Amplitude-level NLO predictions for exclusive $J/\psi+\gamma$ production in ultraperipheral collisions

In the subatomic world, matter is not a solid block but a dynamic interplay of forces and fleeting particles. Among the most elusive of these are heavy quarkonium states, such as the J/psi particle, which are essentially tightly bound pairs of a heavy charm quark and its antimatter twin. To understand how these particles form and interact, physicists often turn to "ultraperipheral collisions." Imagine two massive atomic nuclei, like lead or gold, speeding toward each other at nearly the speed of light. Because they are so large and charged, they generate immense clouds of electromagnetic energy. When these nuclei pass each other without actually smashing into one another, their surrounding clouds of light—photons—can collide. These photon collisions act as a clean laboratory, free from the messy debris of a direct crash, allowing scientists to observe how light transforms into heavy matter. The specific process of interest here involves two photons colliding to produce a J/psi particle and a single, high-energy photon. This reaction is a precise test of quantum chromodynamics, the theory describing how the strong force binds quarks together, and it requires calculations that account for the complex, invisible fluctuations of the quantum vacuum.

A team of researchers has revisited the theoretical predictions for this specific reaction, uncovering a significant flaw in how these calculations were previously handled. For decades, the standard method for predicting the outcome of such collisions involved adding the basic, most likely outcome to a correction term that represented the interference between the basic outcome and a more complex, one-step loop calculation. However, the researchers found that this conventional approach has a critical weakness: in certain high-energy scenarios, it predicts negative probabilities. In the physical world, a probability cannot be negative; it is a measure of how likely an event is to happen, and a negative number implies a breakdown in the mathematical model rather than a real physical phenomenon. By applying a more rigorous method that treats the entire quantum process as a single, coherent wave and then squares the total result, the team demonstrated that the predictions remain physically sensible and always non-negative, even where the conventional method fails.

The study focused on collisions occurring at the Large Hadron Collider and future heavy-ion colliders, examining photon energies ranging from 80 to 160 gigaelectronvolts. When the researchers used the traditional calculation method, they discovered that for high-energy collisions, the predicted number of J/psi particles produced dropped below zero in specific momentum ranges. This negative result appeared even when using the most standard settings for the calculation, suggesting that the conventional method was missing a crucial piece of the puzzle. The missing piece was the "complete square" of the quantum amplitude. In quantum mechanics, the likelihood of an event is determined by the square of a complex mathematical quantity called an amplitude. The old method only kept the first part of this square, effectively ignoring the contribution of the loop correction squared on its own. The new approach retains this full squared term, which includes contributions from the imaginary parts of the quantum loops that were previously discarded.

When the researchers applied this complete calculation, the negative probabilities vanished in the total rates, and the predictions remained non-negative across all kinematic regions. However, the new method did not simply make the differential spectra positive everywhere; it revealed that while the conventional spectrum becomes negative over finite high-momentum intervals, the complete square preserves non-negativity. Furthermore, this more complete picture changed the predicted numbers significantly. For proton-proton collisions at 14 teraelectronvolts and lead-lead collisions at 5.52 teraelectronvolts, the new method increased the predicted total number of events by between 11.6 percent and 18.4 percent compared to the old method. This increase is not uniform; it varies depending on the angle at which the particles emerge and their transverse momentum, which is the momentum perpendicular to the direction of the beam. The researchers also found that the new method made the predictions more stable for the total integrated rates, reducing the range of uncertainty by between 36 percent and 67 percent. However, this improvement in stability does not apply universally; for differential spectra at low transverse momentum, the uncertainty bands did not necessarily shrink, indicating that the complete square does not guarantee narrower error margins in every specific kinematic region.

The implications of this work extend beyond a simple correction of numbers. The study highlights that in exclusive processes, where no other particles are created, the full quantum mechanical structure must be preserved to avoid unphysical results. The researchers showed that while the total number of particles produced might still increase overall due to positive contributions from other angles, the detailed distribution of those particles is fundamentally altered. In the old model, the high-momentum tail of the distribution was incorrectly predicted to be negative, whereas the new model shows a smooth, non-negative curve. However, even with the new model, the differential cross section in these high-momentum regions can remain below the level predicted by the basic Born approximation, rather than rising above it. This suggests that previous experimental comparisons might have been flawed if they relied on the incomplete model. By ensuring that the mathematical description respects the fundamental rule that probabilities must be non-negative, the team has provided a clearer, more accurate map for physicists to navigate the complex landscape of heavy quark production. The work serves as a reminder that in the quantum realm, even the parts of a calculation that seem like minor corrections can hold the key to a physically consistent reality.