⚛️ phenomenology

Two-loop $gg\to AA$ amplitudes near threshold and the double parton scattering singularity

This paper presents a closed-form expansion of two-loop $gg\to AA$ amplitudes near threshold in powers of pseudoscalar velocity, revealing the cancellation of inverse and odd powers, identifying the leading logarithmic behavior with the double parton scattering singularity, and providing a numerically accurate series for soft-gluon resummation of the pair invariant-mass distribution.

Original authors: Arunima Bhattacharya

Published 2026-10-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Arunima Bhattacharya

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the heart of the Large Hadron Collider, protons smash together at nearly the speed of light, shattering into a spray of new particles. Among the most elusive targets are pairs of heavy, invisible particles called pseudoscalars, which are predicted by theories extending our current understanding of physics. To find them, scientists must predict exactly how often they are produced and what their energy distribution looks like. This requires calculating the complex interactions of the strong nuclear force, the glue that holds atomic nuclei together. When two gluons—the carriers of this force—collide to create a pair of these particles, the calculation involves intricate loops of virtual particles that appear and vanish in the blink of an eye. These calculations are notoriously difficult, especially when the produced particles move very slowly, a condition known as the threshold region. In this slow-motion regime, the mathematical expressions describing the collision become so unwieldy that standard computer programs struggle to evaluate them, often obscuring the very patterns scientists need to see.

A researcher has now untangled this mathematical knot for a specific type of collision where two gluons fuse to create a pair of pseudoscalars. They focused on a challenging region where the particles are produced with very little speed relative to each other. By expanding the complex equations into a series of simpler terms, they managed to strip away the noise and reveal the underlying structure of the interaction. The result is a set of compact formulas that accurately describe the collision process when the particles are moving slowly, replacing a massive, difficult-to-use file with a much smaller, faster, and more transparent set of instructions. This work not only provides a precise tool for future experiments but also uncovers a hidden connection between two seemingly different ways particles can interact.

The researcher began with a massive collection of mathematical expressions that described the collision at a very high level of precision, known as the two-loop level. These expressions were so large they occupied hundreds of megabytes of computer memory, and they contained many terms that canceled each other out almost perfectly, making it hard to see the physical behavior near the threshold. The researcher systematically expanded these expressions in terms of the velocity of the produced particles. They discovered that when they organized the terms correctly, a surprising amount of cancellation occurred. All the confusing, inverse powers of velocity and the odd powers vanished, leaving behind a clean series where every term behaved in a predictable, symmetric way. This confirmed that the mathematics respected the fundamental symmetries of nature, even in this difficult limit.

A key discovery emerged from this clean series: the behavior of the collision is dominated by a specific mathematical term related to the transverse momentum of the particles. This term, which grows as the particles move slower, was identified as a signature of a phenomenon called double parton scattering. In this scenario, a single parton from one proton splits into two, and a single parton from the other proton does the same, with the four resulting particles then interacting to form the final pair. The researcher found that the strength of this effect in their calculation matched a theoretical prediction derived from a much simpler, one-loop calculation. They tested this by looking at the collision in the forward direction, where the particles move almost parallel to the beam, and found that the velocity dependence of their complex two-loop result matched the simpler prediction with an accuracy of three parts in a quadrillion. This confirmed that the complex, multi-loop calculation was correctly capturing the physics of this double-splitting mechanism.

The study also revealed how the behavior changes depending on the spin orientation of the colliding gluons. When the gluons have the same spin, the interaction grows rapidly as the particles slow down, following a pattern that matches predictions from a highly symmetric theoretical model known as super Yang-Mills theory. However, when the gluons have opposite spins, a different pattern emerges. While this configuration showed no sign of the double-splitting effect at the simpler one-loop level, the two-loop calculation showed that it develops a new, significant logarithmic growth at this higher level of precision. This suggests that the double-splitting mechanism becomes relevant for a wider range of interactions than previously thought, even though the exact origin of this new growth remains an open question for future study.

The practical impact of this work is immediate for experimentalists analyzing data from the collider. The new, simplified formulas allow for the precise calculation of the collision rate near the production threshold, a region where the particles are just barely created. This is crucial for distinguishing between standard background noise and potential signals of new physics. The researcher provided the new formulas and the computer code to evaluate them, making it possible to include these high-precision corrections in the analysis of real data. By doing so, they have provided a vital piece of the puzzle for understanding how heavy particles are born in the most energetic collisions in the universe, ensuring that future searches for new physics are built on the most solid theoretical foundation possible.

Technical Summary: Two-loop $gg \to AA$ amplitudes near threshold and the double parton scattering singularity

Problem Statement
The paper addresses the calculation of two-loop QCD amplitudes for the production of pseudoscalar pairs ( $gg \to AA$ ) via gluon fusion, specifically focusing on the "Type II" diagram class where each pseudoscalar is emitted from its own effective gluon vertex. While closed-form expressions for these amplitudes exist (Ref. [9]), they are analytically complex, containing polylogarithms of weight up to four and inverse powers of the Gram determinant. These expressions become numerically unstable and difficult to evaluate near the pair production threshold ( $m_{AA} \approx 2m_A$ ) due to large cancellations between terms. Furthermore, the analytic structure of these amplitudes in the limit of vanishing transverse momentum ( $p_T \to 0$ ), which corresponds to a Double Parton Scattering (DPS) singularity, is obscured by the complexity of the full expressions. Understanding this singular region is crucial for precision predictions of differential cross sections and for soft-gluon resummation near threshold.

