Inclusive hadroproduction of $χ_{c1}(3872)$, $X_b$ and pentaquarks

This paper utilizes Born--Oppenheimer effective field theory factorization to provide genuine, fit-free predictions for the inclusive hadroproduction cross sections of the $\chi_{c1}(3872)$, its bottomonium partner, and various charmonium and bottomonium pentaquark states.

The Gist

Imagine the universe as a giant, bustling construction site. For decades, physicists have known the basic blueprints: how to build simple houses (protons and neutrons) using standard bricks (quarks). But recently, the construction workers at the Large Hadron Collider (LHC) started finding strange, weird structures that didn't fit the standard blueprints. These are the "XYZ" particles—exotic shapes like four-brick towers (tetraquarks) and five-brick structures (pentaquarks).

This paper is like a team of master architects trying to figure out how these weird structures are being built in the first place, and how often they appear when two high-speed trains (protons) crash into each other.

Here is the breakdown of their work, using some everyday analogies:

1. The Problem: The "Ghost" in the Machine

The team focuses on a specific weird particle called the $\chi_{c1}(3872)$. It's a bit of a mystery. It looks like it's made of four quarks stuck together, but it's also very close to the mass of two separate particles stuck together. Is it a tight knot of four bricks, or a loose molecule of two pairs?

The physicists also look at pentaquarks (five-quark particles) and their "cousins" made of heavier bottom quarks instead of charm quarks.

The big question is: If we smash protons together at the LHC, how many of these weird particles should we expect to see?

2. The Tool: The "Born-Oppenheimer" Blueprint

To answer this, the authors use a special mathematical tool called Born-Oppenheimer Effective Field Theory (BOEFT).

The Analogy:
Imagine you are trying to describe a dance between a heavy elephant (the heavy quark) and a swarm of tiny, fast bees (the light quarks and gluons).

• The elephant moves slowly and heavily.
• The bees zip around incredibly fast.

Instead of trying to track every single bee's movement in real-time (which is impossible), the BOEFT method says: "Let's assume the elephant moves so slowly that the bees instantly adjust to wherever the elephant is."

This allows the physicists to treat the heavy quarks as if they are moving through a "landscape" or a "potential field" created by the bees. They can then solve a simple equation (like a Schrödinger equation) to figure out the shape of the final particle, just like figuring out the shape of a cloud based on the wind patterns.

3. The Strategy: The "Factory" and the "Universal Part"

The paper uses a clever trick called factorization. Think of the production of these particles as a two-step factory process:

1. The Short-Distance Step (The Factory Floor): This is the violent crash where the heavy quarks are created. This part is fast, violent, and can be calculated precisely using standard math.
2. The Long-Distance Step (The Assembly Line): This is where the heavy quarks cool down and grab the light quarks to form the final weird shape. This part is messy and hard to calculate directly.

The Magic: The authors realized that the "Assembly Line" part (the messy part) is actually universal. It doesn't matter if you are building a charm-quark pentaquark or a bottom-quark pentaquark; the way the light quarks assemble around the heavy ones is governed by the same rules.

So, they did this:

• They used data from B-meson decays (a different type of particle decay) to measure how "efficient" the assembly line is.
• They plugged that efficiency number into their model.
• Then, they used that same number to predict how many of these particles should be created in the LHC crashes.

4. The Results: Predictions vs. Reality

The team made two main sets of predictions:

• The $\chi_{c1}(3872)$ and its Bottom Cousin ($X_b$):
  They calculated how often the $\chi_{c1}(3872)$ should be made. When they compared their prediction to the actual data from the CMS, ATLAS, and LHCb experiments, it matched!

  • The Twist: They also predicted the existence of a "bottom" version of this particle ($X_b$), which hasn't been seen yet. They gave a specific prediction for how often it should appear, essentially saying, "Look for it here, and it should show up about this many times."

• The Pentaquarks:
  There are two main theories (Scenarios I and II) about how the five-quark pentaquarks are arranged. The authors ran the math for both scenarios.

