Testing the CKM unitarity at high energy via the $W^+W^-$ production at the LHC and future colliders

This paper proposes a novel method to test the unitarity of the CKM matrix by analyzing the high-energy behavior of $W^+W^-$ production cross sections at the LHC and future colliders, deriving current constraints from ATLAS data and forecasting improved sensitivity for upcoming experiments.

The Gist

The Big Picture: Checking the "Rulebook" of the Universe

Imagine the Standard Model of particle physics as the ultimate Rulebook for how the universe works. One of the most important rules in this book is called Unitarity.

Think of Unitarity like a perfectly balanced budget. If you have a certain amount of money (probability) to spend on different types of transactions, the total must always equal 100%. You can't create money out of thin air, and you can't make it disappear.

In the world of subatomic particles, there is a "budget" called the CKM Matrix. It describes how different types of quarks (the building blocks of matter) change into one another when they interact. The rule says: No matter how these quarks mix, the total probability must always add up to exactly 1.

Recently, scientists noticed a tiny glitch in this budget. Some measurements suggest the numbers don't quite add up to 100%. This is called the "Cabibbo Angle Anomaly." It's like finding a missing penny in your bank account that might mean there's a hidden thief (New Physics) or just a calculation error.

The New Idea: A High-Speed Crash Test

The authors of this paper propose a new way to check if this budget is really broken. Instead of counting pennies in a quiet bank (which is what previous experiments did), they want to watch the quarks crash into each other at high speeds at the Large Hadron Collider (LHC).

The Analogy: The Traffic Intersection
Imagine a busy intersection where cars (quarks) are trying to turn into different lanes (other quarks).

• The Standard Model (The Rulebook): Says that if you look at all the cars turning left, right, and straight, the total number of cars leaving the intersection must equal the number entering. The traffic flow is perfectly balanced.
• The Violation: If the "Rulebook" is broken, the traffic flow gets chaotic. At low speeds, it looks fine. But if you send a fleet of supersonic jets (high-energy collisions) through the intersection, the chaos explodes.

The paper suggests that if the CKM matrix is not perfectly balanced (unitarity is violated), the number of W boson pairs (a specific type of particle collision) produced at high energies will grow wildly and unpredictably.

How the Test Works

1. The Setup: Scientists smash protons together at the LHC. Inside these protons, quarks collide.
2. The Signal: They look for the production of two W bosons ($W^+W^-$).
3. The "Quadratic" Growth:
   • If the universe follows the rules (Unitarity holds), the number of these collisions grows slowly and predictably as energy increases.
   • If the rules are broken (Unitarity violation), the number of collisions grows quadratically.
   • Analogy: Imagine driving a car. If you follow the rules, your speed increases linearly with how hard you press the gas. If the rules are broken, pressing the gas a little bit harder makes you go exponentially faster, like a rocket launching. The paper looks for this "rocket launch" behavior in the data.

What They Found

The team used data from the ATLAS experiment at the LHC (which has been smashing particles for years) to check for this "rocket launch."

• The Result: So far, the traffic looks normal. The data fits the "Rulebook" very well.
• The Constraint: They calculated how much the budget could be off before we would have seen the "rocket launch."
  • For the first two generations of quarks (the common ones), the budget can only be off by about 2% to 4%.
  • For the heavier, rarer quarks, the allowed error is larger (up to 30%), because they are harder to see.

Looking to the Future: The 100 TeV Collider

The paper also looks ahead to a future machine called the FCC-hh, which will smash particles at 100 TeV (much more powerful than the current LHC).

• The Analogy: If the current LHC is a speed camera on a city street, the FCC-hh will be a speed camera on a rocket track.
• Because the "rocket launch" effect (the quadratic growth) gets much stronger at higher energies, this future machine will be incredibly sensitive.
• The Prediction: The FCC-hh could test the budget with a precision of 0.01%. This would be just as good as, or even better than, the most precise financial audits we currently have in the world of particle physics.

Why Does This Matter?

Currently, we check the CKM matrix by measuring individual pieces (like measuring the length of one side of a triangle). This paper proposes a new method: measuring the whole triangle at once by watching how it behaves under extreme stress (high energy).

• If they find a violation: It means the Standard Model is incomplete, and we have discovered New Physics (perhaps new particles or forces we haven't seen yet).
• If they don't: It confirms that the "Rulebook" is solid, and the current "missing penny" (the anomaly) is likely just a measurement error or a statistical fluke.

Summary in One Sentence

This paper proposes using high-speed particle crashes to check if the universe's "probability budget" is balanced; so far, the budget looks good, but a future, super-powerful collider could check the math with such extreme precision that it might finally reveal if there is a hidden thief in the system.

Technical Summary

1. Problem Statement

The Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes quark flavor mixing in the Standard Model (SM), is required to be unitary. While current experimental tests of CKM unitarity (specifically the "Cabibbo angle anomaly" involving $\delta_u$) rely on the precise measurement of individual matrix elements and their subsequent summation, this methodology is limited by the precision of each individual measurement and potential correlations.

Furthermore, existing tests are indirect. The authors propose a direct, high-energy test of CKM unitarity. The core problem addressed is whether the CKM matrix remains unitary at high energy scales, or if new physics (such as heavy vector-like quarks or fourth-generation particles) introduces a violation that manifests as an anomalous growth in scattering cross-sections.

2. Methodology

Theoretical Framework

The authors analyze the production of $W^+W^-$ pairs in proton-proton collisions ($pp \to W^+W^-$) via quark-antiquark fusion ($q\bar{q} \to W^+W^-$).

