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!
Perhaps not, but it’s a useful way to organize your thinking about the calculation.
Sorry for not making much sense on Friday. I wrote that quickly before I had to leave for other matters of a social nature. I was mentioning some ideas which might tie in some possible threads. Maybe I am wrong, but this theory does appear to involve gauge fields in some ways which might be generalizable to E_8 unification.
As for spin Hall quantization and EP violation the article by Gosselin, Berard, Mohrbach
http://arxiv.org/PS_cache/hep-th/pdf/0603/0603227v4.pdf
is what I had in mind. A spin hall quantization has been found to induce a differential index of refraction and a spin-dependent displacement in photon paths as found experimentally by Hosten & Kwiat “Science,” vol 8 8-Feb 2008, and this is transferable in principle to a possible mechanism for EP violations. Photons in spacetime can have a helicity or spin dependency in their geodesic path or deviation from such.
With respect to Chern-Simons lagrangians and Hall effects, the paper by G. Dunne “Aspects of Chern-Simons Theory”
arxiv:hep-th/9902115v1
illustrates the connection to anyons and this sort of quantization of magnetic flux with Hall quantization. It struck me that maybe this sort of physics was at play here, at least if there is a connection between this approach to a violation of EP and the spin quantum Hall approach. Further the Lisi E_8 irrep indicates how in B-F actions one arrives at C-S boundary terms.
So I was simply pointing out some possible threads. Nothing here is anything which I would call even a conjecture, but just pondering on some things 😉
Lawrence B. Crowell
Spurious results – the G,B term at the bottom is probably unnecessary as it wouldn’t be a dominant term in any likely Feynman diagram. The 1PI picture described is probably unphysical.
” Of course, we didn’t actually calculate this diagram”
Of course you didn’t – especially since you have no idea how to determine the scaling factors and integrals needed at the vertices of the diagram. What exactly are the Feynman diagram rules for Higgs bosons and dark matter? What precisely is the dark matter propagator? Is the dark matter field scalar, vector, spinor, Grassmannian? No one knows for sure.
Nice try – it probably “pulls the wool” over amateurs , but please don’t think all of us are fooled by this guff.
“The limit of a very heavy WIMP is not one that is especially relevant to the real world.”
Then why are we bothering to discuss this model? The fact is WIMPS are still hypothetical – not even one has ever been observed in a collider experiment. They are only postulated to “explain” certain problems in cosmological models. Do we really require their existence? The “missing mass” could just as easily be more mundane particles such as neutrinos ( which do have a nonzero rest mass and are manifestly abundant in the universe).
Jack:
This alone shows you have absolutely no clue what you are talking about. Any more troll-like comments and I’m going to delete them without warning or comment.
I’d appreciate it if others didn’t bother replying to his comments.
The GB interactions means that the spinor field could be quarks and the Yukawa terms g(q-bar)phi q will contribute significantly. This then appears to put the g/m_i in the region in diagram 1 that is “WIMP mediated.” I think that without the hypercharge this might not work very well. If the WIMP lacks hypercharge the 3-loop induced interaction is needed. The point appears to be that you need a large coupling g — which is given by the B or G-B.
I am trying to wrap my mind around this paper, and there is a lot of background things one needs to refer to here. Yet it appears one needs these stronger fields in order to get the coupling constant large enough so that you get the vertical and horizontal lines in figure 1.
The dark matter or WIMP field is of course an unknown. One can only make some conjectures about this and see if anything at all works. My idea of connecting this to the EP violations due to the Hall effect is that we might see parallels between equation 1 in arxiv:0807.4363 and with the last equation in
http://arxiv.org/PS_cache/hep-th/pdf/0603/0603227v4.pdf
which I suggested above. Here the angle deviation for photons in an EP is given by L/r_0, for L the wavelength and the r_0 the distance from the static field. This deviation is due to an Ahranov-Bohm phase
$latex
exp(ieoint Acdot dx~=~exp(ie^2/kappa)
$
This angle deviation is then determined by this anyonic exchange phase
$latex
Deltatheta~=~frac{e^2}{4pikappa}
$
There are some connections between this and Higgsian induced masses, though that involves some depth of discussion. So there might be connections here between quantum spin-Hall effect violation of EP and the WIMP, which might indicate what this WIMP field really is.
Lawrence B. Crowell
Okay, we are going to fall on our swords and admit that Tim and Jacques are right about the three-loop diagram. (And they will be gratefully acknowledged in a revised version of the paper.)
If there is any remaining pedagogical value to be squeezed out of our mistake, I can explain why we mixed ourselves up. A two-loop effect is what you would expect at first blush, and indeed that’s what we calculated for the case where the dark matter carries hypercharge. But in the case without hypercharge we talked ourselves into believing that you couldn’t get a two-loop coupling of a left-handed fermion to a right-handed fermion, because the Higgs insertion would have to be on an external leg, and therefore should count as part of the self-energy and be amputated. But that’s not right, and once you convince yourself it’s not right there are various ways to say it. If you don’t like the equation-of-motion argument that Jacques gave above, and you want to be persnickety about gauge invariance, you can just calculate the operator $latex phi psi_L H psi_R$, and then just give the Higgs a vev.
Happily, none of the qualitative conclusions of our paper are changed; one 10^{-9} becomes 10^{-8}, and the yellow band of predictions on the first graph will narrow a bit. (Which is good news.) I’ll update the post later tonight.
But sadly, my first paper with a three-loop diagram will have to wait until later.
Hi Sean,
Glad that its cleared up and sorry I was offline and missed most of the responses.
