[1]Xiaoxin Yin

# “Turing Tests” For An AI Scientist

###### Abstract

The rapid advancements in deep learning have demonstrated the potential for AI
agents to perform tasks previously limited to humans, including scientific
research. While LLMs have shown impressive capabilities in solving math or
coding problems, the ability to make scientific discoveries remains a distinct
challenge. This paper proposes a "Turing test for an AI scientist" to assess
whether an AI agent can conduct scientific research independently, without
relying on human-generated knowledge.

Drawing inspiration from the historical development of science, we propose
seven benchmark tests that evaluate an AI agent’s ability to make
groundbreaking discoveries in various scientific domains. These tests include
inferring the heliocentric model from celestial observations, discovering the
laws of motion in a simulated environment, deriving the differential equation
governing vibrating strings, inferring Maxwell’s equations from
electrodynamics simulations, inventing numerical methods for initial value
problems, discovering Huffman coding for data compression, and developing
efficient sorting algorithms.

To ensure the validity of these tests, the AI agent is provided with
interactive libraries or datasets specific to each problem, without access to
human knowledge that could potentially contain information about the target
discoveries. The ultimate goal is to create an AI scientist capable of making
novel and impactful scientific discoveries, surpassing the best human experts
in their respective fields. These "Turing tests" serve as intermediate
milestones, assessing the AI agent’s ability to make discoveries that were
groundbreaking in their time.

If an AI agent can pass the majority of these seven tests, it would indicate
significant progress towards building an AI scientist, paving the way for
future advancements in autonomous scientific discovery. This paper aims to
establish a benchmark for the capabilities of AI in scientific research and to
stimulate further research in this exciting field.

###### keywords:

Artificial Intelligence, Benchmark, Deep Learning

##  1 Introduction

The recent advances in deep learning, especially those in large language
models, have shown the possibility of an AI agent performing any task a human
can perform, including scientific research. Recent studies have shown that
LLMs such as GPT-4[1], Microsoft Copilot[2], and CodeLlama[3] can solve
competition-level coding problems [4], and LLMs such as GPT-4 and Llemma[5]
can solve some high-school-level competition math problems (including some
IMO-level problems). These LLMs can certainly help researchers solve some
problems they encounter in their daily research.

However, being able to solve a type of well-defined problems is very different
from making discoveries in scientific research. For instance, in order to
train an LLM to solve coding problems, a general-purpose LLM is often fine-
tuned on all public code on GitHub, and also fine-tuned on hundreds of
thousands of coding problems from various platforms such as CodeForce and
LeetCode. For example, CodeLlama-Python underwent fine-tuning with 100 billion
tokens of Python code. The LLM simply learns how to write code given the
coding problem (which is the prompt), by learning to predict the next token in
its code given the prompt and tokens it has generated. This is essentially the
same methodology used to train a model to write novels after reading millions
of novels. It does not have the capability of discovering what it has not been
taught, making it unable to make scientific discoveries like a scientist would
do.

This makes it necessary to define a “qualification test for an AI scientist”.
If an AI agent can finish this test without help from human, we can conclude
that this agent qualifies as a scientist and can conduct scientific research
on its own.

This resembles the Turing Test, which was proposed by Alan Turing in 1950 and
serves as a foundational concept in the field of artificial intelligence,
challenging whether machines can exhibit human-like intelligence. Turing’s
seminal paper, “Computing Machinery and Intelligence"[6], introduced the idea
of an imitation game where a human interrogator would attempt to distinguish
between a computer and a human through a series of text-based questions. The
inability of the interrogator to consistently identify the machine is
considered a measure of the machine’s intelligence. This test not only sparked
decades of philosophical debate but also drove technological advances in AI
research, shaping the development of intelligent systems.

Unlike today’s LLMs which are trained on a very large corpus in order to
perform similar tasks, science is about discoveries, especially in new areas
that have not been explored. In order to define a Turing test for an AI
scientist, let us first review the development of science in its early stage.

The night sky played an essential role in the transition to modern scientific
methodologies, largely through the efforts of astronomers such as Johannes
Kepler and Galileo Galilei. Kepler’s laws of planetary motion, derived from
meticulous observations of the night sky, laid the groundwork for the
heliocentric model of the solar system and ultimately for Newton’s theory of
gravitation. His reliance on empirical data and systematic experimentation
marked a significant departure from the speculative philosophies that had
previously dominated the scientific arena. Galileo’s method of integrating
experimental evidence with mathematical analysis is a cornerstone of the
scientific method, earning him the title “father of modern science." His work
exemplifies how observations of the night sky were instrumental in shaping the
development of science in its modern form.

Therefore, the first “Turing test” for an AI scientist should be the discovery
of the heliocentric model through the observations of the night sky. This
requires an AI agent to discover laws governing the motions of celestial
objects, and fit them into a mathematical framework. It also requires the AI
agent to make groundbreaking conjectures such as the earth is similar to the
planets in the night sky. Both requirements are necessities for a scientist.

In order to be a good benchmark test for an AI scientist, a test needs to
provide a very large amount of data or an interactive environment. For
example, one can access the location of any observable celestial object at any
moment of time through the AstroPy library[7].

Based on the above two standards we choose the following seven tests as the
Turing tests for an AI scientist. In each test the AI agent cannot be trained
on human knowledge, but is accessible to math tools such as SymPy[8] and
NumPy[9], and any other datasets that do not “leak information”, i.e.,
containing clues of target discoveries to be made.

  1. 1.

_Heliocentric Model_ : Given an interactive python library[7] that provides
the coordinates of any observable celestial object in the night sky at any
given moment, check if an AI agent can infer Kepler’s three laws and conclude
that all planets orbit the sun. A bonus question is that the earth orbits the
sun but it is not required.

  2. 2.

_Laws of Motions_ : Given an interactive library that controls Minecraft[10],
check if an AI agent can discover the Law of Inertia and the Law of
Acceleration (only for gravity).

  3. 3.

_Vibrating Strings_ : Vibrating strings is one of the most important problems
that drove the development of differential equations[11]. Given a Python
library that provides the position of each point on a vibrating string of many
different initial conditions, check if an AI agent can infer the differential
equation governing the motion:

$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial
x^{2}}$

where $u(x,t)$ is the displacement of the string, $c$ is the speed of wave
propagation in the string, $t$ is time, and $x$ is the spatial coordinate
along the string. Please note the AI agent should not have any prior knowledge
about calculus, and has to define differential equations on its own.

  4. 4.

_Maxwell’s Equations_ : Maxwell’s equations are often considered to be the
most beautiful equations in physics. Given a Python-based electrodynamics
simulator[12], check if an AI agent can infer the Maxwell’s equations or their
equivalent forms. Again the agent cannot use any prior knowledge about
calculus.

  5. 5.

_Initial Value Problem (IVP)_ : IVP is probably the most important problem in
numerical computing, and the Runge-Kutta method[13] invented at the end of the
19th century is still widely used today. Given math tools such as SymPy[8] and
NumPy[9] that can calculate integrals of functions both symbolically and
numerically, check if an AI agent can invent a method for IVP that is at least
as accurate as the fourth-order Runge-Kutta method.

  6. 6.

_Huffman Coding_ : Huffman coding[14] is a most important piece of work in
information theory. Given a large corpus of ascii characters, and Python
functions to operate on bits, check if an AI agent can discover Huffman coding
when working towards the goal of minimizing storage under the constraint that
each character be represented by a specific sequence of 0’s and 1’s.

  7. 7.

_Sorting Algorithm_ : Sorting is probably the most studied problem in computer
science. Given a very large number of examples of sorting integer arrays and a
Python environment, check if an AI can discover a sorting algorithm that runs
in expected $O(n\log{n})$ time.

Please note that each test selected only requires data or interaction within a
well-defined scope (such as a dataset or an interactive library). This makes
it possible for an AI agent to make discoveries without being trained on
human-written documents, which may leak information about the target
discoveries. For the same reason we do not select any tests from many most
important disciplines, such as chemistry, biology, and geology, because they
either require interacting with the physical world or have a limited amount of
observations. In order to make important discoveries in these disciplines, it
is inevitable to use knowledge outside a small predefined scope, which may
leak key information to the AI agent.

The ultimate goal for an AI scientist should be making novel and impactful
scientific discoveries that no one has made before. Then why do we still need
these “Turing tests” which have been discovered decades or centuries ago? The
reason is that the “ultimate goal” is very challenging because the AI agent
needs to be better than the best human experts in the world. It is analogical
to building an AI agent that can beat the best GO player in the world, while
our benchmark is like beating a top GO player a thousand years ago when GO was
in its early age, or beating an amateur GO player today. If we could build an
AI agent that passes the majority of the above seven tests, we can conclude
that we are in the right direction of building an AI scientist, and it should
evolve into someone who can make important scientific discoveries in the
foreseeable future.

##  2 Related Work

The idea of automating scientific research activities dates back to the early
days of computer science. An article on _Science_ in 2009 [15] provides a
great overview on the early explorations. Also in 2009 a “Robot Scientist”
named Adam was released [16]. The authors developed specialized hardware for
conducting basic experiments, such as tracking yeast growth with varying gene
deletions and metabolites. This was paired with logic programming software for
selecting experiments. The software keeps track of various hypotheses and
chooses experiments likely to refute many of them at once. These experiments
are automatically performed, and their results guide the next experiment’s
selection. Adam effectively identified the functions of multiple genes,
requiring fewer experiments compared to other experiment-selection methods
like cost-based choices. [17] presents a research that utilizes special
hardwares to automatically learn the effects of different drugs upon the
distribution of different proteins within mammalian cells.

Very recently a breakthrough was brought by DeepMind [18], in which the
authors created a large language model that learned geometry on one billion
generated problems, in order to discover geometry properties, and train itself
to prove these properties. The model was tested on 30 IMO geometry and got 25
of them correct, which outperforms the majority of IMO participants. This is
the first time a neural network model learns to master a discipline of science
on its own, and it will not be surprising if the same methodology can be
extended to other disciplines such as number theory and combinatorics.

Our goal is to let AI make scientific discoveries on its own. There are two
routes towards this goal. The first is to build an AI agent that can make
novel and impactful scientific discoveries that have not been made before.
This is our ultimate goal. But it is very challenging because the AI agent
needs to be better than the best human expert in a field.

The alternative route is to build an AI agent that can make some of the most
important scientific discoveries in the history, without reading human
knowledge that may contain key information to these discoveries. We believe
this is an easier route because some of such discoveries can be inferred from
abundant data and a scientific methodology. Comparing with the first goal
which is analogical to building an AI agent that can beat the best GO player
in the world, the second goal is like building an AI agent that can beat an
amateur GO player. We believe the second goal is a good starting point for
building an AI scientist, which should eventually evolve into someone who can
make new and important scientific discoveries.

##  3 The Seven Qualification Tests for an AI Scientist

###  3.1 Selection Criteria

An ideal “Turing” test for an AI scientist should satisfy the following three
criteria:

  1. 1.

It is the key to an important discovery in the development of science.

  2. 2.

It is possible to be discovered digitally, without interaction with the
physical world.

  3. 3.

The discovery is possible based on data or interaction within a well-defined
scope (such as a dataset or a set of interactive libraries).

The first two criteria are straight-forward, and here we explain why we need
the third criterion. Each important scientific discovery has deep impact in
our civilization, and may have become common sense (e.g., the earth orbits the
sun). Both the discovery itself and the facts and technologies depending on it
can be documented here and there in our written corpus. It is impossible to
create a generic training set for a model without including such knowledge.
Therefore, we have to confine the scope of the data and/or interactive tools
an AI can access, to avoid any possible information leak.

