Lorentz and CPT violation and the hydrogen and antihydrogen molecular ions I -- rovibrational states

This paper analyzes the rovibrational spectrum of hydrogen and antihydrogen molecular ions within a low-energy effective theory to demonstrate that these systems offer enhanced sensitivity to Lorentz and CPT violation in the proton sector—scaling as $O(m_p/m_e)$ compared to atomic transitions—thereby providing a promising avenue for high-precision tests of fundamental symmetries.

The Gist

The Big Picture: Testing the Rules of the Universe

Imagine the universe is a giant, perfectly symmetrical dance floor. For decades, physicists have believed in two fundamental rules that keep the dance going smoothly:

1. Lorentz Invariance: The laws of physics are the same no matter which way you are facing or how fast you are moving. (If you spin around or run, the music doesn't change).
2. CPT Symmetry: If you take a particle, flip its charge (making it "anti"), reverse its time, and mirror its position, it should behave exactly like the original particle. (Matter and antimatter are perfect dance partners).

However, scientists suspect that maybe, just maybe, the dance floor isn't perfectly smooth. There might be tiny, invisible bumps or cracks in the floor that cause the music to change slightly depending on your direction or speed. These "bumps" are called Lorentz and CPT violations.

The goal of this paper is to find a way to detect these tiny bumps with extreme precision.

The Problem: The Atomic "Ruler" Isn't Long Enough

To find these tiny bumps, scientists usually look at Hydrogen (the simplest atom, one proton and one electron) and Antihydrogen (one antiproton and one positron). They measure the energy of the "jumps" electrons make between energy levels.

Think of Hydrogen as a metronome. It ticks very regularly. If the universe has a bump, the metronome might speed up or slow down slightly depending on which way the Earth is pointing.

The problem is that the "ticks" of a single atom are very fast and hard to distinguish from other noise. Also, in a single atom, the electron and the proton are tangled together. If you see a weird tick, you don't know if it's the electron acting up or the proton acting up. It's like trying to hear a whisper in a crowded room; you can't tell who is speaking.

The Solution: The Molecular "Tuning Fork"

The author, Graham Shore, proposes a new tool: The Hydrogen Molecular Ion ($H_2^+$).

Instead of one proton and one electron, this is a molecule with two protons and one electron. Imagine it as a dumbbell (the two protons) with a tiny electron buzzing around it like a bee.

This molecule has a special feature: Rovibrational States.

• Vibration: The two protons can bob up and down like a spring.
• Rotation: The whole dumbbell can spin.

The Analogy:
Think of the single Hydrogen atom as a guitar string. It vibrates at a high pitch.
Think of the Hydrogen Molecular Ion as a giant, heavy tuning fork. It vibrates much slower and more steadily. Because the two protons are heavy, they move slowly, creating "narrow linewidths." This means the "note" they play is incredibly pure and precise, like a laser beam compared to a flashlight.

Why This is a Game-Changer

The paper argues that using this molecular tuning fork gives us two massive advantages:

1. Isolating the Proton (The "Hadron" Sector)

In a single atom, the electron and proton are mixed up. In this molecule, the electron acts as the "glue" holding the two protons together, but the protons are doing their own heavy lifting (vibrating and rotating).

• The Metaphor: Imagine trying to test if a heavy truck (the proton) has a flat tire. If you put the truck on a bouncy trampoline (the electron), it's hard to tell if the bounce is from the truck or the trampoline.
• The Breakthrough: In this molecule, the "bounce" (vibration) is dominated by the heavy truck. The author shows that the sensitivity to violations in the proton sector is boosted by a factor of 1,000 ($m_p/m_e$) compared to atomic tests. It's like switching from a feather to a bowling ball; the heavy ball feels the "bumps" in the floor much more intensely.

2. Separating the Rules

The paper shows that by measuring different types of transitions (spinning the dumbbell vs. vibrating the spring), we can separate the effects on the electron from the effects on the proton.

• In atomic tests, we usually only get a "mixed bag" result.
• In this molecular test, we can say, "Okay, the electron part is fine, but the proton part is wobbling." This allows us to pinpoint exactly where the laws of physics might be breaking.

The "Antimatter" Twist

The ultimate test is to compare Hydrogen ($H_2^+$) with Antihydrogen ($\bar{H}_2^-$).

• If CPT symmetry holds, the "tuning fork" made of matter and the one made of antimatter should play the exact same note.
• If they play slightly different notes, it means the universe treats matter and antimatter differently. This would be a revolutionary discovery, potentially explaining why the universe is made of matter instead of being empty (since matter and antimatter should have annihilated each other at the Big Bang).

