graviton-graviton coupling

The graviton in bulk is dual to a a scalar glueball or $G^{2}$ on the
boundary. The effective action for the graviton is given by the standard
Einstein-Hilbert action

$\frac{1}{2\tilde{g}_{5}^{2}}\int d^{5}x\sqrt{g}e^{-2\phi}\sqrt{R}$ (122)

with $R$ the scalar curvature and $2\tilde{g}_{5}^{2}=2\kappa^{2}=16\pi
G_{N}={8\pi^{2}}/{N_{c}^{2}}$. To extract the bulk gravitational coupling, we
linearize the metric around the flat metric
$\eta_{\mu\nu}\rightarrow\eta_{\mu\nu}+\sqrt{2\kappa^{2}}h_{\mu\nu}$. In de
Donder gauge with
$\partial_{\alpha}h^{\alpha}_{\mu}=\frac{1}{2}\partial_{\mu}f$ and
$f=h_{\mu}^{\mu}$, the expansion of (122) yields the quadratic contributions

$(\sqrt{2\kappa^{2}})^{2}\bigg{(}\frac{1}{8\tilde{g}_{5}^{2}}\,g^{MN}\eta^{\alpha\bar{\alpha}}\eta^{\beta\bar{\beta}}\partial_{M}h_{\alpha\bar{\alpha}}\,\partial_{N}h_{\beta\bar{\beta}}-\frac{1}{4\tilde{g}_{5}^{2}}\,g^{MN}\eta^{\alpha\bar{\alpha}}\eta^{\beta\bar{\beta}}\partial_{M}h_{\alpha\beta}\partial_{N}h_{\bar{\alpha}\bar{\beta}}\bigg{)}$
(123)

and the cubic contributions

$(\sqrt{2\kappa^{2}})^{3}\bigg{(}-\frac{1}{8\bar{g}_{5}^{2}}\,g^{MN}\eta^{\alpha\bar{\alpha}}\eta^{\beta\bar{\beta}}f\,\partial_{M}h_{\alpha\bar{\beta}}\,\partial_{N}h_{\bar{\alpha}\beta}-\frac{1}{4\bar{g}_{5}^{2}}\,g^{MN}\eta^{\alpha\bar{\alpha}}\eta^{\beta\bar{\beta}}\,\partial_{M}f\,h_{\alpha\bar{\beta}}\,\partial_{N}h_{\bar{\alpha}\beta}\bigg{)}$
(124)

where only the couplings through the tracefull $f$ and traceless
$h_{\alpha\beta}$ were retained. Note that $f(z=0,x)$ couples to
$T^{\mu}_{\mu}(x)$ of the gauge theory on the boundary, hence the
identification with the dual of $G^{2}$.

The induced interaction between the scalar and the graviton on the boundary,
follows from the bulk decomposition $h_{\mu\nu}(x,z)=h(z)h_{\mu\nu}(x)$ and
$f(x,z)=s(z)f(x)$, in the form

$\displaystyle\int
d^{4}x\,\eta^{\alpha\bar{\alpha}}\eta^{\beta\bar{\beta}}\bigg{(}$
$\displaystyle{\bf
C}^{+}_{1}\,f(x)\,\eta^{\mu\nu}\partial_{\mu}h_{\alpha\bar{\beta}}(x)\partial_{\nu}h_{\bar{\alpha}\beta}(x)$
$\displaystyle+$ $\displaystyle{\bf
C}^{+}_{2}\,f(x)\,h_{\alpha\bar{\beta}}(x)h_{\bar{\alpha}\beta}(x)+{\bf
C}_{3}^{+}\,\eta^{\mu\nu}\partial_{\mu}f(x)h_{\alpha\bar{\beta}}(x)\partial_{\nu}h_{\bar{\alpha}\beta}(x)\bigg{)}$
(125)

with the induced bulk coefficients

$\displaystyle{\bf C}_{1}^{+}=$
$\displaystyle-\frac{(\sqrt{2\kappa^{2}})^{3}}{4\tilde{g}_{5}^{2}}\int
dz\sqrt{g}e^{-\phi}\bigg{(}\frac{z^{2}}{2}s(z)h^{2}(z)\bigg{)}$
$\displaystyle{\bf C}_{2}^{+}=$
$\displaystyle+\frac{(\sqrt{2\kappa^{2}})^{3}}{4\tilde{g}_{5}^{2}}\int
dz\sqrt{g}e^{-\phi}\bigg{(}\frac{z^{2}}{2}s(z)h^{\prime
2}(z)+\frac{z^{2}}{2}s^{\prime}(z)(h^{2})^{\prime}(z)\bigg{)}$
$\displaystyle{\bf C}_{3}^{+}=$
$\displaystyle-\frac{(\sqrt{2\kappa^{2}})^{3}}{4\tilde{g}_{5}^{2}}\int