  • They found that while the internal arrangement might differ, the production rate (how often they are made) is surprisingly similar in both cases.
  • They provided a "range" of expected numbers for the LHC to look for.

5. Why This Matters

This paper is a "genuine prediction." They didn't just tweak their math to fit the data they already had. They used a fundamental theory to predict the numbers, and those numbers turned out to be correct.

The Big Takeaway:
The authors argue that even if these particles are "loosely bound molecules" (like a fragile house of cards), they are still being produced efficiently in high-energy crashes. This is because the "construction" happens at the very moment of the crash (short distance), where the heavy quarks are forced into a specific configuration that naturally leads to these exotic shapes.

In a nutshell:
The physicists built a universal "assembly manual" for exotic particles. They tested it on a known weird particle, and it worked. Now, they are handing that manual to experimentalists, saying, "Here is exactly how many of the heavier, stranger cousins you should find in your next experiment."

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of calculating the inclusive hadroproduction cross sections for exotic hadrons, specifically:

• The tetraquark state $\chi_{c1}(3872)$ and its bottomonium partner $X_b$.
• The hidden-charm pentaquark states $P_c\bar{c}(4312)^+$, $P_c\bar{c}(4457)^+$, $P_c\bar{c}(4380)^+$, and $P_c\bar{c}(4440)^+$, along with their hypothetical bottomonium counterparts.

Standard Non-Relativistic QCD (NRQCD) factorization relies on fitting Long-Distance Matrix Elements (LDMEs) to data. However, for exotic states, the nature of the wave function (molecular vs. compact tetraquark) and the relevant color configurations are not fully settled, leading to large uncertainties. The authors aim to provide genuine predictions (without fitting to prompt hadroproduction data) by leveraging the Born-Oppenheimer Effective Field Theory (BOEFT) to factorize the production mechanism into universal non-perturbative elements and calculable wave functions.

2. Methodology

The authors employ a multi-step theoretical framework combining NRQCD, potential NRQCD (pNRQCD), and BOEFT:

• Factorization Framework:

  • They utilize the NRQCD factorization formula where the cross section is a sum of short-distance coefficients (SDCs) and LDMEs.
  • For exotic states, they extend pNRQCD to BOEFT. In this framework, the heavy quark-antiquark pair ($Q\bar{Q}$) moves adiabatically in the presence of light degrees of freedom.
  • Crucially, they identify that for $\chi_{c1}(3872)$ and pentaquarks, the production at leading order in velocity ($v$) is dominated by the color-octet configuration of the heavy quark pair, as the light degrees of freedom form an adjoint meson at short distances.

• Wave Function Calculation:

  • The radial wave functions are obtained by solving coupled Schrödinger equations derived from BOEFT.
  • The potentials used in these equations are constrained by lattice QCD results and symmetry arguments (e.g., mixing between $\Sigma_g^+$, $\Sigma_g^{'+}$, and $\Pi_g$ states for $\chi_{c1}(3872)$).
  • They perform a unitary transformation to a "diabatic basis" to separate the centrifugal barrier, identifying dominant S-wave and D-wave components.

• LDME Factorization:

  • The LDMEs are factorized into the square of the wave function at the origin ($|\phi(0)|^2$) and a universal matrix element ($M$) that depends only on light degrees of freedom and is independent of the heavy quark mass.
  • Constraint Strategy: Since the universal matrix elements cannot be calculated perturbatively, the authors constrain them using:
    1. Theoretical Upper Limits: Derived from the completeness of the sum over intermediate color-octet states (related to $N_c^2-1$).
    2. Experimental Lower Limits: Extracted from $B$-hadron decay branching ratios (e.g., $B \to \chi_{c1}(3872) + X$ and $\Lambda_b \to P_c + X$).

• Scenarios for Pentaquarks:

  • The analysis is performed under two distinct scenarios regarding the Born-Oppenheimer potentials and quantum numbers ($J^P$) of the pentaquarks, based on previous literature (Scenario I from [44] and Scenario II from [45]).