_Test_ |  _Discipline_ |  _Significance_
---|---|---
Heliocentric Model |  Astronomy |  Laid the foundation of scientific method
Motion Laws |  Physics (Mechanics) |  Revolutionized understanding of the physical world
Vibrating Strings |  Mathematics & Physics |  Drove the development of differential equations
Maxwell’s Equation |  Physics (Electromagnetism) |  United electricity and magnetism
Initial Value Problem |  Numerical computing |  Most studied problem in numerical computing
Huffman Coding |  Information theory |  Cornerstone in the development of information theory
Sorting Algorithm |  Computer science |  Most studied problem in algorithms
Table 1: The seven tests for an AI scientist, and the significance of each
test in the development of science.

Table 1 summarizes our seven tests and their significance in the history of
science. We do not select any test from many most important disciplines, such
as chemistry, biology, and geology, because they either require interacting
with the physical world or have a limited amount of observations.

###  3.2 The Heliocentric Model Test

The exploration of the night sky was pivotal in the evolution to modern
scientific methods, primarily driven by the contributions of astronomers like
Johannes Kepler and Galileo Galilei. Kepler’s laws of planetary motion,
derived from his observations, established the foundation for the heliocentric
solar system model, paving the way for Newton’s theory of gravity. Similarly,
Galileo’s approach of blending experimental data with mathematical analysis
became a fundamental element of the scientific method, earning him the title
"Father of Modern Science."

Thus, a suitable initial "Turing test" for an AI scientist might involve
rediscovery of the heliocentric model using only observations of the night
sky. This would require an AI to derive laws that govern celestial motion and
integrate these into a mathematical model, including making revolutionary
conjectures, such as suggesting Earth and other celestial bodies have similar
properties.

For such a test to effectively assess an AI scientist, it should involve a
vast dataset and/or an interactive environment. For instance, the position of
celestial bodies at specific times could be determined using the AstroPy
library[7].

Here is our first test, the _Heliocentric Model Test_ : Given an interactive
Python library like AstroPy, which provides the coordinates of any observable
celestial objects at any moment, the test would see if an AI agent can derive
Kepler’s three laws and acknowledge that planets orbit the sun. An additional
challenge could involve recognizing that Earth orbits the sun, although it is
optional.

Here is an example of using AstroPy to get the location of a celestial object
at a certain moment.

⬇

1 from astropy.coordinates import SkyCoord

2 from astropy.time import Time

3 import astropy.units as u

4

5 # Define the name of the star and the observation time

6 star_name = "Betelgeuse"

7 observation_time = Time("2024-05-18 22:00:00")

8

9 # Get the coordinate of the star using its name

10 star_coord = SkyCoord.from_name(star_name)

11

12 # Calculate the position of the star at the given time

13 altaz = star_coord.transform_to(’altaz’, obstime=observation_time)

14

15 # Print the altitude and azimuth

16 print(f"Altitude: {altaz.alt:.2f}, Azimuth: {altaz.az:.2f}")

An AI agent can easily get the locations of all observable celestial objects
at every minute. To go deeper, it may use symbolic regression tools such as
PySR[19] to extract the mathematical formulae behind the trajectories of
objects, and use mathematical tools such as SymPy[8] to simplify and possibly
generalize the various formulae, in order to infer simple rules based on
Occam’s Razor. This is only one possible route, and different AI agents may
find different routes towards the final goal.

###  3.3 The Motion Laws Test

Our second test, _Motion Laws Test_ , aims at rediscovering the fundamental
principles of motion. It is non-trivial for an AI agent to interact with the
real world objects. Fortunately the virtual worlds such as Minecraft offers a
platform for exploration in kinetics. This test would assess the AI’s ability
to derive the Law of Inertia, and the Law of Acceleration under the influence
of gravity, solely from interactions and observations within the game and a
few mathematics tools such as PySR and SymPy.

In this test, the AI would need to manipulate and measure the dynamics of
various objects under different conditions within the game. For example, the
AI could alter the mass of blocks, apply forces, and observe the trajectories.
By analyzing these observations (using tools such as PySR and SymPy), the AI
would need to derive the formula corresponding to the Law of Inertia and the
Law of Acceleration due to gravity.

One can use _Minecraft: Pi edition API Python Library_[10] to control objects
in Minecraft in Python. As shown in the example below, one can set a block in
the air and observe its position after one second.

⬇

1 import mcpi.minecraft as minecraft

2 import mcpi.block as block

3 import time

4

5 # Connect to Minecraft

6 mc = minecraft.Minecraft.create()

7

8 # Set the coordinates for the block (for example, 10 units above the
player’s current position)

9 x, y, z = mc.player.getTilePos()

10 y += 10

11

12 # Place a block in the air

13 mc.setBlock(x, y, z, block.STONE.id)

14

15 # Wait for a second

16 time.sleep(1)

17

18 # Get the position of the block

19 block_pos = mc.getBlock(x, y, z)

20

21 # Print the position and the type of the block

22 print(f"Block placed at: ({x}, {y}, {z})")

23 print(f"Block type at position: {block_pos}")

###  3.4 The Vibrating Strings Test

The problem of vibrating strings significantly influenced the development of
differential equations during the 17th and 18th centuries, especially in the
context of music and acoustics. In his seminal work in 1747, Jean le Rond
d’Alembert formulated the one-dimensional wave equation to describe the motion
of a vibrating string. This equation, expressed in trigonometric functions,
suggested that the string’s vibrations could be depicted as a sum of
sinusoidal waves of various frequencies and amplitudes.

The intense debate on the correct solution to the vibrating string problem
among mathematicians like Daniel Bernoulli and Leonhard Euler fueled advances
in differential equations. Bernoulli’s advocacy for representing vibrations as
a series of harmonic motions led to the principle of superposition in wave
theory, while Euler explored different boundary conditions. Their collective
efforts advanced the field of differential equations by developing techniques
like separation of variables, and applied these methods to practical mechanics
and beyond.

In the _Vibrating Strings Test_ , an AI agent would be assessed by its
capability to derive the simple and elegant different equation for vibrating
strings:

$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial
x^{2}}$ (1)

where $u(x,t)$ is the displacement of the string, $t$ is time, and $x$ is the
spatial coordinate along the string. It is not required for the AI to infer
that $c$ is the speed of wave propagation in the string, and the AI can
replace $c^{2}$ with a positive constant.

Please note the AI is not allowed to use prior knowledge about calculus,
because that would reduce this problem to a simple symbolic regression on
second derivatives. Instead, we expect the AI to discover the concept of
"differentiation" on it own, possibly through exploring a large variety of
possible concepts.

One can use the python package for simulating vibrating strings in [20] to
create infinite examples, which should allow the AI to apply all kinds of
hypotheses, in order to discover the simplest one that is consistent with the
observations.

###  3.5 The Maxwell’s Equations Test

Since proposed in 1862, Maxwell’s equations have been celebrated for their
mathematical elegance, encapsulating the fundamentals of electromagnetism in a
set of concise, interrelated equations. Here are the four equations formed as
differential equations:

Gauss’s Law for Electricity:

$\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}}$

Gauss’s Law for Magnetism:

$\nabla\cdot\mathbf{B}=0$

Faraday’s Law of Induction:

$\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$

Ampere’s Law with Maxwell’s Addition:

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\epsilon_{0}\frac{\partial\mathbf{E}}{\partial
t}$

In the _Maxwell’s Equations Test_ , an AI will be assessed by whether it can
derive some or all of the four equations (or their equivalent forms), given an
interactive library for simulating electrodynamics. Again the AI should not
have prior knowledge of calculus.

One can use PyCharge[21] (downloadable at [22]) for such simulations. Fig. 1
shows an example of using PyCharge to simulate the electromagnetic field of an
oscillating charged particle. Below is a code segment that can be used to
generate this simulation, with the full code at
https://github.com/MatthewFilipovich/pycharge/blob/master/examples/paper_figures/figure5.py.

⬇

1# Calculate and plot E and B

2charges = (pc.OscillatingCharge((0, 0, 0), (1, 0, 0), 2e-9, omega, q=e),

3pc.OscillatingCharge((0, 0, 0), (-1, 0, 0), 2e-9, omega, q=-e))

4

5simulation = pc.Simulation(charges)

6coord = np.linspace(-lim, lim, grid_size)

7x, y, z = np.meshgrid(coord, coord, 0, indexing=’ij’)

8

9Ex, Ey, _ = simulation.calculate_E(0, x, y, z, ’Acceleration’)

10_, _, Bz = simulation.calculate_B(0, x, y, z, ’Acceleration’)

Figure 1: An examples simulation by PyCharge of the electromagnetic field of
an oscillating charged particle.

###  3.6 The Initial Value Problem Test

An initial value problem (IVP) involves solving a differential equation
subject to specific initial conditions. The development of IVP, particularly
in the context of differential equations, is a cornerstone of modern numerical
computing. During the 18th and 19th centuries, mathematicians like Leonhard
Euler, Joseph-Louis Lagrange, and Carl Friedrich Gauss further developed
methods to solve differential equations arising in physics and astronomy.
Euler’s method, developed in the 1760s, is one of the earliest numerical
methods for solving initial value problems. Consider the initial value problem
(IVP) for the differential equation:

$\frac{dy}{dt}=f(t,y),$ (2)

with an initial condition $y(t_{0})=y_{0}$. Euler’s method approximates the
solution at subsequent points using:

$y_{n+1}=y_{n}+hf(t_{n},y_{n}),$ (3)

where $y_{n}$ is the current approximate value of $y$, $h$ is the step size,
and $t_{n}$ is the current time. One can start with the initial value $y_{0}$,
and keep updating $y_{n+1}$ using the above formula.

Given a very large set of initial value problems (each containing a
differential equation in the form of $\frac{dy}{dt}=f(t,y)$ and the numerical
result of its solution), and mathematical libraries such as SymPy and NumPy,
it should not be very challenging for an AI to come up with something similar
to Euler’s method. For example, an AI could explore a huge number of random
equations, in order to find Equation (3).

Euler’s method could easily be improved to increase its precision, and the
Runge-Kutta method[13] invented at the end of the 19th century is a milestone
and still widely used today. It works as follows:

$\displaystyle k_{1}$ $\displaystyle=f(t_{n},y_{n}),$ $\displaystyle k_{2}$
$\displaystyle=f\left(t_{n}+\frac{h}{2},y_{n}+\frac{h}{2}k_{1}\right),$
$\displaystyle k_{3}$
$\displaystyle=f\left(t_{n}+\frac{h}{2},y_{n}+\frac{h}{2}k_{2}\right),$
$\displaystyle k_{4}$ $\displaystyle=f(t_{n}+h,y_{n}+hk_{3}),$ $\displaystyle
y_{n+1}$ $\displaystyle=y_{n}+\frac{h}{6}(k_{1}+2k_{2}+2k_{3}+k_{4}).$

Here $k_{1}$, $k_{2}$, $k_{3}$ and $k_{4}$ are intermediate values used to
calculate $y_{n+1}$, which is the next approximation of the solution. Please
note this is the fourth-order Runge-Kutta method, meaning its global
truncation error is of the order $O(h^{4})$, where h is the step size. One can
choose Runge-Kutta methods (or alternatives) with higher orders, which usually
have lower errors.

In the _Initial Value Problem Test_ , an AI is assessed by its capability in
inventing a numerical method that is at least as precise as the fourth-order
Runge-Kutta method. This probably requires the AI to go beyond simple try and
error, and learn from its own exploration (e.g., with reinforcement learning).

