Resolved photoproduction of the $B_c$ meson in electron-proton collisions

This paper presents a systematic NRQCD-based study of $B_c$ meson photoproduction at electron-proton colliders, revealing that while the direct $\gamma+g$ channel dominates, the resolved $g+g$ contribution provides a significant $\mathcal{O}(10\%)$ correction in the low-$p_T$ region and grows with collision energy.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles zoom around at nearly the speed of light. Physicists are the race officials, trying to understand the rules of the game by smashing these particles together and seeing what new, exotic creatures pop out of the collision.

This paper is about a very specific, rare, and exotic creature called the $B_c$ meson.

The Exotic Creature: The $B_c$ Meson

Think of the $B_c$ meson as a "hybrid" car in a world of standard vehicles.

• Most heavy particles are like a family of identical twins (two heavy quarks of the same type, like two bottom quarks).
• The $B_c$ meson is unique because it's a mismatched pair: it's made of one heavy "bottom" quark and one heavy "charm" quark. It's the only known particle made of two different heavy flavors.
• Because it's so heavy and complex, making one is like trying to bake a cake that requires two different rare ingredients to appear in the oven at the exact same millisecond. It's incredibly difficult to produce.

The Race Track: Electron-Proton Colliders

The authors are studying how to make these $B_c$ mesons in a specific type of racetrack called an electron-proton collider (like the old HERA, or future ones like the LHeC and FCC-ep).

In these collisions, a high-speed electron crashes into a proton. But the electron doesn't just hit the proton directly; it acts like a flashlight, shooting out a beam of light (a photon) that then hits the proton.

The Two Ways to Make the Cake (The Main Discovery)

The paper investigates two different ways this "photon beam" can create the $B_c$ meson. The authors used a complex mathematical framework (NRQCD) to calculate the odds of these two methods happening.

1. The "Direct Hit" (The Direct Channel)

• The Analogy: Imagine the photon beam is a laser pointer. It hits the proton, and the laser energy instantly transforms into the $B_c$ meson and some debris.
• The Result: This is the dominant method. It's like the main highway; almost all the traffic (about 90% or more) goes this way. The photon acts like a solid, point-like particle.

2. The "Unpacking the Suitcase" (The Resolved Channel)

• The Analogy: Here is where the paper gets interesting. At very high energies, the photon isn't just a simple point of light. It's more like a suitcase that has been packed with smaller particles (gluons and quarks) inside it.
• When the photon hits the proton, it doesn't just hit with its "skin"; it opens up, and the gluons inside the photon collide with the gluons inside the proton to make the $B_c$ meson.
• The Result: The authors found that while this is a smaller road, it's not a dead end.
  • At lower energies, this "suitcase" method is tiny (about 2% of the traffic).
  • But as the racetrack gets faster and more energetic (like the future FCC-ep collider), this method becomes much more important, accounting for up to 10-15% of all the $B_c$ mesons produced.
  • It's like a side road that gets busier the faster you drive. If you ignore it, your map of the traffic is wrong.

3. The "Ghost Road" (The Quark Channel)

• There was a third possibility where particles inside the photon (quarks) collide with particles in the proton.
• The Result: This road is practically empty. It contributes almost nothing (less than 1%). The authors basically said, "We checked this, but you can ignore it."

Why Does This Matter?

The authors calculated these numbers for several future and past colliders. Here is the takeaway in plain English:

1. The Main Story: The "Direct Hit" is still the king. It produces the most $B_c$ mesons.
2. The Plot Twist: The "Resolved" method (where the photon acts like a suitcase full of particles) is significant. If you are building a future super-collider, you cannot ignore this 10-15% contribution, especially for particles moving at lower speeds.
3. The Future: As we build bigger, faster colliders (like the FCC-ep), the "suitcase" effect gets stronger. Understanding this helps physicists:
   • Predict exactly how many $B_c$ mesons they will see.
   • Learn more about the "insides" of the photon (its partonic structure), which is a bit like discovering what's actually inside that suitcase.

The Bottom Line

This paper is a detailed traffic report for a very rare type of particle. It confirms that while the "Direct Hit" is the main way to make these particles, the "Resolved" method (where the photon reveals its internal structure) is a crucial side road that becomes increasingly busy at high speeds. Ignoring it would be like trying to navigate a city while ignoring a major highway that opens up during rush hour.

The authors also noted that the calculations have some uncertainty (like a weather forecast), but the trend is clear: The photon's internal structure matters more than we thought, especially at the highest energies.

Technical Summary

1. Problem Statement

The $B_c$ meson is a unique heavy-flavored hadron composed of a bottom quark and a charm antiquark. While its production at hadron colliders (like the LHC and Tevatron) is well-studied via strong interactions, its production in electron-proton ($ep$) collisions via photoproduction has received less attention regarding resolved photon contributions.

Previous studies of $B_c$ photoproduction have predominantly focused on the direct channel ($\gamma + g \to B_c + X$), where the photon acts as a point-like particle. However, at high energies, the photon can fluctuate into a hadronic state with a non-trivial partonic structure (quarks and gluons). The paper addresses the gap in understanding the magnitude and kinematic dependence of these resolved photon contributions (specifically $g+g$ and $q+\bar{q}$ subprocesses) across various future $ep$ collider configurations.

2. Methodology

The authors employ a systematic theoretical framework combining Non-Relativistic QCD (NRQCD) factorization with the Weizsäcker–Williams Approximation (WWA).

