Resummation of threshold double logarithms in quarkonium fragmentation functions

This paper develops a perturbative formalism to resum threshold double logarithms in heavy quarkonium fragmentation functions, thereby resolving unphysical negative cross sections arising from fixed-order calculations and ensuring positive-definite results without relying on nonperturbative models.

The Gist

Imagine you are trying to predict how often a specific type of heavy, exotic car (called a "quarkonium") is created when two high-speed particle beams crash into each other. Physicists use a set of mathematical rules called "fragmentation functions" to describe how a tiny, fast-moving piece of debris (a parton) slows down and turns into this heavy car.

For a long time, the math used to calculate these rules had a major glitch. When the debris was moving at speeds very close to the maximum possible limit (the "threshold"), the equations would break down. They would spit out numbers that were negative. In the real world, you can't have a "negative number of cars" or a "negative probability" of an event happening. This made the predictions unreliable, especially for high-speed collisions.

The problem was caused by "soft gluons." Think of a gluon as a tiny, invisible thread of energy that holds particles together. When a particle is about to reach its maximum speed, it tends to emit a lot of these soft threads. In the old calculations, these threads created a mathematical "singularity"—a point where the numbers went wild and infinite, similar to trying to divide by zero.

The Solution: Resummation

The authors of this paper developed a new way to handle these runaway numbers. Instead of trying to calculate the effect of these soft threads one by one (which leads to the negative numbers), they grouped them all together and calculated their combined effect all at once. They call this process "resummation."

Here is an analogy to understand what they did:
Imagine you are trying to predict the total noise level in a room where people are whispering. If you try to add up the whispers one by one, you might get confused by the overlapping sounds and make a mistake. But if you realize that all the whispers together create a specific, predictable "hum," you can calculate the total hum directly. This new method calculates the "hum" of the soft gluons directly, smoothing out the mathematical bumps that caused the negative numbers.

How They Did It

The team broke the problem down into two parts, like separating the engine of a car from its wheels:

1. The Hard Part: The actual creation of the heavy particle.
2. The Soft Part: The messy cloud of soft gluons radiating out.

They proved that all the trouble (the singularities that caused negative numbers) was hiding entirely in the "Soft Part." By isolating this soft cloud and using a special mathematical trick called "exponentiation" (which is like stacking the effects of the soft gluons on top of each other in a neat, predictable tower), they were able to tame the infinities.

The Result

After applying this new method, the fragmentation functions became "positive definite." This means they always give a positive number, which makes physical sense. The jagged, broken edges of the old math were smoothed out into a nice, continuous curve that behaves well right up to the speed limit.

Why It Matters (According to the Paper)

The paper states that this fix is crucial for understanding how heavy quarkonia (like the $J/\psi$ particle) are produced at very high speeds in particle colliders. Without this fix, the predictions for how many of these particles are made at high speeds were wrong and could even suggest impossible negative rates. With the new "resummed" formulas, physicists can now accurately describe these high-speed production rates and compare them with real-world data from experiments like those at the Large Hadron Collider.

The authors also note that this method works not just for one type of particle, but for various different states of these heavy particles, including those that are spinning or polarized. They provided the detailed mathematical "recipe" for how to do this calculation, ensuring that future predictions will be physically sensible and free of the negative-number glitch.

Technical Summary

1. Problem Statement

In the Non-Relativistic QCD (NRQCD) factorization formalism, the production of heavy quarkonia (such as $J/\psi$, $\psi(2S)$, and $\chi_{cJ}$) at large transverse momentum ($p_T$) is described by fragmentation functions (FFs). However, fixed-order perturbative calculations of these FFs suffer from threshold singularities arising from the radiation of soft gluons.

• Nature of the Singularity: As the momentum fraction $z \to 1$ (the kinematic threshold where the fragmenting parton carries almost all the momentum), the FFs develop singular distributions, specifically threshold double logarithms of the form $\alpha_s^n \left[ \frac{\ln^{2n-1}(1-z)}{1-z} \right]_+$.
• Consequences:
  • Loss of Positivity: In fixed-order calculations (e.g., Next-to-Leading Order, NLO), these singularities cause the fragmentation functions to become negative in certain kinematic regions. This leads to unphysically negative cross-sections for quarkonium production at very large $p_T$.
  • Inconsistency: Fixed-order truncations treat threshold logarithms inconsistently across different color and angular momentum channels (e.g., including NLO corrections for the color-octet $^3S_1^{[8]}$ channel while neglecting them for $P$-wave channels), leading to unreliable predictions for production rates.
  • Theoretical Control: Without resummation, the theoretical description of large-$p_T$ quarkonium production remains unstable and cannot be reliably compared with experimental data (e.g., from ATLAS or LHCb).

2. Methodology

The authors develop a rigorous formalism to resum these threshold double logarithms to all orders in perturbation theory. The methodology relies on soft factorization and the Grammer-Yennie approximation.

