New Physics and the CKM Matrix

Educated lay people who are even passingly familiar with particle physics know about the six kinds of quarks, three kinds of electrons, three kinds of neutrinos, three weak particle force carriers, photons, gluons and if they are particularly at the top of their game, the way that anti-particles, color charge, parity and polarization permutate these basic components of the Standard Model.

Most pertinently for this discussion, there are two kinds of quarks that make up ordinary protons and neutrons. The up which has charge +2/3 and the down which has the charge -1/3. A proton is made up two ups and one down. A neutron is made of two downs and one up. A wide variety of other combinations can be formed, but none are stable. There are heavier versions of each kind of quark that fall into three "generations." One set from lightest to heaviest is up, charm, top. The other set from lightest to heaviest is down, strange, bottom. Particles with second or third generation quarks, on average, rapidly decay via the weak force to lighter particles.

Less widely known, because it doesn't appear in the chart illustrating the different fundamental particles that go into the Standard Model is the Cabibbo–Kobayashi–Maskawa matrix, which is universally known as the CKM matrix, because spelling out proper names is just not what physics majors like to do. (The parallel matrix for electrons and neutrinos is the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix) for which good estimates for all of the constants have not been determined.)

Basically, the CKM matrix is a three by three matrix that summarizes the constants in the Standard Model that determine probability that the weak force will convert one kind of quark into another and predicts that a certain amount of CP violation will exist. The entries in the matrix correspond to and have values according to experiments conducted to date of approximately:

Up-Down (0.9743) : Up-Strange (0.225) : Up-Bottom (0.0035)

Charm-Down (0.225) :Charm-Strange (0.9735) : Charm-Bottom (0.041)

Top-Down (0.0086) : Top-Strange (0.040) : Top-Bottom (0.99915)

The matrix has a number of properties that cause the nine different entry values to be derivable from less than nine constants. Lincoln Wolfensten's parameterization derives its values with formulas that have four parameters, with their approximate values as follows: λ (0.226), A (0.81), ρ (0.14), and η (0.35). In Wolfenstein's model four of the entries (ud, us, cd, cs) depend only on λ, entry tb is exactly equal to one, two entries (ts and cb) depend on both λ and A, and two entries (td and ub) depend on all four constants of his parameterization.

As explained in a recent post at the blog Résonaances:

During the last decade the Standard Model description of flavor transitions has been put to multiple tests, especially in the B-meson sector. [Ed.The B-meson is composed of a bottom antiquark and either an up, down, strange, or charm quark.] The overall agreement between theory and experiment is excellent. . . . Here and there, however, one finds a few glitches - most likely experimental flukes or underestimated theory errors but intriguing enough . . . . This year there has been a lot of commotion about the . . observation of the same sign di-muon asymmetry, since the Standard Model predicts this effect should be well below the current experimental precision. If the . . . result is confirmed, it would be a clear indication of new physics contribution to CP violation in the mixing of neutral B-mesons. Another, less publicized 3-sigma blip is the tension between:

* the CP asymmetry in the Bd meson decay into J/ψ + kaon,

* the branching fraction of the decay of a charged B meson into a tau lepton and a tau neutrino. . . .

The parameters λ and A are well measured in several different ways that yield consistent results. Therefore one is more interested in constraints on the remaining two parameters called ρ and η.

The B-meson processes that have recently been producing beyond the Standard Model experimental CP violations that don't match the theoretical prediction are dependent upon parameters called ρ and η, which are numbers that must be between zero and one that influence the td and ub values in the CKM matrix. If Wolfenstein's parameterization is correct, the ub value should be (ρ-iη), where i is the square root of negative one, times A times λ cubed, while the td value should be (1-ρ-iη) times A times λ cubed.

It is possible to adjust values of ρ and η in the Standard Model to fit individual anamolous CP value results from experiments, but it is not quite possible to choose values for these parameters that predict results within three standard deviations of all of the experimental data, although if one believes that the published error estimates are underestimates, it is possible to get consistent values for these parameters that fit all of the data. The gap between the values that fit the experiments for ρ are comparatively modest -- the experimental best fits for individual experiments are between 0.1 and 0.2 and have significant error margins, so they almost reconcile. The canonical value of 0.135 for ρ is pretty close to an overlap of the error bars on existing measurements. The gap betwen the values that fit the experiments for η, which controls the magnitude of quark sector CP violations in the Standard Model, are bigger -- the best fit for one experiment for η is a little more than 0.45, while the best fit for another is a little less than 0.35.

If the experimental data's error bars are underestimated, then more experiments will reach consensus values for ρ and η that fit and make new physics in the quark flavor changing process unnecessary. But, if more experiments make it clear that there is no way that ρ and η can be chosen to fit the data, than Wolfenstein's parameterization is incorrect, the CKM matrix is "broken" and we need beyond the Standard Model physics to explain what we are observing.

For example, a "broken" CKM matrix could be an indication that there are actually four, rather than merely three generations of quarks, a possibility explored, for exampe, here. A four generation Standard Model would, if it existed, have a number of theoretical attractions:

An additional fourth generation (SM4) is one of the simplest extensions of the SM, and retains all of its essential features: it obeys all the SM symmetries and does not introduce any new ones. At the same time, it can give rise to many new effects, some of which may be observable even at the current experiments. Even though the fourth-generation quarks may be too heavy to have been produced at the pre-LHC colliders, they may still affect low-energy measurements through their mixing with the lighter quarks. The up-type quark t′ would contribute to b → s and b → d transitions at the 1-loop level, while the down-type quark b′ would contribute similarly to c → u and t → c. The addition of a fourth generation to the SM leads to a 4 × 4 quark mixing matrix CKM4, which is an extension of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix in the SM. The parametrization of this unitary matrix requires six real parameters and three phases. The additional phases can lead to increased CP violation, and can provide a natural explanation for the deviations from the SM predictions seen in some measurements of CP violation in the B-meson system. A heavy fourth generation can play a crucial role in the dynamical generation of the electroweak (EW) symmetry breaking [8]. The large Yukawa couplings of the fourth generation quarks, together with the possible large phases, can help efficient EW baryogenesis. In addition, the SM4 is consistent with the SU(5) gauge couplings, and hence can be unified without supersymmetry.