Quantum tomography of $H \to ZZ, WW$ beyond leading order

This paper demonstrates that consistent quantum tomography of $H \to ZZ$ and $H \to WW$ decays requires the subtraction of higher-order corrections to ensure physical spin density operators, while also highlighting the potential to observe parity-violating effects in $H \to WW$.

The Gist

Imagine you are a detective trying to solve a mystery at a high-energy particle collider. The mystery? What is the "personality" (spin state) of a Higgs boson just before it explodes into two other particles?

In the world of quantum physics, particles like the Higgs don't just sit there; they spin and dance. When a Higgs boson decays into two Z bosons or two W bosons (which then decay into smaller particles like electrons and muons), the way those smaller particles fly out tells us how the Higgs was spinning. This process is called Quantum Tomography. Think of it like taking a 3D CT scan of a ghost: you can't see the ghost directly, but by looking at the shadows it casts from different angles, you can reconstruct its shape.

The Problem: The "Blurry" Photo

For a long time, physicists had a simple rulebook (called Leading Order or LO) to interpret these shadows. It worked perfectly in theory. However, the real world is messy.

When you look at the actual data, there are "higher-order corrections." In our detective analogy, imagine you are trying to photograph a spinning top.

• Leading Order (LO): You take a perfect, crisp photo in a dark room. You can clearly see the spin.
• Higher-Order Corrections (NLO): Suddenly, a fan turns on, blowing dust around, and a flash of light (a photon) bounces off the top. Now, your photo is blurry. The dust and the extra light mess up the angles.

The paper by Aguilar-Saavedra and Giardino says: "If you try to use the old rulebook on this blurry photo, you get nonsense."

Specifically, if you try to calculate the "spin density" (the 3D shape of the ghost) using the messy NLO data, the math breaks. It produces a "spin" that is physically impossible (like a probability that is negative). It's like trying to reconstruct a face from a photo where the nose is in the wrong place; the result isn't a face, it's a monster.

The Failed Fixes

The authors tested two common tricks to fix the blur:

1. The "Effective Lens" Trick (for Z bosons):
   Scientists tried to tweak their camera lens (changing a parameter called "spin-analyzing power") to compensate for the blur.

   • Result: It helped a little, but the photo was still too blurry to be a valid face. The math still broke.

2. The "No Flash" Rule (Photon Veto):
   They tried to ban any events where a bright photon (flash of light) was detected, hoping to get a clean photo.

   • Result: Even without the flash, the dust (virtual corrections) was still there. The photo was still too blurry to be physically valid.

The Solution: Subtracting the Noise

So, how do we get a clear picture? The authors propose a clever subtraction method.

Imagine you have a blurry photo of a spinning top, and you also have a separate photo of just the dust and the fan noise (the "background").

• The Old Way: Try to guess the top's shape from the blurry photo. (Fails).
• The New Way: Take the blurry photo and subtract the noise photo from it.
  • Mathematically: Real Data minus Calculated Noise = Clean Signal.

By mathematically removing the "extra" effects (the higher-order corrections) from the data, they can recover the clean, physical "spin state" of the Higgs boson. This allows them to perform the quantum tomography correctly.

The Surprise: A Ghostly Twist

While solving this, they found something exciting. In the case of the Higgs decaying into W bosons ($H \to WW$), the "noise" (the extra photon) actually reveals a hidden secret: Parity Violation.

• Parity is like looking in a mirror. Usually, physics works the same in a mirror.
• Parity Violation means the mirror image looks different.
• The authors found that when a photon is involved, the Higgs decay shows a tiny "handedness" (it prefers left over right, or vice versa) that wasn't visible before. It's like finding out the spinning top has a slight wobble that only appears when the fan is on. This is a rare and exciting discovery in Higgs physics.

Is This Important Right Now?

The authors check the "resolution" of our current cameras (the LHC experiments).

