# test

Zongqiang Pang, Liping Bai Nanjing Unversity of Posts and Telecommunications,
College of Automation & College of Artificial Intelligence, Nanjing,
Jiangsu,210000 China<EMAIL_ADDRESS>
# Coaching with Controllers: A Novel Paradigm for Merging Classic Control with
Learning

Zongqiang Pang, Liping Bai Nanjing Unversity of Posts and Telecommunications,
College of Automation & College of Artificial Intelligence, Nanjing,
Jiangsu,210000 China<EMAIL_ADDRESS>
###### Abstract

In this paper, we propose a controller based coaching scheme in order to
accelerate reinforcement learning (RL). Increasingly, control community and RL
community have realized that they are complimentary additions to one another.
Existing control strategies would make the learning process more efficient
while learning extends control to less scripted scenarios. We believe one
productive way to merge classic control with learning is through the concept
of coaching. World class atheletes forge their skillsets with a team of
coaches, who provide conditions for the atheletes to collect critical
experiences. This relationship can be constructed between the classical
controllers and RL agents as well. While classical controllers might be
inferior to RL agents in terms of control strategies, they can provide
guidance for expedited data collection. We devise and implemente a method for
"Controller Based Coaching", derived from our observations of the athletic
training process. Experiments are conducted in Mujoco locomotion environments
and we conclude from the data that when the coaching structure is set at its
goldilocks spot, agents training can be accelerated, in some cases
significantly, yielding uncompromised training results in the mean time. This
is an important proof of concept that classical controller based coaching can
be a novel and effective paradigm for merging control with learning and
warrants further investigations along this direction. All the code and data
can be found at: github.com/BaiLiping/ControllerBasedCoaching

###### Index Terms:

Reinforcement Learning, Control, L4DC

##  I Introduction

Reinforcement Learning (RL) is a cyle between interaction with the environment
based on current understandings and refinement of that understanding based on
observations. RL agents methodically extracting information from experiences,
gradually bounding system models, policy distributions or cost-to-go
approximations to maximize the expected rewards along a sequence of actions.
While the field of RL routinely generate jaw-dropping results
[Mnih2013PlayingAW][Hausknecht2015DeepRQ][Andrychowicz2020LearningDI][Kalashnikov2018QTOptSD][Lee2020LearningQL],
there are aspects of its methodology that are inherently inefficient and the
focus of current researches is to reformulate RL into its more effective form.

There are three angles of attack in order to address the inefficiencies.
First, RL researchers who are well versed in statistics and information
theories approach the problem by devising ever more sophisticated statistical
processes to push information extraction to its
limit.[Ho2016GenerativeAI][Finn2016UnsupervisedLF][Pathak2017CuriosityDrivenEB][Burda2019LargeScaleSO][Finn2017ModelAgnosticMF][Mishra2018ASN][Nachum2018DataEfficientHR][Vezhnevets2017FeUdalNF][Blundell2015WeightUI][Gal2017ConcreteD][Dasgupta2019CausalRF][Zhang2020DesigningOD]
Second, control researchers who are capable of wielding control theories and
optimization techniques provide a systematic perspective to RL which is a
dearth of analytical
frameworks.[Weinan2017APO][Dupont2019AugmentedNO][Betancourt2018OnSO][Nachum2020ReinforcementLV][Luo2019ADR][Wu2020DataDrivenDL][Shi2019NeuralLS][Hewing2020LearningBasedMP][Mohan2020EmbeddingHP][Lusch2018DeepLF][Bai2019DeepEM][BelbutePeres2020CombiningDP]
Third, a immitation or teaching structure to incorporate the best control
strateges into RL training process.
[Knox2009InteractivelySA][Knox2010CombiningMF][Peng2018DeepMimicED][Peng2020LearningAR][Paine2018OneShotHI]

In this paper, we propose a fourth angle of attack: classical controllers
based coaching. Professional athletes don’t get where they are via trial-and-
error. Their skillsets are forged through a series of theoretically and
empirically proven training techniques that seek to maximize training
efficiency. Based on this observation, we construct a coaching relationship
between classical controllers and RL agents.

The objective of a classical controller based coach is to facilitate data
collection on critical states, much like a coach guides athletes towards
essential experiences instead of repeating steps that merely pave the way. We
ascribe the observed training acceleration brought about by our approach to
this expedited data collection on critical states. We would elaborate on our
ideas further in the subsequent sections.

We implemented the proposed active boundary scheme on various OpenAI Gym
environments as a proof of concept and we concluded that when the boundaries
are set appropriately, accelerated learning can be achieved and the resulting
agents have indistinguishable evaluation scores compare to the agents trained
with normal methods. We have confidence in our conclusion since while
randomness might underline one set of data, it is highly unlikely that
randomness alone can explain the same observations made in a different set of
experiments. Our result indicated the feasibility of adopting human training
techniques into RL setup and could be the beginning of other fruitful
researches in this direction.

In section II we introduce reinforcement learning in the language of control.
In section III we detail our proposed active boundary scheme for accelerated
RL agent training. In section IV, we present the results of all our
experiments. While the experiments are conducted in OpenAI Gym simulated
environment, we consciously designed things in a way such that it can be
conveniently implemented in the physical world as well. In the final section,
we conclude what we have learned from our experiments and layout directions
for further research.

##  II Reinforcement Learning For Control Researchers

###  II-A Note on Notation and Nomenclature

The first barrier stands between control researchers and RL is the jungle of
notations. At this point, just about every researcher has his own symbolic
system. As noted by Warren Powell in his 2014 introductory paper on
Approximate Dynamic Programming: "Somewhat frustratingly, these communities
have also developed different vocabularies and different notational systems."
Yet in the same breath, Powell introduced a new set of customized notation.
[Powell2009WhatYS] Here, we try to point out the differences to make life
easier for control researchers. Control communities use the notation system
introduced by Lev Pontryagin. The state is denoted by $\mathcal{X}$,
conventionally a letter representing the unknown. Action is denoted by
$\mathcal{U}$, the first letter of Russian for action. The dynamics and
stochasticity is captured by physical model constraints
$x_{t+1}=f(x_{t},u_{t},e_{t})$ where e denotes the noise of a system. The
objective is usually to minimize the cost function $\mathcal{J(.)}$.
Reinforcement Learning communities adapt their notation system from the field
of Operations Research. State is denoted $\mathcal{S}$, Action is denoted
$\mathcal{A}$. The system dynamics and stochasticity is captured via
transition matrix $\mathcal{P}$ of a Markov Decision Process. The objective of
RL is the maximize the reward function $\mathcal{R(.)}$.

Richard Sutton uses terminologies that can be overloaded and confusing, such
as Monte Carlo, Temporal Difference(TD), in his book Introduction to
Reinforcement Learning [Sutton1998IntroductionTR]. Although the control
community also has to deal with the sequential aspect of a problem, the
terminologies used are much more down-to-earth and straightforward, such as
one-step look ahead, n-steps look ahead. In the context of RL, Monte Carlo
simply means update with data collected from an entire episode. Temporal
Difference means update with data collected with one step. TD($\lambda$) means
update with data collected with $\lambda$ steps.

###  II-B From Optimization To Learning

Reinforcement Learning is not a new subject, control researchers probably know
it by the name Approximate Dynamic Programming (ADP) or Neuro Dynamic
Programming (NDP). When the first batch of reasonable RL results was
introduced, it was met with coldness by people in control. The preface of
Bertseka’s book Neuro Dynamic Programming provides a good sample of how RL was
perceived by the control community back in 1996: "…These methods
(Reinforcement Learning) were aiming to provide effective suboptimal solutions
to complex problems of planning and sequential decision making under
uncertainty, that for a long time were thought to be intractable. Our first
impression was that the new methods were ambitious, overly optimistic, and
lacked a firm foundation. Three years later, after a lot of study, analysis,
and experimentation, we believe that our initial impressions were largely
correct." [Bertsekas1996NeuroDynamicP]

Today, the theoretical foundation of RL is as bleak as it was in 1996, but its
deficiencies can be easily papered over with the advant of Neural Networks and
ever more impressive computational capacities. Before 2010, the predominant
function approximators are handcrafted kernels, where feature spaces transform
nonlinear functions into linear space. Today, the default function
approximator is Deep Neural Network and its variants. In the computation
front, Graphic Process Unit (GPU) based parallelization is accessible to
anyone who is proficient with C/C++[Nickolls2008ScalablePP], and software
packages such as PyTorch[Paszke2017AutomaticDI] and
Tensorflow[Abadi2016TensorFlowAS] made this task even more trivial.

In order to better introduce RL to control researchers, we built on Benjamin
Recht’s presentation[Recht2018ATO], framed RL in the language of optimization.
Reinforcement Learning is an umbrella term, which encompasses distinctive
genres of learning. Learning in the context of RL means updating parameters of
the function approximators. Categorically speaking, there are three avenues
where learning finds its way into optimization as shown in Figure1.

