Heavy Flavour Physics


Last week I spent a few days in Oxfordshire to attend the 2007 Cosener’s Forum on Heavy Flavour Physics. This is a brief report of some background I was able to pick up about the field. The conference included a mix of experimentalists and theorists and focused mainly on flavour physics in the near and mid-ranged future. The trip was a chance to touch base with my superviser in Durham and place my new research project in context.

Cosener’s House, pictured below, is a conference site owned by the Science and Technology Facilities Council. It’s situated in Abingdon, a small town between Oxford and the Rutherford Appleton Laboratory, along a distributary of the Thames.

I’ll try to give a report, though I was a neophyte to the conference and can only provide a student’s-eye-view. I apologise ahead of time for any inaccuracies. Most of the information below Any opinions are only mine and reflect a limited (but growing!) background. The forum program and slides are available online.

Background: The CKM Matrix

The flavour structure of the Standard Model comes from the Cabibbo-Kobayashi-Maskawa (CKM) matrix that mediates inter-generation transitions between quarks. If we disregard the Yukawa terms, the Standard Model lagrangian obeys an [SU(3)]^5 flavour symmetry. One can rotate each of the five types of fermion in `flavour space.’

Parenthetical note: recall the Standard Model fermion structure

  • Q, the left-handed quark doublets (up- and down-type)
  • u, the right-handed up-type quarks
  • d, the right-handed down-type quarks
  • L, the left-handed lepton doublets (charged and neutrino)
  • e, the right-handed charged leptons

(The Standard Model is a chiral theory, so these are different particle representations. Also, we’re ignoring the right-handed neutrino… i.e. the pre-1998 Standard Model.)

The Yukawa sector contains terms of the form:

(left handed doublet) (Higgs doublet) (right handed fermion)

When the Higgs takes a non-zero vacuum expectation value, these become mass terms. The fermions carry flavour indices, so that the coefficient of these terms are actually 3\times 3 flavour matrices. Consider the \lambda_{ij}Q_iHu_j term where I’ve written the flavour indices explicitly. Let us write this in matrix notation as H (Q\lambda u). We can use the [SU(3)]^5 symmetry to rotate the Q and u fields into Q'_i = A_{ij}Q_j and u'_i = B_{ij}u_j such that H Q'\lambda u' = H Q (A^\dagger \lambda B) u where A^\dagger \lambda B is a diagonal matrix.

Parenthetical statement: \lambda can be decomposted into \lambda = M_H U where M_H is hermitian and U is unitary. Thus we can diagonalise \lambda by sandwiching it between the appropriate unitary matrices.

Great: we can diagonalise the up-type quark Yukawa terms. Thus, giving the Higgs field a vacuum expectation value, we can diagonalise the up-type quark mass terms. This means that the propagating fields are also flavour eigenstates.

But we run into a problem with the down-type quark masses. The relevant term is H (Q\eta d) where \eta is the Yukawa coupling matrix. We can try to diagonalise \eta just as we did for \lambda, but we no longer have any freedom to rotate Q since it was fixed when we diagonalised \lambda. We can still rotate d to `partially’ diagonalise \eta, but since Q is fixed we end up with a left over unitary matrix multiplying a diagonal matrix. This matrix is the CKM matrix that mediates the off-diagonal (i.e. flavour changing) currents.

In slightly more physical words: the CKM matrix represents the mixing between physical propagating states (mass eigenstates) versus the flavour states in which the interactions occur. One can think about it this way: in a Feynman diagram, propagators are mass eigenstates, while vertices are flavour eigenstates.

Motivation for Flavour Research

The old motivation for flavour research was CP violation, i.e. the asymmetry between particles and their charge-parity conjugate states. CP violation was first observed in the flavour-oscillations of the kaon system. At least in the quark sector, CP violation and nontrivial flavour structure go hand-in-hand. Hence flavour physicists could tell the general public that the fact we are here indicates that there is more matter than anti-matter in the universe. Interesting flavour structure is required for CP violation which is, in turn, required for the matter excess (c.f. Sakharov conditions). Thus we can easily explain to funding agencies why it is important to fund accelerators that study B-physics, where we can experimentally probe the structure of the CKM matrix. This was great because the Standard Model matrix doesn’t appear to account for the required CP violation to generate the observed matter asymmetry. (See this excellent post on matter/antimatter asymmetry by Mark Trodden on his previous blog.)

Unfortunately, we can no longer just point to baryon asymmetry from CP violation as a `guaranteed’ reason to study flavour physics. The discovery of nonzero neutrino masses in 1998 opened the door to a new mechanism to generate matter/antimatter asymmetry: leptogenesis. In this model a lepton-antilepton asymmetry in the early universe turns into a baryon-antibaryon asymmetry due to solitonic `sphaeleron’ gauge field configurations. For those interested in further reading, I’ve foundnice review articles by Buchmuller/Peccei/Yanagida and by Mu-Chun Chen.