• Current Data: The "blur" (NLO corrections) is smaller than the "fuzziness" of our current cameras. So, for now, we can get away with the old, simple methods. The noise isn't big enough to ruin the picture yet.
• Future Data (HL-LHC): As we get more data and our cameras get sharper, that "blur" will become the biggest problem. We will need this subtraction method to keep our physics accurate.

Summary in a Nutshell

1. The Goal: Reconstruct the quantum spin of a Higgs boson.
2. The Obstacle: Real-world physics adds "noise" (extra particles and corrections) that makes the math break if you ignore it.
3. The Discovery: Simple fixes (changing lenses or blocking flashes) don't work.
4. The Fix: You must mathematically subtract the noise from the data to get a valid result.
5. The Bonus: This process reveals a tiny, previously hidden "mirror-breaking" effect in how the Higgs decays.

This paper is essentially a manual for how to clean up the "static" on the radio so that, in the future, we can hear the Higgs boson's true voice clearly.

Technical Summary

Here is a detailed technical summary of the paper "Quantum tomography of H → ZZ, WW beyond leading order" by J. A. Aguilar-Saavedra and Pier Paolo Giardino.

1. Problem Statement

Quantum tomography in high-energy physics aims to reconstruct the spin density operators of intermediate resonances (like the Higgs boson decaying into $ZZ$ or $WW$) to study quantum correlations such as entanglement.

• The Limitation: Standard tomography relies on a one-to-one correspondence between angular distributions of decay products and spin operators, which holds strictly only at Leading Order (LO).
• The Issue: At Next-to-Leading Order (NLO), virtual corrections and real radiation (e.g., photon emission) introduce amplitudes that cannot be mapped onto well-defined spin degrees of freedom.
• Consequence: If one naively applies LO tomography formulas to NLO angular distributions, the resulting spin density operators are non-physical (i.e., not positive semi-definite, possessing negative eigenvalues).
• Previous Attempts: Previous work suggested using an "effective spin-analyzing power" ($\eta_\ell$) for $H \to ZZ$ or imposing photon vetoes to fix this. The authors investigate whether these methods are sufficient to restore physicality.

2. Methodology

The authors perform a rigorous analysis using MadGraph5_aMC@NLO to calculate differential cross-sections for $H \to ZZ \to e^+e^-\mu^+\mu^-$ and $H \to WW \to e^\pm\nu\mu^\mp\nu$ at both LO and NLO.

• Angular Parameterization: They expand the angular distribution in terms of spherical harmonics ($Y_L^M$) to extract coefficients ($a_{LM}, c_{L_1M_1L_2M_2}$).
• Density Operator Reconstruction: They map these coefficients to the spin density operator $\rho_{S_1S_2}$ using the standard tree-level correspondence relations.
• Test Scenarios: They evaluate the physicality (positive semi-definiteness) of the reconstructed density matrices under three conditions:
  1. Naive NLO: Using NLO coefficients directly with LO mapping.
  2. Effective $\eta_\ell$: Using an effective weak mixing angle to redefine the spin-analyzing power (specifically for $ZZ$).
  3. Photon Veto: Excluding events with energetic photons (exclusive selection, $E_\gamma < 10$ GeV).
• Statistical Analysis: They estimate statistical uncertainties using pseudo-experiments for Run 2+3 (350 fb$^{-1}$) and the High-Luminosity LHC (HL-LHC, 3 ab$^{-1}$) to determine if NLO effects are distinguishable from experimental noise.
• Subtraction Scheme: They propose a formal subtraction method where higher-order corrections ($\Delta_{NLO} = \sigma_{NLO} - \sigma_{LO}$) are treated as a background to be subtracted from data to recover a physical LO-like observable.

3. Key Contributions

A. Failure of Naive Remedies

The study definitively proves that neither the use of an effective spin-analyzing power (for $ZZ$) nor a photon veto is sufficient to render the naively constructed NLO density operators physical.

