Renormalization of axial anomaly in SU(N)$\times$U(1)

The Gist

Imagine you are trying to build a perfect, high-precision model of the universe using a set of mathematical blueprints. These blueprints describe how tiny particles interact. For decades, physicists have used a specific tool called Dimensional Regularization to handle the messy math that pops up when particles move at near-light speeds. It's like a universal translator that helps them make sense of equations that would otherwise break.

However, there is one stubborn piece of the puzzle that this translator struggles with: a mathematical object called $\gamma_5$ (gamma-five).

The Problem: The "Chiral" Glitch

Think of $\gamma_5$ as a special switch that determines a particle's "handedness" (whether it spins left or right). In our everyday 4-dimensional world, this switch works perfectly. But the mathematical tool the physicists use (Dimensional Regularization) forces them to imagine the universe having a slightly different number of dimensions (like 3.99 dimensions) to do the calculations.

In this weird, slightly-different dimension, the "handedness" switch gets jammed. It stops working the way it should, causing the mathematical "laws of conservation" (called Ward identities) to break. It's like trying to drive a car with a steering wheel that occasionally spins 90 degrees on its own; the car goes off the road.

The Old Fix: The "Patch"

A physicist named Larin came up with a clever workaround. He said, "Okay, let's just admit the switch is broken in this math, but we'll add a special 'patch' (a renormalization constant) to fix the steering wheel every time we take a step."

For a long time, physicists knew how to make this patch for the most common interactions (Pure QCD, or the strong force). They had the patch up to four steps deep. But the universe is more complex. We also need to understand how the strong force mixes with the electromagnetic force (the force behind light and electricity). This is the Mixed SU(N) × U(1) sector.

The problem? The old patches didn't work for this mixed scenario. The "steering wheel" was still jamming when both forces were involved.

The New Solution: A Novel Technique

In this paper, the authors (Tanmoy Pati and Narayan Rana) propose a new way to find the right patches for this mixed scenario.

Instead of trying to fix the steering wheel by looking at the broken parts directly, they look at the footprints the car leaves behind. In physics, these footprints are called Form Factors.

Here is their creative trick:

1. The Universal Footprint: They realized that no matter what kind of car (interaction) you have, the way it leaves "dust" (mathematical infinities called Infrared divergences) on the road is universal. Everyone leaves the same kind of dust pattern.
2. The Cleanup: By calculating the total dust pattern and then mathematically "sweeping it away" (subtracting the universal part), they are left with a clean, finite result.
3. The Patch: From this clean result, they can reverse-engineer exactly what the "patch" (the renormalization constant) needs to be to fix the broken $\gamma_5$ switch.

What They Found

Using this "footprint" method, they did two major things:

• They verified their tool: They first used their method on the known "Pure QCD" problem. It worked perfectly, matching all previous results. This proved their new technique is reliable.
• They solved the unknown: They calculated the necessary patches for the Mixed SU(N) × U(1) scenario for the first time, going up to three loops (three levels of complexity).

They also discovered something interesting about a shortcut called "Abelianization." Physicists used to think they could just take the results for the strong force and slightly tweak them to get the results for the mixed force. The authors showed that at this high level of complexity (three loops), that shortcut fails. You can't just tweak the old numbers; you have to do the hard work of calculating the new ones from scratch.

The Bottom Line

The authors have provided the essential mathematical "patches" needed to fix the broken "handedness" switch when calculating how particles interact with both the strong and electromagnetic forces.

They didn't just guess these numbers; they built a new, robust method to find them. This work is a crucial step toward making the theoretical predictions for future particle colliders (like the High-Luminosity LHC) precise enough to match the incredible accuracy of the data those machines will collect. Without these patches, the theoretical predictions would be slightly off, making it impossible to tell if new physics is hiding in the data or if it was just a math error.

Technical Summary

Technical Summary: Renormalization of Axial Anomaly in SU(N)×U(1)

Problem Statement
The definition of the chirality matrix $\gamma_5$ within dimensional regularization (DR) presents a fundamental obstacle in higher-order perturbative calculations. A fully anticommuting $\gamma_5$ is incompatible with the Dirac algebra in $d \neq 4$ dimensions. While Larin's prescription resolves this by extending the four-dimensional definition of $\gamma_5$ via the Levi-Civita tensor into $d = 4 - 2\epsilon$ dimensions, it explicitly breaks anticommutativity. Consequently, standard Ward identities are violated. To restore these identities and ensure the renormalization-group invariance of axial currents, additional finite renormalization constants ($Z_5$) must be introduced alongside standard ultraviolet (UV) counterterms.