###  3.7 The Huffman Coding Test

Huffman coding[14] is a most important piece of work in information theory. It
generates variable-length codes where each code’s length is inversely
proportional to the likelihood of the symbol it represents. This aligns
directly with Shannon’s source coding theorem[23], a fundamental principle in
information theory. The theorem states that in an optimal code, the average
length of the symbols should be close to the entropy of the source. Huffman
coding achieves this by ensuring that the most frequent symbols have the
shortest codes, thereby minimizing the overall expected code length needed to
represent each symbol.

Our sixth test is the _Huffman Coding Test_. Given a large corpus of ascii
characters, and Python functions to operate on bits, check if an AI agent can
discover Huffman coding when working towards the goal of minimizing storage
under the constraint that each character be represented by a specific sequence
of 0’s and 1’s.

Given the above constraint, an AI could create many random assignments of
codes for various characters. It then needs to discover the _Prefix-free
Property_ (i.e., no code is a prefix of another code), in order to create
valid codings. Then it needs to observe the efficiency of each coding, and
learns from the exploration of various codings.

###  3.8 The Sorting Algorithm Test

Sorting is probably the most studied problem in computer science, with
numerous great algorithms proposed. Given a very large set of examples (e.g.,
arrays of integers and the sorted version of them), it should be trivial for a
large model to be trained to generate the sorted array based on the original
array. However, a black-box model is not what we want. Our goal is to develop
an efficient sorting algorithm that can run on a simple single-threaded
manner.

Our last test is the _Sorting Algorithm Test_ , which assesses whether an AI
can come up with a sorting function in Python that runs in expected $O(nlogn)$
time, given a very large number of examples of sorting integer arrays. To
avoid leaking the answer, the AI should not be aware of any human-written
programs. However, it should know Python’s syntax and be able to generate
valid (but random) Python code, without understanding its meaning.

One possible route is to let the AI generate a huge number of random Python
code and run them on the given arrays. In this way it should be able to learn
what kind of code converts an array into another array. Then it can generate a
huge number of such random Python functions, and observes which of them can
successfully sort a (possibly small) input array. As it keeps learning from
its exploration, it should be able to generate various types of sorting
functions. Its final step should be learn to predict the running time of each
sorting function, in order to generate more efficient algorithms.

##  4 Discussions

###  4.1 Can an AI possibly conquer these tests?

Making scientific discoveries is different from training LLMs because it would
not be useful to simply feed the model with a very large set of human written
corpus. Instead, we will require the AI to explore on its own and learns from
the exploration, just like what a human scientist would do.

However, we probably still need to use large language models to accomplish
such tasks, and therefore a key question is what information can be used to
train a model. The answer is exploration, probably similar to how a
reinforcement learning model learns to play StarCraft[24]. An AI scientist
must be able to explore, either using an interactive tool or a very large
dataset, to gain knowledge about how to accomplish a particular goal.

Let us take the fifth test, initial value problem, as an example. Given a
large variety of math functions and the solutions to their initial value
problems (i.e., curves of their integrals), an AI agent should start from
randomly exploring tools at hand, such as SymPy and NumPy, to get closer to
the standard answer. For example, the agent should soon find that
$y_{1}=y_{0}+f(x_{0})\cdot\Delta x$, which can be its first answer. Then it
should keep exploring, and possibly find that
$y_{1}=y_{0}+\frac{f(x_{0})+f(x_{1})}{2}\Delta x$ is a better solution. After
many rounds of exploration, it should gradually transit from random
exploration to more informed exploration, either through online learning or
reinforcement learning. This process ends when it finds a solution that is at
least as good as the fourth-order Runge-Kutta method[13].

Learning from exploration is just one possible route to pass such tests.
Another key method is to use Occam’s razor, which prefers simpler
explanations. To be more exact, it prefers explanations that posit fewer
entities, or fewer kinds of entities, with other things equal. On the other
hand, we do hope that an AI agent can develop its own methods in solving these
tests.

###  4.2 Why do we need these tests?

The ultimate goal for an AI scientist is to make novel and impactful
scientific discoveries that no one has made before. Then why do we need these
“Turing tests” which have been discovered decades or centuries ago? There are
two main reasons.

The first reason is that we need a benchmark, just like we need ImageNet[25]
for studies in computer vision. Suppose a great AI scientist has been built
and it makes some new discoveries that have not been made before. Different
people probably have different assessments on the importance of the new
discovery, and it is hard to measure the level of human involvement in the
process of research. With a well-defined benchmark, including both the targets
and the scope of data and tools that can be used, it is much easier to measure
the capability of an AI scientist.

The second reason is that the ultimate goal of making important novel
discoveries is very challenging, as it requires the AI agent to be better than
the best human experts in the world. It is analogical to building an AI agent
that can beat the best GO player in the world. While passing some of our tests
is like beating a top GO player a thousand years ago when GO was in its early
age, or beating an amateur GO player today. If we could build an AI agent that
passes the majority of the above seven tests, we can conclude that we are in
the right track of building an AI scientist, and it should evolve into someone
who can make important scientific discoveries in the foreseeable future.

##  5 Conclusions and Future Work

Recent advancements have enabled LLMs to solve complex problems, highlighting
their potential as tools in daily scientific research. However, the ability to
solve predefined problems is completely different from pioneering scientific
discoveries. This distinction prompts the need for a “qualification test for
an AI scientist" to determine whether an AI can independently conduct
scientific research without human assistance.

The proposed framework for such a test is analogous to the Turing Test, which
assesses whether machines can exhibit human-like intelligence. Unlike LLMs
that learn from extensive datasets, scientific innovation often stems from
exploring uncharted territories. We propose a series of "Turing tests for an
AI scientist" based on key historical scientific breakthroughs such as the
heliocentric model and Maxwell’s equations, which were derived from empirical
data and critical reasoning about the natural world.

Seven such tests are outlined, ranging from astronomy to information theory,
each designed to evaluate the AI’s ability to derive fundamental scientific
principles from raw data. These tests require the AI to engage with
interactive environments or large datasets without prior exposure to human-
derived solutions in these fields.

This approach not only aims to gauge an AI’s ability to generate scientific
insights but also seeks to set a benchmark for AI capabilities in scientific
thinking and discovery. The ultimate goal is to develop an AI that not only
replicates but also innovates, paving the way for AIs that contribute uniquely
to scientific progress.

#### Conflict of Interest Statement

The authors did not receive support from any organization for the submitted
work. The authors have no relevant financial or non-financial interests to
disclose.

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# Hadron spectroscopy using the light-front holographic Schrödinger equation
and the ’t Hooft equation

Mohammad Ahmady<EMAIL_ADDRESS>Department of Physics, Mount Allison
University, Sackville, New Brunswick, Canada, E4L 1E6 Satvir Kaur
<EMAIL_ADDRESS>Department of Physics, Dr. B.R. Ambedkar National
Institute of Technology, Jalandhar 144011, India Sugee Lee MacKay
<EMAIL_ADDRESS>Department of Physics, Mount Allison University, Sackville,
New Brunswick, Canada, E4L 1E6 Chandan Mondal<EMAIL_ADDRESS>Institute
of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China $\&$
School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, China Ruben Sandapen<EMAIL_ADDRESS>Department of Physics, Acadia University, Wolfville, Nova-Scotia, Canada, B4P
2R6

###### Abstract

Light-front holographic QCD provides a successful first approximation to
hadron spectroscopy in the chiral limit of $(3+1)$-dim light-front QCD, where
a holographic Schrödinger-like equation, with an emerging confining scale,
$\kappa$, governs confinement in the transverse direction. In its
supersymmetric formulation, light-front holography predicts that each baryon
has two superpartners: a meson and a tetraquark, with their degenerate masses
being generated by the same scale, $\kappa$. In nature, this mass degeneracy
is lifted by chiral symmetry breaking and longitudinal confinement. In this
paper, we show that the latter can be successfully captured by the ’t Hooft
equation of $(1+1)$-dim, large $N_{c}$, QCD. Together, the holographic
Schrödinger equation and the ’t Hooft equation, provide a good global
description of the data across the full hadron spectrum with a universal
$\kappa$.

##  I Introduction

In light-front $(3+1)$-dim QCD, the mass of a quark-antiquark meson is given
by Brodsky et al. (2015)

$M^{2}=\int\mathrm{d}x\mathrm{d}^{2}\mathbf{b}_{\perp}\Psi^{*}(x,\mathbf{b}_{\perp})\left[-\frac{\nabla_{b_{\perp}}^{2}}{x(1-x)}+\frac{m^{2}_{q}}{x}+\frac{m^{2}_{\bar{q}}}{1-x}\right]\Psi(x,\mathbf{b}_{\perp})+\mathrm{interactions}\;,$
(1)

where $\Psi(x,\mathbf{b}_{\perp})$ is the meson light-front wave function, $x$
the light-front momentum fraction carried by the quark, and
$\mathbf{b}_{\perp}$ (or $b_{\perp}e^{i\varphi}$ in polar representation) the
transverse distance between the quark and the antiquark. In Equation (1),
$m_{q}$ and $m_{\bar{q}}$ are the effective quark and antiquark masses
respectively Brodsky et al. (2015). The “interactions” contain all the
complicated QCD dynamics of a bound state, including the effect of higher Fock
sectors on the valence sector.

Introducing the light-front variable,

$\boldsymbol{\zeta}=\sqrt{x(1-x)}\mathbf{b}_{\perp}\;\,$ (2)

allows a factorization of the wave function into a transverse mode,
$\phi(\zeta)$, and a longitudinal mode, $X(x)$:

$\Psi(x,\zeta,\varphi)=\frac{\phi(\zeta)}{\sqrt{2\pi\zeta}}e^{iL\varphi}X(x)\,$
(3)

with $L=|L_{z}^{\mathrm{max}}|$ being the light-front orbital angular
momentum, and $X(x)=\sqrt{x(1-x)}\chi(x)$. Equation (1) then becomes

$M^{2}=M_{\perp}^{2}+M_{\parallel}^{2}\,$ (4)

where

$M_{\perp}^{2}=\int\mathrm{d}^{2}\boldsymbol{\zeta}\phi^{*}(\zeta)\left[-\frac{d^{2}}{d\zeta^{2}}+\frac{4L^{2}-1}{4\zeta^{2}}+U_{\perp}(\zeta)\right]\phi(\zeta)\;,$
(5)

and

$M_{\parallel}^{2}=\int\mathrm{d}x\chi^{*}(x)\left[\frac{m_{q}^{2}}{x}+\frac{m_{\bar{q}}^{2}}{1-x}+U_{\parallel}(x)\right]\chi(x)\;,$
(6)

with

$\int\mathrm{d}x|\chi(x)|^{2}=\int\mathrm{d}^{2}\boldsymbol{\zeta}|\phi(\zeta)|^{2}=1\;.$
(7)

Here the QCD bound-state dynamics are encoded in the potentials
$U_{\perp}(\zeta)$ and $U_{\parallel}(x)$. Their exact derivation from first
principles remains an open question.