After leaving on Friday, I came up with the better argument that Jacques also touched on in his discussiont:
“Let me be stupid and just imagine I am computing fermion-fermion scattering at low momentum transfer Imediated by phi’s (the low momentum transfer is so I won’t be tempted to assume EW symmetry – I’ll just pick the Unitary gauge and go ahead and compute)… then I will get a result which is proportional to the fermion mass (squared in my example, once per vertex), but it will still be four loops total, or two per vertex.”
Or the way you said it above (“just calculate the operator , and then just give the Higgs a vev”) – sigh, I need to learn how you do equations – is the way I would have thought about it.
And yeah, no big deal… it just changes an estimate slightly! And I’m glad to be of help. I read blogs infrequently and I’m kind of tickled that I could actually contribute to science through one..
Tim
Tim, thanks. And doing equations is easy!
I’m really enjoying this discussion. I took QFT a few years ago, but my thesis research hasn’t required me to draw a single Feynman diagram, and I’ve forgotten a lot. So this has finally given me the kick in the pants I needed to go back and try to relearn this stuff.
Sean wrote:
TimG– it’s not the gamma matrices that are to blame, it’s the SU(2) generators that are traceless. Each vertex involving a non-abelian gauge field comes with a matrix, the generator of that gauge group. For U(1) that would be trivial, but for SU(n) they are traceless Hermitian matrices.
Thanks, Sean. I forgot that you said I was right in initially thinking it’s related to the non-abelian gauge group — so of course I can’t figure it out by analogy to QED.
Checking my QFT textbook, I find that the vertex factors in non-abelian gauge theory are $latex iggamma^{mu}t^a$, where I take it the $latex t^a$’s are the traceless SU(2) generators. Makes sense.
I assume that the reason the gamma matrices aren’t a problem is because the propagator of $latex chi$ contains a p-slash in the numerator (like the electron propagator), and thus you can get an even number of gamma matrices inside the trace. Is that right?
If so, does this mean you wouldn’t get a contribution from such a diagram if you instead considered bosonic dark matter? And is there any reason to expect dark matter to be fermionic? (E.g., do we know whether the lightest superpartner should be a fermion?)
I guess you can forget that question about whether the diagram cancels for a boson loop. First, I’m pretty sure there wouldn’t even be a gamma on the vertex in that case, so my concern about that was groundless. But more importantly I forgot that the diagram we’re looking at has two W’s, so everything in the vertex factor appears twice.
And of course, even in the fermion case I don’t need to worry about how we get an even number of gammas, since we already knew we needed an even number of $latex Wchi^2$vertices to get an even number of $latex t^a$’s.
Sorry for the repeated comments — I wish there was a way to edit (or, for that matter, to preview).
Sean — a little off-topic, but can I ask how you make those pretty pictures with the circles and the wavy lines? (This might be a little too technical to answer via blog comments, I know you like to keep the discussion here light).
TimG– dark matter certainly could be bosonic, although many of the most popular candidates for WIMP’s are fermions.
Alex F– Two crucial ingredients: (1) Adobe Illustrator, (2) Long, boring plane flights.
TimG, the sneutrino is an example of a bosonic DM candidate that has been investigated, see e.g. here.
Thanks again to everyone for the very informative answers. Is there any more that can be said about why we didn’t have to worry about the coupling to the Higgs being amputated after all? (See the second half of my post #35 for more detail on why I’m confused.)
It seems possible to have an odd number of vertices, we just can’t have one. The action S = log det(iD + m), for D = gam^a(&_a + W_a), an expansion gives a second order Tr[(1/i& + m)W(1/i& + m)W] and higher. The first order term in W is tadpole —> 0 and the zeroth order is subtracted. The vertices will contain traceless terms gam*(tW) which will contribute to a quadratic term in t which has a trace. Yet a third order term may not be zero as it appears there might be a Levi-Civita Tr(gam^a gam^b gam^c) = eps^{abc}. Might this then contribute to a a nonvanishing cubic term in the group elements t?
Is suppose that if the mass of the chi-field or WIMP is large enough then chi-loop divergence can be handled by introducing mass counter terms as the loop contracts to zero or large momentum. I am not sure how this would be done.
Lawrence B. Crowell
Furry’s theorem says that an odd number of vertices will yield zero.
I managed to convince myself this would indeed vanish. It appears that this is the case for SU(2). I’d have to check to see if this is the case for SU(3).
L. C.
Shoot. I was worried it was Illustrator. I had a trial version of it once and it was great… but it’s a $600 program. I’d been hoping you could point me to an awesome free program, since I’d only ever use about $2 of the $600 worth… (And no, Inkscape is no substitute).
There is a free program, specifically for making Feynman diagrams: Jaxodraw. I’ve never been able to get it to work, although I’ve admittedly not tried to hard, as I like Illustrator.
The jaxodraw disk image for OS X just seems to work for me – it is a pretty nice program!
You can also make reasonably pretty Feynman diagrams using root-
http://root.cern.ch/
(the documentation is not great, but search for ‘Feynman’) though of course root itself is a much more sophisticated and general tool. I use it for all of my other plotting needs as well, so it works well for me.
Going back to the Higgs insertion, it doesn’t carry any momentum, and in fact you shouldn’t think of it as a real interaction at all… Think of any number of such insertions as having been already resummed into our definition of the fermion. That language is very useful in order to keep track of fermion chiralities (which in this example is what tells you the whole graph is proprtional to the fermion mass), but it is not really intended for serious calculations.
Oh, and Furry’s theorem works well for QED, but it has loopholes that apply to non-Abelian gauge theories, or a theory with chiral interactions (such as the electroweak theory of the Standard Model).
Tim