The current `party line’ for quark flavour physics is a little more subtle. Barring unnatrual fine-tuning, we strongly suspect that there is new physics at the TeV-scale. In order to address the hierarchy problem (see previous link), TeV-scale physics must couple to the Standard Model, which has a nontrivial flavour structure. Hence the new physics must also have a nontrivial flavour structure. (This is a bit handwavy, sorry.) Unfortunately this message isn’t as sexy for the general public. But the point is that we should expect either (i) new flavour structure beyond the Standard Model or (ii) some good reason why there wouldn’t be new flavour structure (see “minimal flavour violation,” below).

The selling point for flavour physics machines (e.g. BaBar and Belle, see below, “B-physics”) is the discovery potential in flavour-violating processes. The great feature of some of the interesting flavour systems—i.e. kaons, B-mesons, and D-mesons—is that one can compare processes that are dominated by tree level processes versus loop processes.

The Standard Model makes predictions for the branching ratio of both types of processes. But if there is new physics beyond the standard model at, say, the TeV scale, then we would expect new particles to contribute as off-shell internal lines in the loop diagrams. Hence deviations in the branching ratio of loop diagrams are signatures of new physics. These loop diagrams are called penguin diagrams.

Because internal lines (i.e. the particles in the loop) needn’t be on-shell, the point is that penguin diagrams can probe physics at a scale much higher than the centre-of-mass energy of the interaction. For example, even though the centre-of-mass energy of collisions at current B-meson machines is on the order of 10 GeV (and parton interactions are even less), they have a discovery reach of perhaps 10 TeV. The trade off, of course, is that this is very different from a precision machine where on-shell new physics is produced. The trade-off for greater reach is much less information. “Penguin probes” of high scales are really looking for flavour structure at high scales, so it’s possible for “flavour-trivial” new physics to be missed by such searches. Also, given a deviation from the Standard Model prediction, one is rather limited in what one can say about the possible new particles contributing to the deviation.

The punchline is that our experimental exploration of the Terascale will rely on complimentarity between the high-energy collisions in general purpose detectors (ATLAS, CMS) and the lower-energy flavour factories (BaBar, Belle, LHCb).

A point of language: most phenomenologists seem to use the acryonym BSM (beyond the standard model) to refer to any unexpected discoveries in particle physics. Flavour physicists seem to use the acronym NP (new physics) instead. I’m not sure why this is.

B Physics

There are a few favoured systems that physicists use to study heavy flavour physics. (The `heavy,’ I suspect, is meant to distinguish from the neutrino sector.) As mentioned before, the kaon system was the first observation of CP-violation. The charmed countarparts, the D-mesons, are also constraining other components of the CKM matrix. For now, however, the bread-and-butter of an experimental flavour physicist are the B-mesons.

The current machines of interest are BaBar at SLAC (California) and Belle at KEK (Japan). Because they produce B-mesons at a certain rate, these machines are called B-factories. The name BaBar (B\bar{B}), by the way, is a pun of the fictional elephant of children’s literature. The BaBar collaboration obtained permission to use the elephant as their logo and mascot.

Image from BaBar workbook.

These B factories collide electrons and positrons asymetrically. For example, the PEP-II electron-positron circulation ring that feeds into BaBar is composed of a 9 GeV electron beam colliding with a 3.1 GeV positron beam. In the centre-of-mass frame, the collision of these two beams is just on-shell for producing \Upsilon(4S) particles, which then decays into a B-\bar{B} pair. Because the beams are asymmetric, the lab frame is different from the centre-of-mass frame and the B-mesons are produced with nonzero momenta. Thus they drift a bit before they decay.

Now here’s the beautiful part: the clever experimentalists can now tag the decay products of the B and \bar{B}. They can then reconstruct the kinematics of each decay and identify the distance each meson travelled before it decayed. By dividing the distance travelled by the speed at which the meson was travelling (and then inverting), one can determine the decay rate of the meson into a given set of final products. Nature is generous, and there are some decays with final-state products that are the same for both the B and anti-B mesons. By comparing the decay rates for these processes, one can measure CP asymmetry. (If CP were a good symmetry, then the B and anti-B decay rates to the same final states should be identical.) Ta-da!

The Unitarity Triangle(s)

The B-factories have been very sucessful in testing the flavour structure of the Standard Model. From here, the flavour community has done an excellent job of presenting the constraints on the CKM matrix in terms of the unitarity triangle below.

