Axial triangles in $q\bar{q}\to Zγ$ at two loops in QCD directly in four dimensions

This paper presents a numerical evaluation of the two-loop QCD squared matrix element for $q\bar{q}\to Z$ and $q\bar{q}\to Z\gamma$ processes involving heavy quark axial couplings, demonstrating that infrared, ultraviolet, and threshold singularities can be subtracted directly in four-dimensional loop momentum space to avoid the complexities of handling $\gamma^5$ in dimensional regularization.

The Gist

Imagine you are trying to predict the outcome of a very complex dance between subatomic particles. Specifically, you want to know what happens when a quark and an anti-quark (tiny building blocks of matter) smash together to create a Z boson (a heavy force carrier) and sometimes a photon (a particle of light).

Physicists usually calculate these dances using a mathematical tool called "Dimensional Regularization." Think of this tool as a translator that speaks a very strange, abstract language (involving extra, invisible dimensions) to help solve equations that would otherwise break. However, this specific dance involves a tricky character: the $\gamma_5$ (gamma-five) matrix. In the abstract language of extra dimensions, $\gamma_5$ is like a chameleon that changes its rules depending on the dimension, making the translation incredibly difficult and prone to errors.

The Big Idea: "Just Do It in 4D"
The authors of this paper, Dario Kermanschah and Matilde Vicini, decided to skip the abstract translator entirely. They wanted to do the calculation directly in our real world—four dimensions (three of space, one of time).

Think of it like this: Usually, to solve a puzzle with a piece that doesn't quite fit, you build a special jig (the extra dimensions) to hold it in place. These authors said, "Let's just force the piece to fit right here on the table." They managed to do this by carefully arranging the pieces so that the "glitches" (mathematical infinities) cancel each other out locally, right where they happen.

The "Triangle" Problem

In this particle dance, there are specific steps called axial triangles. Imagine a triangle made of three particles circulating in a loop.

• The Rule of Cancellation: In the Standard Model of physics, for every generation of particles (like the "up" and "down" quarks), there is a rule that says their contributions should cancel each other out, like two people pushing a car from opposite sides with equal force.
• The Loophole: This cancellation only works perfectly if the two particles have the exact same mass. But in reality, the Top quark is a giant (very heavy), and the Bottom quark is much lighter. Because they are different sizes, they don't push equally. The cancellation isn't perfect, leaving a tiny, but measurable, "leftover" force.

The paper focuses on calculating this tiny leftover force when heavy Top and Bottom quarks are circulating in that triangular loop.

The "Noise" Problem (Singularities)

When you try to calculate these loops, you run into three types of mathematical "noise" that make the numbers blow up to infinity:

1. UV Noise (Ultraviolet): Problems happening at incredibly tiny distances.
2. IR Noise (Infrared): Problems happening when particles move very slowly or have zero mass.
3. Threshold Noise: Problems that happen when the energy is just enough to create new particles, like a car engine stalling right at the moment it tries to shift gears.

The Solution: Local Subtraction
Instead of trying to fix the whole equation at once, the authors use a "local subtraction" method.

• Analogy: Imagine you are trying to measure the temperature of a room, but there are three different heaters blowing hot air and three air conditioners blowing cold air, creating chaotic drafts.
• Old Way: You try to model the whole room's airflow in a super-complex simulation (Dimensional Regularization).
• This Paper's Way: You stand right next to each heater and air conditioner and put a small shield (a counterterm) that blocks the draft right there. You do this for every single source of noise. Once you block the drafts locally, the room becomes calm, and you can take a simple, accurate measurement.

By doing this, they can perform the calculation using standard computer math (Monte Carlo integration) without needing the weird "extra dimensions" or the tricky $\gamma_5$ rules.

What Did They Find?

They ran a massive computer simulation (using the Euler cluster at ETH Zurich) to calculate the "squared matrix element." In plain English, this is the probability of this specific dance happening.

• They verified their method works by comparing it to known results for just the Z boson (without the photon).
• They then provided brand new results for the Z boson plus a photon ($Z\gamma$).
• They confirmed that the "leftover" force from the Top and Bottom quarks is real and calculable, and that the "noise" (infinities) cancels out perfectly when you treat the Top and Bottom quarks together.

Why Does This Matter?

1. Precision: As our particle colliders (like the LHC) get more powerful, we need to predict particle behavior with extreme precision. Tiny effects like these "axial triangles" are now becoming important enough to matter.
2. Simplicity: This paper proves that you don't always need the most complex, abstract mathematical tools (Dimensional Regularization) to solve hard physics problems. Sometimes, if you are clever enough to cancel out the errors locally, you can solve the problem directly in the real world (4 dimensions).
3. Future Proofing: This method makes it easier to include other complex effects in future calculations, potentially speeding up our understanding of the universe.

In a nutshell: The authors found a clever way to calculate a tricky particle interaction by fixing the math errors right where they happen, allowing them to work in our familiar 4D world without getting tangled in the confusing rules of extra dimensions. They successfully measured the tiny "echo" left behind when heavy Top quarks and lighter Bottom quarks dance together.

Technical Summary

Here is a detailed technical summary of the paper "Axial triangles in $\bar{q}q \to Z\gamma$ at two loops in QCD directly in four dimensions" by Dario Kermanschah and Matilde Vicini.

1. Problem Statement

The paper addresses the calculation of two-loop QCD corrections for the processes $\bar{q}q \to Z$ and $\bar{q}q \to Z\gamma$. Specifically, it focuses on contributions arising from triangular fermion loops involving heavy quarks (top and bottom).

