Planar master integrals for two-loop NLO electroweak light-fermion contributions to $g g \rightarrow Z H$

This paper presents an analytic computation of the master integrals for the planar light-fermion contributions to the two-loop NLO electroweak corrections of $gg \rightarrow ZH$, utilizing a canonical differential-equation framework to express most results in terms of Goncharov polylogarithms while handling remaining nested square roots via one-fold integrals.

The Gist

Imagine the universe as a giant, complex machine where tiny particles constantly collide and transform. One of the most important jobs at the Large Hadron Collider (LHC) is smashing particles together to create a specific, rare combination: a Z boson (a heavy carrier of the weak force) and a Higgs boson (the particle that gives others mass).

While most of these collisions happen in a straightforward way, there is a sneaky, complicated side channel where two invisible "gluons" (particles that hold atomic nuclei together) smash together to create this Z-Higgs pair. This process is like a secret backdoor entry into the machine. Even though it happens less often than the main door, it's significant enough that if we ignore it, our map of how the universe works will be slightly off.

This paper is about calculating the "blueprints" for that secret backdoor entry with extreme precision. Here is the breakdown of what the authors did, using simple analogies:

1. The Problem: A Maze of Infinite Possibilities

When physicists try to calculate what happens when particles collide, they have to account for every possible way the particles can wiggle, loop, and interact during the split second of the crash. These interactions are drawn as Feynman diagrams (think of them as flowcharts of particle traffic).

For this specific collision ($gg \to ZH$), there are 132 different flowcharts (diagrams) involving light particles (like electrons and light quarks) looping around. Trying to solve the math for all 132 at once is like trying to drink from a firehose; it's too messy.

2. The Solution: Finding the "Master Keys"

The authors realized that all 132 flowcharts are actually built from a smaller set of fundamental building blocks. They used a mathematical tool called Integration-by-Parts (IBP) to break the massive problem down.

Think of it like a complex Lego castle. You don't need to calculate the shape of every single brick individually. Instead, you identify the Master Integrals (MIs)—the unique, essential brick shapes that, when combined in different ways, can build the entire castle.

• They found that for the "planar" (flat, non-tangled) diagrams, there are 62 unique master keys for one type of interaction and 59 for another.
• Once you know the value of these master keys, you can instantly figure out the value of the entire castle.

3. The Method: The "Canonical" Map

To solve for these master keys, the authors used a technique called the Canonical Differential-Equations Method.

• The Analogy: Imagine you are lost in a foggy forest (the math problem). You know the trees (the variables) are changing, but you don't know the path. Instead of guessing, they built a perfect GPS map (the canonical basis) that tells you exactly how the path changes as you move.
• They used a mathematical trick called the Magnus expansion to straighten out the map. This turned a messy, tangled set of equations into a clean, orderly list where every step is predictable.

4. The Obstacle: The "Nested Square Roots"

As they tried to write down the final answers, they hit a wall. The math involved square roots (like $\sqrt{2}$ or $\sqrt{x}$).

• In simple cases, you can get rid of these square roots easily, turning the answer into a neat list of standard functions (called Goncharov Polylogarithms or GPLs). Think of these as standard "words" in the language of physics.
• However, in this specific problem, some square roots were nested inside other square roots (like a Russian nesting doll). It was like trying to untangle a knot where the string is wrapped around itself in a way that makes it impossible to pull straight all at once.
• The Result: For most of the master keys, they found a clean "word" solution. But for a few of the most complicated ones (the ones with the nested knots), they couldn't untangle it completely. Instead, they had to leave them as one-fold integrals.
  • Analogy: Instead of giving you a finished sentence, they gave you a sentence with a "fill-in-the-blank" spot that requires a tiny, specific calculation to complete. It's not a full, clean word, but it's a precise instruction on how to finish the sentence.

5. The Verification: The "Double Check"

To make sure they didn't make a mistake in their complex algebra, they compared their hand-written "blueprints" against a supercomputer simulation called AMFlow.

• They picked a specific test point in the "Euclidean region" (a safe, theoretical zone where the math is stable) and ran the numbers.
• The Outcome: Their analytic formulas matched the computer's numerical results perfectly, down to 30 decimal places. This is the mathematical equivalent of two people measuring a table and agreeing on the length down to the width of an atom.

Summary

This paper doesn't tell us how to build a new particle accelerator or cure a disease. Instead, it provides the essential, high-precision mathematical ingredients needed to understand a specific, rare particle collision at the LHC.

By solving the "master integrals" for the light-fermion contributions, the authors have cleared the fog from a specific part of the Standard Model. They have provided the exact formulas physicists need to predict what happens when gluons create a Z boson and a Higgs boson, ensuring that future experiments can spot any tiny deviations that might hint at new physics beyond what we currently know.

Technical Summary

1. Problem Statement

The paper addresses the calculation of Next-to-Leading Order (NLO) Electroweak (EW) corrections to the associated production of a Higgs boson ($H$) and a $Z$ boson via gluon-gluon fusion ($gg \to ZH$) at the Large Hadron Collider (LHC).

• Context: While QCD corrections to this process are well-studied, NLO EW corrections remain largely uncomputed. These corrections are crucial for future high-precision predictions, as they contribute approximately 4% to the cross-section (based on analogous processes) and become significant in the boosted regime.
• Specific Challenge: The calculation involves two-loop virtual diagrams induced by light-fermion loops. These diagrams introduce a new mass scale ($m_W$) alongside the kinematic invariants ($s, t, u$) and other masses ($m_Z, m_H$), leading to complex Feynman integrals.
• Scope: The authors focus specifically on the planar topologies among the eight distinct topologies arising from light-quark loops (four involving the $WWH$ vertex and four involving the $ZZH$ vertex). The non-planar topologies are excluded from this specific analytic computation.