3. Key Contributions

• Genuine Predictions for $\chi_{c1}(3872)$: The authors derive a factorization formula where the LDME is determined by the wave function at the origin (calculated from BOEFT) and a universal factor constrained by $B$-decay data. This allows for a prediction of the prompt hadroproduction cross section without fitting to LHC production data.
• Prediction of the Bottomonium Partner ($X_b$): By exploiting the universality of the matrix element $M_S$, they predict the production cross section for the $X_b$ state, a partner to $\chi_{c1}(3872)$ in the bottomonium sector.
• Pentaquark Production in Two Scenarios: They provide the first systematic calculation of inclusive pentaquark production cross sections for both Scenario I and Scenario II, deriving specific LDMEs for each state.
• Bottomonium Pentaquarks: They extend the predictions to the bottomonium sector ($P_{b\bar{b}}$ analogues), providing the first theoretical estimates for these yet-to-be-discovered states.
• Resolution of the "Molecule vs. Compact" Controversy: The paper demonstrates that the BOEFT framework naturally accommodates a state that is loosely bound (molecular-like) at large distances but produced via a compact color-octet mechanism at short distances, reconciling the large radius of $\chi_{c1}(3872)$ with its efficient production in $pp$ collisions.

4. Results

• $\chi_{c1}(3872)$ Production:

  • The predicted differential cross section ($d\sigma/dp_T$) at LHC energies (7, 8, and 13 TeV) agrees with experimental data from CMS, ATLAS, and LHCb within uncertainties.
  • The theoretical prediction lies slightly above the experimental central values but is consistent. The inclusion of the BOEFT upper bound significantly reduces the uncertainty bands compared to standard NRQCD fits.
  • The extracted universal matrix element is $M_S \approx 1.70^{+0.81}_{-0.78}$.

• $X_b$ Production:

  • Using the same universal $M_S$ and the bottom quark mass ($m_b \approx 4.74$ GeV), they predict the $X_b$ cross section. The wave function at the origin is significantly larger than for the charm case ($|\phi_S(0)|^2 \approx 0.158$ GeV$^3$), leading to a higher production rate relative to the mass scaling.

• Pentaquark Production:

  • Scenario I: Predicts cross sections for $P_c(4312)^+$, $P_c(4457)^+$, $P_c(4380)^+$, and $P_c(4440)^+$. The results show comparable magnitudes to $\chi_{c1}(3872)$ but with larger uncertainties due to the range of allowed matrix elements ($M_S$ and $M_{(1/2)_g}$).
  • Scenario II: Yields similar cross sections to Scenario I within the error bands, suggesting that current experimental precision cannot yet distinguish between the two quantum number assignments.
  • Bottomonium Pentaquarks: The predicted cross sections for the bottomonium analogues are provided, showing distinct $p_T$ distributions compared to the charm sector.

5. Significance

• Theoretical Advancement: This work establishes a rigorous, first-principles framework for calculating exotic hadron production, moving beyond phenomenological fits to data. It validates the BOEFT as a powerful tool for connecting the internal structure of exotic states (potentials, wave functions) to their production rates.
• Phenomenological Impact: The predictions for $X_b$ and bottomonium pentaquarks provide concrete targets for future experiments (LHCb, CMS, ATLAS) to search for these states.
• Clarification of Production Mechanisms: The paper clarifies that the production of loosely bound exotic states is governed by short-distance color-octet dynamics, resolving the apparent paradox of how large-radius molecules can be produced efficiently in high-energy collisions.
• Future Directions: The authors highlight that the primary limitation is the lack of lattice QCD determinations for pentaquark potentials. Future lattice calculations would allow for a definitive discrimination between the proposed scenarios and precise determination of wave functions without relying on the current constraints.

In summary, the paper successfully applies Born-Oppenheimer EFT to compute inclusive production cross sections for a wide range of exotic hadrons, providing genuine predictions that align with existing data and offering a roadmap for discovering new states in the bottomonium sector.