Quantum Diaries

`Yesterday marked the beginning of the HEP summer conference season with EPS-HEP 2011, which is particularly exciting since the LHC now has  enough luminosity (accumulated data) to start seeing hints of new physics. As Ken pointed out, the Tevatron’s new lower bound on the B_s → μμ decay rate seemed to be a harbinger of things to come (Experts can check out the official paper, the CDF public page, and the excellent summaries by Tommaso Dorigo and Jester.).

Somewhat unfortunately, the first LHCb results on this process do not confirm the CDF excess, though they are not yet mutually exclusive. Instead of delving too much into this particular result, I’d like to give some background to motivate why it’s interesting to those of us looking for new physics. This requires a lesson in “the birds and the Bs”—of course, by this I mean B mesons and the so-called ‘penguin’ diagrams.

The B_s meson: why it’s special

It's a terrible pun, I know.

A B_s meson is a bound state of a bottom anti-quark and strange quark; it’s sort of like a “molecule” of quarks. There are all sorts of mesons that one could imagine by sticking together different quarks and anti-quarks, but the B_s meson and it’s lighter cousin, the B_d meson, are particularly interesting characters in the spectrum of all possible mesons.

The reason is that both the B_s and the B_d are neutral particles, and it turns out that they mix quantum mechanically with their antiparticles, which we call the B_s and B_d. This mixing is the exact same kind of flavor phenomenon that we described when we mentioned “Neapolitan” neutrinos and is analogous to the mixing of chiralities in a massive fermion. Recall that properties like “bottom-ness” or “strangeness” are referred to as flavor. Going from a B_s to a B_s changes the “number of bottom quarks” from -1 to +1 and the “number of strange quarks” from +1 to -1, so such effects are called flavor-changing.

To help clarify things, here’s an example diagram that encodes this quantum mixing:

The ui refers to any up-type quark.

Any neutral meson can mix—or “oscillate”—into its antiparticle, but the B mesons are special because of their lifetime. Recall that mesons are unstable and decay, so unlike neutrinos, we can’t just wait for a while to see if they oscillate into something interesting. Some mesons live for too long and their oscillation phenomena get ‘washed out’ before we get to observe them. Other mesons don’t live long enough and decay before they have a chance to oscillate at all. But B mesons—oh, wonderful Goldilocks B mesons—they have a lifetime and oscillation time that are roughly  of the same magnitude. This means that by measuring their decays and relative decay rates we can learn about how these mesons mix, i.e. we can learn about the underlying flavor structure of the Standard Model.

The CDF and LHCb results: B_s → mu mu

It turns out that there are interesting flavor-changing effects even without considering meson mixing, but rather in the decay of the B meson itself. For example, we can modify the previous diagram to consider the decay of a B_s meson into a muon/anti-muon pair:

This is still a flavor-changing decay since the net strangeness (+1) and bottom-ness (-1) is not preserved; but note that the lepton flavor is conserved since the muon/anti-muon pair have no net muon number. (As an exercise: try drawing the other diagrams that contribute; the trick is that you need W bosons to change flavor.) You could also replace muons by electrons or taus, but those decays are much harder to detect experimentally. As a rule of thumb muons are really nice final state particles since they make it all the way through the detector and one has a decent shot at getting good momentum measurements.

It turns out that this decay is extremely rare. For the B_s meson, the Standard Model predicts a dimuon branching ratio of around 3 × 10^-9, which means that a B_s will only decay into two muons 0.0000003% of the time… clearly in order to accurately measure the actual rate one needs to produce a lot of B mesons.

In fact, until recently, we simply did not have enough observed B meson decays to even estimate the true dimuon decay rate. The ‘B factories’ of the past decade were only able to put upper limits on this rate. In fact, this decay is one of the main motivations for LHCb, which was designed to be the first experiment that would be sensitive enough to probe the Standard Model decay rate. (This means that if the decay rate is at least at the Standard Model rate, then LHCb will see it.)

The exciting news from CDF last week was that—for the first time—they appeared to have been able to set a lower bound on the dimuon decay rate of the B_s meson. (The B_d meson has a smaller decay rate and CDF was unable to set a lower bound.) The lower bound is still statistically consistent with the Standard Model rate, but the suggested (‘central value’) rate was 1.8 × 10^-8. If this is true, then it would be a fairly strong signal for new physics beyond the Standard Model. The 90% confidence level range from CDF is:

4.6 × 10^-9 < BR(B_s → μ⁺μ^–) < 3.9 × 10^-8.

Unfortunately, today’s new result from LHCb didn’t detect an excess with which it could set a lower bound and could only set a 90% confidence upper bound,

BR(B_s → μ⁺μ^–) < 1.3 × 10^-8.

This goes down to 1.2 × 10^-8 when including 2010 data. The bounds are not yet at odds with one another, but many people were hoping that LHCb would have been able to confirm the CDF excess in dimuon events. The analyses of the two experiments seem to be fairly similar, so there isn’t too much wiggle room to think that the different results just come from having different experiments.

More data will clarify the situation; LHCb should accumulate enough data to prove branching ratios down to the Standard Model prediction of 3 × 10^-9. Unfortunately CDF will not be able to reach that sensitivity.

New physics in loops

Now that we’re up to date with the experimental status of B_s → μμ, let’s figure out why it’s so interesting from a theoretical point of view. One thing you might have noticed from the “box” Feynman diagrams above is that they involve a closed loop. An interesting thing about closed loops in Feynman diagrams is that they can probe physics at much higher energies than one would naively expect.