• For $H \to ZZ$: Even with an effective $\eta_\ell$ and a photon veto, the resulting density matrix has a significant negative eigenvalue ($\lambda_{min} \approx -0.082$).
• For $H \to WW$: The situation is similar; the naive NLO density operator remains non-physical ($\lambda_{min} \approx -0.0056$ with veto, $-0.0128$ inclusive).
• Conclusion: A direct subtraction of the NLO corrections is necessary to perform consistent quantum tomography.

B. Discovery of Parity Violation in $H \to WW$

A striking by-product of the analysis is the identification of Parity (P)-violating effects in $H \to WW$ at NLO.

• In the CP-self-conjugate final state $W^+W^-$, P and CP are equivalent. Since CP violation in Higgs decays is negligible ($\sim 10^{-5}$), P violation was expected to be zero.
• However, the presence of an extra photon (real radiation) breaks this symmetry. The authors find non-zero NLO coefficients ($a_{10}, a_{20}, c_{1M2-M}, c_{2M1-M}$) that correspond to P-violating contributions.
• While the photon veto suppresses these effects, they remain a distinct feature of the NLO calculation.

C. Statistical Relevance

The authors calculate the expected statistical precision for angular coefficients:

• Current Data (Run 2+3): The difference between LO and NLO is generally smaller than or comparable to current statistical uncertainties. NLO corrections are not yet critical for $H \to WW$ due to dominant systematic uncertainties (neutrino reconstruction).
• Future Data (HL-LHC): For $H \to ZZ$, NLO effects will become relevant as statistical precision improves. For $H \to WW$, systematic uncertainties are expected to remain the limiting factor.

D. Formal Subtraction Framework

The paper formalizes a method to recover physical density operators by defining pseudo-observables. By subtracting a gauge-invariant subset of higher-order corrections ($\Delta$) from both experimental data and theoretical predictions, one can isolate the LO angular structure required for valid quantum tomography. This approach is analogous to the definition of pseudo-observables at the Z-pole.

4. Key Results (Numerical)

• Density Matrices: The LO density matrices are positive semi-definite and sparse (following tree-level relations). The NLO matrices, when constructed naively, acquire non-zero entries in previously zero positions (breaking tree-level relations) and possess negative eigenvalues.
• Coefficients:
  • For $H \to ZZ$, the difference between LO and NLO coefficients (e.g., $c_{1010}$) is significant enough to invalidate the LO interpretation without subtraction.
  • For $H \to WW$, P-violating coefficients appear at NLO (e.g., $a_{10} \approx -0.03$) but are suppressed by photon vetoes.
• Uncertainties: Statistical uncertainties for Run 2+3 are estimated to be $\sim 0.2$ for $ZZ$ coefficients and $\sim 0.01-0.05$ for $WW$ coefficients. The NLO shifts are often within $1\sigma$for current data but will be significant for HL-LHC$ZZ$ measurements.

5. Significance

• Theoretical Rigor: The paper establishes that quantum tomography in the presence of higher-order corrections cannot be performed by simply "tuning" parameters (like $\eta_\ell$) or cutting events. A rigorous subtraction scheme is required to ensure the reconstructed quantum state is physically valid (positive semi-definite).
• Experimental Guidance: It provides a roadmap for future analyses at the HL-LHC. As statistical precision increases, neglecting NLO corrections will lead to unphysical interpretations of entanglement and spin correlations.
• New Physics Sensitivity: By clarifying the Standard Model NLO background (including the subtraction scheme), the paper ensures that any future observation of non-standard quantum correlations or parity violation is not a misinterpretation of SM higher-order effects.
• Parity Violation: The identification of P-violating effects in $H \to WW$ via photon radiation opens a new, albeit challenging, avenue for testing the chiral structure of the Higgs sector.

In summary, the authors demonstrate that while current experimental precision may not yet demand NLO corrections for $H \to WW$, the theoretical framework for quantum tomography must be updated to include subtraction of higher-order effects to remain valid for future high-precision measurements, particularly for $H \to ZZ$.