Light-front holography Brodsky and de Teramond (2006); de Teramond and Brodsky
(2005, 2009); Brodsky et al. (2015) considers the chiral limit, i.e., massless
quarks, and neglects longitudinal confinement, thus implying that
$M_{\parallel}=0$. Remarkably, the form of the transverse confinement
potential, $U_{\perp}(\zeta)$, is uniquely fixed by the underlying conformal
symmetry and a holographic mapping to anti-deSitter $\mathrm{AdS}_{5}$,
resulting in Brodsky and de Teramond (2006); de Teramond and Brodsky (2005,
2009); Brodsky et al. (2015)

$U_{\perp}^{\mathrm{LFH}}(\zeta)=\kappa^{4}\zeta^{2}+2\kappa^{2}(J-1)\;$ (8)

where $J=L+S$. In this holographic mapping, the variable $\zeta$ maps onto the
fifth dimension of $\mathrm{AdS}_{5}$. Equation (5), which can be rewritten as

$\left(-\frac{d^{2}}{d\zeta^{2}}+\frac{4L^{2}-1}{4\zeta^{2}}+U^{\mathrm{LFH}}_{\perp}(\zeta)\right)\phi(\zeta)=M_{\perp}^{2}\phi(\zeta)\;,$
(9)

maps onto the wave equation for the amplitude of spin-$J$ modes propagating in
$\mathrm{AdS}_{5}$ modified by a quadratic dilaton field. In light-front
holography, the longitudinal mode, $X(x)$, is hence not dynamical, i.e.,
undetermined by Equation (6). It is instead fixed by the holographic mapping
of the electromagnetic (or gravitational) pion form factor in physical
spacetime and $\mathrm{AdS}_{5}$ Brodsky and de Teramond (2008a, b), resulting
in $\chi(x)=1$, i.e.,

$X(x)=\sqrt{x(1-x)}\;.$ (10)

The supersymmetric formulation of light-front holography Dosch et al. (2015,
2017); Nielsen and Brodsky (2018); Nielsen et al. (2018); Brodsky et al.
(2016) provides a unified description of baryons and mesons/tetraquarks where
each baryon (viewed as a quark-diquark system) possesses two superpartners: a
(quark-antiquark) meson and a (diquark-antidiquark) tetraquark. This
supersymmetric connection stems from the fact that a diquark can be in the
same $\textrm{SU}_{c}(3)$ representation as an antiquark, and an antidiquark
can be in the same $\textrm{SU}_{c}(3)$ representation as a quark. The
supersymmetric holographic Schrödinger equation reads Nielsen and Brodsky
(2018)

$H\ket{\phi}=M_{\perp}^{2}\ket{\phi}$ (11)

where

$H=\begin{pmatrix}-\frac{d^{2}}{d\zeta^{2}}+\frac{4L_{M}^{2}-1}{4\zeta^{2}}+U_{M}(\zeta)&0\\\
0&-\frac{d^{2}}{d\zeta^{2}}+\frac{4L_{B}^{2}-1}{4\zeta^{2}}+U_{B}(\zeta)\end{pmatrix}$
(12)

with

$U_{M}(\zeta)=\kappa^{4}\zeta^{2}+2\kappa^{2}(L_{M}+S_{M}-1)\,,$ (13)
$U_{B}(\zeta)=\kappa^{4}\zeta^{2}+2\kappa^{2}(L_{B}+S_{D})$ (14)

where $S_{M}$ is the total quark-antiquark spin and $S_{D}$ is the diquark
spin. The $4$-plet, $\ket{\phi}$, is given by

$\ket{\phi}=\begin{pmatrix}\phi_{M}(L_{M}=L_{B}+1)&\psi^{-}(L_{B}+1)\\\
\psi^{+}(L_{B})&\phi_{T}(L_{T}=L_{B})\end{pmatrix}$ (15)

where $\psi^{+}$ and $\psi^{-}$ are the two components of the baryon wave
function and $\phi_{M/T}$ is the meson/tetraquark wave function. Equation (11)
admits analytical solutions, with its eigenvalues given by

$M_{\perp,M}^{2}=4\kappa^{2}\left(n_{\perp}+L_{M}+\frac{S_{M}}{2}\right)\;,$
(16)
$M_{\perp,B}^{2}=4\kappa^{2}\left(n_{\perp}+L_{B}+\frac{S_{D}}{2}+1\right)\;,$
(17)

and

$M_{\perp,T}^{2}=4\kappa^{2}\left(n_{\perp}+L_{T}+\frac{S_{T}}{2}+1\right)\,,$
(18)

where $n_{\perp}$ is the principal quantum number that emerges when solving
the holographic Schrödinger equation, and $S_{T}$ is total diquark-antidiquark
spin.

It follows that baryons with quantum numbers, $L_{B}=L_{M}-1$ and
$S_{D}=S_{M}$, are superpartners to mesons with quantum numbers, $L_{M}$ and
$S_{M}$, and tetraquarks with quantum numbers $S_{T}=S_{D}$ and $L_{T}=L_{B}$.
Hence, within a given family of superpartners, we can write $S\equiv
S_{D}=S_{M}=S_{T}$. Note that the lowest-lying mesons with $n_{\perp}=L_{M}=0$
do not have a baryonic superpartner. Furthermore, pseudoscalar mesons with
$n_{\perp}=L_{M}=S_{M}=0$, like the pion and the kaon, are predicted to be
massless, just as expected in the chiral limit of QCD.

##  II Chiral symmetry breaking and longitudinal confinement

The effects of nonzero quark masses were originally taken into account using a
prescription by Brodsky and de Téramond (BdT) Brodsky and de Teramond (2009),
which relies on the fact that the holographic ground state wave function
depends on the invariant mass of the $q\bar{q}$ pair. Indeed, in momentum
space,

$\tilde{\phi}(x,\mathbf{k})\propto\frac{1}{\sqrt{x(1-x)}}\exp\left(-\frac{M_{q\bar{q}}^{2}}{2\kappa^{2}}\right)$
(19)

where

$M_{q\bar{q}}^{2}=\frac{\mathbf{k}^{2}}{x(1-x)}$ (20)

is the invariant mass of the $q\bar{q}$ pair, with $\mathbf{k}$ being the
transverse momentum of the quark. For massive quarks, the invariant mass
becomes

$M_{q\bar{q}}^{2}=\frac{\mathbf{k}^{2}}{x(1-x)}+\frac{m^{2}_{q}}{x}+\frac{m^{2}_{\bar{q}}}{1-x}\;.$
(21)

Thus, after Fourier transforming back to position space, the BdT prescription
amounts to a modification of the longitudinal mode:

$X_{\mathrm{BdT}}(x)=\sqrt{x(1-x)}\chi_{\mathrm{BdT}}(x)$ (22)

where

$\chi_{\mathrm{BdT}}(x)=\exp\left(-\frac{(1-x)m_{q}^{2}+xm^{2}_{\bar{q}}}{2\kappa^{2}x(1-x)}\right)\,.$
(23)

Treating the kinetic energy of massive quarks as a first-order perturbation to
the holographic potential, Equation (8), the resulting first-order shift to
$M=M_{\perp}$ due to quark masses is then given by

$\Delta
M_{\mathrm{BdT}}^{2}=\int\mathrm{d}x\chi_{\mathrm{BdT}}^{2}(x)\left(\frac{m_{q}^{2}}{x}+\frac{m_{\bar{q}}^{2}}{1-x}\right)\,,$
(24)

so that the pion mass, $M_{\pi}=\Delta M_{\mathrm{BdT}}$. Similarly, the kaon
mass, $M_{K}=\Delta M_{\mathrm{BdT}}$ when the strange quark is taken into
account. Equation (24) can be generalized for baryons and tetraquarks Brodsky
et al. (2016):

$\Delta
M_{\mathrm{BdT}}^{2}=\frac{\lambda^{2}}{F}\frac{\mathrm{d}F[\lambda]}{\mathrm{d}\lambda}$
(25)

where

$F[\lambda]=\int_{0}^{1}\prod_{i=1}^{n}\mathrm{d}x_{i}e^{-\frac{1}{\lambda}\sum_{j}^{n}m_{j}^{2}/x_{j}}\delta\left(\sum_{j}x_{j}-1\right)$
(26)

with $\lambda=\kappa^{2}$. Note that Equation (25) coincides with Equation
(24) for $n=2$.

Strictly speaking, Equation (24) is only accurate for light hadrons in their
ground states. However, as a first approximation, it is not unreasonable to
assume the same correction for excited states. At the same time, this
guarantees that the predicted Regge trajectories remain linear. Indeed,
Equation (24), together with Eqs. (16), (17), and (18), have been used to fit
the light hadron spectrum in Ref. Brodsky et al. (2016), resulting in a
universal $\kappa=0.523\pm 0.024$ GeV, with $m_{u/d}=0.046$ GeV and
$m_{s}=0.357$ GeV. The use of Equation (24) has also been extrapolated to
heavy quarks in order to predict the heavy-light and heavy-heavy hadron
spectra in Refs. Dosch et al. (2017); Nielsen et al. (2018); Dosch et al.
(2015), with the requirement that $\kappa$ is no longer universal in the heavy
hadron sector. In an attempt to protect the universality of $\kappa$ in the
heavy sector, Ref. Branz et al. (2010) introduces another mass scale (instead
of $\kappa$) in Equation (23).

A theoretical shortcoming of Equation (24) is that, in the chiral limit, it
predicts Li and Vary (2021) $M_{\pi}^{2}\propto
2m_{u/d}^{2}(\ln\kappa^{2}/m_{u/d}^{2}-\gamma_{E})$ where $\gamma_{E}\approx
0.577216$ is Euler’s constant, which is not consistent with the Gell-Mann-
Oakes-Renner (GMOR) relation Gell-Mann et al. (1968) which states that
$M_{\pi}^{2}\propto m_{u/d}$ in the chiral limit. An improved ansatz,
consistent with the GMOR relation, has been proposed in Ref. Gutsche et al.
(2013). Recently, this shortcoming has been more rigorously addressed in Refs.
Li and Vary (2021); De Téramond and Brodsky (2021) by solving Equation (6)
with a longitudinal confining potential Sheckler and Miller (2021); Li et al.
(2016)

$U_{\parallel}(x)=-\sigma^{2}\partial_{x}(x(1-x))\partial_{x}$ (27)

where $\sigma$ is a mass scale. Reference Li and Vary (2021) studies light
mesons including their excited states while Ref. De Téramond and Brodsky
(2021) focuses on the ground states of light and heavy mesons. While using
Equation (27), Ref. De Téramond and Brodsky (2021) also discusses the relation
of their approach to the ’t Hooft equation. The idea to use the ’t Hooft
equation to go beyond the BdT prescription was first proposed in Ref.
Chabysheva and Hiller (2013), with the unique goal of predicting the meson
decay constants while leaving the predicted mass spectrum unchanged. The
latter constraint is imposed by subtracting $M^{2}_{\parallel}$ from the ’t
Hooft potential.

In this paper, we use the ’t Hooft equation (without shifting the ’t Hooft
potential as in Ref. Chabysheva and Hiller (2013)), together with the
holographic Schrödinger equation, to compute the full hadron spectrum, i.e.,
the masses of the ground and excited states of mesons, baryons, and
tetraquarks, including those with one or two heavy quarks. In Ref. Ahmady et
al. (2021), only the meson spectrum was studied. We shall also compare our
results with the mass spectrum obtained using the alternative longitudinal
potential given by Equation (27).

