Measuring new forces with molecular vibrations

[BillKet]

Hello! I noticed in several papers describing high resolution vibrational measurements in diatomic molecules, such as this one, in the conclusion section, that they mention that we can search for new forces (or deviation from gravity inverse square law) between the 2 nuclei by measuring well enough the vibrational levels of the molecule. In principle, on the theoretical side, one just needs to add a correction term $V'(r)$ to the electronic Hamiltonian which gives the potential energy curve, upon which the vibrational levels are calculated.

However, the theory can't predict these curves very well (maybe at the 1% uncertainty level?). And given that these new forces are a lot smaller than 1% relative to the normal electrostatic contribution, I am not sure how we can compare experimental measurements (regardless of how accurate they are), with theoretical calculations, given that the deviations we are looking for are far below the theoretical uncertainties.

In the cases where they search for parity and/or time reversal violations, they usually search for an effect that would be zero in the absence of these violations. But in this case I am not sure how one can isolate that small contribution from the normal electrostatic interaction. Can someone point me towards some papers that discuss this in more detail (or tell me their understanding of this)? Thank you!

 

[Twigg]

Good question!

My gut feeling on this (take this with a grain of salt!) is that the estimated electronic potentials should have a pretty accurate functional form (e.g., Lennard-Jones) with the slop in the constant coefficients. Keep in mind that Tanya Zelevinsky's $\mathrm{Sr}_2$ molecular lattice clock uses very highly excited vibrational states. I think the clock state is like $\nu = 62$ or so? Or somewhere in the sixties. They could choose to do precision spectroscopy on a wide range of different vibrational states. Having spectroscopy for a wide range of vibrational states gives you information about $V(R)$. You could search for deviations from the theoretical model that cannot be reproduced by the theoretical functional form of $V(R)$ (e.g., a $1/R^2$ dependence cannot be reproduced by a Lennard-Jones curve). If you find a statistically significant deviation from the model function, you could call that evidence of a new force.
You'd also need a way to, for example, tell electromagnetic $1/R^2$ terms from gravitational $1/R^2$ terms. I think that'd be identified by looking for shifts across different isotopes. For sure, the math gets hairy.

 

[BillKet]

Good question!

My gut feeling on this (take this with a grain of salt!) is that the estimated electronic potentials should have a pretty accurate functional form (e.g., Lennard-Jones) with the slop in the constant coefficients. Keep in mind that Tanya Zelevinsky's $\mathrm{Sr}_2$ molecular lattice clock uses very highly excited vibrational states. I think the clock state is like $\nu = 62$ or so? Or somewhere in the sixties. They could choose to do precision spectroscopy on a wide range of different vibrational states. Having spectroscopy for a wide range of vibrational states gives you information about $V(R)$. You could search for deviations from the theoretical model that cannot be reproduced by the theoretical functional form of $V(R)$ (e.g., a $1/R^2$ dependence cannot be reproduced by a Lennard-Jones curve). If you find a statistically significant deviation from the model function, you could call that evidence of a new force.
You'd also need a way to, for example, tell electromagnetic $1/R^2$ terms from gravitational $1/R^2$ terms. I think that'd be identified by looking for shifts across different isotopes. For sure, the math gets hairy.

Thank you! This make sense. I am not very familiar with reconstructing the potential energy curves from the vibrational measurements this way (in the papers I read they just fit to a finite number of Dunham-like parameters). Are we confident enough that the functional form should match the actual molecule exactly (given the measurement accuracy), in the absence of these extra forces? Can't it be that the model breaks down, even if we have just electrostatic interactions? For example if one naively assumes a Morse potential, we would get deviations from the measurements at a certain level of accuracy. But they are just from breaking down of the model, not from new physics.

 

[Twigg]

I'm not versed with Dunham parameters, but the picture I had in mind was that the vibrational state wavefunctions define a somewhat orthogonal basis of functions of internuclear distance. By doing the precision spectroscopy on vibrational level $\nu$, you're projecting $V(R)$ onto $\psi_\nu (R)$. If you measure the projection for each element in the basis (as many $\nu$'s as you can), you get a more complete approximation of $V(R)$. You could see how a theoretical model function would turn into a vector subspace, and any measured component of $V(R)$ that goes orthogonal to that subspace would be evidence for new physics (or old physics that was missing from the model).
As far as generating a complete model function, I don't know how difficult it is to capture all the old physics. I think you can narrow these things down a little by their expected isotope dependence.
Edit: Also I just wanted to add, like our discussion about effective Hamiltonians, you might be able to get by without a complete model of all R dependence. All you really need is the knowledge of how your scientifically interesting signal is distinguished from the background.