• Unitarity Violation Parameter: They introduce diagonal parameters $\delta_q$ (for $q=u,d,s,c,b$) defined as the deviation of the sum of squared CKM elements from unity:
  $$\delta_q \equiv \sum_{j} |V_{qj}|^2 - 1 \quad (\text{for up-type } q)$$
  In the SM, $\delta_q = 0$.
• Amplitude Analysis: The tree-level amplitude for $q\bar{q} \to W^+W^-$ involves $s$-channel and $t/u$-channel diagrams. In the SM, gauge invariance and CKM unitarity ensure the cancellation of terms that grow quadratically with energy ($s$).
• Anomalous Behavior: If unitarity is violated ($\delta_q \neq 0$), this cancellation is incomplete. The unpolarized squared amplitude acquires an anomalous contribution:
  $$|\mathcal{M}_{q\bar{q}}|^2(\delta_q) = |\mathcal{M}_{q\bar{q}}^{SM}|^2 + \delta_q \Delta \mathcal{M}_1 + \delta_q^2 \Delta \mathcal{M}_2$$
  Crucially, the leading anomalous term $\Delta \mathcal{M}_1$ grows linearly with $s/M_W^2$, causing the total cross-section to grow quadratically with the diboson invariant mass ($m_{WW}$) relative to the SM prediction.

Experimental Strategy

• Signal: The differential cross-section $d\sigma/dm_{WW}$ is analyzed. The signal of unitarity violation is an excess of events at high $m_{WW}$ (specifically $m_{WW} > 1$ TeV).
• Backgrounds:
  • Vector Boson Fusion (VBF): The process $pp \to W^+W^-jj$ is a potential background. The authors estimate this using the Effective Vector Boson Approximation (EVBA) and conclude it constitutes only a few percent of the signal at the LHC and up to 20% at 100 TeV, but can be suppressed via kinematic cuts (excluding forward jets).
  • Theoretical Uncertainties: The analysis incorporates NNLO QCD and NLO EW corrections via a multiplicative $K$-factor. The uncertainty is estimated at 2-3%.
• Statistical Analysis: The authors perform a $\chi^2$ test comparing experimental data (or projected yields) against the SM prediction plus a unitarity violation term. They assume one non-vanishing $\delta_q$ at a time to maintain model independence.

3. Key Contributions

1. Novel High-Energy Probe: The paper proposes using the high-energy behavior of the $W^+W^-$ cross-section as a direct probe of CKM unitarity, bypassing the need for precise individual flavor measurements.
2. Quadratic Sensitivity: The authors demonstrate that the sensitivity to $\delta_q$ scales quadratically with the invariant mass of the $W$ pair, making high-energy colliders uniquely suited for this test.
3. Angular Distribution Insight: They show that the anomalous contribution peaks in the central detector region ($\theta \approx \pi/2$), suggesting that angular cuts could further enhance sensitivity.
4. Comprehensive Forecasts: The study provides constraints based on current LHC Run 2 data, projections for the High-Luminosity LHC (HL-LHC), and forecasts for the Future Circular Collider (FCC-hh) at 100 TeV.

4. Results

The authors set 95% Confidence Level (CL) bounds on the parameters $\delta_q$:

• LHC Run 2 ($\sqrt{s}=13$ TeV, $L=140$ fb$^{-1}$):

  • $\delta_u, \delta_d$: Constrained to $\sim 2\text{--}4\%$.
  • $\delta_s, \delta_c$: Constrained to $\sim 10\text{--}20\%$.
  • $\delta_b$: Constrained to $\sim 30\%$.
  • Note: These are currently weaker than flavor physics constraints but provide an independent cross-check.

• High-Luminosity LHC (HL-LHC, $L=3000$ fb$^{-1}$):

  • Sensitivity improves by a factor of $\sim 4\text{--}5$ for $\delta_{u,d,s}$ and $\sim 3\text{--}4$ for $\delta_{c,b}$.
  • Expected bounds: $\delta_{u,d} \sim 0.3\text{--}0.6\%$.

• Future Circular Collider (FCC-hh, $\sqrt{s}=100$ TeV, $L=30$ ab$^{-1}$):

  • The quadratic growth of the cross-section allows for unprecedented precision.
  • Expected bounds:
    • $\delta_{u,d} \sim 10^{-4}$
    • $\delta_s \sim 3 \times 10^{-4}$
    • $\delta_c \sim 4 \times 10^{-4}$
    • $\delta_b \sim 7 \times 10^{-4}$
  • These projections indicate that the FCC-hh could test CKM unitarity at a level comparable to or exceeding current precision from low-energy flavor physics experiments.

5. Significance

• Independence: This method offers a model-independent test of CKM unitarity that does not rely on the cancellation of individual matrix element measurements, addressing potential systematic biases in global fits.
• New Physics Reach: It provides a direct window into high-scale new physics (e.g., heavy vector-like quarks) that might mix with SM quarks, which would otherwise be invisible at lower energies.
• Complementarity: While current flavor physics bounds are tighter for $\delta_u$ and $\delta_d$, the high-energy approach is crucial for the $b$-sector and provides a necessary consistency check. If the "Cabibbo angle anomaly" is due to new physics, this high-energy test could confirm or refute it by observing the specific energy-dependent deviation predicted.
• Future Potential: The results highlight the critical importance of 100 TeV colliders for precision electroweak physics, suggesting they can probe unitarity violations at the $10^{-4}$ level, a regime previously thought accessible only through low-energy precision experiments.