Figure 1: From Optimization to Learning

Model-Based or Model-Free learning refers to whether or not learning is used
to approximate the system dynamics function. If there is an explicit action
policy, it is called on-policy learning. Otherwise, the optimal action would
be implicitly captured by the Q value function and that would be called off-
policy learning instead. Importance sampling allows "limited off-policy
learning" capacity, which allows data reuse in a trusted region. Online
learning means interleaving data collection and iterative network parameters
update. Offline learning means the data is collected in bulk first and then
the network parameters are set with regression computation. Batch learning, as
the name suggested, is in between online and offline learning. An agent would
first generate data that fill its batch memory and then sample from the batch
memories for iterative parameter update. New data would be generated with the
updated parameters to replace older data in the memory. This taxonomy is
somewhat outdated now. When Richard Sutton wrote his book, the algorisms he
had in mind fall nicely into various categories. Today, however, the popular
algorisms would combine more than one route to derive superior performance,
and can’t be pigeonholed.

We will not go into the details of various RL algorithms and the zoo or
acronyms that come with it, as that is beyond the scope of this paper. PPO
agents were chosen for all our experiments because the learning curves
generated by PPO agents are smoother compare to other agents. The goal of our
paper is to indicate the proposed active boundary method works, which is
easier to be shown with smoother curves. However, we would like to point out
two crucial concepts that undergird the feasibility of reinforcement learning.

####  II-B1 Bellman Update

Reasoning backward or Dynamic Programming forms the basis of reinforcement
learning. If we know the desired final state and system dynamics then we can
simply compute backward on the penultimate step. Set the penultimate step as
the final step, then repeat the process until the action strategy for the
initial state is obtained. This approach, Bellman Update, would work if two
prerequisites are met: first, the system dynamics function is known; second,
the dimensions of observation space and action space are small. Neither would
be true for the problems we encounter in the real world.

When the dynamics are unknown yet the dimensions of the problem are
manageable, the cost-to-go lookup table can be constructed with sampling
methods. As mentioned earlier, the way that the lookup table is updated with
the sample can be categorized by the horizon of the data collection. If the
lookup table is updated with data collected from an entire episode, it is
called the Monto Carlo method. If the lookup table is updated every step, it
is called Temporal Difference method, and there are various methods in between
these two extremes. When the dimensions of the problem are large in addition
to unknown dynamics, function approximators are used for the system model, the
policy function, or the cost-to-go function.

Immediately, control researchers can spot the deficiency with the latter case.
For backward reasoning to work properly, the notion of optimality has to be
satisfied either through known dynamics function or robust sampling methods,
yet when the dynamics are unknown and the dimensions large, the notion of
optimality can no longer be guaranteed through sampling. This is where the
expertise of statisticians and information theoriests comes in and the
research topic of exploration-exploitation trade-off.

####  II-B2 Bootstrap

Another fundamental concept for RL is convergence through bootstrap. Instead
of asymptotically approaching a known target function, bootstrap methods
approach an assumed target first and then update the target assumption based
on collected data. This process is illustrated by Figure2. Intuitively, when
estimation functions are evaluated with assumption instead of the true value,
things could just run around in circles and never converge. That is indeed the
case before the introduction of DQN[Osband2016DeepEV]. As a matter of fact.
the stability and convergence achieved in that paper are more of a reflection
on the techniques and know-how of the DeepMind team. Initially, those results
can not be easily replicated by other labs. Even today, instability in RL is
still a major research focus.

(a) With Known Evaluation Function

(b) Bootstrap

Figure 2: Convergence through Bootstrap

##  III Expedited Data Collection on the Critical States with Active Boundary

As we pointed out in the introduction, human has amassed vast amounts of
empirically proven techniques for accelerated skill acquisition. In this
paper, we focus on the idea of expedited data collection on critical states
with active boundaries, where critical states refer to the ones that are
essential for skill acquisition.

Focused training with critical states is the bread and butter for athletes.
Chess players start their training with endgames because that allows them to
dwell on critical steps which illustrate the reasoning behind chess tactics,
instead of wasting time on the "mechanical/maintenance moves" that lead up to
a strategic position; Tennis coaches feed athlete balls with a specific angle
and spin speed to force he with a backhand or forehand response. Examples like
this are abundant in the world of sports. Active boundary is one way to
implement focused training with critical states.

When the athlete is about to deviate from optimal action, an attentive coach
would apply force to course-correct on behalf of the athlete. This way, the
athlete would have experienced the right way to handle that state as opposed
to just falling. As indicated by Figure 3, a coach can implement an active
boundary because he can apply the appropriate force in due time. A safety net,
on the other hand, is a passive boundary, since it does nothing other than
catch people during a fall and no information on the critical state is
generated in that process. The distinction between active and passive boundary
is important because while prevention of premature termination is an implicit
outcome of active boundary, it is by no means its primary goal. The protective
property of active boundary stems from the fact that when a critical state is
handled poorly, a terminal state would ensue. Therefore, the protective
property is a by-product of directing the agents towards critical states. We
would further accentuate the distinctions in the following section where we
provide details of our implementation of active boundary.

(a) Active Boundary

(b) Passive Boundary

Figure 3: Difference Between Active Boundary and Passive Boundary

##  IV Implement active Boundary in OpenAI Gym

We implemented our active boundary experiments in OpenAI
Gym[Brockman2016OpenAIG] as a proof of concept. We are cognizant of the fact
that eventually, things should apply to a real-world scenario. Therefore, we
designed the active boundary in a manner such that it can be swiftly adapted
to physical implementation. Agents are implemented with
tensorforce[tensorforce]. We sorted out the observation space and action space
for each environment and recorded them in the code of each experiment.

As we suggested in the previous section when agents reach the critical state,
"correct actions" would be applied by the active boundary to the agent, hence
the name "active". Much like a coach would guide athletes through the most
crucial step of the training. We also designed a penalty term associated with
active boundary to discourage its use. If the agent act flawlessly at the
critical state, no penalty would be applied. On the other hand, if the agent
requires active boundary intervention, a penalty term would be registered.

The positions and penalty terms are set in an ad hoc fashion in our
experiments. At this point, we do not have an analytical frame that can guide
us in making such decisions. We choose three positions for the boundary during
each experiment and 4-6 penalty terms ranging from 0 upwards. The rule of
thumb we used for the penalty is that the punishment score should be
comparable to the reward.

Our experiments are conducted over the following environments: CartPole, with
discrete action space; Inverted Pendulum, with continuous action space;
Inverted Double Pendulum; Hopper; Walker2D. We did not implement our approach
to more sophisticated environments such as HalfCheetah, Humanoid, and
HumanroidStandup due to hardware constraints. Yet we feel that for the data we
collected, our approach has shown conclusive results. The following are our
experiment setup and results.

The dashed black lines are the training results of agents with normal
training, which functions as the control group and the benchmark. The colored
lines are the results of agents trained with varying penalty terms. The
average evaluation scores of the agents are presented at the upper left corner
of the graph.

###  IV-A CartPole and Inveretd Pendulum

CartPole’s Observation Space is the vector: [Cart Position, Cart Velocity,
Pole Angle, Pole Angular Velocity]. The discrete action space for cartpole is:
[Push Cart to the left, Push cart to the right]. The continuous action space
for Inverted Pendulum is an action ranging from -1 to 1, where -1 means the
actuator moves the cart to the left with maximum power and 1 means the
actuator moves the cart to the right with maximum power. The maximum episode
step number for cartpole and inverted pendulum are 500 steps and 300 steps
respectively. The terminal state for cartpole is angle of absolute value
greater than 24 degrees. The terminal state for the inverted pendulum is an
angle of absolute value greater than 0.2 radius. The reward for cartpole and
inverted pendulum are both 1 for each step and 0 for the terminal step.

The active boundary is implemented as a ring anchored on the cart, at the
height of the middle of the pole. Touch sensors would be installed in order to
detect contact between the pole and the ring. We propose three sets of rings
each allows a maximum angle of 0.5, 0.7, and 0.9 of the terminal state angle
in the cartpole environment and 0.1, 0.4, 0.5 radii in the inverted pendulum
environment, as shown by Figure 4.

Figure 4: CartPole/Inverted Pendulum active Boundary

When contact between the pole and the active boundary is detected by the
sensor, the cart would be pushed towards the appropriate direction depending
on the sign of the angle. This step can be easily programmed by a
microcontroller in physical implementation, and for implementation in
simulation, it is just a matter of specifying a premeditated action. For
detailed code, please refer to the code depository. Note that should the
boundary is set without touch sensors and the corresponding actions, it would
just be a passive boundary instead of an active one.

A penalty term, which is chosen from vector [0,-5,-7,-10,-12,-15], would incur
every time the pole contacts the active boundary. Our experiment result on
cartpole are shown by Figure 5. The experiment result on inverted pendulum is
shown by figure 6. As the data suggest, when the boundary is set too
restrictively, it severely hinders the training. The best results is seen when
the boundary is set at 10.8 degrees. After the agents are done training, we
evaluate their performance in normal environments, the average evaluation
scores are presented at the upper left part of the graphs. Our data suggest
that during the evaluation, agents with accelerated training perform
comparably to the agents with normal training.

Figure 5: CartPole Experiments

Figure 6: Inverted Pendulum Experiments

###  IV-B Inverted Double Pendulum

The inveretd double pendulum has observation space of the following: [x
position of the cart, sin($\theta_{1}$),
sin($\theta_{2}$),cos($\theta_{1}$),cos($\theta_{2}$),velocity of x, angular
velocity of $\theta_{1}$, angular velocity of $\theta_{2}$, constraint force
on x, constraint force on $\theta_{1}$, constraint force on $\theta_{2}$].
$\theta_{1}$ and $\theta_{2}$ are the angles of the upper and lower pole
perspectively.

