Physical remnant of electroweak theta angles

The paper demonstrates that the Standard Model contains a physical, chiral-invariant electroweak theta angle (identified with the QED theta angle) that acts as an independent parameter potentially observable in spacetimes with non-simply connected topological features.

The Gist

Imagine the Standard Model of physics as a massive, intricate recipe book for the universe. It tells us how particles like quarks and electrons interact to build everything we see. For a long time, physicists knew this recipe had a few "secret ingredients" called theta angles.

Think of a theta angle like a hidden knob on a machine. Turning this knob changes the physics of the universe, specifically how it behaves regarding "handedness" (left vs. right) and charge (positive vs. negative).

The Known Problem: The Strong Knob

We already know about one famous knob: the QCD theta angle (related to the strong nuclear force).

• The Analogy: Imagine you have a radio that is supposed to be perfectly silent. But if you turn this specific knob even a tiny bit, the radio starts screaming with static (violating CP symmetry).
• The Reality: Experiments show the universe is incredibly quiet. The knob is turned almost all the way to zero. Why? We don't know. This mystery is called the "Strong CP Problem." Some scientists think there's a hidden "axion" particle acting as a mute button to keep the volume down.

The New Discovery: The Electroweak Knob

This paper by James Brister and colleagues asks: Are there other hidden knobs we missed?

In the Standard Model, there are two other forces mixed together: the Weak force (responsible for radioactivity) and the Electromagnetic force (light and electricity). Physicists thought the "knobs" for these forces could be twisted away and made to disappear by simply rearranging the particles (a mathematical trick called a "chiral rotation"). They thought these knobs were fake—like a dial on a toy that doesn't actually control anything.

The authors' breakthrough: They realized that while you can twist the knobs for the Weak force and the Electromagnetic force individually, there is a special combination of them that cannot be twisted away.

The "Ghost" in the Machine

Here is the core of their discovery, explained with an analogy:

Imagine you have two dials on a control panel:

1. Dial A (Weak Force)
2. Dial B (Electromagnetic Force)

If you try to turn Dial A to zero, Dial B shifts. If you try to turn Dial B to zero, Dial A shifts. You can't get rid of both. However, if you look at a specific mixture of the two (let's call it the "Electroweak Smoothie"), you find that no matter how you twist the particles around, this Smoothie stays exactly the same.

The paper proves that this "Smoothie" is a real, physical thing. It is a new, independent parameter of the universe. It's a leftover "ghost" of the original settings that survived the Big Bang.

Where Can We See It?

Here is the catch: This new knob is very shy.

• The Strong Knob (QCD): We can see its effects right here in our lab (like in the shape of a neutron).
• The New Knob (QED/Electroweak): Its effects are incredibly tiny and only show up in very specific, weird environments.

The Analogy of the Donut:
Imagine the universe is a flat sheet of paper. On a flat sheet, this new knob does nothing. But, imagine the universe is shaped like a donut (a shape with a hole in the middle, or "non-simply connected").

• If you wrap a string around the hole of the donut, the string "remembers" the twist.
• The authors suggest that if our universe (or a part of it) has a shape like a donut, or if we create a "donut-like" magnetic field in a lab, this hidden knob will start to turn.

Why Does This Matter?

1. It's a New Fundamental Constant: Just like the speed of light or the mass of an electron, this "Electroweak Theta Angle" might be a fundamental setting of our universe that we haven't measured yet.
2. Lab Experiments: The paper suggests we might be able to detect this in a laboratory using clever setups with magnets and conductors (like a giant, high-tech version of the Aharonov-Bohm experiment).
3. Cosmology: It might explain weird things happening in the far reaches of the universe, beyond what we can currently see.

The Bottom Line

The authors found a hidden, unremovable setting in the universe's code. While we can't see it in everyday life, it's there, waiting to be discovered if we look at the universe through the right "lens" (either by looking at the shape of the cosmos or building a very specific magnetic experiment in a lab). It's a reminder that even in a theory as complete as the Standard Model, there are still secret dials waiting to be turned.

Technical Summary

1. Problem Statement

The Standard Model (SM) allows for CP-violating topological $\theta$-terms in the Lagrangian for the $SU(3)_C$, $SU(2)_L$, and $U(1)_Y$ gauge groups.

• QCD ($\theta_3$): The QCD $\theta$-angle is a well-known physical parameter constrained to be extremely small ($|\bar{\theta}_{QCD}| < 10^{-10}$) by neutron electric dipole moment (EDM) measurements, constituting the "Strong CP Problem."
• Electroweak ($\theta_2, \theta_1$): Traditionally, the $SU(2)_L$ and $U(1)_Y$ $\theta$-angles are considered unphysical.
  • The $SU(2)$ $\theta$-angle can be removed via chiral rotations of quarks and leptons (which do not affect masses in the absence of BSM operators).
  • The $U(1)$ $\theta$-angle is usually deemed unphysical because $U(1)$ instantons do not exist in standard Minkowski spacetime.
• The Gap: The authors investigate whether a physical, invariant combination of the electroweak $\theta$-angles survives Electroweak Symmetry Breaking (EWSB) and chiral rotations, potentially manifesting in non-trivial spacetime topologies.

2. Methodology

The authors employ a rigorous analysis of chiral transformations and their effects on the path integral measure and mass terms within the Standard Model.