##  III The ’t Hooft Equation

The ’t Hooft equation is derived from the QCD Lagrangian in $(1+1)$-dim and in
the large $N_{c}$ approximation where only planar diagrams contribute. The
result is ’t Hooft (1974):

$\left(\frac{m_{q}^{2}}{x}+\frac{m_{\bar{q}}^{2}}{1-x}\right)\chi(x)+\frac{g^{2}}{\pi}\mathcal{P}\int{\rm
d}y\frac{|\chi(x)-\chi(y)|}{(x-y)^{2}}=M^{2}_{\parallel}\chi(x)\;,$ (28)

where $g=g_{s}\sqrt{N_{c}}$ is the (finite) ’t Hooft coupling and
$\mathcal{P}$ denotes the principal value prescription. Note that Equation
(28) can also be derived in the continuum limit of discretized $(1+1)$-dim
light-front QCD Hornbostel et al. (1990). We use the ’t Hooft equation for
baryons and tetraquarks by making the transformation $\bar{q}\to[qq]$ for
baryons, followed by $q\to[\bar{q}\bar{q}]$ for tetraquarks. Since the color
interaction is invariant under these transformations, the longitudinal
confinement scale, $g$, remains the same within a family of superpartners. The
’t Hooft equation has been extensively studied in the literature Bars and
Green (1978); Zhitnitsky (1985); Grinstein and Lebed (1998); Grinstein and
Mende (1992); Bergknoff (1977); Ji et al. (2021); Lebed and Uraltsev (2000);
Ma and Ji (2021); Umeeda (2021) and it is worth noting that, in the conformal
limit, it possesses a gravity dual on $\mathrm{AdS}_{3}$ Katz and Okui (2009).
Here, we only highlight how it is both complementary and consistent with the
holographic Schrödinger equation

First, assuming a ’t Hooft wave function of the form De Téramond and Brodsky
(2021)

$\chi(x)\approx x^{\beta_{1}}(1-x)^{\beta_{2}}$ (29)

then, near the endpoints, $x\to 0,1$,

$\frac{\pi m_{i}^{2}}{g^{2}}-1+\pi\beta_{i}\cot(\pi\beta_{i})=0\;.$ (30)

Equation (30) implies that, in the chiral limit, $m_{i}\to 0$,

$\beta_{i}=\sqrt{3m_{i}^{2}/\pi g^{2}}$ (31)

and

$M_{\pi}^{2}=g\sqrt{\frac{\pi}{3}}(m_{u}+m_{d})+\mathcal{O}(m_{u}+m_{d})^{2}\,.$
(32)

Since, via Equation (16), the holographic Schrödinger equation predicts a
massless pion, it follows that the only contribution to the pion mass is
generated by the ’t Hooft equation. Thus, together, the holographic
Schrödinger equation and the ’t Hooft equation correctly predicts the GMOR
relation, $M_{\pi}^{2}\sim m_{u/d}$. Notice also that in the chiral limit,
$\beta_{i}\to 0$ and $\chi(x)=1$ as in light-front holography.

Second, as already noted in Ref. Ahmady et al. (2021), the ’t Hooft potential
is consistent with the holographic potential in that they both correspond to a
linear confining potential in the non-relativistic limit, with $g=\kappa$
Ahmady et al. (2021). To show this, first note that the ’t Hooft potential in
position space reads

$U_{\parallel}^{\mathrm{tHooft}}(x^{-})=\frac{g^{2}}{2}P^{+}|x^{-}|=g^{2}P^{+}b_{\parallel}\;,$
(33)

where $x^{-}=x^{0}-x^{3}$ is the light-front longitudinal distance, and
$P^{+}$ is the light-front longitudinal momentum of the meson. The second
equality follows from the fact that $x^{+}=x^{0}+x^{3}=0$, so that
$|x^{-}|=2|x^{3}|\equiv 2b_{\parallel}$. Note that $\tilde{z}=P^{+}x^{-}$ is
the frame-independent longitudinal coordinate of Ref. Miller and Brodsky
(2020).

Using the general relation between a frame-invariant light-front (LF)
potential and a center of mass (CM) frame instant-form (IF) potential
Trawiński et al. (2014):

$U_{\mathrm{LF}}=V_{\mathrm{IF}}^{2}+4mV_{\mathrm{IF}}\;,$ (34)

it follows from Equation (34) that the (chiral-limit) quadratic light-front
holographic potential, Equation (8), corresponds to a linear instant form
potential. The light-front ’t Hooft potential, Equation (33), also corresponds
to a linear instant form potential in the heavy quark (non-relativistic) limit
where $m_{Q}\gg g$ and $P^{+}\to 2m_{Q}$, where $m_{Q}$ is the heavy quark
mass. In this non-relativistic limit, it is also appropriate to take $x\to
1/2$ in Equation (8). Therefore, we see that, in the non-relativistic limit,
the holographic potential and the ’t Hooft potential are both equivalent to
CM-frame linear instant-form potentials:

$V_{\perp}\to\frac{1}{2}\kappa^{2}b_{\perp}\,,$ (35)

and

$V_{\parallel}\to\frac{1}{2}g^{2}b_{\parallel}\;.$ (36)

It then follows that $V_{\perp}=V_{\parallel}$, i.e., rotational symmetry is
restored in the CM frame of a heavy-heavy meson if $g=\kappa$.

Third, when solving the ’t Hooft equation, an additional quantum number, which
we shall label as $n_{\parallel}$, emerges. Thus, each hadronic state is
identified by four quantum numbers, $n_{\perp},L,S,n_{\parallel}$, with their
squared masses given by

$M_{M}^{2}=M^{2}_{\perp,M}(n_{\perp},L_{M},S_{M},\kappa)+M^{2}_{\parallel,M}(n_{\parallel},m_{q},m_{\bar{q}},g)\;,$
(37)
$M_{B}^{2}=M^{2}_{\perp,B}(n_{\perp},L_{B},S_{D},\kappa)+M^{2}_{\parallel,B}(n_{\parallel},m_{q},m_{[qq]},g)\;,$
(38)
$M_{T}^{2}=M^{2}_{\perp,B}(n_{\perp},L_{T},S_{T},\kappa)+M^{2}_{\parallel,T}(n_{\parallel},m_{[\bar{q}\bar{q}]},m_{[qq]},g)\,,$
(39)

and their parity given by

$P=(-1)^{L_{M}+1}=(-1)^{L_{B}}=(-1)^{L_{T}}\;.$ (40)

In addition, where appropriate, the charge conjugation of mesons/tetraquarks
is given by

$C=(-1)^{n_{\parallel}+L_{M}+S_{M}}=(-1)^{n_{\parallel}+L_{T}+S_{T}-1}\;.$
(41)

À posteriori, we find that

$n_{\parallel}\geq n_{\perp}+L$ (42)

i.e., in any hadron, an orbital and/or radial excitation in the transverse
dynamics is always accompanied by an excitation in the longitudinal dynamics.
This is a signature of the underlying link between the holographic Schrödinger
equation and the ’t Hooft equation. This is perhaps not surprising since they
both derive from the QCD Lagrangian, albeit in different dimensions and
limits.

Fourth, the ’t Hooft equation correctly predicts the scaling law of the decay
constant of heavy-light mesons with the heavy quark mass: $f_{M}\propto
1/\sqrt{m_{Q}}$ Grinstein and Lebed (1998). Since the decay constant depends
on the meson wave function at zero transverse separation, it follows that this
remains true if the meson wave function is given by Equation (3).

Unlike the holographic Schrödinger equation, the ’t Hooft equation does not
have exact analytical solutions. We shall solve it numerically using the
matrix method outlined in Ref. Chabysheva and Hiller (2013), and already used
in Ref. Ahmady et al. (2021).

Hadron | $g$ | $m_{u/d}$ | $m_{s}$ | $m_{c}$ | $m_{b}$
---|---|---|---|---|---
Light | 0.128 | 0.046 | 0.357 | - | -
Heavy-light | 0.410 | 0.330 | 0.500 | 1.370 | 4.640
Heavy-heavy | 0.523 | - | - | 1.370 | 4.640
Table 1: Quark masses and longitudinal confinement scale, $g$, in
$\mathrm{GeV}$ used in conjunction with the ’t Hooft potential. Note that we
use $\kappa=0.523$ GeV for all hadrons, and that it coincides with $g$ for
hadrons with two heavy quarks.

##  IV Computing the hadron spectrum

We now use Eqs. (37), (38), (39), (40) and (41) to compute the hadron spectrum
with the quark masses and values of $g$ shown in Table 1, together with
$\kappa=0.523$ GeV for all hadrons. In comparison to Ref. Ahmady et al.
(2021), we allow the light quark masses to vary between light and heavy-light
hadrons, with the goal of achieving a better global description of the full
hadron spectrum. We adopt the simplest assumption that the (anti)diquark mass
is the sum of the (anti)quark masses, i.e., the (anti)diquark is essentially a
cluster of two (anti)quarks. Note that we identify the superpartners as in
Ref. Nielsen and Brodsky (2018). Our results are shown in Table 3 for the
light hadrons, in Table 4 for hadrons with one heavy quark, and in Table 5 for
hadrons with two heavy quarks.

In general, our meson and baryon masses are in good agreement with the data,
with a discrepancy not exceeding $13\%$, except for $\eta^{\prime}(958)$,
where it is $21\%$. For the tetraquark candidates, the discrepancy does not
exceed $18\%$, with notable exceptions for $f_{0}(500)$ and
$f_{0}(980)(a_{0}(980))$, where the discrepancies are very large. The
agreement for mesons and baryons is thus more impressive than for tetraquarks.
In fact, the large discrepancies for the tetraquark candidates, $f_{0}(500)$
and $f_{0}(980)(a_{0}(980))$, are already present before any supersymmetry
breaking by longitudinal dynamics. This can be seen by comparing the values of
$M_{\perp}$ to the physical masses of $f_{0}(500)$ and
$f_{0}(980)(a_{0}(980))$ in Table 3. Thus these discrepancies are mostly
inherited from the original formulation of supersymmetric light-front
holography, and are not alleviated by longitudinal dynamics.

It is instructive to compare our results with those obtained using the
alternative longitudinal potential given by Equation (27). In this case, an
analytical formula can be derived for the meson mass spectrum Li and Vary
(2021):

$M_{\parallel,M}^{2}=\sigma(m_{q}+m_{\bar{q}})(2n_{\parallel}+1)+\sigma^{2}n_{\parallel}(n_{\parallel}+1)+(m_{q}+m_{\bar{q}})^{2}\;.$
(43)

Reference De Téramond and Brodsky (2021) shows that the restoration of
rotational symmetry in heavy-heavy mesons implies the constraint:

$\sigma=\frac{\kappa^{2}}{m_{Q}+m_{\bar{Q}}}\;,$ (44)

which is to be contrasted with the analogous constraint, $g=\kappa$, for the
’t Hooft potential: see Equation (35) and Equation (36). To compute the hadron
masses, we proceed in the same way as with the ’t Hooft potential: within a
family of superpartners, we use Equation (43), with $\bar{q}\to[qq]$ (for the
baryon), followed by $q\to[\bar{q}\bar{q}]$ (for the tetraquark). Again, we
assume that the (anti)diquark mass is twice the quark mass. This means that
our goal is only to investigate the sensitivity of our results to the
analytical form of the longitudinal potential. As can be seen in Tables 6, 7
and 8, the agreement with data is similar in quality to that achieved with the
’t Hooft potential, provided that $\sigma$ remains universal while $\kappa$ is
allowed to vary across the spectrum, subject to the constraint given by
Equation (44). Their numerical values, as well as those of the quark masses,
are given in Table 2.

Hadron |  | $\kappa$ | $m_{u/d}$ | $m_{s}$ | $m_{c}$ | $m_{b}$
---|---|---|---|---|---|---
Light |  | 0.523 | 0.046 | 0.357 | - | -
Heavy-light | 1 $c$-quark | 0.580 | 0.330 | 0.500 | 1.470 | -
1 $b$-quark | 1.000 | 0.330 | 0.500 | - | 4.670
Heavy-heavy | 2 $c$-quarks | 0.600 | - | - | 1.470 | -
2 $b$-quarks | 1.070 | - | - | - | 4.670
Table 2: Quark masses and the transverse confinement scale, $\kappa$, in
$\mathrm{GeV}$ used in conjunction with the alternative longitudinal
potential, Equation (27). Note that we use $\sigma=0.12$ GeV for all hadrons,
and that the constraint, Equation (44), is satisfied for hadrons with two
heavy quarks.

