Lorentz and CPT violation and the hydrogen and antihydrogen molecular ions III -- rovibrational spectrum and the non-minimal SME

This paper presents a comprehensive derivation of the rovibrational spectrum for hydrogen and antihydrogen molecular ions within the non-minimal Standard-Model Extension, demonstrating how high-precision spectroscopy of these systems can test Lorentz and CPT symmetry with sensitivities up to $O(10^{-17})$ by analyzing quantum number dependencies and temporal variations.

The Gist

Imagine the universe as a giant, perfectly smooth dance floor. For decades, physicists have believed this floor is perfectly symmetrical: it looks the same no matter which way you spin, and it doesn't matter if you are standing still or gliding across it at a constant speed. This idea is called Lorentz invariance. There's also a rule called CPT symmetry, which basically says that if you swap matter for antimatter, flip left and right, and reverse time, the laws of physics should still work exactly the same.

But what if the dance floor isn't perfectly smooth? What if there are tiny, invisible bumps or ripples that only show up when you look very, very closely? This is the question physicist Graham Shore is asking in this paper.

Here is a simple breakdown of what the paper is about, using some everyday analogies.

1. The "Test Dummies": Hydrogen and Antihydrogen Molecules

To test if the universe has these invisible bumps, the author suggests using the smallest, simplest dancers possible: Hydrogen molecular ions ($H_2^+$) and their antimatter twins, Antihydrogen molecular ions ($H_2^-$).

Think of these ions as tiny dumbbells made of two protons (or a proton and an antiproton) with an electron (or positron) zipping around them. They don't just sit still; they vibrate and spin, like a spinning top that is also bouncing on a spring. These movements create a specific "song" or spectrum of frequencies.

2. The "Standard Model Extension" (SME): The Map of the Bumps

Physicists have a theory called the Standard Model Extension (SME). Think of this as a massive map that lists every possible way the universe's "dance floor" could be bumpy.

• Some bumps might make the floor feel different if you spin (Lorentz violation).
• Some might make the floor behave differently if you swap a dancer for their mirror-image twin (CPT violation).

The paper is essentially a user manual for how to read the "song" of our hydrogen dumbbells to find these bumps.

3. The "Spherical Tensor" Language: Reading the Music Sheet

In the past, trying to calculate how these bumps affect the molecules was like trying to read a musical score written in a language you don't speak. The author uses a special mathematical tool called spherical tensors.

Imagine the molecule is a spinning globe. The "bumps" on the universe's floor hit this globe from different angles. The spherical tensor method is like a translator that takes the complex, messy angles of the universe and translates them into a clean, organized musical score. This makes it much easier to see exactly how a specific bump would change the pitch of the molecule's song.

4. The "Non-Minimal" Twist: Looking Deeper

Previous versions of this map only looked at the biggest, most obvious bumps. This paper goes deeper. It looks at the "non-minimal" SME, which means it's hunting for the tiniest, most subtle ripples that previous maps ignored.

It's like the difference between looking at a mountain range from a plane (seeing the big peaks) versus looking at it with a microscope (seeing the tiny cracks in the rocks). The author shows that by listening to the molecule's vibrations with extreme precision, we might finally hear these tiny cracks.

5. The "Sidereal" Effect: The Earth as a Moving Camera

Here is the most exciting part. The Earth is constantly spinning and orbiting the Sun.

• Sidereal Variation: As the Earth spins, our laboratory changes its orientation relative to the "bumps" in the universe. It's like holding a radio up to the sky; as you turn around, the static changes.
• Annual Variation: As the Earth orbits the Sun, we move at different speeds and angles.

The paper explains that if the universe has these bumps, the "song" of the hydrogen molecule shouldn't stay the same all day. It should wiggle slightly every 24 hours (as Earth spins) and every year (as Earth orbits). By tracking these tiny wiggles, scientists can pinpoint exactly where the "bumps" are and how strong they are.

6. Why This Matters: The Ultimate Antimatter Test

The paper also highlights a future experiment involving antimatter. If we can make a hydrogen molecule and an antihydrogen molecule and compare their songs, we can test CPT symmetry.