The action space for Inverted Double Pendulum is an action ranging from -1 to
1, where -1 means the actuator moves the cart to the left with maximum power
and 1 means the actuator moves the cart to the right with maximum power.

The Terminal state of double inveretd pendulum is when the y postion of the
upper pole, which can not be observed by the agent, falls below 1.

We design the active boundary to be located at the same height as does the
joint between the upper and the lower pole, as indicated by Figure 7. The
boundary allows the joint to flex for 0.4, 0.5 and 0.6 radii respectively,
each with the penalty parameters of a choice between 0,-3,-4,-5,-6,-7. When
the joint extends to touch the active boundary, an action of -1 or 1 would be
automatically applied to rectify the situation, depending on the sigh of the
angles.

Figure 7: Inveretd Double Pendulum active Boundary

Our experiment result is shown by Figure 8. The conclusion is similar to that
of cartpole and inverted pendulum results. When the active boundary is set at
0.4 radii, it hinders learning rather than facilitate it. When the boundary is
set at 0.5 radii with a penalty parameter of -7, we see the most drastic
acceleration of agent training. However, when the boundary is set at 0.6
radii, it seems to have given the agent too much flexibility since the result
suggests that the acceleration is not as profound compare to that of the
boundary set at 0.5.

Figure 8: Inverted Double Pendulum Experiments

###  IV-C Hopper

The observation space of hopper is the following vector: [z position, y
position, thigh joint angle, leg joint angle, foot joint angle, velocity at
x-axis, velocity at z-axis, velocity at y-axis, angular velocity of thigh
joint, angular velocity of leg joint, angular velocity of foot joint].

The action space of hopper are three continuous action choices for three
actuators [thigh actuator, leg actuator, foot actuator]. The range of for
actuators are -1, which means apply force towards the negative direction with
maximum power, and 1 which means apply force towards the positive direction
with maximum power.

The terminal states for hopper are when the absolute y position is greater
than 0.2.

There are more than one ways to implement active boundary in hopper
environment. Because we want things to be applicable in real-world scenarios,
we chose the implementation with the easiest setup. In our experiment, the
active boundary is defined with the y-position at 0.05, 0.1 and 0.15, each
with the penalty parameters that is chosen from 0,-3,-5,-7,-10 as shown by
Figure 9 When the agent touches the active boundary, we gently guide it
slightly towards the center line, and the velocities terms are all set to
zero. We seek to mimic how a coach would guide an athlete towards the desired
trajectory during training. When the athlete is about to deviate off-course,
the coach would take over and apply the appropriate force to slightly reset
the trajectory. This can be implemented with an automatic harness in a
physical experiment.

Our experiment result is shown by Figure 10. Similar to the conclusion drawn
with previous data, when the active boundary is set too narrowly at 0.05 y
position, it restricts the agent and almost completely halts the agent
training. The best training acceleration is observed when the boundary is set
at 0.15 y position.

Figure 9: Hopper active Boundary

Figure 10: Hopper

###  IV-D Walker2d

The walker environment’s observation space is the following vector: [z
position, y position, right thigh joint angle, right leg joint angle, right
foot joint angle, left thigh joint angle, left leg joint angle, left foot
joint angle, velocity along x axis, velocity along z axis, velocity along y
axix, angular velocity for right thigh joint, angular velocity for right leg
joint, angular velocity for right foot joint, angular velocity for left thigh
joint, angular velocity for left leg joint, angular velocity for left foot
joint]

There are 6 actuators for the right thigh joint, right leg joint, right foot
joint and left thigh joint, left leg joint, left foot joint perspectively. The
range for the motors is -1, which means apply force towards the negative
direction with maximum power and 1, which means apply force towards the
positive direction with maximum power.

The terminal states are when the z position falls below 0.8 or rise above 2 or
the absolute value of y position is greater than 1.

The active boundary is set with the y position at 0.3, 0.7 and 0.9, each with
the penalty parameters that is chosen from 0,-5,-10,-15,-20 as shown by Figure
11. Similar to the active boundary for hopper, when the agent touches the
active boundary in the walker environment, we gently guide it slightly towards
the central line, and the velocities terms are all set to zero in the
meantime.

Our experiment result is shown in Figure 12. Yet again the experiment result
suggests the same conclusion: when the boundary is positioned at 0.1, it acts
as a barrier to training. When it is set at 0.7 positions, it could accelerate
training with the appropriate penalty term.

Figure 11: Walker2d active Boundary

Figure 12: Walker2D

##  V Conclution

In this paper, we introduced the active boundary idea based on our observation
of athletic training. Our proof of concept data suggests when the position and
penalty associated with active boundary are set appropriately, accelerated
learning can be achieved. Future research should focus on devising analytical
tools that can systematically derive the best position and penalty for an
active boundary.

Next step, we plan on extending the active boundary idea to the deep drone
acrobat project [Kaufmann2020DeepDA]. The training of drone acrobat is done in
simulation and then transferred to a real-world drone controller. We think our
active boundary method can greatly accelerate the training of drone acrobats.

We believe our research opens door to a rich reservoir of potential researches
along the direction of adapting proven human training techniques to RL
environments. Human achieves superhuman feats not because of talent, but
because of the meticulously engineered coaching tactics. Current researches in
RL focus solely on the "athlete" side of the equation, i.e. how to build an
efficient RL agent. But we feel "coach" is as important, if not more so. For
instance, the endgames training strategy was deployed to educate chess
players. During the training phase, the agent should start the game middle way
to amass experience with critical states, instead of wasting time on steps
leading up to the critical position. Coaches also challenge athletes, pushing
them to experience difficult situations. "Coach" could even be implemented by
independent neural networks, in addition to the agent, and trained to "teach"
better. All these should be investigated by future researches.


# Color symmetry and confinement as an
underlying superconformal structure
in holographic QCD

Guy F. de Téramond<EMAIL_ADDRESS>Laboratorio de Física Teórica y
Computacional, Universidad de Costa Rica, 11501 San José, Costa Rica  Stanley
J. Brodsky<EMAIL_ADDRESS>SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94309, USA

###### Abstract

Dedicated to the memory of our colleague, Harald Fritzsch, who, together with
Murray Gell-Mann, introduced the color quantum number as the exact symmetry
responsible for the strong interaction, thus establishing quantum
chromodynamics (QCD) as a fundamental non-Abelian gauge theory. A basic
understanding of hadron properties, however, such as confinement and the
emergence of a mass scale, from first principles QCD has remained elusive:
Hadronic characteristics are not explicit properties of the QCD Lagrangian and
perturbative QCD, so successful in the large transverse momentum domain, is
not applicable at large distances. In this article, we shall examine how this
daunting obstacle is overcome in holographic QCD with the introduction of a
superconformal symmetry in anti de Sitter (AdS) space which is responsible for
confinement and the introduction of a mass scale within the superconformal
group. When mapped to light-front coordinates in physical spacetime, this
approach incorporates supersymmetric relations between the Regge trajectories
of meson, baryon and tetraquark states which can be visualized in terms of
specific $SU(3)_{C}$ color representations of quarks. We will also briefly
discuss here the implications of holographic models for QCD color transparency
in view of the present experimental interest. Based on the invited
contribution to the book dedicated to the memory of Harald Fritzsch.

††preprint: SLAC-PUB-17740.

##  I Introduction

Harald Fritzsch, together with Murray Gell-Mann, introduced in 1972 the color
quantum number of quarks as the source of the strong interactions, with the
restriction that all physical states and observables are singlets under the
color group $SU(3)_{C}$[1]. This far reaching assumption provides a simple
explanation of the spin-statistics problem which hindered the quark
model111Previously O. W. Greenberg proposed that quarks should be parafermions
to comply with the Pauli exclusion principle [2] and M.-Y. Han and Y. Nambu
suggested the existence of three integer charged triplets with a broken
$SU(3)\times SU(3)$ symmetry group [3]., but also allowed for the correct
prediction of the observed decay rate of the neutral pion and the ratio $R$
for electron-positron annihilation [4]. While trying to understand why a
single color quark or gluon cannot be observed as a physical state, known as
the confinement problem, Fritzsh and Gell-Mann proposed to use the exact color
symmetry as a gauge group, thus establishing quantum chromodynamics (QCD) as a
non-Abelian gauge theory describing the interactions of quark fields, in the
fundamental triplet representation of the $SU(3)_{C}$ group, and the gluon
color octet in its adjoint representation [1, 5].

The interactions between the fundamental degrees of freedom of the QCD
Lagrangian, the quark and gluons, observed in high energy scattering
experiments is described to high precision by QCD, establishing this theory as
the standard model of the strong interactions [6]. At low momentum transfers,
however, the nonperturbative nature of the strong interactions at large
distances becomes dominant, and the usual techniques of perturbative QCD
become inapplicable. Despite the important advances of Euclidean lattice QCD
[7] and other nonperturbative approaches, a full understanding of color
confinement, the origin of the hadron mass scale and the emergence of hadronic
degrees of freedom from the QCD Lagrangian, has remained a deep unsolved
problem.

Recent theoretical developments aimed at understanding strongly coupled
dynamics are based on AdS/CFT – the correspondence between classical gravity
in a higher-dimensional anti-de Sitter (AdS) space and conformal field
theories (CFT) in physical space-time [8, 9, 10]. In practice, the AdS/CFT
duality provides an effective weakly coupled classical gravity description in
a ($d+1$)-dimensional AdSd+1 space in terms of a flat $d$-dimensional strongly
coupled quantum field theory defined on the AdS asymptotic boundary, the
physical four-dimensional Minkowski spacetime, where the boundary conditions
are imposed [11].

