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 ηOM, 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 ηDM, 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!
Someday, I wish I could know enough physics to actually understand what you just said. Still, it *sounds* like an impressive paper.
Yeah, this post was dashed off rather than making a careful attempt to be explain all the details. But feel free to ask questions!
What Freiddie said.
You did a good job of getting across the heart of the matter, though. I just wish I knew something about Feynmann diagrams.
Oh – and here, have an Umlaut:
A similar paper by Farrar and Bovy:
http://arxiv.org/abs/0807.3060
Correct me if I am wrong but it looks like that calculation would be complex enough that it would be best done computationally.
Looks like a very cool work…
-NM
I understand Feynman diagrams pretty well, but what is wrong with just coupling to left handed matter? Why do you need an interaction that changes the left handed matter to right handed?
Maybe you can walk on over to Kapustin’s office and have him calculate that diagram for you.
j/k of course.
When physicists are casting about for a force with which dark matter may interact with light, why is the weak force so popular? I can see that the darkness of dark matter seems to preclude electromagnetics, but why not model strongly interacting massive particles instead?
LL, Good question. If the dark matter interacted via the strong interaction then its presence would be obvious in ways other than light. A neutron is an example of a strongly interacting massive particle (SIMP?) which does not interact via electromagnetism. Free neutrons would be dark, but neutrons’ existence is obvious because they clump in atomic nuclei (with protons) and are produced in in huge numbers in particle physics experiments. They do these things because the strong force is so strong.
The reason WIMP’s are popular DM candidates is that it’s easy to get the right relic abundance from the early universe. Given that they interact weakly, you can figure out how often they annihilate, and that tells you how many will be left over today. Generally, if the dark matter were strongly-interacting, it would have mostly annihilated away long ago.
Gavin, the interaction we were trying to get was $latex phibarpsipsi = phi psi_L psi_R$, so we needed both right- and left-handed fields.
… and yes, I should have mentioned the Bovy and Farrar paper. They were more interested in implications for direct detection, but it’s a similar theme.
I have just looked at this. I have a few questions.
I am presuming that you need two W^a fields to connect to the WIMP loop in order to avoid a tadpole-like diagram.
Would I be right in saying that the W^a is the SU(2)xU(1) W gauge bosons? The G or B boson appears to be some other gauge field. If G is a gluon then the fermion is a quark, so this would be a part of a QCD.
Finally you need the Higgs H I presume because this is what determines the mass of the WIMP via the scalar interaction with ordinary matter. IOW the WIMP in this model “gets its mass” through this interaction with other fields, such as the fermion above.
Finally, though this “fifth force” interaction is small, are you arguing that the above interaction causes a small violation of the weak EP. Is this because some small amount the mass induced by the phi or the WIMP has no gravitational mass? This part is what eludes me at this time.
Lawrence B. Crowell
Lawrence, don’t be reluctant to look at the actual paper! The W’s are SU(2) gauge bosons; G and B are SU(3) and U(1) gauge bosons, respectively. Unlike the W’s, they couple both to left and right-handed fermions. The reason why you need two W’s was given above — the analogous graph with just one W would vanish by gauge invariance.
And you need the Higgs because we are trying to couple a single scalar boson φ to a left-handed fermion and its right-handed partner; in the Standard Model, the only way to do that is through the Higgs. The Higgs gives mass to the Standard Model fermions, but we don’t assume that the dark matter gets its mass from the Higgs. (Nor would it matter, one way or another.)
Yes, a model like this leads to a violation of the equivalence principle. The φ boson is massless, so it gives rise to a long-range force; with a bit of work, you can show that the only long range force that is consistent with the EP is gravity (mediated by a massless spin-2 boson, the graviton). So we are basically guaranteed to get EP violation (in other words, composition-dependent forces). If you like, you can think of the force as arising from the virtual cloud of dark matter particles in an atomic nucleus; roughly, the more ordinary matter, the more virtual dark matter, but not exactly.
I have downloaded the paper, but have yet to get to crunch time on it. Thanks, what you explained clears things up and suggests I was partially right on my initial assessment of some of this.
L C.
Hi Sean:
Could you elaborate on the experimental limits that would exclude such a model. IOW how much improvement in current fith force experimental acuuracy would be necessary to exclude this model if at all.
“Of course, we didn’t actually calculate this diagram…”
read: We were too nice to push the evil, mind-numbing calculation onto our grad students.
cecil– The model doesn’t make a unique prediction, but the yellow band shown above. So any improvement has a chance of detecting it, but there is no threshold. However, if you do detect a fifth force in one of the experiments, you automatically make predictions for where it should be found in the others.
sean, a fascinating post and topic, and interesting discussion so far, thanks
so this does not address the possible existence of “short range” dark matter physics, at least vaguely analogous to the strong force, right? presumably because there is maybe not much to be said about such things due or our lack of knowledge, or simply to narrow the focus?
curious also because I noted that this paper
http://arxiv.org/abs/0808.0472
described “dark matter heating” via wimp annihilation, which made me wonder how complex the “nuclear physics” of dark matter might actually be
I have given this a quick reading, though I intend to read this more closely tomorrow. One of the things which I find a bit interesting about the Feynman diagram above is that it seems to fit in with the Garret Lisi E_8 irrep. The G&B gauge bosons are SU(3) and U(1), which are decomposed from G_2. The weak stuff W^a is then in the Patti-Salam model SO(4), which exists along SO(3,1) in SO(7,1). So this WIMP theory appears to have GUT or GUT-like physics behind it.