• If the songs are identical, the universe is perfectly symmetrical.
• If there is even a tiny difference, it would be a massive discovery, proving that matter and antimatter are fundamentally different in a way we never expected.

The Bottom Line

This paper is a sophisticated instruction manual for the next generation of physics experiments. It tells scientists:

1. How to calculate the tiny effects of a "bumpy" universe on the simplest molecules.
2. What to look for: specific changes in the molecule's vibration frequency that happen as the Earth spins and moves.
3. Why it's worth it: If we find these changes, we might rewrite the fundamental laws of physics and understand why the universe is made of matter instead of antimatter.

It's like giving a detective a new, ultra-sensitive microphone and a map of the city, telling them, "Listen closely to the wind; if you hear a specific hum, you've found the secret of the universe."

Technical Summary

1. Problem Statement

The paper addresses the theoretical framework required to test Lorentz invariance and CPT symmetry using high-precision rovibrational spectroscopy of the hydrogen molecular ion ($H_2^+$) and its antimatter counterpart, the antihydrogen molecular ion ($H_2^-$).

While previous work (Papers I and II) established the basics, this paper aims to:

• Provide a comprehensive derivation of the rovibrational spectrum within the Standard Model Extension (SME), an effective field theory incorporating Lorentz and CPT violation.
• Extend the analysis to the non-minimal SME, which includes non-renormalizable operators (higher-dimensional operators).
• Systematically derive the dependence of energy levels on quantum numbers ($v, N, J, M_J$) and background magnetic fields (both high and low).
• Analyze sidereal (daily) and annual variations in transition frequencies caused by Earth's rotation and orbital motion, which are crucial for isolating specific SME couplings.

2. Methodology

The authors employ a multi-step theoretical approach combining quantum mechanics, effective field theory, and group theory:

• Framework: The analysis is based on the Standard Model Extension (SME) Lagrangian. The couplings are represented as spherical tensors ($V_{njm}, T_{njm}$) rather than Lorentz tensors, which is better suited for the symmetries of atomic and molecular systems.
• Born-Oppenheimer Approximation: The full Schrödinger equation is factorized into an electron equation (determining the inter-nucleon potential) and a nucleon equation (determining rovibrational states).
  • The electron is assumed to be in the $1s\sigma_g$ ground state.
  • The analysis focuses on para-$H_2^+$ (total nuclear spin $I=0$, even rotational quantum number $N$).
• Reference Frames:
  • MOL: Molecular frame (aligned with the molecular axis).
  • EXP: Experimental frame (quantization axis aligned with the background magnetic field $\mathbf{B}$).
  • SUN: Solar-centered frame (standard frame for comparing SME constraints).
  • LAB: Local laboratory frame.
  • The paper derives explicit transformation rules (using Wigner matrices) to translate couplings between these frames.
• Perturbation Theory:
  • The SME contributions are treated as perturbations to the hyperfine-Zeeman energy levels.
  • The energy spectrum is expanded in powers of $(v + 1/2)$ and $N(N+1)$, with coefficients depending on SME couplings.
  • Calculations are performed for both weak magnetic fields (hyperfine eigenstates $|v N J M_J\rangle$) and strong magnetic fields (Zeeman eigenstates $|v N M_J M_S\rangle$).
• Non-Minimal Extension: The derivation includes terms up to $O(p^4)$ and $O(P^4)$ (where $p$ is electron momentum and $P$ is proton momentum), allowing sensitivity to non-minimal SME couplings.

3. Key Contributions

A. Spherical Tensor Formalism for Molecular Dynamics

The paper provides a complete first-principles derivation of the molecular dynamics using spherical tensor representations of SME couplings.

• It explicitly handles the lack of full spherical symmetry in the $H_2^+$ molecule (which only possesses cylindrical symmetry). This allows sensitivity to couplings with angular momentum indices $j \neq 0$, which would vanish in spherically symmetric atoms.
• It derives the inter-nucleon potential $V_{SME}(R)$ including both spin-independent ($V_{njm}$) and spin-dependent ($T_{njm}$) couplings.

B. Extension to Non-Minimal SME

The authors systematically extend previous results to include non-minimal operators ($n=4$ terms).