The "Sidereal" Signal

The paper also mentions looking for seasonal changes.

• The Analogy: Imagine the "bumps" in the universe are fixed in space, like stars in the sky. As the Earth spins on its axis and orbits the Sun, our laboratory changes direction relative to these bumps.
• If the "note" of the molecular ion changes slightly over the course of a day (sidereal variation) or a year (annual variation), it proves that the laws of physics depend on our direction of motion. This is a smoking gun for Lorentz violation.

Summary

This paper is a blueprint for a new, ultra-precise experiment.

1. The Tool: Use the heavy, slow vibration of a hydrogen molecule ($H_2^+$) instead of the fast, messy jumps of a single atom.
2. The Benefit: It acts like a magnifying glass that makes the "proton" part of the universe 1,000 times easier to test.
3. The Goal: To compare this with antimatter to see if the universe has a hidden bias against antimatter, or if the "dance floor" of the universe has tiny cracks we haven't seen before.

If successful, this could rewrite our understanding of the fundamental symmetries of nature, potentially solving one of the biggest mysteries in physics: Why are we here?

Technical Summary

1. Problem Statement

The fundamental principles of locality, Lorentz invariance, and CPT (Charge, Parity, Time) symmetry are cornerstones of the Standard Model. While high-precision experiments on atomic hydrogen ($H$) and antihydrogen ($\bar{H}$) have placed stringent bounds on violations of these symmetries, there remains a need for independent, high-precision tests that can isolate specific sectors of the Standard Model Extension (SME).

Specifically, current atomic tests (e.g., the 1S-2S transition) primarily constrain SME couplings involving electrons and protons simultaneously, often with the proton contribution suppressed by the electron mass ($m_e$). Furthermore, atomic transitions sensitive to certain CPT-odd couplings (specifically those with non-zero orbital angular momentum, like 1S-2P) suffer from broader natural linewidths, limiting their precision compared to the 1S-2S transition.

The paper addresses the potential of using the molecular hydrogen ion ($H_2^+$) and its antimatter counterpart ($\bar{H}_2^-$) to:

1. Test Lorentz and CPT invariance with unprecedented precision (potentially reaching $10^{-17}$).
2. Isolate violations in the proton (hadron) sector from the electron sector.
3. Achieve an enhancement in sensitivity to proton-sector CPT violation by a factor of $O(m_p/m_e) \sim 10^3$ compared to atomic transitions.

2. Methodology

The author employs a low-energy effective field theory framework known as the Standard Model Extension (SME). The methodology involves a systematic perturbative analysis of the $H_2^+$ molecular ion using the Born-Oppenheimer approximation.

A. Theoretical Framework

• SME Lagrangian: The QED Lagrangian is extended to include Lorentz tensor operators with fixed background couplings. The analysis focuses on spin-independent couplings ($c_{\mu\nu}$ and $a_{\mu\nu\lambda}$), which are less constrained than spin-dependent ones.
• Hamiltonian: A non-relativistic SME Hamiltonian is derived, containing terms proportional to momentum squared ($p_i p_j$) weighted by SME coefficients ($E_{ij}$).

B. Born-Oppenheimer Separation

The 3-body problem (2 protons + 1 electron) is separated into two Schrödinger equations:

1. Electron Equation: Solved for a fixed internuclear separation $R$. The energy eigenvalues $E_e(R)$ define the inter-nucleon potential $V_M(R)$.
   • The SME couplings for the electron modify this potential, introducing an orientation-dependent term $V^e_{SME}(R, \theta, \phi)$.
   • The calculation uses a variational ansatz for the electron wavefunction (a sum of hydrogenic 1s states with a scaling factor $\gamma(R)$) to accurately model the potential and its derivatives.
2. Nucleon Equation: Describes the rovibrational motion of the two protons in the potential $V_M(R) + V^e_{SME}(R)$.
   • Crucially, the proton SME couplings ($E^p_{ij}$) appear directly in the nucleon kinetic energy term ($P_i P_j / m_p^2$).
   • This direct appearance is the source of the enhanced sensitivity to the proton sector.