• Soft Factorization: The authors isolate the source of infrared (IR) singularities by applying the Grammer-Yennie approximation to the Feynman diagrams. This approximates soft-gluon attachments to the heavy quark ($Q$) and antiquark ($\bar{Q}$) lines as eikonal interactions.
• Wilson Lines: The soft dynamics are encapsulated in soft functions, defined as vacuum expectation values of products of path-ordered Wilson lines (timelike for the heavy quarks and lightlike for the fragmenting gluon) and field-strength tensors.
• Projectors: The formalism projects the final-state $Q\bar{Q}$ pair onto specific color ($^1S_0, ^3S_1, ^3P_J$) and angular momentum states using spin and color projectors. This allows the derivation of specific soft factorization formulas for channels like $^3S_1^{[8]}$, $^3P_J^{[8]}$, and $^3P_J^{[1]}$.
• Calculation of Soft Functions:
  • Leading Order (LO): The authors verify that the soft factorization reproduces the singular distributions (Dirac delta and plus distributions) of the LO fragmentation functions.
  • Next-to-Leading Order (NLO): They compute the soft functions at NLO in the strong coupling $\alpha_s$. A key finding is that the threshold double logarithms arise exclusively from planar diagrams involving the exchange of a virtual gluon between the timelike and lightlike Wilson lines.
  • UV vs. IR: The double logarithms are traced back to double ultraviolet (UV) poles in the soft functions (specifically $1/\epsilon_{UV}^2$ in dimensional regularization), which are tied to the cusp anomalous dimension.

3. Key Contributions

• Derivation of Resummation Formalism: The paper provides the first detailed derivation of the resummation formula for quarkonium fragmentation functions, previously only presented in a phenomenological context (Ref. [24]).
• Identification of the Source: It rigorously proves that threshold double logarithms in quarkonium FFs originate solely from specific planar soft-gluon exchanges, allowing for an all-orders resummation via exponentiation.
• Channel Independence of Coefficients: The analysis shows that while the leading-order behavior differs between channels, the coefficient of the threshold double logarithm is determined by the cusp anomalous dimension in the adjoint representation ($C_A = N_c$).
  • For the $^3S_1^{[8]}$ channel, the correction factor is proportional to $-\frac{C_A \alpha_s}{\pi} \ln^2 N$.
  • For $P$-wave channels ($^3P_J^{[8]}$ and $^3P_J^{[1]}$), the coefficient is scaled by a factor of $4/3$ relative to the $^3S_1^{[8]}$ case due to the different leading-order behavior.
• Polarized Fragmentation: The formalism is extended to polarized fragmentation functions, demonstrating that the resummation factor is independent of the helicity state of the produced quarkonium.

4. Results

The authors derive the all-orders resummed fragmentation function in Mellin space ($N$-space):

$$\tilde{D}^{\text{resum}}_{g \to Q\bar{Q}(N)}(N) = \exp\left[ J_N \right] \tilde{D}^{\text{LO}}_{g \to Q\bar{Q}(N)}(N)$$

Where the kernel $J_N$ captures the leading double logarithms:
$$J_N \approx -\frac{C_A \alpha_s}{\pi} \ln^2 N \quad (\text{for } ^3S_1^{[8]})$$
$$J_N \approx -\frac{4}{3} \frac{C_A \alpha_s}{\pi} \ln^2 N \quad (\text{for } P\text{-waves})$$

• Positivity Restoration: The exponential suppression $\exp[-\ln^2 N]$ ensures that the resummed fragmentation functions vanish smoothly at the threshold $z=1$ in $z$-space. This eliminates the negative regions found in fixed-order calculations, thereby restoring the positivity of the cross-sections.
• Smooth Distributions: The resummed FFs are regular functions at the kinematic endpoint, unlike the singular distributions in fixed-order perturbation theory.
• Consistency: The formula is consistent with the NRQCD matching procedure, as it resums logarithms while maintaining a fixed hard renormalization scale, avoiding the inconsistencies of evolving the scale with $z$.

5. Significance

• Theoretical Resolution: This work resolves the long-standing issue of negative cross-sections in fixed-order NRQCD calculations for heavy quarkonium production at large $p_T$.
• Phenomenological Impact: The resummed formalism is crucial for interpreting experimental data. It has been successfully applied to:
  • Explain the large-$p_T$ production rates of prompt $J/\psi$ (ATLAS data).
  • Validate the tetraquark hypothesis for $\chi_{c1}(3872)$ by correcting overestimations in fixed-order cross-sections.
  • Describe the transverse momentum distributions of quarkonia within jets.
• Universality: The derivation confirms that the mechanism for threshold resummation in quarkonium production is universal and analogous to Sudakov resummation in standard perturbative QCD, governed by the cusp anomalous dimension.
• Future Directions: The paper sets the stage for higher-order resummation (single logarithms) and extensions to next-to-leading power in the $1/N$ expansion, which are necessary for precision phenomenology.

In summary, this paper provides the rigorous theoretical foundation required to make reliable QCD predictions for heavy quarkonium production at high energies, ensuring physical consistency and enabling precise comparisons with modern collider data.