Holographic models motivated by the AdS/CFT correspondence have provided a
useful semiclassical approximation which captures essential features of QCD
and thus can give important insights into its nonperturbative regime. Our
approach to holographic QCD on the light-front (HLFQCD) is based on the
embedding of Dirac’s relativistic front form of dynamics [12] into AdS space.
This precise mapping between semiclassical LF Hamiltonian equations in QCD and
wave equations in AdS space [13], leads to relativistic boost-invariant wave
equations in physical space-time, similar to the Schrödinger equation in
atomic physics, providing an effective computational framework for hadron
properties [14]. Further extensions of HLFQCD incorporate the exclusive-
inclusive connection in QCD and provides nontrivial interconnections between
the dynamics of form factors and quark and gluon distributions [15, 16, 17]
with pre-QCD nonperturbative approaches such as Regge theory and the Veneziano
model.

In our work with Hans Günter Dosch, we found that a remarkable property of
holographic light front QCD is the emergence of a superconformal algebraic
structure [18, 19, 20]: It is realized in the holographic coordinate $z$ of
anti de Sitter (AdS) space and is responsible for the introduction of a mass
scale within the algebra. The superconformal symmetry determines uniquely the
confining interaction in the semiclassical Hamiltonian equations as well as
the corresponding modification of AdS space in the infrared region. It gives
rise to striking connections between the Regge trajectories of mesons, baryons
and tetraquarks [21, 22, 23] which can be visualized in terms of the
fundamental $SU(3)_{C}$ representation of the constituent quarks as a quark-
diquark cluster configuration222The idea to apply an effective supersymmetry
to hadron physics is certainly not new [24, 25, 26], but had failed to account
for the special role of the pion.. The pion is identified with the zero mode
which appears in the mass spectrum of the superconformal Hamiltonian equations
and has no baryonic supersymmetric partner according to the Witten index333The
Witten index counts the difference between the number of bosonic and fermionic
zero energy states in a supersymmetric quantum system and it is a topological
invariant [27].. This characteristic pattern is observed across all hadron
families. The resulting picture has similarities to the duality approach
described in [5], where quark and gluons are not introduced initially,
provided that mesons and baryons turn out to behave as if they were composite
objects of quarks and gluons.

We shall give in this contribution an overview of relevant aspects of the
semiclassical approximation to QCD quantized in the light front and its
holographic embedding in AdS space, with an emphasis on the superconformal
structure in the dual gravity theory for hadron spectroscopy and confinement.
We will also discuss briefly the predictions of holographic QCD for the onset
of color transparency in QCD, namely the reduced absorption of a hadron
propagating in a nucleus when produced at high-momentum transfer [28, 29], a
subject of renewed experimental interest. Other relevant aspects and
applications of the light-front holographic approach to hadron physics have
been described in the recent review [6].

##  II Semiclassical approximation to light-front QCD

A semiclassical approximation to QCD has been obtained using light-front (LF)
quantization, where the initial surface is the null plane,
$x^{+}=x^{0}+x^{3}=0$ tangent to the light cone, thus without reference to a
specific Lorentz frame [12]. Evolution in LF time $x^{+}$ is given by the
Hamiltonian equation [30]

$\displaystyle i\frac{\partial}{\partial
x^{+}}|\psi\rangle=P^{-}|\psi\rangle,\quad
P^{-}|\psi\rangle=\frac{\mathbf{P}_{\perp}^{2}+M^{2}}{P^{+}}|\psi\rangle,$ (1)

for a hadron with 4-momentum $P=(P^{+},P^{-},\mathbf{P}_{\\!\perp})$,
$P^{\pm}=P^{0}\pm P^{3}$, where $P^{-}$ is a dynamical generator and $P^{+}$
and $\mathbf{P}_{\\!\perp}$ are kinematical. Hadron mass spectra can be
computed from the LF invariant Hamiltonian

$\displaystyle P^{2}|\psi\rangle=M^{2}|\psi\rangle,$ (2)

where $P^{2}=P_{\mu}P^{\mu}=P^{+}P^{-}\\!-\mathbf{P}_{\perp}^{2}$. The linear
evolution in Hilbert space with light-front time $x^{+}$ and the quantum-
mechanical probabilistic interpretation of hadron states in terms of the
eigenfunctions of the LF Hamiltonian (2) in a Fock component basis,
$|\psi\rangle=\sum_{n}\psi_{n}|n\rangle$, leads to wave equations for the
light-front wave functions (LFWFs) $\psi_{n}$ similar to the usual Schrödinger
equation.

In practice, solving the actual eigenvalue problem (2) in QCD is a formidable
computational task for a non-abelian quantum field theory beyond 1 + 1
dimensions [31]. In 1 + 1 dimensions, for example, the QCD coupling $g$ has
dimensions of mass. In this case, the theory can be solved for any number of
constituents and colors using discretized light-cone quantization (DLCQ)
methods [32, 33, 34]: All physical quantities can be computed in terms of the
basic 1 + 1 Lagrangian parameters, the QCD coupling and the quark masses, thus
no emergent phenomena occurs. In contrast, in 3+1 physical space the coupling
$g$ is dimensionless and, in the limit of massless quarks, the QCD Lagrangian
is conformally invariant. Thus, in addition to the formidable computational
complexity in this case, there is no indication of the origin of the hadron
mass scale from the QCD Lagrangian.444Renormalization of the theory introduces
a scale $\Lambda_{\rm QCD}$ which breaks conformal invariance and leads to a
logarithmic decreasing coupling $g(\mu)$, which depends on $\Lambda_{\rm
QCD}$, and to asymptotic freedom [35, 36] for large values of the momentum
transfer $\mu^{2}$. The parameter $\Lambda_{\rm QCD}$ is determined in high
energy experiments and its value is renormalization scheme dependent. The link
between the dimensionful parameter $\Lambda_{\rm QCD}$ and the dimensionless
coupling $g$ is known as dimensional transmutation.

Starting from the QCD Lagrangian, one can express the Hamiltonian operator
$P^{-}$ in terms of dynamical fields, the Dirac field $\psi_{+}$, and the
transverse gauge field $\mathbf{A}$ in the $A^{+}=0$ gauge [30]

$\displaystyle
P^{-}={\textstyle\frac{1}{2}}\int\\!dx^{-}d^{2}\mathbf{x}_{\perp}\bar{\psi}_{+}^{\dagger}\frac{\left(i\mathbf{\nabla}_{\\!\perp}\right)^{2}+m_{q}^{2}}{i\partial^{+}}\psi_{+}+~{}{\rm
interactions},$ (3)

where the interaction terms in (3) vanish in the limit $g\to 0$ and the
integral is over the initial surface $x^{+}=0$, $x^{\pm}=x^{0}\pm x^{3}$. For
simplicity, we have omitted in (3) the contribution from the gluon field
$\mathbf{A}$.

The mass spectrum is computed from the hadronic matrix element
$\langle\psi(P^{\prime})|P_{\mu}P^{\mu}|\psi(P)\rangle=M^{2}\langle\psi(P^{\prime})|\psi(P)\rangle$.
For a $q\bar{q}$ bound state we factor out the longitudinal $X(x)$ and orbital
$e^{iL\theta}$ dependence from the LFWF $\psi$

$\displaystyle\psi(x,\zeta,\varphi)=e^{iL\theta}X(x)\frac{\phi(\zeta)}{\sqrt{2\pi\zeta}},$
(4)

where $\zeta^{2}=x(1-x)b_{\perp}^{2}$ is the invariant transverse separation
between two quarks and $L$ their relative LF orbital angular momentum. The
relative impact variable $\mathbf{b}_{\perp}$ is conjugate to the relative
transverse momentum $\mathbf{k}_{\perp}$, and $x$ is the longitudinal momentum
fraction $x$. To reduce further the dynamical problem to a single variable we
take the chiral limit of zero quark masses, $m_{q}\to 0$, to obtain [13]

$\displaystyle
M^{2}=\int\\!d\zeta\,\phi^{*}(\zeta)\sqrt{\zeta}\left(-\frac{d^{2}}{d\zeta^{2}}-\frac{1}{\zeta}\frac{d}{d\zeta}+\frac{L^{2}}{\zeta^{2}}\right)\frac{\phi(\zeta)}{\sqrt{\zeta}}+\int\\!d\zeta\,\phi^{*}(\zeta)U(\zeta)\phi(\zeta),$
(5)

where the effective potential $U$ has units of $M^{2}$ and should enforce
confinement at some IR scale. It acts on the valence state and comprises all
interactions, including those from higher Fock states. The Lorentz invariant
LF equation (2) thus becomes a LF wave equation for $\phi$ [13]

$\displaystyle\left(-\frac{d^{2}}{d\zeta^{2}}-\frac{1-4L^{2}}{4\zeta^{2}}+U(\zeta)\right)\phi(\zeta)=M^{2}\phi(\zeta),$
(6)

where the critical value of the LF orbital angular momentum $L=0$ corresponds
to the lowest possible stable solution, the ground state of the light-front
Hamiltonian. Eq. (6) is relativistic and frame-independent: It has a structure
similar to wave equations in AdS, provided that one identifies $z=\zeta$, the
holographic variable [13].

