# Dark Matter and Fifth Forces

I promised (myself) that I would post something every time I submitted a paper, but have been falling behind. An exciting glimpse into How Science Is Done!

So here is arxiv:0807.4363:

Dark-Matter-Induced Weak Equivalence Principle Violation

Sean M. Carroll, Sonny Mantry, Michael J. Ramsey-Musolf, Christopher W. StubbsA long-range fifth force coupled to dark matter can induce a coupling to ordinary matter if the dark matter interacts with Standard Model fields. We consider constraints on such a scenario from both astrophysical observations and laboratory experiments. We also examine the case where the dark matter is a weakly interacting massive particle, and derive relations between the coupling to dark matter and the coupling to ordinary matter for different models. Currently, this scenario is most tightly constrained by galactic dynamics, but improvements in Eotvos experiments can probe unconstrained regions of parameter space.

The idea of a long-range “fifth force” is a popular one, although it’s hard to make compelling models that work. In this paper we focused in on one particular idea: imagine that there were a new long-range force that directly coupled only to dark matter. (An old idea: see Frieman and Gradwohl, 1993.) After all, there is a lot more dark matter than ordinary matter, and we don’t know much about the physics in the dark sector, so why not? But then we can also imagine that the dark matter itself interacts, via the weak interactions of the Standard Model, with ordinary matter — i.e., that the dark matter is a Weakly Interacting Massive Particle (WIMP). Then, through the magic of quantum field theory, the fifth force would automatically interact with ordinary matter, as well.

So we scoped out the possibilities and wrote a short paper; a longer one that goes into more details about the field theory is forthcoming. The punchline is this graph:

You can think of the horizontal axis as “strength with which the new force couples to ordinary matter,” and the vertical axis as “strength with which the new force couples to dark matter.” Then you have various experimental constraints, and a band representing a range of theoretical predictions. The excluded blue region to the right, labeled η_{OM}, comes from direct searches for fifth forces coupled to ordinary matter, by measuring tiny composition-dependent accelerations of test bodies in the lab. The excluded red region on top, labeled β and involving only dark matter, comes from purely astrophysics, namely the fact that dark matter and ordinary matter seem to move in concert in the Sagittarius tidal stream. The diagonal green region at top right which doesn’t actually independently exclude anything, labeled η_{DM}, comes from searching for anomalous accelerations in the direction of the galactic center, where the source would mostly be dark matter. If the experimental sensitivity improves by enough, that constraint will become independently useful. The yellow diagonal band is the prediction of our models, in which the fifth force only interacts with ordinary matter via its coupling to WIMP’s. The length comes from the fact that the direct coupling of the new force to WIMP’s is a completely free parameter, and the thickness comes from the fact that the WIMP’s can couple to ordinary matter in different ways, depending on things like hypercharge, squarks, etc.

It was a fun paper to write — a true collaboration, in that none of the authors would ever have written a paper like this all by themselves. Part of our goal was to use particle physicist’s techniques on a problem that gets more attention from astrophysicists and GR types.

[**Update**: this part of the post is edited from the original, as will become clear.] Amusing technical sidelight: the way that you actually get a coupling between the fifth force and Standard Model particles can depend on details, as we show in the paper. For example, if there are “sfermions” (scalar partners with the same quantum numbers as SM fermions) in the theory, you can induce a coupling at one loop. But if you stick just to the WIMP’s themselves, the coupling first appears at two loops:

You certainly need at least one WIMP loop (that’s χ), by hypothesis. You might think that you could just have a single SU(2)_{L} or U(1) hypercharge gauge boson connect that loop to the Standard Model fermion ψ, but that vanishes by gauge invariance; you need two gauge bosons, and thus two loops. But the the interaction you are looking for couples left- and right-handed fermions, so you need to insert a Higgs coupling. At low energies the Higgs gets a vacuum expectation value, and acts like a mass term, converting the left-handed fermion into a right-handed fermion, which is what you want.

In the original version of this post (and in the original version of our paper), I claimed that you would need a *three* loop diagram in the case where the dark matter had zero hypercharge (so you had to use SU(2)_{L} gauge bosons, which couple only to the left-handed fermions). It was just the diagram shown above, with an extra gauge boson connecting the final leg to the segment between the existing gauge bosons. Fortunately, Tim Tait and Jacques Distler convinced us otherwise, in the comments of this very blog. (Fortunately for the integrity of the scientific method, anyway; for us personally, we would rather have figured it out ourselves.) You can read my version of an explanation here. The internet works!