What’s going on in this colourful image? Recall that the CKM matrix is unitary. (Sanity check: it represents the mixing matrix between states so that it must be unitary.) Then for a unitary matrix V_{ij}, we know $1 = VV^\dagger$. And hence for off diagonal elements, such as the d-b element:

0=V_{du}V^\dagger_{ub}+V_{dc}V^\dagger_{cb}+ V_{dt}V^\dagger_{tb}

0=V_{du}V^*_{bu}+V_{dc}V^*_{bc}+ V_{dt}V^*_{bt}

(Recall that u-d, c-s, t-b are the diagonal elements.) Then dividing by V_{dc}V^*_{bc}, we get the following relation:

0= \frac{V_{du}V^*_{bu}}{V_{dc}V^*_{bc}}+1+ \frac{V_{dt}V^*_{bt}}{{V_{dc}V^*_{bc}}}

We can plot each term in the sum as a vector in the complex plane. The fact that they sum to zero means that we can join the heads and tails of these vectors to form the unitarity triangle, pictured above. The colours in the plot represent experimental constraints.

The punch line is that all of the angles appear to match the Standard Model predictions. That’s a bummer for those looking for signals of new physics, but we take what nature gives us. To be fair, the current experimental bounds do leave wiggle room for new discoveries. In his talk at the forum, Amarjit Soni (quite the optimist) exhorted us to remember neutrino masses — for a long time we suspected neutrinos were massless, but it turned out that that they were just mysteriously small.

So is that it? Are we left waiting for experimentalists to further confirm the Standard Model while quietly crossing our fingers and hoping for something different? The situation is at least slightly more interesting than this. As I mentioned earlier, we believe there is new physics with nontrivial flavour structure at the TeV scale. If this physics contributes no new flavour structure to the Standard Model, then the TeV scale physics must also be given by the same CKM matrix. This is the Minimal Flavour Violation scenario, to be discussed briefly in the next section.

One note, though: there appears to be a small 2.6 \sigma discrepancy in the measured value of \sin (2\beta), where \beta is one of the angles in the unitarity triangle. All measured values are slightly less than the expected value. However, the general sentiment I found at the forum was that it is too early too trust this result. The uncertainties are still rather large, coming both from experimental and theoretical sources (theoretical predictions are based on lattice calculations). Further, the discrepancy is based on a `statistically dubious’ averaging of many measured values. For now this is only a hint of a signal for new physics.

Minimal Flavour Violation

The current constraints (according to Adrian Bevan’s talk) on the b-u and t-d transitions are:

  • \beta measured to within 1^ \circ
  • \alpha measured to within 7^ \circ
  • \gamma just beginning to be constrained

We check consistency of these values by doing experiments that overconstrain the CKM matrix. There are 6 unitarity triangles associated with the independent elements of a unitary 3\times 3 matrix. The next generation of flavour machines will, indeed, further constrain these measurements. However, if one were to make a bet today about the final word on flavour structure at the TeV scale, the Nevada bookies would be betting on Minimal Flavour Violation. In this scenario, the flavour structure of the new physics at the TeV scale is goverend by the Standard Model CKM matrix.

Flavour probes of new physics are a bit of a coarse tool. The corresponding theoretical technique is to parametrise TeV physics as effective operators. The Minimal Flavour Violation ansatz is included by promoting the Yukawa matrices to `spurion fields.’ That is to say that we pretend the Yukawa matrices \lambda_{ij} as \mathbf{\bar{3}}\times\mathbf{3} fields. At the end of the day we’ll remember that these aren’t really physical fields, this is just a book keeping tool. With this tool even the Yukawa terms in the Standard Model lagrangian — the ones that originally violated flavour — obey the [SU(3)]^5 flavour symmetry. We can then impose this flavour symmetry on all higher-dimensional effective operators. Upon remembering that the spurion fields are really constants, we see that the flavor-violating higher-dimensional operators are those which included the Yukawa spurions. And hence, bada bing bada boom, we have parameterised new physics where all flavour-violation comes from the Standard Model; i.e. Minimal Flavour Violation for general new physics.

Some follow-up reading:

New Machines and New Physics

The Cosener’s Forum was also meant to provide a framework for future planning for the UK flavour community. Looking towards the next generation of B-factory, LHCb, the key emphasis was the complimentarity between flavour machines and `general purpose detectors’ (GPDs) like the better-known ATLAS and CMS. (If the LHC is the couch on Friends, ATLAS and CMS are studly Joey and sexy Rachel, while LHCb is geeky Ross.)

We would like to connect the direct measurements of new physics at the general purpose detectors with the indirect measurements at the flavour machines. In a nutshell, the goal is to connect the experimental results from one sector to the theoretical parameters of the other. This, however, is apparently more non-trivial than it sounds. A good reference is the relevant page on the CERN twiki: https://twiki.cern.ch/twiki/bin/view/Main/ColliderAndFlavour

Some further background on flavour physics:

Thanks to Science and Technology Facilities Council for supporting the 2007 Coseners Forum on Heavy Flavour Physics. Special thanks to all who spent time who chatted with me and offered advice and insight for a generally confused student.

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