##  III Higher spin wave equations in AdS

Anti-de Sitter AdSd+1 is the maximally symmetric $d+1$ space with negative
constant curvature and a $d$-dimensional flat space boundary, Minkowski
spacetime. In Poincaré coordinates
$x^{M}=\left(x^{0},x^{1},\cdots,x^{d-1},z\right)$ the asymptotic boundary of
AdS space is given by $z=0$. The line element is

$\displaystyle ds^{2}$ $\displaystyle=g_{MN}dx^{M}dx^{N}$
$\displaystyle=\frac{R^{2}}{z^{2}}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dz^{2}\right),$
(7)

where $\eta_{\mu\nu}$ is the usual Minkowski metric in $d$ dimensions and $R$
is the AdS radius. Five-dimensional anti-de Sitter space, AdS5, has 15
isometries which induce in the Minkowski spacetime boundary the symmetry under
the conformal group with 15 generators in four dimensions [11]. This conformal
symmetry implies that there can be no scale in the boundary theory and
therefore no discrete spectrum.

The variable $z$ acts like a scaling variable in Minkowski space: different
values of $z$ correspond to different energy scales at which a measurement is
made. Short space-time intervals map to the boundary in AdS space-time near
$z=0$. This corresponds to the ultraviolet (UV) region of AdS space, the
conformal boundary. On the other hand, a large four-dimensional object of
confinement dimensions of the hadronic size maps to the large infrared (IR)
region of AdS. Thus, in order to incorporate confinement and discrete
normalizable modes in the gravity dual, the conformal invariance must be
broken by modifying AdS space in the IR large $z$ region, by introducing, for
example, a sharp cut-off at the IR border, as in the “hard-wall” model of Ref.
[37], or by using a “soft-wall” model [38] to reproduce the observed linearity
of Regge trajectories.

###  III.1 Integer-spin wave equations

The holographic embedding of the semiclassical LF bound-state wave equation in
AdS space allows us to extend (6) to arbitrary integer spin $J$ [13, 39]. To
this end, we start with the action in AdSd+1 space for a tensor-$J$ field
$\Phi_{J}=\Phi_{N_{1}\dots N_{J}}$ in the presence of a dilaton profile
$\varphi(z)$ responsible for the confinement dynamics

$\displaystyle S=\int
d^{d}x\,dz\sqrt{g}\,e^{\varphi(z)}\left(D_{M}\Phi_{J}D^{M}\Phi_{J}-\mu^{2}\Phi_{J}^{2}\right),$
(8)

where $g$ is the determinant of the metric tensor $g_{MN}$, $\mu$ is the AdS
mass, $d$ is the number of transverse coordinates, and $D_{M}$ is the
covariant derivative which includes the affine connection. The variation of
the AdS action leads to the wave equation

$\displaystyle\left[-\frac{z^{d-1-2J}}{e^{\varphi(z)}}\partial_{z}\Big{(}\frac{e^{\varphi(z)}}{z^{d-1-2J}}\partial_{z}\Big{)}+\frac{(\mu\,R)^{2}}{z^{2}}\right]\Phi_{J}(z)=M^{2}\Phi_{J}(z),$
(9)

after a redefinition of the AdS mass $\mu$, plus kinematical constraints which
eliminate lower spin from the symmetric tensor $\Phi_{N_{1}\dots N_{J}}$ [39].

By substituting $\Phi_{J}(z)=z^{(d-1)/2-J}e^{-\varphi(z)/2}\,\phi_{J}(z)$ in
(9), we find for $d=4$ the semiclassical light-front wave equation (6) with

$\displaystyle
U_{J}(\zeta)=\frac{1}{2}\varphi^{\prime\prime}(\zeta)+\frac{1}{4}\varphi^{\prime}(\zeta)^{2}+\frac{2J-3}{2\zeta}\varphi’(\zeta),$
(10)

as long as $z\mapsto\zeta$. This precise mapping allows us to write the LF
confinement potential $U$ in terms of the dilaton profile which modifies the
IR region of AdS space to incorporate confinement [14], while keeping the
theory conformal invariant in the UV boundary of AdS provided that
$\varphi(z)\to 0$ for $z\to 0$. The separation of kinematic and dynamic
components, allows us to determine the mass in the AdS action in terms of
physical kinematic quantities with the AdS mass-radius $(\mu
R)^{2}=L^{2}-(2-J)^{2}$, consistent with the AdS stability bound [13, 39, 40].

###  III.2 Half-integer-spin wave equations

A similar derivation follows from the Rarita-Schwinger action for a spinor
field $\Psi_{J}\equiv\Psi_{N_{1}\dots N_{J-1/2}}$ in AdSd+1 for half-integral
spin $J$ [39]. However, in this case, the dilaton term does not lead to an
interaction [41], and an effective Yukawa-type coupling to a potential $V$ in
the action has to be introduced instead [39, 42, 43]:

$\displaystyle
S=\int\,d^{d}x\,dz\sqrt{g}\,\overline{\Psi}_{J}\left(i\Gamma^{A}e_{A}^{M}D_{M}-\mu+\frac{z}{R}V(z)\right)\Psi_{J},$
(11)

where $e^{M}_{A}$ is the vielbein, and the covariant derivative $D_{M}$ on a
spinor field includes the affine connection and the spin connection. The
tangent space Dirac matrices, $\Gamma^{A}$, obey the usual anticommutation
relations $\\{\Gamma^{A},\Gamma^{B}\\}=2\eta^{AB}$. Factoring out the four-
dimensional plane-wave and spinor dependence, $\Psi_{J}(x,z)_{\pm}=e^{iP\cdot
x}z^{(d+1)/2-J}\,\psi_{\pm}(z)\,u^{\pm}_{\nu_{1}\cdots\nu_{J-1/2}}({P})$, we
find from (11) the coupled linear differential equations for the chiral
components $\psi_{\pm}$

$\displaystyle-\frac{d}{dz}\psi_{-}-\frac{\nu+\textstyle\frac{1}{2}}{z}\psi_{-}-V(z)\psi_{-}$
$\displaystyle=M\psi_{+},$ (12)
$\displaystyle\frac{d}{dz}\psi_{+}-\frac{\nu+\textstyle\frac{1}{2}}{z}\psi_{+}-V(z)\psi_{+}$
$\displaystyle=M\psi_{-},$ (13)

with $|\mu R|=\nu+\textstyle\frac{1}{2}$ and equal probability

$\displaystyle\int dz\,\psi_{+}^{2}(z)=\int dz\,\psi_{-}^{2}(z).$ (14)

Eqs. (12) and (13) are equivalent to the second-order equations [39]

$\displaystyle\left(-\frac{d^{2}}{dz^{2}}-\frac{1-4\nu^{2}}{4z^{2}}+U^{+}(z)\right)\psi_{+}\\!$
$\displaystyle=M^{2}\psi_{+},$ (15)
$\displaystyle\left(-\frac{d^{2}}{dz^{2}}-\frac{1-4(\nu+1)^{2}}{4z^{2}}+U^{-}(z)\right)\psi_{-}$
$\displaystyle=M^{2}\psi_{-},$ (16)

with

$\displaystyle U^{\pm}(z)=V^{2}(z)\pm V’(z)+\frac{1+2\nu}{z}\,V(z).$ (17)

Embedding semiclassical light-front Hamiltonian equations in AdS space gives
important insights into the nonperturbative structure of semiclassical bound
state equations in QCD for arbitrary spin, but it does not answer the question
of how the effective confinement dynamics is actually determined, and how it
can be related to the symmetries of QCD itself. An important clue, however,
comes from the realization that the potential $V(z)$ in Eq. (17) plays the
role of the superpotential in supersymmetric quantum mechanics (QM) [27].

##  IV Superconformal symmetry and emergence of a mass scale

We follow Fubini and Rabinovici [19] and consider a one-dimensional quantum
field theory invariant under conformal and supersymmetric transformations: It
is a superconformal symmetry which determines uniquely the superpotential
$V(z)$ and, consequently, the IR deformation of AdS space.

As a first step, we examine supersymmetric QM which is based on a graded Lie
algebra consisting of two anti-commuting supercharges $Q$ and $Q^{\dagger}$,
$\\{Q,Q\\}=\\{Q^{\dagger},Q^{\dagger}\\}=0$, which commute with the
Hamiltonian $H$ [27]

$\displaystyle\tfrac{1}{2}\\{Q,Q^{\dagger}\\}$ $\displaystyle=H,$ (18)
$\displaystyle[Q,H]$ $\displaystyle=[Q^{\dagger},H]=0.$ (19)

If the state $|E\rangle$ is an eigenstate with energy $E$,
$H|E\rangle=E|E\rangle$, then, it follows from the commutation relations (19)
that the state $Q^{\dagger}|E\rangle$ is degenerate with the state $|E\rangle$
for $E\neq 0$, but for $E=0$ we have $Q^{\dagger}|E=0\rangle=0$, namely the
zero mode has no supersymmetric partner [27], a key result for deriving the
supermultiplet structure and the pattern of the hadron spectrum.