Interestingly, it seems that either the longitudinal or the transverse
confinement scale can be universal across the spectrum, depending on whether
we choose Equation (27) or the ’t Hooft potential to model the longitudinal
dynamics. The ’t Hooft potential offers the advantage that its connection to
the QCD Lagrangian is known. For this reason, it remains our focus in this
paper.

We now proceed to show the Regge trajectories obtained using the ’t Hooft
potential: Figs. 1 and 2 for the light hadrons, Figs. 3 and 4 for hadrons with
one heavy quark, and Figs. 5 and 6 for hadrons with two heavy quarks. As
expected, the agreement is best for mesons and baryons. Except for the lowest-
lying pseudoscalar mesons for which $M_{\perp}=0$, $M_{\perp}\gg
M_{\parallel}$ for light hadrons. On the other hand, $M_{\parallel}\gg
M_{\perp}$ for hadrons with one or two heavy quarks. It is therefore
legitimate to investigate the sensitivity of our Regge trajectories to a small
variation in $g$ (for light hadrons) or $\kappa$ (for heavy hadrons), while
keeping the quark masses fixed. In Fig. 7, we show the effect of changing $g$
from $0.128$ to $0.200$ GeV on the $\pi$ and $\rho$ Regge trajectories. As can
be seen, the slopes of the trajectories are sensitive to this variation in
$g$. Note that, as expected, the pion mass also changes from $0.140$ to
$0.163$ GeV but this is not visible on the mass scale of Fig. 7. Meanwhile, in
Fig. 8, we illustrate the effect of changing $\kappa$ from $0.523$ to $0.700$
GeV on the $\eta_{c}$ and $J/\psi$ Regge trajectories. We can see that the
Regge slopes are significantly changed with this variation in $\kappa$. These
observations are also typical for the other Regge trajectories. Therefore, the
precise slopes and locations of Regge trajectories, for a given set of quark
masses, are sensitive to both $g$ and $\kappa$. It is worth highlighting that,
in our approach, $\kappa$ remains universal across the full hadron spectrum.

In Figs. 9 and 10, we compare our results for all baryons, including those
with no identified superpartners from the PDG. Using $\kappa=0.500\pm 0.024$
GeV, we achieve very good agreement with the data.

##  V Conclusions

We have shown that the ’t Hooft equation is complementary to the holographic
Schrödinger equation in predicting the full hadron spectrum. While the
holographic Schrödinger equation generates the hadronic mass in the chiral
limit of QCD with a universal emerging transverse confinement scale, the ’t
Hooft equation generates the contribution to the hadronic mass due to nonzero
quark masses and longitudinal confinement. Agreement with spectroscopic data
spectra is generally good, while the disagreement for the tetraquark
candidates, $f_{0}(500)$ and $f_{0}(980)(a_{0}(980))$, already present when
neglecting longitudinal dynamics, persists.

##  VI Acknowledgements

We thank Stan Brodsky and Guy de Téramond for useful discussions. R.S. and
M.A. are supported by individual Discovery Grants No. SAPIN-2020-00051 and No.
SAPIN-2021-00038 from the Natural Sciences and Engineering Research Council of
Canada (NSERC). C. M. is supported by new faculty start up funding by the
Institute of Modern Physics, Chinese Academy of Sciences, Grant No.
E129952YR0. C.M. also thanks the Chinese Academy of Sciences President’s
International Fellowship Initiative for the support via Grant No. 2021PM0023.

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Meson | Baryon | Tetraquark
---|---|---
$J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$
$0^{-+}$ | $\pi(140)$ | 140 | 0 | 140 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{1}(1170)$ | 335 | 1046 | 1098 | $(1/2)^{+}$ | $N(940)$ | 188 | 1046 | 1063 | $0^{++}$ | $f_{0}(500)$ | 335 | 1046 | 1098
$2^{-+}$ | $\eta_{2}(1645)$ | 460 | 1479 | 1549 | $(3/2)^{-}$ | $N(1520)$ | 296 | 1479 | 1508 | $1^{-+}$ | - | 408 | 1479 | 1534
$1^{--}$ |  $\rho(770)$, $\omega(780)$ | 140 | 740 | 753 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ |  $a_{2}(1320)$, $f_{2}(1270)$ | 335 | 1281 | 1324 | $(3/2)^{+}$ | $\Delta(1232)$ | 188 | 1281 | 1295 | $1^{++}$ | $a_{1}(1260)$ | 235 | 1281 | 1302
$3^{--}$ |  $\rho_{3}(1690)$, $\omega_{3}(1670)$ | 460 | 1654 | 1717 | $(3/2)^{-}$ | $\Delta(1700)$ | 296 | 1654 | 1680 | $1^{-+}$ | $\pi_{1}(1600)$ | 335 | 1654 | 1688
$4^{++}$ |  $a_{4}(1970)$, $f_{4}(2050)$ | 559 | 1957 | 2035 | $(7/2)^{+}$ | $\Delta(1950)$ | 372 | 1957 | 1992 | - | - | - | - | -
$0^{-}$ | $\bar{K}(495)$ | 456 | 0 | 456 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{K}_{1}(1270)$ | 550 | 1046 | 1182 | $(1/2)^{+}$ | $\Lambda(1115)$ | 500 | 1046 | 1159 | $0^{+}$ | $K^{*}_{0}(1430)$ | 546 | 1046 | 1180
$2^{-}$ | $K_{2}(1770)$ | 617 | 1479 | 1603 | $(3/2)^{-}$ | $\Lambda(1520)$ | 588 | 1479 | 1592 | $1^{-}$ | - | 633 | 1479 | 1609
$0^{-}$ | $\bar{K}(495)$ | 456 | 0 | 456 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{K}_{1}(1270)$ | 550 | 1046 | 1182 | $(1/2)^{+}$ | $\Sigma(1190)$ | 500 | 1046 | 1159 | $0^{++}$ | $a_{0}(980)$ | 920 | 1046 | 1393
|  |  |  |  |  |  |  |  |  |  | $f_{0}(980)$ |  |  |
$1^{-}$ | $K^{*}(890)$ | 456 | 740 | 869 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $K^{*}_{2}(1430)$ | 550 | 1281 | 1394 | $(3/2)^{+}$ | $\Sigma(1385)$ | 500 | 1281 | 1375 | $1^{+}$ | $K_{1}(1400)$ | 546 | 1281 | 1392
$3^{-}$ | $K^{*}_{3}(1780)$ | 617 | 1654 | 1765 | $(3/2)^{-}$ | $\Sigma(1670)$ | 588 | 1654 | 1755 | $2^{-}$ | $K_{2}(1820)$ | 633 | 1654 | 1771
$4^{+}$ | $K^{*}_{4}(2045)$ | 672 | 1957 | 2069 | $(7/2)^{+}$ | $\Sigma(2030)$ | 650 | 1957 | 2062 | - | - | - | - | -
$0^{-+}$ | $\eta^{\prime}(958)$ | 759 | 0 | 759 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{1}(1380)$ | 883 | 1046 | 1369 | $(1/2)^{+}$ | $\Xi(1320)$ | 805 | 1046 | 1320 | $0^{++}$ | $f_{0}(1370)$ | 920 | 1046 | 1393
|  |  |  |  |  |  |  |  |  |  | $a_{0}(1450)$ |  |  |
$2^{-+}$ | $\eta_{2}(1870)$ | 968 | 1479 | 1768 | $(3/2)^{-}$ | $\Xi(1620)$ | 876 | 1479 | 1719 | $1^{-+}$ | - | 969 | 1479 | 1768
$1^{--}$ | $\Phi(1020)$ | 759 | 740 | 1060 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $f^{\prime}_{2}(1525)$ | 883 | 1281 | 1556 | $(3/2)^{+}$ | $\Xi^{*}(1530)$ | 805 | 1281 | 1513 | $1^{++}$ | $f_{1}(1420)$ | 850 | 1281 | 1537
|  |  |  |  |  |  |  |  |  |  | $a_{1}(1420)$ |  |  |
$3^{--}$ | $\Phi_{3}(1850)$ | 968 | 1654 | 1916 | $(3/2)^{-}$ | $\Xi(1820)$ | 876 | 1654 | 1872 | - | - | - | - | -
$2^{++}$ | $f_{2}(1640)$ | 883 | 1281 | 1556 | $(3/2)^{+}$ | $\Omega(1672)$ | 1114 | 1281 | 1698 | $1^{+}$ | $K_{1}(1650)$ | 1159 | 1281 | 1728

Table 3: Computed masses (in MeV), using the ’t Hooft potential, for the light
hadrons compared to the PDG data Zyla et al. (2020).

