Dark Matter and Fifth Forces

By Sean Carroll | August 14, 2008 12:39 pm

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. Stubbs

A 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!

  • http://freiddy.blogspot.com Freiddie

    Someday, I wish I could know enough physics to actually understand what you just said. Still, it *sounds* like an impressive paper.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Yeah, this post was dashed off rather than making a careful attempt to be explain all the details. But feel free to ask questions!

  • Sili

    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.

  • Sili

    Oh – and here, have an Umlaut:

  • Count Iblis

    A similar paper by Farrar and Bovy:


  • Nicholas

    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…


  • Gavin Polhemus

    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?

  • joe

    Maybe you can walk on over to Kapustin’s office and have him calculate that diagram for you.

    j/k of course.

  • http://lablemminglounge.blogspot.com/ Lab Lemming

    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?

  • Gavin Polhemus

    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.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • Lawrence B. Crowell

    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

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • Lawrence B. Crowell

    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.

  • cecil kirksey

    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.

  • JerseyBoy

    “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.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • Tyler

    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
    described “dark matter heating” via wimp annihilation, which made me wonder how complex the “nuclear physics” of dark matter might actually be

  • Lawrence B. Crowell

    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

  • Tim Tait

    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?


  • TimG

    Sean wrote:

    You might think that you could just have a single SU(2) gauge boson connect that loop to the Standard Model fermion ?, but that vanishes by gauge invariance

    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 . . .

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    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).

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    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.”

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    The limit of a very heavy WIMP is not one that is especially relevant to the real world.

    Perhaps not, but it’s a useful way to organize your thinking about the calculation.


  • Lawrence B. Crowell

    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


    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”


    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

  • Jack

    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.

  • Jack

    “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).

  • http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark


    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).

    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.

  • Lawrence B. Crowell

    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


    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

    exp(ieoint Acdot dx~=~exp(ie^2/kappa)

    This angle deviation is then determined by this anyonic exchange phase


    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

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • Tim Tait

    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..


  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Tim, thanks. And doing equations is easy!

  • TimG

    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:


  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • TimG

    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?)

  • TimG

    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.

  • TimG

    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).

  • Alex F

    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).

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • Count Iblis

    TimG, the sneutrino is an example of a bosonic DM candidate that has been investigated, see e.g. here.

  • TimG

    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.)

  • Lawrence B. Crowell

    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

  • Count Iblis

    Furry’s theorem says that an odd number of vertices will yield zero.

  • Lawrence B. Crowell

    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.

  • Alex F

    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).

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

    The jaxodraw disk image for OS X just seems to work for me – it is a pretty nice program!

  • Tim Tait

    You can also make reasonably pretty Feynman diagrams using root-


    (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).


  • TimG

    Tim Tait wrote:

    Going back to the Higgs insertion, it doesn

  • Tim Tait

    That’s right. What I mean is that one can just think of the four-component fermion $latex psi$ which contains both $latex psi_L$ and $latex psi_R$. In that language there is no fermion propagator and no Higgs insertion – the effect of the Higgs has been summed into our definition of the massive fermion.

    And you’re right, the denominator of that propagator looks like it is zero otherwise, which is from various points of view: why we redefine the two massless fermions into a single massive one as above, why we amputate diagrams with self-energy corrections to external legs (but that is the same thing I just said in a more general language), and (I think implicitly) what Sean and collaborators were worrying about in the original (three loop) version of the diagram.

  • Tim Tait

    Sean: You’re right… Equations ARE easy…!

  • OneForce

    Say, I don’t know if this is the proper venue for a fellow to just drop in and pop a question or not but I have one so I’ll ask it and let the chips fall where they may.

    Question: Is it fair to say in our mysteriously accelerating and occasionally preposterous Universe that Gravitation as a strictly Attractive force fails to sufficiently explain what’s going on? That there must be a corresponding Repulsion force at play or the whole interpretation goes up in flames? Based on the relatively new evidence we’ve accumulated about not just our expanding universe but a rapidly expanding one, in that it expands at ever greater rates as our conception of Time passes?

    Feel free to shoo me away if this is an annoying question.

  • http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

    OneForce, you might try searching our archive for “cosmic acceleration”. Here are a couple of other posts from our earlier blogs that might help further:



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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .


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