As a second step, and following Fubini and Rabinovici, we consider the scale-
deformed supercharge operator,

$\displaystyle R_{\lambda}=Q+\lambda S,$ (20)

a superposition of supercharges within the extended graded algebra [19], where
$S$ is related to the generator of special conformal transformations $K$. The
generator $R_{\lambda}$ is also nilpotent,
$\\{R_{\lambda},R_{\lambda}\\}=\\{R_{\lambda}^{\dagger},R_{\lambda}^{\dagger}\\}=0$,
and gives rise to a new scale-dependent Hamiltonian $G$, which also closes
under the extended algebra and is a compact operator

$\displaystyle\tfrac{1}{2}\\{R_{\lambda},R_{\lambda}^{\dagger}\\}$
$\displaystyle=G,\quad\tfrac{1}{2}\\{S,S^{\dagger}\\}=K,$ (21)
$\displaystyle[R_{\lambda},G]$ $\displaystyle=[R_{\lambda}^{\dagger},G]=0.$
(22)

The fermion supercharges $R_{\lambda}$ and $R^{\dagger}_{\lambda}$ have the
matrix form

$\displaystyle R_{\lambda}=\left(\begin{array}[]{cc}0&r_{\lambda}\\\
0&0\end{array}\right),\quad
R_{\lambda}^{\dagger}=\left(\begin{array}[]{cc}0&0\\\
r^{\dagger}_{\lambda}&0\end{array}\right),$ (27)

which is realized as operators in the holographic $z$-coordinate:

$\displaystyle r_{\lambda}=-\partial_{z}+\frac{f}{z}+\lambda z,\quad\quad
r^{\dagger}_{\lambda}=\partial_{z}+\frac{f}{z}+\lambda z.$ (28)

The parameter $f$ is dimensionless and $\lambda$ has the dimension of
[$M^{2}$], since $z$ has dimensions [$M^{-1}$]. Therefore, a mass scale is
introduced in the Hamiltonian within the conformal group.

The Hamiltonian equation $G|M\rangle=M^{2}|M\rangle$ leads to the wave
equations

$\displaystyle\left(\\!-\frac{d^{2}}{dz^{2}}-\frac{1-4(f+\textstyle\frac{1}{2})^{2}}{4z^{2}}+\lambda^{2}\,z^{2}+2\lambda\left(f-\textstyle\frac{1}{2}\right)\right)\\!\phi_{+}=M^{2}\phi_{+},~{}~{}~{}~{}~{}~{}$
(29)
$\displaystyle\left(\\!-\frac{d^{2}}{dz^{2}}-\frac{1-4(f-\textstyle\frac{1}{2})^{2}}{4z^{2}}+\lambda^{2}\,z^{2}+2\lambda\left(f+\textstyle\frac{1}{2}\right)\right)\\!\phi_{-}=M^{2}\phi_{-},~{}~{}~{}~{}~{}~{}$
(30)

which have the same form as the Euler-Lagrange equations obtained from the AdS
action, but here, the interaction potential is completely fixed by the
superconformal symmetry [22, 23].

###  IV.1 Light-front mapping and baryons

Upon mapping (29) and (30) to the semiclassical LF wave equations (15) and
(16) using the substitutions
$f\mapsto\nu+\textstyle\frac{1}{2},~{}\phi_{+}\mapsto\psi_{-}~{}{\rm
and}~{}\phi_{-}\,\mapsto\psi_{+}$, we obtain the result

$\displaystyle U^{+}(z)$ $\displaystyle=\lambda^{2}z^{2}+2\lambda(\nu+1),$
(31) $\displaystyle U^{-}(z)$ $\displaystyle=\lambda^{2}z^{2}+2\lambda\nu.$
(32)

Finally, thereafter the mapping $z\mapsto\zeta$, and $\nu\mapsto L$, we find
the light-front semiclassical wave equation for the components $\psi_{+}$ and
$\psi_{-}$[22]

$\displaystyle\left(-\frac{d^{2}}{d\zeta^{2}}-\frac{1-4L^{2}}{4\zeta^{2}}+\lambda^{2}\zeta^{2}+2\lambda(L+1)\right)\psi_{+}\\!$
$\displaystyle=M^{2}\psi_{+},$ (33)
$\displaystyle\left(-\frac{d^{2}}{d\zeta^{2}}-\frac{1-4(L+1)^{2}}{4\zeta^{2}}+\lambda^{2}\zeta^{2}+2\lambda
L\right)\psi_{-}$ $\displaystyle=M^{2}\psi_{-},$ (34)

which correspond to nucleons with LF orbital angular momentum $L$ and $L+1$.

Figure 1:  Model predictions for the orbital and radial positive-parity
nucleons (left) and positive and negative parity $\Delta$ families (right)
compared with the data from Ref. [44]. The values of $\sqrt{\lambda}$ are
$\sqrt{\lambda}=0.485$ GeV for nucleons and $\sqrt{\lambda}=0.498$ GeV for the
deltas.

The solutions of (33) and (34) give the eigenfunctions

$\displaystyle\psi_{+}(\zeta)$
$\displaystyle\sim\zeta^{\frac{1}{2}+L}e^{-\lambda\zeta^{2}/2}L_{n}^{L}(\lambda\zeta^{2}),$
(35) $\displaystyle\psi_{-}(\zeta)$
$\displaystyle\sim\zeta^{\frac{3}{2}+L}e^{-\lambda\zeta^{2}/2}L_{n}^{L+1}(\lambda\zeta^{2}),$
(36)

with eigenvalues

$\displaystyle M^{2}=4\lambda(n+L+1).$ (37)

The polynomials $L_{n}^{L}(x)$ are associated Laguerre polynomials, where the
radial quantum number $n$ counts the number of nodes in the wave function. We
compare in Fig. 1 the model predictions with the measured values for the
positive parity nucleons [44] for $\sqrt{\lambda}=0.485$ GeV. The LF
semiclassical wave equations (33) and (34) are $J$-independent equations
resulting in the absence of spin-orbit coupling, in agreement with the
observed degeneracy in the baryon spectrum for a fixed value of $L$ along a
given Regge trajectory, as illustrated in Fig. 1.

###  IV.2 Superconformal meson-baryon symmetry

In the usual applications of supersymmetry the supercharges connect bosonic to
fermionic states. In the present holographic context superconformal quantum
mechanics leads to a relation of meson and baryon wave functions, therefore to
an effective hadronic supersymmetry which underlies the $SU(3)_{C}$
representation properties, since a diquark cluster can be in the same color
representation as an antiquark, namely $\bf{\bar{3}}\in{\bf 3}\times{\bf 3}$,
thus incorporating the dynamics of color. Supersymmetry at the hadronic level
is realized at larger distances, where the diquark cluster appears effectively
as an antiquark. At larger momentum transfer, however, where single quarks are
resolved individually, this effective supersymmetry is expected to be only an
approximate symmetry. For example, the recent analysis in Ref. [16] of the
$u$-quark distribution in the proton shows a deviation by 10–15% from a quark-
diquark configuration 555See also Sec. 5.4.13 of Ref. [6].

The specific meson–nucleon connection follows from the substitution
$\lambda\mapsto\lambda_{B}=\lambda_{M},\ f\mapsto
L_{M}-\textstyle\frac{1}{2}=L_{B}+\textstyle\frac{1}{2},\
\phi_{+}~{}\mapsto~{}\phi_{M},~{}\phi_{-}\mapsto\phi_{B}$ and $z\to\zeta$ in
the superconformal equations (29) and (30). We find the LF meson (M)–baryon
(B) bound-state equations [23]

$\displaystyle\left(-\frac{d^{2}}{d\zeta^{2}}-\frac{1-4L_{M}^{2}}{4\zeta^{2}}+\lambda_{M}^{2}\,\zeta^{2}+2\lambda_{M}(L_{M}-1)\right)\phi_{M}$
$\displaystyle=M^{2}\,\phi_{M},$ (38)
$\displaystyle\left(-\frac{d^{2}}{d\zeta^{2}}-\,\frac{1-4L_{B}^{2}}{4\zeta^{2}}\,+\lambda_{B}^{2}\,\zeta^{2}+\,2\lambda_{B}(L_{B}+1)\right)\phi_{B}$
$\displaystyle=M^{2}\,\phi_{B}.$ (39)

Figure 2:  This schematic figure from Ref. [23] illustrates the meson-nucleon
superconformal connection. The predicted value of $M^{2}$ in units of
$4\lambda$ for $S=0$ mesons and $S=\frac{1}{2}$ baryons is plotted vs the
orbital angular momentum $L$. The $\pi$-meson has no baryonic partner.

The superconformal structure imposes the condition
$\lambda=\lambda_{M}=\lambda_{B}$ and the remarkable relation $L_{M}=L_{B}+1$,
where $L_{M}$ is the LF angular momentum between the quark and antiquark in
the meson, and $L_{B}$ between the active quark and spectator cluster in the
baryon. Likewise, the equality of the Regge slopes embodies the equivalence of
the ${\bf 3}-{\bf\bar{3}}$ color interaction in the $q\bar{q}$ meson with the
${\bf 3}-{\bf\bar{3}}$ interaction between the quark and diquark cluster in
the baryon. The mass spectrum from (38) and (39) is

$\displaystyle M^{2}_{M}$ $\displaystyle=4\lambda(n+L_{M}),$ (40)
$\displaystyle M^{2}_{B}$ $\displaystyle=4\lambda(n+L_{B}+1).$ (41)

The pion has a special role as the unique state of zero mass and, since
$L_{M}=0$, the pion does not have a baryon partner, as illustrated
schematically in figure 2. This remarkable result underlines the dual role
played by the pion in light-front holographic QCD since, as a unique state of
zero energy, it formally plays the same role as the unique lowest state in a
supersymmetric quantum field theory [27, 18, 19, 23]. In the light front, the
supersymmetric constraint on the lowest state is realized by the exact
cancellation in Eq. (38) of the confinement potential and the quark LF kinetic
energy for massless quarks.