2. Methodology

The authors employ the Canonical Differential-Equations (DE) Method to solve the master integrals (MIs) analytically. The workflow involves:

• Integral Reduction: Using the Kira package (implementing the Laporta algorithm), the authors reduce the 132 irreducible two-loop Feynman diagrams to a finite set of Master Integrals.
  • Branch B1 (WWH vertex): 62 MIs.
  • Branch B2 (WWH vertex): 59 MIs.
• Canonical Basis Construction: Instead of solving the DEs in a generic basis, they construct a canonical basis $\vec{g}$ where the differential equations take the $\epsilon$-factorized form:
  $$d\vec{g} = \epsilon \, dA(\vec{x}) \, \vec{g}$$
  This is achieved using the Magnus expansion method, starting from a linear basis of integrals.
• Alphabet and Symbol Letters: The matrix $dA$ is expressed in terms of $d\log$ forms, $dA = \sum M_a d\log(\omega_a)$. The set of functions $\omega_a$ (symbol letters) constitutes the "alphabet" of the solution.
  • The alphabets involve non-trivial radicals (square roots) arising from the kinematics and mass scales.
  • B1: Involves two square roots, $r_1$ and $r_2$.
  • B2: Involves five square roots, including nested square roots ($r_4, r_5$), making simultaneous rationalization impossible.
• Radical Structure Analysis: To handle the radicals, the authors introduce a radical index (a binary vector) to classify MIs based on their dependence on specific square roots. They construct "tropical trees" to determine the minimal subsystems required for each MI.
• Rationalization and Solution:
  • For subsystems with simple radical structures, variable transformations are applied to rationalize the square roots.
  • Solutions are expressed in terms of Goncharov Polylogarithms (GPLs) up to order $\mathcal{O}(\epsilon^4)$.
  • For the most complex subsystems (specifically in B2 at $\mathcal{O}(\epsilon^3)$ and $\mathcal{O}(\epsilon^4)$), where nested radicals prevent full rationalization, the results are represented as one-fold integrals over GPLs.
• Boundary Conditions: The integration constants are fixed using:
  • Regularity conditions at singular points (Landau singularities and spurious singularities).
  • Vanishing conditions at specific kinematic limits.
  • Permutation symmetries between topologies.
  • Analytic limits (e.g., $m_W \to m_Z$) and numerical matching via the PSLQ algorithm.

3. Key Contributions

1. Analytic Computation: The first analytic evaluation of the planar master integrals for the two-loop NLO EW corrections to $gg \to ZH$ involving light fermions.
2. Handling Nested Radicals: A systematic framework for dealing with integrals containing nested square roots. The authors demonstrate that while full rationalization is impossible for the B2 branch at higher orders, the integrals can still be systematically reduced to one-fold integrals over GPLs.
3. Canonical Bases: Explicit construction of canonical bases for two complex integral families (B1 and B2) containing 62 and 59 MIs respectively.
4. Extension to ZZH Vertex: Derivation of the canonical MIs for the $ZZH$ vertex (Branches C1 and C2) by taking the limit $m_W \to m_Z$ ($z \to -1$) of the $WWH$ results, handling the resulting divergences analytically.
5. High-Precision Results: Providing analytic expressions up to $\mathcal{O}(\epsilon^4)$, which is necessary for the complete NLO EW calculation (since the tree-level amplitude is $\mathcal{O}(\epsilon^0)$, the loop expansion requires higher orders in $\epsilon$).

4. Results

• B1 Branch (WWH): The 62 MIs are classified into four subsets based on radical structures ($00, 10, 01, 11$). All are expressible in terms of GPLs after appropriate rationalization transformations.
• B2 Branch (WWH): The 59 MIs are classified into three subsets.
  • Subsets with simple radical structures are expressed as GPLs.
  • The subset with the most complex structure ($S_2(11111)$) contains MIs that, at $\mathcal{O}(\epsilon^3)$ and $\mathcal{O}(\epsilon^4)$, require representation as one-fold integrals over GPLs due to the presence of nested radicals ($r_4, r_5$) that cannot be simultaneously rationalized.
• C1/C2 Branches (ZZH): The authors successfully derive the analytic expressions for the $ZZH$ vertex by taking the mass limit, verifying the consistency of the singular parts and the regularity of the remaining integrals.
• Numerical Validation: The analytic results were cross-checked against independent numerical calculations using the AMFlow package (Auxiliary Mass Flow method). The agreement was verified to 30 significant digits at a benchmark Euclidean point, confirming the correctness of the analytic expressions.

5. Significance

• Precision Phenomenology: These results provide the essential building blocks (Master Integrals) required to compute the complete NLO EW corrections for $gg \to ZH$. This is vital for reducing theoretical uncertainties in Higgs physics at the LHC and future colliders.
• Methodological Advancement: The paper demonstrates a robust strategy for tackling multi-loop integrals with multiple mass scales and nested radicals. The "tropical tree" approach for classifying radical dependencies and the use of one-fold integrals for non-rationalizable cases offer a template for similar complex calculations in the Standard Model and Beyond.
• New Physics Probes: By improving the theoretical precision of the $gg \to ZH$ cross-section, these calculations enhance the ability to probe deviations from the Standard Model, particularly in the Higgs-gauge boson couplings and potential new physics contributions in the loop.

In summary, this work represents a significant step forward in the theoretical description of Higgs production, bridging the gap between QCD and Electroweak precision calculations for a key LHC process.