Odd Physics Off the Diagonal: Constraining CP-violating SMEFT with Quantum Tomography

This paper proposes a novel approach using quantum tomography to reconstruct the spin density matrix of diboson systems, enabling the simultaneous and superior constraint of both CP-even and CP-violating SMEFT operators by exploiting full Beyond-Standard-Model signatures, including pure quadratic new physics terms that are typically degenerate in traditional polarisation-blind observables.

The Gist

The Big Picture: Why Are We Here?

Imagine the Universe is a giant party. When the party started (the Big Bang), there should have been an equal number of "matter" guests and "antimatter" guests. But somehow, the antimatter guests vanished, leaving only matter behind. We exist because of this imbalance.

Physicists know the "Standard Model" (the rulebook of particle physics) has some rules about how matter and antimatter behave differently (called CP violation), but the rulebook doesn't have enough "weirdness" to explain why the universe is so lopsided. They suspect there are hidden, new rules (New Physics) that we haven't found yet.

This paper is about a new, high-tech way to hunt for those hidden rules using the Large Hadron Collider (LHC).

The Problem: The "Blindfold" Effect

To find these new rules, scientists smash protons together to create pairs of heavy particles called W and Z bosons (think of them as the "heavyweights" of the particle world).

Usually, scientists look at how these particles fly apart. They measure angles and speeds. However, there's a problem:

• The "Blindfold": If you just look at the general spray of particles (ignoring their internal spin), the "new physics" looks exactly the same as the "old physics." It's like trying to tell the difference between a left-handed and a right-handed glove just by looking at a blurry photo of a pile of gloves.
• The "Interference" Trick: Scientists found a way to lift the blindfold slightly by looking at specific angles (like the azimuthal angle $\phi$). This creates an "interference pattern" (like ripples in a pond) that reveals some new physics. But, this trick has a limit. It works well for some types of new physics but fails to distinguish between two specific "flavors" of new rules, especially when the new physics gets very strong.

The Solution: Quantum Tomography (The "3D X-Ray")

The authors propose a new method called Quantum Tomography.

The Analogy: The Spinning Top
Imagine the W and Z bosons aren't just point-like particles; they are like spinning tops.

• Traditional Method: You take a photo of the spinning top from one angle. You can see it's spinning, but you don't know exactly how it's tilted or if it's wobbling in a specific way.
• Quantum Tomography: Instead of one photo, you take a full 3D scan (like a CT scan at a hospital). You reconstruct the Spin Density Matrix (SDM). This is a mathematical map that describes the entire spin state of the system, including how the two tops are spinning in relation to each other.

The "Odd" Discovery: Real vs. Imaginary

The paper's title mentions "Odd Physics Off the Diagonal." Here is what that means in our analogy:

The Spin Density Matrix is a grid of numbers (a table).

1. The Diagonal: The numbers running from the top-left to the bottom-right. These tell you the basic probability of the particles being in certain states.
2. The Off-Diagonal: The numbers everywhere else. These tell you about the correlations and interference between different states.

The authors discovered a magical split:

• CP-Even Physics (The "Normal" Weirdness): When this new physics interferes with the Standard Model, it shows up as Real numbers in the off-diagonal part of the grid.
• CP-Odd Physics (The "Odd" Weirdness): When this specific type of new physics interferes, it shows up as Imaginary numbers in the off-diagonal part.

Why is this cool?
It's like having a secret code. If you see a "Real" number in a specific spot, you know it's one type of new physics. If you see an "Imaginary" number in that same spot, you know it's the other type. Traditional methods (looking at angles) couldn't tell them apart because they were "degenerate" (looking identical). This method separates them perfectly.

The Challenge: The Missing Neutrino

In the real world, there's a catch. When the W boson decays, it spits out a neutrino. Neutrinos are ghosts; they pass through the detector without being seen.

• The Analogy: Imagine you are trying to reconstruct a spinning top, but one of its legs (the neutrino) is invisible. You have to guess where that leg is based on where the other parts flew.
• The Result: This guessing game creates a "two-fold ambiguity" (two possible answers). It makes the data a bit fuzzier, like looking through a slightly foggy window.

The Good News: Even with this fog, the authors showed that their "3D X-Ray" (Quantum Tomography) still works better than the old "blurry photo" methods. It can still tell the difference between the "Real" and "Imaginary" signals, even with the missing neutrino.

The Conclusion: Why This Matters

The paper concludes that by using this full "3D scan" of the particle spins:

1. We can see more of the new physics than before.
2. We can distinguish between different types of "New Physics" that used to look identical.
3. We can even see the "pure" new physics effects (the quadratic terms) that traditional methods miss entirely.

In short: Instead of just guessing the shape of a shadow, this new method builds a 3D model of the object casting the shadow. It allows physicists to finally separate the "odd" CP-violating physics from the "even" kind, bringing us one step closer to solving the mystery of why the Universe is made of matter.

Technical Summary

1. Problem Statement

The observed matter-antimatter asymmetry of the Universe requires sources of Charge-Parity (CP) violation beyond those provided by the Standard Model (SM) CKM matrix. The Standard Model Effective Field Theory (SMEFT) offers a model-independent framework to search for these new sources via higher-dimensional operators.

However, a significant challenge exists in constraining CP-odd operators (specifically dimension-six operators like $O_{\tilde{W}}$) using collider data:

• Degeneracy in Polarisation-Blind Observables: In high-energy limits, the interference between SM and CP-odd amplitudes often vanishes in inclusive measurements.
• Limitations of Angular Observables: While the azimuthal decay angle ($\phi$) can "resurrect" interference effects by selecting specific polarisation states, it fails to distinguish between CP-even ($O_W$) and CP-odd ($O_{\tilde{W}}$) operators in the quadratic (pure New Physics squared) terms. In these quadratic regimes, the $\phi$ distributions for both operators become largely degenerate.
• Incomplete Information: Traditional analyses often rely on 1D angular projections or kinematic tails (like $p_T$), which discard the full spin-correlation information inherent in the diboson system.