###  IV.3 AdS warped metrics and superconformal symmetry

Instead of introducing a dilaton profile $\varphi(z)$ in the AdS action as in
Eq. (8) (the string frame), one can modify the the AdS metric $g_{MN}$ (the
Einstein frame) by introducing a $J$-independent warp factor: $g_{MN}\to
g_{MN}e^{2f(z)}$. Both descriptions are equivalent, provided that
$f(z)=\varphi(z)/(d-1)$ [14]. The dilaton profile can be determined by
integrating Eq. (10) for a given effective potential $U_{J}(z)$. To simplify
the actual computation, we introduce the function $h(z)=e^{\varphi(z)/2}$ in
(10) to obtain

$\displaystyle
U_{J}(z)=\frac{1}{h(z)}z^{-\gamma-1}\partial_{z}\left(z^{\gamma}z\partial_{z}h(z)\right),$
(42)

with $\gamma=2J-d$: It can be expressed as the linear differential equation

$\displaystyle\Big{[}z^{2}\partial_{z}^{2}-(d-1-2J)z\partial_{z}-z^{2}U_{J}(z)\Big{]}h(z)=0,$
(43)

amenable to analytic solution.666Eq. (42) was found previously in the context
of a renormalization group flow study in Ref. [45] in terms of the one-
particle irreducible effective action in AdS.

The meson potential for $J=L$ (no internal spin) follows from (38): It is
given by

$\displaystyle U(z)=\lambda^{2}z^{2}+2\lambda(J-1),$ (44)

and the corresponding solution of (43) by

$\displaystyle h(z)=A\,e^{\lambda z^{2}/2}+B\,\Gamma\left(2-J,\lambda
z^{2}\right)e^{-\lambda z^{2}/2},$ (45)

where $\Gamma(a,b)$ is the incomplete Gamma function. In the semiclassical
approximation (neglecting backreaction from the metric), the dilaton profile
should depend only on the modification of AdS space, therefore independent of
$J$. This condition implies that $B=0$, and since $\varphi(z)\to 0$ for $z\to
0$ we set $A=1$. Thus the solution

$\displaystyle\varphi(z)=2\log[h(z)]=\lambda z^{2},$ (46)

imposed by the superconformal algebraic symmetry: It has the same form as the
result found in Ref. [46] using a different procedure. Since the dilaton
profile (46) is $J$-independent, an equivalent description can be given in the
Einstein frame.

###  IV.4  Spin interaction and diquark clusters

The LF bound-state equations in AdS space allows us to extend the
superconformal Hamiltonian to include the spin-spin interaction, a problem not
defined in the chiral limit by standard procedures. Since the dilaton profile
$\varphi(z)=\lambda z^{2}$ is valid for arbitrary $J$, it leads from (9) to
the additional term $2\lambda\mathcal{S}$ in the effective potential for
mesons. We are thus led to extend the original superconformal Hamiltonian by
writing

$\displaystyle
G=\tfrac{1}{2}\\{R_{\lambda},R_{\lambda}^{\dagger}\\}+2\lambda\mathcal{S},$
(47)

which maintains the meson-baryon supersymmetry [47]. The spin
$\mathcal{S}=0,1$, is the total internal spin of the meson, or the spin of the
diquark cluster of the baryon partner. The effect of the spin term is an
overall shift of the quadratic mass,

$\displaystyle M^{2}_{M}$
$\displaystyle=4\lambda(n+L_{M})+2\lambda\mathcal{S},$ (48) $\displaystyle
M^{2}_{B}$ $\displaystyle=4\lambda(n+L_{B}+1)+2\lambda\mathcal{S},$ (49)

as depicted in Fig. 3 for the spectra of the $\rho$ mesons and $\Delta$
baryons by shifting one unit the value of $L_{B}$ [23].

Figure 3:  Supersymmetric vector meson and $\Delta$ partners from Ref. [23].
The experimental values of $M^{2}$ from Ref. [44] are plotted vs
$L_{M}=L_{B}+1$ for $\sqrt{\lambda}\simeq 0.5$ GeV. The $\rho$ and $\omega$
mesons have no baryonic partner, since it would imply a negative value of
$L_{B}$.

For the $\Delta$ baryons the total internal spin $S$ is related to the diquark
cluster spin $\mathcal{S}$ by $S=\mathcal{S}+\frac{1}{2}(-1)^{L}$, and
therefore, positive and negative $\Delta$ baryons have the same diquark spin,
$\mathcal{S}=1$. As a result, all of the $\Delta$ baryons lie, for a given
$n$, on the same Regge trajectory, as shown in Fig. 1. Positive parity
nucleons are assigned $\mathcal{S}=0$ and are well described by the
holographic model as shown in Fig. 1. For negative parity nucleons, both
$\mathcal{S}=0$ and $\mathcal{S}=1$ are possible, but the precise comparison
with data is not as successful as for the $\Delta$ baryons and positive parity
nucleons.

###  IV.5 Completing the supersymmetric hadron multiplet

Besides mesons and baryons, the supersymmetric multiplet

$\displaystyle\Phi=\begin{pmatrix}\phi_{M}&\phi_{B}^{-}\\\
\phi_{B}^{+}&\phi_{T}\end{pmatrix},$ (50)

contains a further bosonic partner, a tetraquark, which, follows from the
action of the SUSY operator $R^{\dagger}_{\lambda}$ (27) on the negative-
chirality component of a baryon [47], as illustrated in Fig. 4. A clear
example is the SUSY positive parity $J^{P}$ multiplet
$2^{+},\frac{3}{2}^{+},1^{+}$ of states
$f_{2}(1270),~{}\Delta(1232),~{}a_{1}(1260)$, where the $a_{1}$ is interpreted
as a tetraquark.

Figure 4:  The meson-baryon-tetraquark supersymmetric 4-plet
$\\{\phi_{M},\phi_{B}^{+},\phi_{B}^{-},\phi_{T}\\}$ follows from the two-step
action of the supercharge operator $R^{\dagger}_{\lambda}$ (27): $\bar{3}\to
3\times 3$ on the pion, followed by $3\to\bar{3}\times\bar{3}$ on the negative
chirality component of the nucleon.

Unfortunately, it is difficult to disentangle conventional hadronic quark
states from exotic ones and, therefore, no clear-cut identification of
tetraquarks for light hadrons, or hadrons with hidden charm or beauty, is
possible [47, 49, 48]. The situation is, however, more favorable for
tetraquarks with open charm and beauty which may be stable under strong
interactions and therefore easily identified [50]. Our prediction [51] for a
doubly charmed stable boson $T_{cc}$ with a mass of 3870 MeV has been observed
at LHCb a year later at 3875 MeV [52], and it is a member of the positive
parity $J^{P}$ multiplet $2^{+},\frac{3}{2}^{+},1^{+}$ of states
$\chi_{c2}(3565),~{}\Xi_{cc}(3770),~{}T_{cc}(3875)$.777For the inclusion of
quark masses and longitudinal dynamics in holographic models see Sec. 5.5.9 of
Ref. [6]. In the limit of heavy quark masses HLFQCD leads to linear
confinement, see Ref. [53]. The occurrence of stable doubly beautiful
tetraquarks and those with charm and beauty is also possible [50, 51].

##  V Color transparency

A striking property of QCD phenomenology is color transparency (CT) [28], the
reduced absorption of a hadron as it propagates through nuclear matter, if it
is produced at high transverse momentum in a hard exclusive process, such as
elastic lepton-proton scattering. The nuclear absorption reflects the
effective size of the propagating hadron; i.e., the separation between its
color constituents. CT has been confirmed in many experiments, such as semi-
exclusive hard electroproduction, $eA\to e^{\prime}\pi X$ for mesons produced
at $Q^{2}>3~{}{\rm GeV}^{2}$. However, a recent JLab (Jefferson Laboratory)
measurement for a proton electroproduced in carbon $e\,{\rm C}\to
e^{\prime}pX$ fails to observe CT at $Q^{2}$ up to 14.2 GeV2 [54].