• Chiral Rotations: They define general chiral rotations for fermions ($\psi_L \to e^{i\alpha}\psi_L, \psi_R \to e^{-i\alpha}\psi_R$). They analyze how these rotations shift the $\theta$-angles (via the chiral anomaly) and the phases of fermion mass matrices.
• Anomaly Calculation: Using the Fujikawa method (functional measure change) and triangle diagrams, they calculate the divergence of the chiral current, linking it to the topological charge density ($F\tilde{F}$).
• Parameter Counting: They systematically count the independent chiral rotation phases available in the quark and lepton sectors (CKM and PMNS matrices) to determine how many $\theta$-angles can be "rotated away" versus how many remain invariant.
• Symmetry Breaking: They apply Electroweak Symmetry Breaking (EWSB) to rotate the gauge fields ($W^A_\mu, B_\mu$) into the physical basis ($W^\pm_\mu, Z_\mu, A_\mu$) to express the $\theta$-terms in terms of physical fields.
• Topological Considerations: They analyze the physical observability of these terms in spacetimes with non-simply connected features (e.g., compact manifolds or specific laboratory backgrounds like interferometers with magnetic helicity).

3. Key Contributions

The paper establishes two primary theoretical breakthroughs:

1. Identification of an Invariant Combination: The authors prove that while individual $\theta_2$ and $\theta_1$ can be shifted by chiral rotations, there exists a specific linear combination of the SM $\theta$-angles and the phases of fermion mass determinants that is invariant under arbitrary chiral rotations of quarks and leptons.

   • The invariant QCD combination is $\bar{\theta}_3 = \theta_3 + \phi_u + \phi_d$ (the standard $\bar{\theta}_{QCD}$).
   • The new invariant electroweak combination is:
     $$\bar{\theta}_{21} = \frac{1}{2}\theta_2 + \theta_1 + \frac{4}{3}\phi_u + \frac{1}{3}\phi_d + \phi_e$$
     (where $\phi$ represents the phases of the determinants of the mass matrices).

2. Physical Interpretation as $\bar{\theta}_{QED}$: Upon EWSB, the authors demonstrate that this invariant combination $\bar{\theta}_{21}$ maps directly to the coefficient of the electromagnetic field strength term ($F_{\mu\nu}\tilde{F}^{\mu\nu}$) in the effective Lagrangian.

   • They identify $\bar{\theta}_{21}$ as the effective QED theta angle ($\bar{\theta}_{QED}$).
   • Unlike the $SU(2)$ term which can be removed, and the $U(1)$ term which is usually trivial, this specific combination survives as a physical parameter of the SM.

4. Results

• Lagrangian Decomposition: After EWSB, the total $\theta$-term Lagrangian decomposes into terms involving gluons ($G$), photons ($A$), $Z$ bosons, and $W$ bosons.
  $$\mathcal{L}_\theta \supset \frac{e^2}{32\pi^2} \left(\frac{1}{2}\theta_2 + \theta_1\right) \epsilon^{\mu\nu\kappa\lambda} F_{\mu\nu} F_{\kappa\lambda} + \dots$$
  The coefficient of the photon term is exactly the invariant combination $\bar{\theta}_{21}$.
• Suppression Mechanisms:
  • QCD ($\bar{\theta}_{QCD}$): Physical effects are non-perturbative ($\sim e^{-8\pi^2/g_3^2}$) but exist in Minkowski spacetime.
  • QED ($\bar{\theta}_{QED}$): Physical effects are also non-perturbative ($\sim e^{-8\pi^2/e^2}$) but are exponentially more suppressed due to the smallness of the electromagnetic coupling $e$. Furthermore, these effects only manifest in spacetimes with non-simply connected features (e.g., compact dimensions or specific topological backgrounds).
• Massive vs. Massless Fields: Terms involving massive $Z$ and $W$ bosons are further suppressed by powers of their masses (Higgs VEV) compared to the massless photon term. Thus, the photon term dominates the physical observability of $\bar{\theta}_{QED}$.
• Independence from Neutrino Mass: The existence and invariance of $\bar{\theta}_{21}$ hold regardless of whether neutrinos have Dirac or Majorana masses.

5. Significance

• New SM Parameter: The paper argues that $\bar{\theta}_{QED}$ should be viewed as an independent parameter of the Standard Model, distinct from the QCD $\theta$-angle.
• Observability: While the QCD $\theta$-angle is constrained by EDMs, the QED $\theta$-angle is currently unconstrained but potentially observable in:
  • Cosmology: If the universe has non-simply connected topological features beyond the visible horizon.
  • Laboratory Experiments: In effective backgrounds with non-trivial topology, such as interferometer setups with non-zero magnetic helicity (analogous to the Aharonov-Bohm effect) or parallel conductor plates with uniform magnetic fields.
• Theoretical Consistency: The coincidence of the coefficients in the chiral rotation invariance condition and the field strength trace indices suggests a deep constraint on the gauge group representations of SM fermions, potentially offering a new avenue for testing the consistency of the SM gauge structure.

In conclusion, the authors demonstrate that the electroweak sector of the Standard Model contains a physical, CP-violating theta angle associated with electromagnetism, which survives chiral rotations and symmetry breaking, offering a new window for testing the topological structure of the SM.