Meson | Baryon | Tetraquark
---|---|---
$J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$
$0^{-}$ | $D(1870)$ | 1861 | 0 | 1861 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $D_{1}(2420)$ | 2135 | 1046 | 2377 | $(1/2)^{+}$ | $\Lambda_{c}(2290)$ | 2191 | 1046 | 2428 | $0^{+}$ | $\bar{D^{*}_{0}}(2400)$ | 2510 | 1046 | 2719
$2^{-}$ | $D_{J}(2600)$ | 2326 | 1479 | 2756 | $(3/2)^{-}$ | $\Lambda_{c}(2625)$ | 2460 | 1479 | 2870 | $1^{-}$ | - | 2751 | 1479 | 3123
$0^{-}$ | $\bar{D}(1870)$ | 1861 | 0 | 1861 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{D}_{1}(2420)$ | 2135 | 1046 | 2377 | $(1/2)^{+}$ | $\Sigma_{c}(2455)$ | 2191 | 1046 | 2428 | $0^{+}$ | $D^{*}_{0}(2400)$ | 2510 | 1046 | 2719
$1^{-}$ | $D^{*}(2010)$ | 1861 | 740 | 2003 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $D^{*}_{2}(2460)$ | 2135 | 1281 | 2490 | $(3/2)^{+}$ | $\Sigma^{*}_{c}(2520)$ | 2191 | 1281 | 2538 | $1^{+}$ | $D(2550)$ | 2510 | 1281 | 2818
$3^{-}$ | $D^{*}_{3}(2750)$ | 2326 | 1654 | 2854 | $(3/2)^{-}$ | $\Sigma_{c}(2800)$ | 2460 | 1654 | 2964 | - | - | - | - | -
$0^{-}$ | $D_{s}(1968)$ | 2025 | 0 | 2025 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $D_{s1}(2460)$ | 2283 | 1046 | 2511 | $(1/2)^{+}$ | $\Xi_{c}(2470)$ | 2348 | 1046 | 2570 | $0^{+}$ | $\bar{D}^{*}_{s0}(2317)$ | 2676 | 1046 | 2873
$2^{-}$ | $D_{s2}?$ | 2464 | 1479 | 2874 | $(3/2)^{-}$ | $\Xi_{c}(2815)$ | 2586 | 1479 | 2979 | $1^{-}$ | - | 2908 | 1479 | 3262
$1^{-}$ | $D^{*}_{s}(2110)$ | 2025 | 740 | 2156 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $D^{*}_{s2}(2573)$ | 2283 | 1281 | 2618 | $(3/2)^{+}$ | $\Xi^{*}_{c}(2645)$ | 2348 | 1281 | 2675 | $1^{+}$ | $D_{s1}(2536)$ | 2676 | 1281 | 2967
$3^{-}$ | $D^{*}_{s3}(2860)$ | 2464 | 1654 | 2968 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{D}_{s1}?$ | 2283 | 1046 | 2511 | $(1/2)^{+}$ | $\Omega_{c}(2695)$ | 2524 | 1046 | 2732 | $0^{+}$ | - | 2845 | 1046 | 3031
$2^{+}$ | $D^{*}_{s2}?$ | 2283 | 1281 | 2618 | $(3/2)^{+}$ | $\Omega_{c}(2770)$ | 2524 | 1281 | 2830 | $1^{+}$ | - | 3012 | 1281 | 3273
$0^{-}$ | $\bar{B}(5280)$ | 5130 | 0 | 5130 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{B}_{1}(5720)$ | 5385 | 1046 | 5486 | $(1/2)^{+}$ | $\Lambda_{b}(5620)$ | 5460 | 1046 | 5559 | $0^{+}$ | $B_{J}(5732)$ | 5775 | 1046 | 5869
$2^{-}$ | $\bar{B}_{J}(5970)$ | 5560 | 1479 | 5753 | $(3/2)^{-}$ | $\Lambda_{b}(5920)$ | 5714 | 1479 | 5902 | $1^{-}$ | - | 5999 | 1479 | 6179
$0^{-}$ | $B(5280)$ | 5130 | 0 | 5130 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $B_{1}(5720)$ | 5385 | 1046 | 5486 | $(1/2)^{+}$ | $\Sigma_{b}(5815)$ | 5460 | 1046 | 5559 | $0^{+}$ | $\bar{B}_{J}(5732)$ | 5775 | 1046 | 5869
$1^{-}$ | $B^{*}(5325)$ | 5130 | 740 | 5183 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $B^{*}_{2}(5747)$ | 5385 | 1281 | 5535 | $(3/2)^{+}$ | $\Sigma^{*}_{b}(5835)$ | 5460 | 1281 | 5608 | $1^{+}$ | $B_{J}(5840)$ | 5775 | 1281 | 5915
$0^{-}$ | $B_{s}(5365)$ | 5292 | 0 | 5292 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $B_{s1}(5830)$ | 5528 | 1046 | 5626 | $(1/2)^{+}$ | $\Xi_{b}(5790)$ | 5610 | 1046 | 5707 | $0^{+}$ | $\bar{B}^{*}_{s0}?$ | 5936 | 1046 | 6027
$1^{-}$ | $B^{*}_{s}(5415)$ | 5292 | 740 | 5343 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $B^{*}_{s2}(5840)$ | 5528 | 1281 | 5674 | $(3/2)^{+}$ | $\Xi^{*}_{b}(5950)$ | 5610 | 1281 | 5754 | $1^{+}$ | $B_{s1}?$ | 5936 | 1281 | 6073
$1^{+}$ | $B_{s1}?$ | 5528 | 1046 | 5626 | $(1/2)^{+}$ | $\Omega_{b}(6045)$ | 5791 | 1046 | 5885 | $0^{+}$ | - | 6110 | 1046 | 6199

Table 4: Computed masses (in MeV), using the ’t Hooft potential, for hadrons
with one heavy quark, compared to the PDG data Zyla et al. (2020).

Meson | Baryon | Tetraquark
---|---|---
$J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$
$0^{-+}$ | $\eta_{c}(2984)$ | 2927 | 0 | 2927 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{c}(3525)$ | 3440 | 1046 | 3596 | $(1/2)^{+}$ | $\Xi^{\rm SELEX}_{cc}(3520)$ | 3254 | 1046 | 3418 | $0^{++}$ | $\chi_{c0}(3415)$ | 3864 | 1046 | 4003
|  |  |  |  |  | $\Xi^{\rm LHCb}_{cc}(3620)$ |  |  |  |  |  |  |  |
$1^{--}$ | $J/\psi(3096)$ | 2927 | 740 | 3019 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{c2}(3556)$ | 3440 | 1281 | 3671 | $(3/2)^{+}$ | $\Xi^{\rm LHCb}_{cc}(3620)$ | 3254 | 1281 | 3497 | $1^{++}$ | $\chi_{c1}(3510)$ | 3580 | 1281 | 3802
$1^{--}$ | $\psi^{\prime}(3686)$ | 3440 | 1281 | 3671 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{c2}(3927)$ | 3794 | 1654 | 4139 | $(3/2)^{+}$ | $\Xi^{*}_{cc}?$ | 3751 | 1654 | 4099 | $1^{++}$ | $X(3872)$ | 4062 | 1654 | 4386
|  |  |  |  |  |  |  |  |  | $1^{+-}$ | $Z_{c}(3900)$ | 4240 | 1654 | 4551
$0^{-+}$ | $\eta_{b}(9400)$ | 9424 | 0 | 9424 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{b}(9900)$ | 9776 | 1046 | 9832 | $(1/2)^{+}$ | $\Xi_{bb}?$ | 9750 | 1046 | 9806 | $0^{++}$ | $\chi_{b0}(9860)$ | 10285 | 1046 | 10338
$1^{--}$ | $\Upsilon(9460)$ | 9424 | 740 | 9453 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{b2}(9910)$ | 9776 | 1281 | 9860 | $(3/2)^{+}$ | $\Xi_{bb}?$ | 9750 | 1281 | 9834 | $1^{++}$ | $\chi_{b1}(9893)$ | 10081 | 1281 | 10162
$1^{--}$ | $\Upsilon(2S)(10020)$ | 9776 | 1281 | 9860 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{b2}(10270)$ | 10024 | 1654 | 10160 | $(3/2)^{+}$ | $\Xi_{bb}?$ | 10100 | 1654 | 10234 | $1^{++}$ | $X_{b}?$ | 10425 | 1654 | 10555

Table 5: Computed masses (in MeV), using the ’t Hooft potential, for hadrons
with two heavy quarks, compared to the PDG data Zyla et al. (2020).

Meson | Baryon | Tetraquark
---|---|---
$J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$
$0^{-+}$ | $\pi(140)$ | 140 | 0 | 140 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{1}(1170)$ | 387 | 1046 | 1115 | $(1/2)^{+}$ | $N(940)$ | 189 | 1046 | 1063 | $0^{++}$ | $f_{0}(500)$ | 359 | 1046 | 1106
$2^{-+}$ | $\eta_{2}(1645)$ | 629 | 1479 | 1607 | $(3/2)^{-}$ | $N(1520)$ | 312 | 1479 | 1512 | $1^{-+}$ | - | 480 | 1479 | 1555
$1^{--}$ |  $\rho(770)$, $\omega(780)$ | 140 | 740 | 753 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ |  $a_{2}(1320)$, $f_{2}(1270)$ | 387 | 1281 | 1338 | $(3/2)^{+}$ | $\Delta(1232)$ | 189 | 1281 | 1295 | $1^{++}$ | $a_{1}(1260)$ | 236 | 1281 | 1302
$3^{--}$ |  $\rho_{3}(1690)$, $\omega_{3}(1670)$ | 629 | 1654 | 1770 | $(3/2)^{-}$ | $\Delta(1700)$ | 312 | 1654 | 1683 | $1^{-+}$ | $\pi_{1}(1600)$ | 359 | 1654 | 1692
$4^{++}$ |  $a_{4}(1970)$, $f_{4}(2050)$ | 870 | 1957 | 2142 | $(7/2)^{+}$ | $\Delta(1950)$ | 434 | 1957 | 2004 |  | - | - | - | -
$0^{-}$ | $\bar{K}(495)$ | 459 | 0 | 459 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{K}_{1}(1270)$ | 580 | 1046 | 1196 | $(1/2)^{+}$ | $\Lambda(1115)$ | 505 | 1046 | 1162 | $0^{+}$ | $K^{*}_{0}(1430)$ | 552 | 1046 | 1183
$2^{-}$ | $K_{2}(1770)$ | 700 | 1479 | 1636 | $(3/2)^{-}$ | $\Lambda(1520)$ | 626 | 1479 | 1606 | $1^{-}$ | - | 672 | 1479 | 1624
$0^{-}$ | $\bar{K}(495)$ | 459 | 0 | 459 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{K}_{1}(1270)$ | 580 | 1046 | 1196 | $(1/2)^{+}$ | $\Sigma(1190)$ | 505 | 1046 | 1162 | $0^{++}$ | $a_{0}(980)$ | 984 | 1046 | 1436
|  |  |  |  |  |  |  |  |  |  | $f_{0}(980)$ |  |  |
$1^{-}$ | $K^{*}(890)$ | 459 | 740 | 871 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $K^{*}_{2}(1430)$ | 580 | 1281 | 1406 | $(3/2)^{+}$ | $\Sigma(1385)$ | 505 | 1281 | 1377 | $1^{+}$ | $K_{1}(1400)$ | 552 | 1281 | 1395
$3^{-}$ | $K^{*}_{3}(1780)$ | 700 | 1654 | 1796 | $(3/2)^{-}$ | $\Sigma(1670)$ | 626 | 1654 | 1768 | $2^{-}$ | $K_{2}(1820)$ | 672 | 1654 | 1785
$4^{+}$ | $K^{*}_{4}(2045)$ | 821 | 1957 | 2122 | $(7/2)^{+}$ | $\Sigma(2030)$ | 747 | 1957 | 2095 | - | - | - | - | -
$0^{-+}$ | $\eta^{\prime}(958)$ | 772 | 0 | 772 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{1}(1380)$ | 1012 | 1046 | 1455 | $(1/2)^{+}$ | $\Xi(1320)$ | 818 | 1046 | 1328 | $0^{++}$ | $f_{0}(1370)$ | 984 | 1046 | 1436
|  |  |  |  |  |  |  |  |  |  | $a_{0}(1450)$ |  |  |
$2^{-+}$ | $\eta_{2}(1870)$ | 1252 | 1479 | 1938 | $(3/2)^{-}$ | $\Xi(1620)$ | 938 | 1479 | 1751 | $1^{-+}$ | - | 1104 | 1479 | 1846
$1^{--}$ | $\Phi(1020)$ | 772 | 740 | 1069 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $f^{\prime}_{2}(1525)$ | 1012 | 1281 | 1632 | $(3/2)^{+}$ | $\Xi^{*}(1530)$ | 818 | 1281 | 1520 | $1^{++}$ | $f_{1}(1420)$ | 864 | 1281 | 1545
|  |  |  |  |  |  |  |  |  |  | $a_{1}(1420)$ |  |  |
$3^{--}$ | $\Phi_{3}(1850)$ | 1252 | 1654 | 2074 | $(3/2)^{-}$ | $\Xi(1820)$ | 938 | 1654 | 1901 | - | - | - | - | -
$2^{++}$ | $f_{2}(1640)$ | 1012 | 1281 | 1632 | $(3/2)^{+}$ | $\Omega(1672)$ | 1129 | 1281 | 1708 | $1^{+}$ | $K_{1}(1650)$ | 1175 | 1281 | 1738

Table 6: Computed masses (in MeV) using the alternative longitudinal potential
given by Equation (27), for light hadrons compared to the PDG data Zyla et al.
(2020).