In a recent paper [29], the onset of CT for different hadrons was determined
by comparing the $Q^{2}$-dependence of the effective hadronic cross sections
for the initial formation of a small color-singlet configuration, a pointlike
configuration (PLC), using the generalized parton distributions (GPDs) from
holographic light-front QCD [15]. To discuss this problem, we can start from
the flavor form factor of a hadron written in terms of its GPD,
$H_{q}(x,t)\equiv H^{q}(x,\xi=0,t)$ at zero skewness, $\xi$,

$\displaystyle F^{q}(t)$ $\displaystyle=\int_{0}^{1}dx\,H^{q}(x,t)$
$\displaystyle=\int_{0}^{1}dx\,q(x)\exp\left[tf(x)\right],$ (51)

where $q(x)$ is the longitudinal parton distribution function (PDF) and $f(x)$
is the profile function. In HLFQCD $f(x)$ is flavor independent and, for
hadron twist $\tau$, the number of hadron constituents in a given Fock
component, the FF has the reparametrization-invariant integral representation,
expressed in terms of Euler’s Beta function [15]:

$\displaystyle F(t)_{\tau}$
$\displaystyle=\frac{1}{N_{\tau}}B\left(\tau-1,1-\alpha(t)\right)$
$\displaystyle=\frac{1}{N_{\tau}}\int_{0}^{1}dx\,w^{\prime}(x)w(x)^{-\alpha(t)}\left[1-w(x)\right]^{\tau-2},$
(52)

where $\alpha(t)=\alpha(0)+\alpha’t$ is the Regge trajectory of the vector
meson which couples to the quark current in the hadron, and $N_{\tau}$ is a
normalization factor. The trajectory $\alpha(t)$ can be computed within the
superconformal LF holographic framework, and the intercept, $\alpha(0)$,
incorporates the quark masses [22, 23]. The function $w(x)$ is a flavor-
independent function with $w(0)=0,\,w(1)=1$ and $w^{\prime}(x)\geq 0$. The
profile function $f(x)$ and the PDF $q_{\tau}(x)$ are determined by $w(x)$
[15]:

$\displaystyle f(x)$
$\displaystyle=\frac{1}{4\lambda}\log\Big{(}\frac{1}{w(x)}\Big{)},$ (53)
$\displaystyle q_{\tau}(x)$
$\displaystyle=\frac{1}{N_{\tau}}w^{\prime}(x)w(x)^{-\alpha(0)}[1-w(x)]^{\tau-2},$
(54)

with $\alpha’=1/4\lambda$. Boundary conditions follow from the Regge behavior
at $x\to 0$, $w(x)\sim x$, and at $x\to 1$ from the inclusive-exclusive
counting rule [55, 56], $q_{\tau}(x)\sim(1-x)^{2\tau-3}$, which fix
$w^{\prime}(1)=0$. These physical conditions, together with the above-given
constraints, basically determine the form of $w(x)$.

One expects from general considerations that the initial formation of a PLC
with a larger number of constituents –the proton for example, with its large
phase space, has a lower probability to fluctuate to a small configuration as
compared with a two-particle bound state, say the pion. Consequently, it
presents a larger transverse impact area as it traverses through the nucleus,
and it will be slowed down or absorbed with greater probability as compared
with a pion projectile with a smaller impact area for the same transverse
momentum square $Q^{2}=-t$. The onset of color transparency will be higher in
$Q^{2}$ for the particle with a larger number of constituents compared to a
hadron with fewer constituents.

The GPDs incorporate the far-off-shell components of the LFWF which controls
the behavior of the FF at large $Q^{2}$ and the power counting rules from the
inclusive-exclusive connection [15]. The GPDs are thus essential to compute
the $Q^{2}$ evolution of the effective transverse impact surface of a hadron
from its physically observable value at $Q^{2}=0$, namely the hadron square
radius, to its high-virtuality PLC configuration. To this end we compute the
transverse impact surface dependence on the momentum transfer, $t=-Q^{2}$,
from the expectation value of the profile function $f(x)$888The procedure
described here to compute the off-shell $Q^{2}$ evolution of the effective
transverse size differs from the procedure in reference [57], given instead in
terms of relative impact variables.:

$\displaystyle\langle 4f(t)\rangle_{\tau}$ $\displaystyle=\frac{\int
dx\,4f(x)H^{q}_{\tau}(x,t)}{\int dxH^{q}_{\tau}(x,t)}$
$\displaystyle=\frac{1}{\lambda}\sum_{j=1}^{\tau-1}\frac{1}{j-\alpha(t)},$
(55)

which depends explicitly on the hadron’s twist, $\tau$, and the properties of
the specific quark current, which couples with the active quark in the hadron,
characterized by the hadron’s Regge trajectory, $\alpha(t)$. For large values
of $t=-Q^{2}$, Eq. (V) leads to

$\displaystyle\langle f(t)\rangle_{\tau}\to\frac{(\tau-1)}{Q^{2}}.$ (56)

It shows that, as expected, the effective transverse size decreases as
$1/Q^{2}$, but the value of $Q^{2}$ required to contract all of the valence
constituents to a color-singlet domain of a given transverse size grows as
$\tau-1$, the number of the spectator constituents.

Figure 5: The transverse impact area (left) $\langle
4\,f\left(t\right)\rangle$ as a function of $Q^{2}$ and the number of
constituents (twist), $\tau$, implies a significant delay in the onset of
color transparency for $\tau>2$. The off-shell $Q^{2}$ evolution (right) also
shows an important dependence on the number of constituents.

We represent in Fig. 5 the critical dependence on the hadron’s twist $\tau$ of
the effective transverse impact surface $\langle 4f(t)\rangle$. It implies an
important delay in the onset of CT in terms of the number of quark
constituents. For example, the effective transverse impact surface for twist-2
at 4 GeV2 is similar to that of twist-3 at 14 GeV2. Likewise, the impact
surface at 4 GeV2 for twist-2 is similar to that of twist-4 at 22 GeV2, thus,
implying an important delay in the onset of CT at intermediate energies in
terms of the number of quark constituents. This is particularly relevant for
the proton since it contains, at approximately equal probability, both
twist-3, but also twist-4, in its LFWF. The proton is thus expected to have a
“two-stage” color transparency with the onset of CT differing for the spin-
conserving (twist-3) Dirac form factor with a higher onset in $Q^{2}$ for the
spin-flip Pauli (twist-4) form factor. This predicts a larger value of $Q^{2}$
in order to produce color transparency for the proton, consistent with the
absence of proton CT in the present kinematic range of the JLab experiment
[54].

Complementary with the results in Ref. [29], we also show in Fig. 5 the PLC
off-shell twist dependence computed from the $Q^{2}$ evolution of the
invariant mass, along similar lines as the computation of the effective
transverse surface in Ref. [29]. The results in this figure clearly indicate
that the pion goes off-shell much more rapidly than the proton. We only
include in this figure the off-shell dependence of the spectator quarks, since
the active quark remains close to its initial value at $Q^{2}=0$, independent
of $Q^{2}$, indicating that most of the transverse momentum in this model is
transferred to the spectator quarks. A comparison of the expansion effects of
the pion and proton formed in a high-momentum transfer collision was carried
out in Ref. [58] using superconformal baryon-meson symmetry in holographic
QCD. No significant difference was found which could explain the JLab results
in terms of the fast hadrons expanding their size, from the PLC configuration,
as they escape the nucleus. Future experiments at higher values of $Q^{2}$ are
critical to test the mechanisms discussed here which predict a significant
twist-dependent delay in $Q^{2}$ for the onset of color transparency [28].

##  VI Summary and outlook

Although it has been half a century since the $SU(3)_{C}$ color symmetry was
introduced by Fritzsch and Gell-Mann [1] as the gauge field theory of the
strong interactions, there has been little progress in understanding the
fundamental physics which confines the colored quarks and gluons within
hadrons. In fact, a basic understanding of the essential features of hadron
physics from first-principles QCD has remained among the most important
unsolved problems of the last 50 years in particle physics. Hadronic
characteristics are not explicit properties of the QCD Lagrangian but large
distance emergent phenomena, where perturbative QCD, so successful in
describing short distance phenomena, is not applicable.

Our present goal is trying to understand how emerging QCD properties would
appear in an effective computational framework of hadron structure and as a
step in this direction, we have reviewed in this article recent developments
in hadron physics which follow from the application of superconformal quantum
mechanics and light-front holography. This includes new insights into the
physics of color confinement and the origin of the mass scale, chiral
symmetry, the spectroscopy and dynamics of hadrons, the remarkable
cancellation of quark kinetic energy and the color confining potential in the
pion, as well as surprising supersymmetric relations between the masses of
mesons, baryons, and tetraquarks. We have also given a description of the
holographic QCD approach to color transparency phenomena in nuclei [29].

There are other aspects and applications of HLFQCD which are not described in
this article but are reviewed in reference [6]. For example, LF holographic
QCD also incorporates important elements for the study of hadron form factors,
such as the connection between the hadron twist and the scaling behavior for
large $Q^{2}$, and the incorporation of vector meson dominance which is
relevant at lower energies. HLFQCD also incorporates features of pre QCD, such
as the Veneziano model and Regge theory. Further extensions incorporate the
exclusive-inclusive connection in QCD and provide nontrivial relations between
hadron form factors and quark distributions [15, 16], including the intrinsic
strange-antistrange [59] and charm-anticharm [60] asymmetry distributions in
the proton. HLFQCD has also been applied successfully to the description of
gravitational form factors, the hadronic matrix elements of the energy
momentum tensor, and gluon distributions in the proton and the pion [17].
Holographic QCD has also given new insights on the infrared behavior of the
strong coupling in QCD [61, 62]. Study of diffraction physics [63] and the EMC
effect in various nuclei [64], constitute other examples of the application of
the holographic light front ideas to QCD. More recently holographic QCD has
been used to study the relation between the gluon density in a hadron and
entanglement entropy to the high energy diffractive scattering behavior of
hadrons [65]. It was shown, for example, that the growth above the classical
geometric cross section in proton-proton scattering at high energies is
directly related to the increase of the internal quantum entropy from the
entangled parton distribution in hadrons [65].

## Acknowledgements

We are grateful to Hans Günter Dosch for his invaluable contributions and to
Alexandre Deur, Tianbo Liu, Raza Sabbir Sufian and Marina Nielsen for their
collaboration on light-front holography and its implications. This work is
supported by the Department of Energy, Contract DE–AC02–76SF00515.

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