The reason for this is that the particles running in the loop do not have their momenta fixed in terms of the momenta of the external particles. You can see this for yourself by assigning momenta (call them p₁, p₂, … , etc.) to each particle line and (following the usual Feynman rules) impose momentum conservation at each vertex. You’ll find that there is an unconstrained momentum that goes around the loop. Because this momentum is unspecified, the laws of quantum physics say that one must add together the contributions from all possible momenta. Thus it turns out that even though the B_s meson mass is around 5 GeV, the dimuon decay is sensitive to particles that are a hundred times heavier.

Note that unlike other processes where we study new physics by directly producing it and watching it decay, in low-energy loop diagrams one only intuits the presence of new particles through their virtual effects (quantum interference). I’ll leave the details for another time, but here are a few facts that you can assume for now:

1. Loop diagrams can be sensitive to new heavy particles through quantum interference.
2. Processes which only occur through loop diagrams are often suppressed. (This is partly why the Standard Model branching ratio for B_s → μμ is so small.)
3. In the Standard Model, all flavor-changing neutral currents (FCNC)—i.e. all flavor-changing processes whose intermediate states carry no net electric charge—only occur at loop level. (Recall that the electrically-charged W bosons can change flavor, but the electrically neutral Z bosons cannot. Similarly, note that there is no way to draw a B_s → μμ diagram in the Standard Model without including a loop.)
4. Thus, processes with a flavor-changing neutral current (such as B_s → μμ) are fruitful places to look for new physics effects that only show up at loop level. If there were a non-loop level (“tree level”) contribution from the Standard Model, then the loop-induced new physics effects would tend to be drowned out because they are only small corrections to the tree-level result. However, since there are no FCNCs in the Standard Model, the new physics contributions have a ‘fighting change’ at having a big effect relative to the Standard Model result.
5. Semi-technical remark, for experts: indeed, for B_s → μμ the Standard Model diagrams are additionally suppressed by a GIM suppression (as is the case for FCNCs) as well as helicity suppression (the B meson is a pseudoscalar, so the final states require a muon mass insertion).

So the punchline is that B_s → μμ is a really fertile place to hope to see some deviation from the Standard Model branching ratio due to new physics.

Introducing the Penguin

I would be remiss if I didn’t mention the “penguin diagram” and its role in physics. You can learn about the penguin’s silly etymology in its Wikipedia article; suffice it for me to ‘wow’ you with a picture of an autographed paper from one of the penguin’s progenitors:

A copy of the original "penguin" paper, autographed by John Ellis.

The main idea is that penguin diagrams are flavor-changing loops that involve two fermions and a neutral gauge boson. For example, the b→s penguin takes the form (no, it doesn’t look much like a penguin)

You should have guessed that in the Standard Model, the wiggly line on top has to be a W boson in order for the fermion line to change flavors. The photon could also be a Z boson, a gluon, or even a Higgs boson. If we allow the boson to decay into a pair of muons, we obtain a diagram that contributes to B_s → μμ.

Thus, in addition to the box diagrams above, there are penguin diagrams which contribute to B_s → μμ. As a nice ‘homework’ exercise, you can try drawing all of the penguins that contribute to this process in the Standard Model. (Most of the work is relabeling diagrams for different internal states.)

[Remark, 6/23: my colleague Monika points out that it’s ironic that I drew the b, s, photon penguin since this penguin doesn’t actually contribute to the dimuon decay! (For experts: the reason is the Ward identity.) ]

Supersymmetry and the B_s → mu mu penguin

Finally, I’d like to give an example of a new physics scenario where we would expect that penguins containing new particles give a large contribution to the B_s → μμ branching ratio. It turns out that this happens quite often in models of supersymmetry or, more generally, ‘two Higgs doublet models.’

If neither of those words mean anything to you, then all you have to know is that these models have not just one, but two independent Higgs particles which obtain separate vacuum expectation values (vevs). The punchline is that there is a free parameter in such theories called tan β which measures the ratio of the two vevs, and that for large values of tan β, the B_s → μμ branching ratio goes like (tan β)⁶ … which can be quite large and can dwarf the Standard Model contribution.

Added 6/23, because I couldn't help it: a supersymmetric penguin. Corny image from one of my talks.

[What follows is mostly for ‘experts,’ my apologies.]

On a slightly more technical note, it’s not often well explained why this branching ratio goes like the sixth power of tan β, so I did want to point this out for anyone who was curious. There are three sources of tan β in the amplitude; these all appear in the neutral Higgs diagram:

Each blue dot is a factor of tan β. The Yukawa couplings at each Higgs vertex goes like the fermion mass divided by the Higgs vev. For the down-type quarks and leptons, this gives a factor of m/v ~ 1/cos β ~ tan β for large tan β. An additional factor of comes from the mixing between the s and b quarks, which also goes like the Yukawa coupling. (This is the blue dot on the s quark leg.) Hence one has three powers of tan β in the amplitude, and thus six powers of tan β in the branching ratio.

Outlook

While the LHCb result was somewhat sobering, we can still cross our fingers and hope that there is still an excess to be discovered in the near future. The LHC shuts down for repairs at the end of next year; this should provide ample data for LHCb to probe all the way down to the Standard Model expectation value for this process. Meanwhile, it seems that while I’ve been writing this post there have been intriguing hints of a Higgs (also via our editor)… [edit, 6/23: Aidan put up an excellent intro to these results]

[Many thanks to the experimentalists with whom I’ve had useful discussions about this.]