Methodology
The author performs a threshold expansion of the known two-loop finite remainders for $gg \to AA$ in terms of the pseudoscalar velocity $\beta_A = \sqrt{1 - 4m_A^2/\hat{s}}$ , keeping the scattering angle fixed. The procedure involves:

Normalization and Decomposition: The amplitudes are decomposed into helicity structures ( $J_z=0$ and $|J_z|=2$ ) and flavor coefficients ( $n_f^0, n_f^1, n_f^2$ ). The author applies a specific normalization to the published remainders to restore renormalization-scale dependence and remove an overall factor of 64 present in the original files.
Series Expansion: The amplitudes are expanded in powers of $\beta_A$ up to $\beta_A^6$ . The expansion variables are transformed from Mandelstam variables to $\beta_A$ and $\cos \theta$ .
Constant Reduction: The boundary values at $\beta_A = 0$ involve thirteen transcendental constants. Using the PSLQ algorithm at 110-digit precision, these are reduced to a basis of eight transcendental constants (including $\zeta_3, \pi^2 \ln 2, \ln^3 2$ , etc.).
Cancellation Verification: The author verifies that all inverse powers of $\beta_A$ and odd powers up to $\beta_A^5$ cancel identically, leaving a series in even powers of $\beta_A$ with logarithmic terms.
Validation: The expanded series are compared against the unexpanded remainders at high precision (up to 300 digits) across a grid of velocities ( $10^{-3} \le \beta_A \le 0.3$ ), angles, and parameter sets. An independent Python code using mpmath is used to validate the evaluation of the original expressions.

Key Contributions and Results

Analytic Structure and Cancellations: The expansion confirms that the amplitude is analytic in $\beta_A^2$ at threshold, with no inverse powers or odd powers surviving after the reduction of constants. The coefficients are even in $\cos \theta$ , satisfying Bose symmetry.
Identification of the DPS Singularity: The leading power terms of the expansion are found to depend on the scattering angle solely through the logarithm $L \approx \ln(2p_T/m_A)$ $L \approx ln (2 p_{T} / m_{A})$ . The author identifies this logarithm as the Double Parton Scattering (DPS) singularity.
- One-Loop: The coefficient of the $\ln p_T$ term for the $J_z=0$ amplitude is shown to follow the factorized formula of Gaunt and Stirling (Ref. [2]). The momentum-fraction dependence is determined by the $g \to gg$ splitting weights and the relative signs of the $g^+g^+ \to A$ and $g^-g^- \to A$ amplitudes. The prediction matches the exact one-loop coefficient in the forward limit to a relative precision of $3 \times 10^{-15}$ . A corresponding calculation for $gg \to HH$ (CP-even) confirms the weight structure for scalar pairs.
- Two-Loops: The equal-helicity ( $J_z=0$ ) amplitude grows as $\ln^3 p_T$ . The ratio of the leading two-loop logarithmic coefficient to the one-loop coefficient matches the leading-logarithmic ratio found in planar $\mathcal{N}=4$ super Yang-Mills theory. The opposite-helicity ( $|J_z|=2$ ) amplitude, which lacks a one-loop DPS logarithm, develops a $\ln^2 p_T$ term at two loops. The coefficient of this term is found to be $-4N_c$ times the one-loop forward amplitude.
Finite Virtual Corrections: The paper calculates the impact of these amplitudes on the finite virtual corrections to the cross section. Due to a $90^\circ$ phase difference between the tree amplitude and the one-loop DPS logarithm, the one-loop logarithm does not contribute to the interference with the tree level. However, at two loops, the cubic logarithm of the amplitude generates a quadratic logarithm in the interference term. The highest logarithmic term in the finite virtual correction $V_2$ (relative to the squared tree amplitude) is identified as $-\frac{7}{4}\pi^2 N_c^2 \ln^2(2p_T/m_A)$ .
Numerical Performance: The expanded series reproduce the unexpanded amplitudes with high accuracy. For $\beta_A \le 0.2$ , the relative error is at most $6.77 \times 10^{-5}$ . The series files are approximately 5000 times smaller than the original closed-form expressions, and their evaluation is significantly faster (up to ~19 times faster for repeated evaluations).

Significance
The paper provides the first explicit two-loop QCD analysis of the DPS singularity in a $2 \to 2$ process with a color-singlet final state. By isolating the singular structure in a threshold expansion, the author demonstrates that the DPS logarithm is indeed present and follows the expected factorization properties, even in the presence of complex loop corrections.

The results serve as a critical input for the hard function in soft-gluon resummation calculations for the pair invariant-mass distribution near $m_{AA} = 2m_A$ . Specifically, they supply the two-loop Type II contribution required for:

NNLO predictions for pseudoscalar production coupled to gluons only via the effective $G\tilde{G}$ operator.
N $^3$ LO predictions in Two-Higgs-Doublet Models (2HDM), where Type II diagrams interfere with the leading Type I amplitude.

The author releases the threshold series, numerical evaluation code, and validation records to facilitate these higher-order calculations. The work also highlights open questions regarding the dynamical origin of the $|J_z|=2$ double logarithm and the exponentiation of these singular terms in QCD.

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