C. Perturbation Theory

The rovibrational energy levels are expanded in terms of the vibrational quantum number $v$ and rotational quantum number $N$. The author develops a systematic perturbation theory to calculate SME corrections to the energy coefficients:

• Anharmonicity: The potential $V_M(R)$ is treated as an anharmonic oscillator (Morse-like). The analysis includes higher-order derivatives of the potential ($V'''_M, V^{(4)}_M$, etc.) to determine coefficients like the anharmonicity constant $x_0$ and rotation-vibration coupling $\alpha_0$.
• SME Corrections: The SME contributions are calculated to first order. The total energy is expressed as:
  $$E_{vNMN} = V^e_{SME} + (1 + \delta_{SME})(v+1/2)\omega_0 - (x_0 + x_{SME})(v+1/2)^2\omega_0 + \dots$$
  where the coefficients ($\delta, B, \alpha, D, x$) are decomposed into electron ($e$) and nucleon ($n$) parts.

3. Key Contributions

1. Derivation of Rovibrational SME Corrections: The paper provides the first detailed derivation of how Lorentz and CPT violating couplings affect the specific rovibrational energy levels of $H_2^+$, including the dependence on the magnetic quantum number $M_N$.
2. Isolation of Proton Sector: The analysis explicitly demonstrates that the proton SME couplings enter the nucleon Schrödinger equation directly. Because the kinetic energy term scales with $1/m_p^2$ (in the Hamiltonian) but the energy levels scale with the reduced mass, the resulting constraint on proton couplings is enhanced by the mass ratio $m_p/m_e$ relative to atomic transitions where the proton effect is mediated through the electron wavefunction.
3. Separation of Couplings: The paper shows that rovibrational transitions allow for the independent extraction of:
   • $(200)$ couplings: Isotropic terms constrained by 1S-2S atomic transitions.
   • $(220)$ couplings: Anisotropic terms that usually require 1S-2P atomic transitions (which have broad linewidths). In molecules, these appear in rotational transitions ($N \neq 0$) which can be measured with high precision.
4. Antimatter Comparison: The framework is set up to compare $H_2^+$ and $\bar{H}_2^-$. Since SME couplings $c_{\mu\nu}$ are CPT-even and $a_{\mu\nu\lambda}$ are CPT-odd, comparing the spectra of matter and antimatter molecules allows for the isolation of CPT-violating parameters.

4. Key Results

• Energy Expansion: The rovibrational energy levels are expanded in a small dimensionless parameter $\lambda \approx 0.027$ (related to the ratio of vibrational energy to binding energy). The SME corrections are derived as perturbations to the coefficients $B_0, x_0, \alpha_0, D_0$.
• Enhanced Sensitivity:
  • For atomic transitions (e.g., 1S-2S), the constraint on proton CPT-odd couplings ($a_{200}^p$) is suppressed by $m_e$.
  • For molecular rovibrational transitions, the constraint on $a_{200}^p$ is proportional to $m_p$.
  • Result: An enhancement factor of $m_p/m_e \approx 1836$ in the precision of constraints on the proton sector.
• Coupling Constraints:
  • Current atomic bounds on CPT-odd couplings are $\sim 10^{-12}$ (1S-2S) and $\sim 10^{-8}$ (1S-2P).
  • The paper argues that molecular ions could improve bounds on the $(220)$ couplings (currently weak due to 1S-2P linewidths) by several orders of magnitude.
  • By measuring a small number of transitions in both $H_2^+$ and $\bar{H}_2^-$, it is theoretically possible to constrain all 8 relevant SME coefficients (4 electron, 4 proton) individually.
• Numerical Values: The paper calculates numerical prefactors for the SME contributions using the numerically determined potential $V_M(R)$ and its derivatives at the equilibrium bond length $R_0 \approx 2.003 \hat{a}_0$.

5. Significance

• New Frontier in Precision Tests: This work establishes molecular ions as a superior platform for testing fundamental symmetries compared to atoms, specifically for isolating hadronic (proton) physics.
• Overcoming Atomic Limitations: It offers a solution to the "broad linewidth" problem of atomic 1S-2P transitions. Rovibrational transitions in molecules have extremely narrow natural linewidths, allowing for the precise measurement of the anisotropic $(220)$ SME couplings that are currently poorly constrained.
• Direct CPT Test: The ability to synthesize and trap $\bar{H}_2^-$ (a future goal for experiments like ALPHA at CERN) would allow for a direct comparison with $H_2^+$, providing a unique, high-precision test of CPT symmetry in the hadron sector, independent of atomic clock comparisons.
• Theoretical Foundation: The paper provides the necessary theoretical toolkit (perturbation theory, effective potentials, and numerical coefficients) for experimentalists to design and interpret future high-precision spectroscopy experiments on molecular ions.

In summary, Shore demonstrates that the unique mass dependence and structural properties of the hydrogen molecular ion make it a "magnifying glass" for Lorentz and CPT violation in the proton sector, potentially improving current bounds by three orders of magnitude and opening a new window for testing the fundamental symmetries of nature.