Meson | Baryon | Tetraquark
---|---|---
$J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$
$0^{-}$ | $D(1870)$ | 1859 | 0 | 1859 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $D_{1}(2420)$ | 1979 | 1160 | 2294 | $(1/2)^{+}$ | $\Lambda_{c}(2290)$ | 2189 | 1160 | 2478 | $0^{+}$ | $\bar{D}^{*}_{0}(2400)$ | 2519 | 1160 | 2774
$2^{-}$ | $D_{J}(2600)$ | 2099 | 1640 | 2664 | $(3/2)^{-}$ | $\Lambda_{c}(2625)$ | 2309 | 1640 | 2833 | $1^{-}$ | - | 2639 | 1640 | 3108
$0^{-}$ | $\bar{D}(1870)$ | 1859 | 0 | 1859 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{D}_{1}(2420)$ | 1979 | 1160 | 2294 | $(1/2)^{+}$ | $\Sigma_{c}(2455)$ | 2189 | 1160 | 2478 | $0^{+}$ | $D^{*}_{0}(2400)$ | 2519 | 1160 | 2774
$1^{-}$ | $D^{*}(2010)$ | 1859 | 820 | 2032 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $D^{*}_{2}(2460)$ | 1979 | 1421 | 2436 | $(3/2)^{+}$ | $\Sigma^{*}_{c}(2520)$ | 2189 | 1421 | 2610 | $1^{+}$ | $D(2550)$ | 2519 | 1421 | 2892
$3^{-}$ | $D^{*}_{3}(2750)$ | 2099 | 1834 | 2788 | $(3/2)^{-}$ | $\Sigma_{c}(2800)$ | 2309 | 1834 | 2949 | - | - | - | - | -
$0^{-}$ | $D_{s}(1968)$ | 2029 | 0 | 2029 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $D_{s1}(2460)$ | 2149 | 1160 | 2442 | $(1/2)^{+}$ | $\Xi_{c}(2470)$ | 2359 | 1160 | 2629 | $0^{+}$ | $\bar{D}^{*}_{s0}(2317)$ | 2689 | 1160 | 2929
$2^{-}$ | $D_{s2}?$ | 2269 | 1640 | 2800 | $(3/2)^{-}$ | $\Xi_{c}(2815)$ | 2479 | 1640 | 2973 | $1^{-}$ | - | 2809 | 1640 | 3253
$1^{-}$ | $D^{*}_{s}(2110)$ | 2029 | 820 | 2189 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $D^{*}_{s2}(2573)$ | 2149 | 1421 | 2576 | $(3/2)^{+}$ | $\Xi^{*}_{c}(2645)$ | 2359 | 1421 | 2754 | $1^{+}$ | $D_{s1}(2536)$ | 2689 | 1421 | 3042
$3^{-}$ | $D^{*}_{s3}(2860)$ | 2269 | 1834 | 2918 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{D}_{s1}?$ | 2149 | 1160 | 2442 | $(1/2)^{+}$ | $\Omega_{c}(2695)$ | 2529 | 1160 | 2783 | $0^{+}$ | - | 2859 | 1160 | 3086
$2^{+}$ | $D^{*}_{s2}?$ | 2149 | 1421 | 2576 | $(3/2)^{+}$ | $\Omega_{c}(2770)$ | 2529 | 1421 | 2901 | $1^{+}$ | - | 2859 | 1421 | 3193
$0^{-}$ | $\bar{B}(5280)$ | 5060 | 0 | 5060 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $\bar{B}_{1}(5720)$ | 5180 | 2000 | 5552 | $(1/2)^{+}$ | $\Lambda_{b}(5620)$ | 5390 | 2000 | 5749 | $0^{+}$ | $B_{J}(5732)$ | 5720 | 2000 | 6059
$2^{-}$ | $\bar{B}_{J}(5970)$ | 5300 | 2828 | 6007 | $(3/2)^{-}$ | $\Lambda_{b}(5920)$ | 5510 | 2828 | 6193 | $1^{-}$ | - | 5840 | 2828 | 6489
$0^{-}$ | $B(5280)$ | 5060 | 0 | 5060 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $B_{1}(5720)$ | 5180 | 2000 | 5552 | $(1/2)^{+}$ | $\Sigma_{b}(5815)$ | 5390 | 2000 | 5749 | $0^{+}$ | $\bar{B}_{J}(5732)$ | 5720 | 2000 | 6059
$1^{-}$ | $B^{*}(5325)$ | 5060 | 1414 | 5254 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $B^{*}_{2}(5747)$ | 5180 | 2449 | 5730 | $(3/2)^{+}$ | $\Sigma^{*}_{b}(5835)$ | 5390 | 2449 | 5920 | $1^{+}$ | $B_{J}(5840)$ | 5720 | 2449 | 6222
$0^{-}$ | $B_{s}(5365)$ | 5230 | 0 | 5230 | - | - | - | - | - | - | - | - | - | -
$1^{+}$ | $B_{s1}(5830)$ | 5350 | 2000 | 5711 | $(1/2)^{+}$ | $\Xi_{b}(5790)$ | 5597 | 2000 | 5908 | $0^{+}$ | $\bar{B}^{*}_{s0}?$ | 5890 | 2000 | 6220
$1^{-}$ | $B^{*}_{s}(5415)$ | 5230 | 1414 | 5418 | - | - | - | - | - | - | - | - | - | -
$2^{+}$ | $B^{*}_{s2}(5840)$ | 5350 | 2449 | 5884 | $(3/2)^{+}$ | $\Xi^{*}_{b}(5950)$ | 5560 | 2449 | 6075 | $1^{+}$ | $B_{s1}?$ | 5890 | 2449 | 6379
$1^{+}$ | $B_{s1}?$ | 5350 | 2000 | 5711 | $(1/2)^{+}$ | $\Omega_{b}(6045)$ | 5730 | 2000 | 6069 | $0^{+}$ | - | 6060 | 2000 | 6381

Table 7: Computed masses (in MeV), using the alternative longitudinal
potential given by Equation (27), for hadrons with one heavy quark, compared
to the PDG data Zyla et al. (2020).

Meson | Baryon | Tetraquark
---|---|---
$J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$ | $J^{P(C)}$ | Name | $M_{\parallel}$ | $M_{\perp}$ | $M$
$0^{-+}$ | $\eta_{c}(2984)$ | 2999 | 0 | 2999 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{c}(3525)$ | 3239 | 1200 | 3454 | $(1/2)^{+}$ | $\Xi^{\rm SELEX}_{cc}(3520)$ | 3329 | 1200 | 3539 | $0^{++}$ | $\chi_{c0}(3415)$ | 3780 | 1200 | 3965
|  |  |  |  |  | $\Xi^{\rm LHCb}_{cc}(3620)$ |  |  |  |  |  |  |  |
$1^{--}$ | $J/\psi(3096)$ | 2999 | 848 | 3117 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{c2}(3556)$ | 3239 | 1470 | 3557 | $(3/2)^{+}$ | $\Xi^{\rm LHCb}_{cc}(3620)$ | 3329 | 1470 | 3639 | $1^{++}$ | $\chi_{c1}(3510)$ | 3660 | 1470 | 3944
$1^{--}$ | $\psi^{\prime}(3686)$ | 3239 | 1470 | 3557 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{c2}(3927)$ | 3479 | 1897 | 3963 | $(3/2)^{+}$ | $\Xi^{*}_{cc}?$ | 3570 | 1897 | 4042 | $1^{++}$ | $X(3872)$ | 3900 | 1897 | 4337
|  |  |  |  |  |  |  |  |  | $1^{+-}$ | $Z_{c}(3900)$ | 4020 | 1897 | 4445
$0^{-+}$ | $\eta_{b}(9400)$ | 9400 | 0 | 9400 | - | - | - | - | - | - | - | - | - | -
$1^{+-}$ | $h_{b}(9900)$ | 9640 | 2140 | 9874 | $(1/2)^{+}$ | $\Xi_{bb}?$ | 9730 | 2140 | 9962 | $0^{++}$ | $\chi_{b0}(9860)$ | 10180 | 2140 | 10402
$1^{--}$ | $\Upsilon(9460)$ | 9400 | 1513 | 9521 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{b2}(9910)$ | 9640 | 2621 | 9990 | $(3/2)^{+}$ | $\Xi_{bb}?$ | 9730 | 2621 | 10077 | $1^{++}$ | $\chi_{b1}(9893)$ | 10060 | 2621 | 10396
$1^{--}$ | $\Upsilon(2S)(10020)$ | 9640 | 1513 | 9990 | - | - | - | - | - | - | - | - | - | -
$2^{++}$ | $\chi_{b2}(10270)$ | 9880 | 3384 | 10443 | $(3/2)^{+}$ | $\Xi_{bb}?$ | 9970 | 3384 | 10528 | $1^{++}$ | $X_{b}?$ | 10300 | 3384 | 10841

Table 8: Computed masses (in MeV), using the alternative longitudinal
potential given by Equation (27), for hadrons with two heavy quarks, compared
to the PDG data Zyla et al. (2020). Figure 1: Regge trajectories for the
supersymmetric meson-baryon-tetraquark partners for the unflavored light-light
sector. The results are compared to the PDG data Zyla et al. (2020). The
mesons, baryons and tetraquarks trajectories are the dot-dashed-blue, solid-
red, and dashed-brown lines, respectively. For the left panels, $S=0$ and for
the right panels, $S=1$. Figure 2: Regge trajectories for the supersymmetric
meson-baryon-tetraquark partners for the light-light sector with at least one
strange quark. The results are compared to the PDG data Zyla et al. (2020).
The mesons, baryons, and tetraquarks trajectories are the dot-dashed-blue,
solid-red, and dashed-brown lines, respectively. For the left panels, $S=0$
and for the right panels, $S=1$. Figure 3: Regge trajectories for the
supersymmetric meson-baryon-tetraquark partners for the heavy-light sector
with one charm quark. The results are compared to the PDG data Zyla et al.
(2020). The mesons, baryons, and tetraquarks trajectories are indicated by
dot-dashed-blue, solid-red, and dashed-brown lines, respectively. For the left
panels, $S=0$ and for the right panels, $S=1$. Figure 4: Regge trajectories
for the supersymmetric meson-baryon-tetraquark partners for the heavy-light
sector with one bottom quark. The results are compared to the PDG data Zyla et
al. (2020). The mesons, baryons, and tetraquarks trajectories are indicated by
dot-dashed-blue, solid-red, and dashed-brown lines, respectively. For the left
panels, $S=0$ and for the right panels, $S=1$. Figure 5: Regge trajectories
for the supersymmetric meson-baryon-tetraquark partners for the doubly charmed
(heavy-heavy) sector. The results are compared to the PDG data Zyla et al.
(2020). The mesons, baryons, and tetraquarks trajectories are indicated by
dot-dashed-blue, solid-red, and dashed-brown lines, respectively. For the left
panels, $S=0$ and for the right panels, $S=1$. Figure 6: Regge trajectories
for the supersymmetric meson-baryon-tetraquark partners for the doubly bottom
(heavy-heavy) sector. The results are compared to the PDG data Zyla et al.
(2020). The mesons, baryons, and tetraquarks trajectories are indicated by
dot-dashed-blue, solid-red, and dashed-brown lines, respectively. For the left
panels, $S=0$ and for the right panels, $S=1$. Figure 7: Regge trajectories
for $\pi$ and $\rho$ families using different values of $g$, e.g., $g=0.128$
GeV (solid-gray) and $g=0.200$ GeV (dashed-dark-red). Figure 8: Regge
trajectories for $\eta_{c}$ and $J/\psi$ families using different values of
$\kappa$, e.g., $\kappa=0.523$ GeV (solid-gray) and $\kappa=0.700$ GeV
(dashed-dark-red). Figure 9: Regge trajectories for the light baryons,
compared to the PDG data Zyla et al. (2020). Figure 10: Regge trajectories for
the heavy baryons involving one charm (bottom) quark, compared to the PDG data
Zyla et al. (2020).