• Physical Context: In processes involving photons only (like diphoton production), triangular fermion loops vanish due to Furry's theorem. However, when a $Z$ boson is involved, the axial current contributes.
• The Anomaly Challenge: The axial couplings within a generation are opposite ($a_u = -a_d$, $a_t = -a_b$). For equal-mass quarks, these contributions cancel pairwise. A non-zero result arises only when quark masses differ (specifically the top and bottom quarks).
• Theoretical Obstacle: Calculating axial couplings involving $\gamma_5$ in Dimensional Regularization (DR) is notoriously difficult due to the non-trivial behavior of $\gamma_5$ in $d \neq 4$ dimensions. This often leads to complications with the Adler-Bell-Jackiw (ABJ) anomaly and requires complex finite renormalization schemes.
• Goal: To compute these two-loop squared matrix elements numerically, handling infrared (IR), ultraviolet (UV), and threshold singularities directly in four spacetime dimensions ($d=4$), thereby bypassing the complexities of $\gamma_5$ in DR.

2. Methodology

The authors extend the numerical framework introduced in Ref. [1] to include these specific axial triangle contributions. The core methodology involves:

• Local Subtraction in Momentum Space: Instead of analytic integration, the calculation is performed via Monte Carlo numerical integration directly in loop momentum space.
• UV Divergence Cancellation:
  • Individual diagrams with top ($t$) or bottom ($b$) quarks exhibit local UV divergences.
  • However, because the UV limits are independent of the quark mass and the axial couplings satisfy $a_t = -a_b$, the sum of the $t$ and $b$ contributions cancels the UV divergences locally.
  • This allows the calculation to remain finite in $d=4$ without introducing additional UV counterterms.
• Anomaly Cancellation: By summing the contributions of the top and bottom quarks (which belong to the same generation), the ABJ anomaly is cancelled locally in loop momentum space. This ensures the theory remains renormalizable and avoids unphysical UV divergences.
• IR Divergence Removal: Infrared singularities are removed using local IR counterterms, consistent with methods used for massless incoming quarks (Refs. [1, 14, 15]). The sum of these counterterms integrates to zero due to Ward identities.
• Threshold Singularity Handling:
  • Threshold singularities (arising from on-shell intermediate states) are identified using Cutkosky cuts (Fig. 2).
  • The authors define singular surfaces $t(\vec{k}_1, \vec{k}_2)$ in momentum space.
  • These singularities are handled by subtracting carefully constructed local threshold counterterms. The dispersive part is evaluated by subtraction, while the absorptive part is recovered by integrating the counterterms back.
• Dimensionality: The entire calculation, including Dirac algebra and integration, is performed in $d=4$. This avoids the need for a specific $\gamma_5$ scheme (like 't Hooft-Veltman or Larin) and the associated finite renormalization constants.

3. Key Contributions

• Extension of Numerical Framework: Successfully adapted the local subtraction method (previously used for diphoton production) to include axial couplings and triangular loops in $Z$ and $Z\gamma$ production.
• Demonstration of $\gamma_5$ Bypass: Provided an explicit demonstration that axial couplings can be treated in the final state within a four-dimensional framework by realizing anomaly cancellation locally in momentum space.
• Validation of Heavy Quark Contributions: Re-computed and validated the two-loop contributions for $Z$ production (benchmarking against analytic results from Ref. [8]) and provided new results for $Z\gamma$ production.
• Absorptive Part Analysis: Proved that the absorptive part of the amplitude (proportional to $a_t v_q$) vanishes in the squared matrix element due to tensor structure constraints (insufficient independent vectors to contract with a single Levi-Civita tensor).

4. Results

The authors presented numerical results for the helicity-summed two-loop squared matrix element, denoted as $F^{(2, \text{tria})}$.

• Benchmarking ($q\bar{q} \to Z$):
  • Results were compared against analytic expressions from Ref. [8].
  • Table 1 shows excellent agreement across various center-of-mass energies ($E_{CM}$ from 100 GeV to 1000 GeV).
  • Relative precision ($\Delta$) was achieved within $0.01% - 0.1%$.
  • Numerical stability checks were implemented; unstable points were successfully rescued using double-double precision.
• New Results ($q\bar{q} \to Z\gamma$):
  • Table 2 presents results for three specific phase-space points at $E_{CM} = 1000$ GeV.
  • The numerical integration achieved a precision of approximately 0.45% - 0.47%.
  • The results confirm that the absorptive contribution is compatible with zero, as predicted theoretically.
• Mass Dependence: The results confirm that contributions vanish for equal masses and are non-zero only due to the mass splitting between the top ($m_t = 172$ GeV) and bottom ($m_b = 4.18$ GeV) quarks.

5. Significance

• Theoretical Robustness: This work validates a powerful alternative to Dimensional Regularization for processes involving axial currents. By working strictly in $d=4$, it eliminates the ambiguities and algebraic complexities associated with $\gamma_5$ in $d$ dimensions.
• Phenomenological Impact: The results provide precise two-loop QCD corrections for $Z$ and $Z\gamma$ production, which are crucial for high-precision tests of the Standard Model at colliders (e.g., LHC, future FCC).
• Methodological Advancement: The successful application of local subtraction for threshold singularities and UV cancellation in a purely numerical, four-dimensional setting demonstrates the maturity of these techniques for complex multi-loop calculations involving massive fermions.
• Future Applications: The framework paves the way for computing similar axial contributions in other processes (e.g., $Z \to q\bar{q}$, Higgs production) without relying on analytic continuation or complex regularization schemes.