Further, I have thought for some time that some form of EP violation might take place in a manner analogous to spin dependencies in spin quantum hall effects. Due to the quantum hall effect there can exist a spin dependency in the refractive index of a photon. In the same footing the same might be the case with general relativity, shich would be a form of EP violation. If we have gravity in a B-F formalism, with a Chern-Simons Lagrangian from the “B part,” then we might get some form of EP violation from an analogue of the quantum hall effect. This would then be something which “mirrors” CP violations in the SO(4) —>SU(2)xSU(2) in the SO(3,1).
I am not sure where the scalar field would come from. It might be a dilaton or some related field. To call this a “fifth force” might be similar to calling the Higgs field a force. Such a theory would involve two forms of symmetry breaking. One which is from the standard Higgsian degenerate vacuum, while the other from a topologically induced charge or mass. This would appear to involves some interplay between them.
Lawrence B. Crowell
Hi Sean,
This was a nice post, about a paper that I noticed when it appeared on the ArXiv, but didn’t have time to look at yet. (The blog equivalent of “well I didn’t actually read it but I heard the talk and…”).
One thing I didn’t understand-
You said the Feynman diagram was three loops because just tacking the Higgs insertion at the end wouldn’t be 1PI. It’s true that it isn’t 1PI, but in terms of the effective action I would have said “so what?”.
Or put another way, let’s say I had calculated the two loop graph including everything except the G,B loop. What I am saying would happen is that when I reduced the integrals I would find the fermion line contains an external momentum sandwiched between the matter fermion spinors. I can always write that external momentum in terms of the momenta of the external spinors, and then using the Dirac equation turn that into the fermion bilinear you want times the fermion mass. (Sorry for the lack of equations but I am too scared to learn how to use the latex features :).
Or for an effective quantum field theorist like myself, the operator you generate has one left-handed and one right-handed fermion, so gauge invariance tells me there is a Higgs as well, but since you didn’t introduce any extra chiral symmetry breaking, the Higgs interaction must have come from a Yukawa coupling.
Having gone either route, I have a term in the effective action which seems just fine to mediate interactions.
Am I missing something?
Tim
Sean wrote:
Can you (or any commenter) explain this in a bit more detail? E.g., is there some choice of gauge for which it’s readily apparent that this vanishes? Is this due to the fact that the gauge group is non-abelian? (I.e., if chi was subject to electromagnetism, you could have just a single photon, couldn’t you?)
I’m sure I should remember this from when I took field theory . . .
Tim T– sorry, I was rushed and a little confusing. I should have said “amputated” rather than “1PI” (it is 1PI, of course, but that’s not the point). If we wanted to calculate a scattering amplitude, for example fermion/phi to fermion/higgs, then the diagram without the extra gauge loop would definitely contribute. But here the Higgs is just playing the role of a vev, although we’re writing things down in the unbroken variables (because we care about helicity). So that Higgs is just a mass insertion, and it should be thought of as part of the external propagator. And the rules (as I understand them; these aren’t my usual stomping grounds, so feel free to set me straight) tell us to not include self-energy corrections to external legs, which is basically what a single Higgs insertion would be. Does that make sense?
TimG– the relevant one-loop graph would basically describe “mixing” between the W gauge bosons and the scalar phi — a W would come in, split into a chi loop, and then turn into a phi. And that part of the graph includes a trace over all the generators of the gauge fields at each vertex; an SU(2) generator at the W vertex, and nothing at all at the phi vertex. So it would vanish, as the SU(2) generators are traceless. And, like you are anticipating, this argument does depend intimately on the fact that the group is non-abelian.
Lawrence– I’m not sure what to say, except that none of your sentences make any sense, and I hope nobody reading them is tricked into thinking otherwise.
It would probably be helpful to think about this in an effective field theory language. If the WIMP is very heavy, we can integrate it out, and obtain an effective interaction between φ and the SU(2)×U(1) gauge bosons, of the form $latex 1/M phi F^2$, where M is the large mass.
I don’t know why it’s reasonable to assume that the WIMP has vanishing hypercharge, but if we do, then we only get such an interaction for the SU(2) gauge bosons.
Inserting this interaction into a loop produces an effective interaction which, at energies well below the electroweak scale looks like $latex 1/M phi overline{psi}^{dot{alpha}}p_{dot{alpha}alpha} psi^alpha$ .
Because of electroweak symmetry-breaking, the on-shell equations of motion turn that momentum into a mass (for the external fermion).
So I don’t really understand what process Sean has in mind that only receives a 3-loop contribution from the diagram drawn, but not a 2-loop contribution (ie, the contribution I just described).
One more point: the two-loop graph, which contributes to the process Sean is interested in, isn’t “really” a two-loop graph at all. At least in the limit of a very heavy WIMP, it’s a pair of “iterated” one-loop graphs.
The WIMP loop produces an effective local interaction (as I said in my previous comment), which one just inserts into a 1-loop graph. You never really have to do a 2-loop integral. The result can be presented, in closed form; you don’t need to “estimate it.”
Jacques– of course you don’t need to assume the WIMP has no hypercharge; that’s just one of the cases we consider. (Seriously, folks, we did write a paper, to which I helpfully linked.) For the case with hypercharge, the leading contribution is clearly at two loops.
The limit of a very heavy WIMP is not one that is especially relevant to the real world. WIMP’s can easily be 100 GeV, and 1 TeV would be really stretching things, so there is no clean separation of scales with the W bosons.
On the last (and most important) point, I’m confused. You’re saying that we should get an effective interaction that couples φ to two left-handed fermions with a factor of momentum, and then we can just “use the equations of motion” to turn the momentum into a mass term, and one of the L fermions into an R? I guess this is just Tim T’s point in different words, and I guess my response would be the same, according to the rules of the game as I understand them. But I could be wrong! As this is not a game I play very often.
So let me think about it. Unfortunately I’m traveling right now and have to run to some non-physics-related activities, so I won’t be able to think soon, but I will.