2. Methodology

The authors propose using Quantum Tomography to reconstruct the full Spin Density Matrix (SDM) of the $WZ$ diboson system in the process $pp \to W^+Z \to e^+\nu_e \mu^+\mu^-$.

• Theoretical Framework:

  • The study focuses on the Warsaw basis of SMEFT, specifically the CP-even operator $O_W$ and the CP-odd operator $O_{\tilde{W}}$.
  • The squared matrix element is expanded into SM, interference (linear in Wilson coefficients $c_i$), and quadratic (quadratic in $c_i$) terms.
  • The authors leverage the property that CP-even interference yields real contributions to the matrix element, while CP-odd interference yields purely imaginary contributions (Eq. 5).

• Tomographic Reconstruction:

  • Single Bosons: The spin state of the $W$ (spin-1) and $Z$ (spin-1) is reconstructed using the Generalized Bloch Vector parametrisation. The SDM is expanded using Gell-Mann matrices ($\lambda_i$).
  • Mapping: Angular observables (decay angles $\theta, \phi$) are mapped to the SDM parameters (Fano coefficients) using "Wigner P" symbols derived from the Wigner-Weyl formalism.
  • Diboson System: The joint SDM ($\rho^{(9)}$) for the two-boson system is constructed, containing single-boson polarisation vectors ($a_i, b_j$) and a correlation matrix ($c_{ij}$) that encodes the entanglement and spin correlations between the $W$ and $Z$.

• Simulation and Analysis:

  • Events were generated using MadGraph5_aMC@NLO at Leading Order (LO) with full spin correlations.
  • Two event selection schemes were tested: an inclusive setup and a fiducial setup mimicking detector acceptance (including neutrino reconstruction via the on-shell $W$-mass constraint).
  • Statistical limits were derived using a $\chi^2$ test statistic comparing the reconstructed SDM (and traditional observables like $\phi$ and $p_T^Z$) against SM and SMEFT hypotheses at $\sqrt{s}=13$ TeV with an integrated luminosity of $140 \text{ fb}^{-1}$.

3. Key Contributions

• Beyond Interference Resurrection: The paper demonstrates that quantum tomography does not merely "resurrect" the suppressed interference term but exploits the full spin structure, including the pure quadratic New Physics terms where traditional angular observables fail.
• Off-Diagonal Discrimination: The authors identify that CP-even and CP-odd operators populate distinct regions of the SDM:
  • CP-even ($O_W$): Dominates the real components of the interference term.
  • CP-odd ($O_{\tilde{W}}$): Generates characteristic imaginary off-diagonal structures in the interference term.
  • Quadratic Terms: Even in the quadratic regime where $\phi$ distributions are degenerate, the off-diagonal elements of the SDM retain distinct signatures for $O_W$ and $O_{\tilde{W}}$.
• Robustness to Reconstruction: The study quantifies the impact of the two-fold ambiguity in neutrino rapidity reconstruction. While this degrades sensitivity to specific Fano coefficients (those involving $\sin 2\phi$), the reconstructed SDM remains a powerful discriminator.

4. Key Results

• Separation of Operators: The SDM provides superior simultaneous sensitivity to CP-even and CP-odd contributions compared to traditional observables.
  • 1D Limits: For the CP-even operator $O_W$, the SDM (shape + yield) provides the strongest 1D bounds, outperforming the kinematic observable $p_T^Z$. For the CP-odd operator $O_{\tilde{W}}$, $p_T^Z$ remains slightly stronger in 1D, but the SDM is competitive.
  • 2D Limits: In the $(c_W, c_{\tilde{W}})$ plane, the SDM produces less degenerate confidence regions (smaller contour areas) than the azimuthal angle $\phi$ or $p_T^Z$ alone. It effectively decouples the two operators, whereas $\phi$ and $p_T^Z$ show significant correlation/degeneracy, particularly in the quadratic regime.
• Quadratic Regime Performance: In the quadratic-only contribution, where $\phi$ distributions for $O_W$ and $O_{\tilde{W}}$ are nearly identical, the SDM successfully distinguishes them via their unique off-diagonal matrix patterns.
• Neutrino Reconstruction Impact: While the fiducial setup with neutrino reconstruction introduces ambiguity that reduces sensitivity to certain SDM components, the method remains phenomenologically robust and effective for separating CP signatures.

5. Significance

This work establishes Quantum Tomography as a critical tool for future LHC analyses in the electroweak sector.

• Maximizing Information: It moves beyond the limitations of 1D angular projections, utilizing the full Hilbert space of the diboson spin state to extract information that is otherwise lost.
• Discovery Potential: By providing a method to distinguish CP-even and CP-odd physics even in the quadratic (high-energy) regime, the SDM approach offers a powerful pathway to discover or constrain new sources of CP violation that are invisible to standard kinematic or angular analyses.
• Future Directions: The authors note that while the current study is idealized (LO, no backgrounds/systematics), the framework is ready for extension to Next-to-Leading Order (NLO) calculations and full experimental analyses including detector effects and background subtraction.

In summary, the paper argues that reconstructing the Spin Density Matrix transforms the search for CP violation from a search for specific angular modulations into a comprehensive structural analysis of the quantum state, offering superior discrimination power for New Physics.