• They derive the contributions of $O(p^4)$ electron couplings and $O(P^4)$ proton couplings to the rovibrational energy levels.
• They identify that non-minimal couplings (e.g., $V_{400}, V_{420}, V_{440}$) introduce new dependencies on quantum numbers that are distinct from minimal SME couplings.

C. Quantum Number and Magnetic Field Dependence

The paper provides explicit formulae for the energy shifts $\Delta E$ as a function of:

• Vibrational quantum number ($v$).
• Rotational quantum number ($N$).
• Total angular momentum ($J$) and its projection ($M_J$).
• Magnetic field strength ($B$).
• The results are presented as a $2 \times 2$ matrix for the mixing of hyperfine states in weak fields, and diagonal in strong fields.

D. Frame Transformations and Time Variations

A significant portion of the work is dedicated to translating results from the EXP frame to the SUN frame to predict time-dependent variations:

• Sidereal Variations: Caused by Earth's rotation. The paper shows that these variations depend on azimuthal components ($m \neq 0$) of the SME couplings, which are averaged out in static measurements.
• Annual Variations: Caused by Earth's orbital velocity (Lorentz boosts).
• Boost Effects: The authors demonstrate that Lorentz boosts mix spherical tensor couplings of different ranks ($n$). For example, a boost acting on a $n=2$ coupling can induce sensitivity to $n=1$ or $n=3$ couplings. This is derived by converting spherical tensors back to Lorentz tensors for the boost calculation and then re-expressing them.

4. Key Results

• Rovibrational Energy Formula: The energy levels are expressed as:
  $$E_{vNMJ} = V_{SME} + (1 + \delta_{SME}) (v + 1/2)\omega_0 - (x_0 + x_{SME})(v + 1/2)^2\omega_0 + (B_0 + B_{SME})N(N+1)\omega_0 - \dots$$
  where the coefficients $\delta, x, B, \alpha$ are explicitly calculated in terms of spherical tensor SME couplings ($V_{njm}, T_{njm}$) and expectation values of momentum operators.
• Sensitivity to Non-Minimal Couplings: The inclusion of $O(p^4)$ and $O(P^4)$ terms reveals sensitivity to a wider range of SME coefficients (specifically those with $n=4$) that were previously inaccessible.
• Proton vs. Electron Contributions:
  • Electron SME couplings modify the inter-nucleon potential.
  • Proton SME couplings contribute directly to the kinetic energy of the nuclei and indirectly via the potential.
  • The paper provides specific numerical coefficients (e.g., $c_{NMJ}, \hat{c}_{NMJ}$) that dictate how different quantum states respond to specific SME parameters.
• Time-Varying Signals: The derived expressions for $\Delta E$ in the SUN frame show explicit dependence on $\cos(m(\omega_\oplus T_\oplus + \alpha))$ and $\sin(m(\omega_\oplus T_\oplus + \alpha))$. This proves that measuring transition frequencies at different times of day and year allows the isolation of individual SME coupling components ($m \neq 0$) and the detection of boost-induced couplings ($n \pm 1$).

5. Significance and Outlook

• Experimental Precision: The theoretical framework supports experimental goals to reach precisions of $O(10^{-17})$ in $H_2^+$ and $H_2^-$ spectroscopy. Current $H_2^+$ measurements are at $O(10^{-12})$, and future $H_2^-$ experiments (using antiprotons from CERN) aim for higher precision.
• Discrimination of Effects: The detailed quantum number dependence allows experimentalists to design combinations of transition frequencies that cancel systematic errors (Zeeman shifts, quadrupole shifts) while maximizing sensitivity to specific SME couplings.
• Detection Strategy: The paper emphasizes that relying solely on time-averaged measurements is insufficient. To fully test Lorentz and CPT violation, experiments must exploit sidereal and annual variations to access couplings with different $n$ and $j$ values and to distinguish between CPT-even and CPT-odd effects.
• Future Work: The authors note that the next step is to use these results to identify specific transition combinations that minimize systematic uncertainties while maintaining maximum sensitivity to Lorentz/CPT violation, a task reserved for forthcoming publications.

In summary, this paper provides the rigorous theoretical "dictionary" and calculation engine required to interpret high-precision molecular ion spectroscopy data as a probe for fundamental symmetry breaking in the Standard Model Extension.