• Factorization Framework: The total cross-section is factorized into:
  1. Photon Flux: Calculated using WWA to describe the spectrum of quasi-real photons emitted by the electron.
  2. Parton Distribution Functions (PDFs):
     • Proton PDFs ($f_{j/P}$).
     • Photon PDFs ($f_{i/\gamma}$), utilizing the Glück–Reya–Schienbein (GRS) parameterization to describe the partonic content of the resolved photon.
  3. Hard Scattering Subprocesses: Perturbative QCD calculations for the partonic cross-sections ($\hat{\sigma}$).
• NRQCD Formalism:
  • The production is treated within the color-singlet model. The authors argue that color-octet contributions are suppressed for $B_c$ mesons (unlike charmonium/bottomonium) due to the simultaneous creation of two heavy-quark pairs ($b\bar{b}$ and $c\bar{c}$) and the small relative velocity ($v^2 \sim 0.1$) of the heavy quarks.
  • Long-Distance Matrix Elements (LDMEs) are related to the wave function at the origin ($|R(0)|^2$) derived from potential models.
• Subprocesses Considered:
  • Direct: $\gamma + g \to B_c + b + \bar{c}$
  • Resolved (Gluon): $g + g \to B_c + b + \bar{c}$
  • Resolved (Quark): $q + \bar{q} \to B_c + b + \bar{c}$ (where $q = u, d, s$)
• Computational Tools: Analytical derivations and numerical evaluations were performed using the Feynman Diagram Calculation (FDC) package.
• Collider Configurations: The study covers a wide range of center-of-mass energies ($\sqrt{S}$) representing:
  • HERA (Historical)
  • LHeC (Large Hadron-electron Collider)
  • FCC-ep (Future Circular Collider - electron-proton)
  • EIC (Electron-Ion Collider)

3. Key Contributions

• Systematic Inclusion of Resolved Channels: This is one of the first comprehensive studies to quantify the resolved photon contributions ($g+g$ and $q+\bar{q}$) specifically for $B_c$ photoproduction, moving beyond the dominant direct channel approximation.
• Kinematic Dependence Analysis: The paper provides a detailed breakdown of how the relative importance of direct vs. resolved channels changes with collision energy ($\sqrt{S}$) and transverse momentum ($p_T$).
• State-Specific Predictions: Calculations include not only ground states ($B_c, B_c^*$) but also excited states ($B_c(2^1S_0), B_c^*(2^3S_1)$), analyzing their feed-down contributions to the prompt $B_c$ yield.
• Uncertainty Quantification: The authors perform a sensitivity analysis regarding heavy-quark masses ($m_c, m_b$) and the renormalization scale ($\mu$), highlighting the theoretical uncertainties inherent in leading-order calculations.

4. Key Results

A. Cross-Section Contributions

• Dominance of Direct Channel: The direct $\gamma + g$ channel remains the leading contribution across all kinematic ranges and collider energies.
• Significance of Resolved $g+g$ Channel:
  • At HERA ($\sqrt{S} = 319$ GeV), the $g+g$ channel contributes only $\sim 1.9\%$.
  • At LHeC-2 ($\sqrt{S} = 1.98$ TeV), the contribution rises to $\sim 5.8\%$.
  • At FCC-ep-2 ($\sqrt{S} = 10.0$ TeV), the $g+g$ channel reaches $\sim 11.4\%$ of the total cross-section.
  • Low-$p_T$ Enhancement: The resolved $g+g$ channel is most significant in the low transverse-momentum region (where most events occur). At FCC-ep-2, it contributes $\sim 15.8\%$ at $p_T = 1$ GeV, dropping to $\sim 5.8\%$ at $p_T = 50$ GeV. This is attributed to the enhancement of gluon densities at small Bjorken-$x$.
• Negligible $q+\bar{q}$ Channel: The quark-initiated resolved channel remains insignificant ($< 1\%$) across all energies due to suppressed quark densities in the photon and lack of dynamical enhancement.

B. Transverse Momentum ($p_T$) Distributions

• All $B_c$ states exhibit distributions peaking at low $p_T$ and falling rapidly.
• The $g+g$ channel spectrum is softer (peaks at lower $p_T$) compared to the direct channel, while the $q+\bar{q}$ channel has a harder spectrum but remains numerically irrelevant.

C. Theoretical Uncertainties

• Mass Sensitivity: The cross-section is highly sensitive to the charm quark mass ($m_c$). A $\pm 0.1$ GeV variation leads to a 20–30% change in the cross-section, whereas the bottom quark mass ($m_b$) variation induces only a 5–10% change.
• Scale Dependence: As a Leading Order (LO) calculation, the results show a strong dependence on the renormalization scale ($\mu$). Varying $\mu$ by a factor of 2 results in cross-section variations of 30% or larger, indicating a need for higher-order QCD corrections in future work.

D. Event Rates

• At the FCC-ep, with projected luminosities of $\sim 1 \text{ ab}^{-1}$, approximately $10^8$ $B_c$ mesons could be produced, making precision measurements of these resolved effects feasible.

5. Significance

• Probing Photon Structure: The study demonstrates that $B_c$ photoproduction serves as a sensitive probe for the gluon content of the photon, particularly in the small-$x$ regime accessible at future high-energy colliders.
• Precision Physics: For future experiments at the FCC-ep and LHeC, neglecting the resolved $g+g$ contribution would lead to a systematic underestimation of the production rate by $\sim 10\%$ in the low-$p_T$ region, which is critical for precision tests of QCD factorization and parton distribution functions.
• Phenomenological Guidance: The results provide essential baseline predictions for experimental collaborations (EIC, LHeC, FCC-ep) aiming to observe $B_c$ mesons and study heavy-quark dynamics in $ep$ collisions.

In conclusion, while the direct photoproduction mechanism dominates $B_c$ production, the resolved gluon-gluon channel is a non-negligible correction that grows with collision energy and is crucial for accurate theoretical predictions in the low-$p_T$ regime.