While these constants are known up to four loops for pure Quantum Chromodynamics (QCD), the sub-percent precision required for High-Luminosity LHC and future collider phenomenology necessitates the inclusion of mixed QCD-electroweak (EW) effects. Currently, the renormalization constants required by Larin's prescription have not been extended to mixed gauge sectors, specifically the $SU(N) \times U(1)$ gauge group. Furthermore, the "abelianization" procedure, which attempts to derive QED or mixed results from pure QCD expressions by transforming color factors, is known to fail at three loops for specific topologies (specifically those involving a single internal fermion loop).

Methodology
The authors propose a novel computational technique to determine the three-loop renormalization constants for mixed $SU(N) \times U(1)$ gauge groups. The core of this method relies on the universality of infrared (IR) divergences in gauge theories.

1. Form Factor Calculation: The authors compute three-loop quark form factors (matrix elements of vector, axial-vector, scalar, and pseudo-scalar currents) using automated multi-loop frameworks. Diagrams are generated via QGRAF, algebraic manipulations are performed in FORM, and integrals are reduced to Master Integrals (MIs) using Kira 3.
2. Handling Massive Bosons: The framework accounts for both massless and massive $U(1)$ gauge bosons. The evaluation of diagrams with massive $U(1)$ fields introduces significant complexity in the Integration-By-Parts (IBP) reduction step, for which the authors utilize MIs evaluated in a previous work [17].
3. IR Subtraction and Extraction:
   • For non-singlet currents, constants are derived by comparing UV-renormalized axial-vector and vector form factors.
   • For the pure-singlet case, the finite renormalization constant $Z_5^s$ is extracted by enforcing the all-order axial-anomaly equation: $\langle q | \partial_\mu J^\mu_5 | q \rangle = a_s n_f T_F \langle q | G\tilde{G} | q \rangle + a_e \langle q | F\tilde{F} | q \rangle$.
   • Since direct algebraic comparison of the anomaly equation fails beyond two loops due to IR poles, the authors employ Catani's factorization framework. They define a universal IR renormalization factor, $Z_{IR}$, which contains all IR divergences. By multiplying the UV-renormalized matrix elements by $Z_{IR}^{-1}$, they isolate finite remainders.
   • The renormalization constants are determined by demanding that the IR-subtracted contributions are finite as $\epsilon \to 0$ and that the anomaly equation holds for these finite remainders.

Key Contributions and Results

• Three-Loop Renormalization Constants: The paper presents the first complete calculation of the three-loop renormalization constants ($Z_{MS}$ and $Z_5$) for axial-vector and pseudo-scalar currents in a mixed $SU(N) \times U(1)$ gauge group. This includes both non-singlet and pure-singlet contributions.
• Pure-Singlet Contributions: The authors provide the pure-singlet contributions to the quark axial-vector form factor at three loops within the mixed gauge setup.
• Validation of Method: The technique is validated by reproducing known three-loop pure QCD results for specific color factors ($C_F^2 n_f T_F$ and $C_F n_f^2 T_F^2$), showing perfect agreement with existing literature.
• Abelianization Limits: The study explicitly confirms that the abelianization procedure fails at three loops for the color factor $C_F^2 n_f T_F$ (originating from single internal fermion loop topologies). The authors demonstrate that mixed QCD-QED results cannot be simply predicted from pure QCD expressions without explicit computation due to the distinct behavior of photons versus gluons regarding flavor charges.
• Massive Gauge Boson: The authors verify that introducing a mass to the $U(1)$ gauge boson does not alter the UV behavior originating from $\gamma_5$, confirming the mass-independent nature of these renormalization constants.

Significance
The paper establishes a crucial theoretical foundation for precision Standard Model phenomenology. By providing the necessary three-loop renormalization constants for mixed gauge sectors, the work enables theoretical predictions to match the statistical precision of future collider data (e.g., High-Luminosity LHC). The proposed technique, utilizing form factors and IR universality, offers a robust alternative to direct diagrammatic evaluation of the anomaly equation, successfully overcoming the IR divergence issues that plague higher-order calculations. The results serve as a necessary first step toward computing full three-loop mixed QCD-EW corrections, addressing a bottleneck in current theoretical precision.