The Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix is a central pillar of the Electroweak Standard Model. The elements of the matrix must be determined from experiment, but the Model requires the matrix itself to be unitary. Any deviation from unitarity would signal the presence of “new physics” beyond the Standard Model, so tests of CKM unitarity have attracted considerable attention. Currently the most precise test is of the normalization of the top row, which has now reached a precision of 0.06% based on measurements of superallowed 0+ nuclear β decay and of kaon semileptonic and leptonic decays. This work overviews the status of the normalization tests and speculates on likely future improvements.
The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a central pillar of the Electroweak Standard Model. Its origins are with the pioneering 1963 work of Cabibbo , who reconciled the concept of vector-current universality with the recently observed difference between strangeness changing and strangeness non-changing weak decays by proposing that universality was manifest only if one considered the total strength of both the strangeness changing and non-changing decays. In modern terminology we would say that he was the first to realize that there was mixing between the first two generation of quarks. Ten years later, Kobayashi and Maskawa , who were faced with the puzzle of CP violation, demonstrated that it could only be explained by the existence of a third generation of quarks with many more mixing possibilities. They argued that the quark-mass eigenstates are not the same as the weak-interaction eigenstates, but are related by a 3× 3 unitary rotation matrix – the CKM matrix.
By convention, the three quarks with charge 2/3 (u, c and t) are unmixed and all the mixing is expressed in terms of the CKM matrix operating on the quarks with charge -1/3 (d, s and b):
The Standard Model itself does not prescribe the elements of the CKM matrix – they have to be determined experimentally – but the Standard Model does insist that the matrix be unitary. This requirement can be tested if the elements of the matrix can be individually measured with sufficient precision. Should the result violate the matrix's unitarity, then that would be very exciting. It would indicate that the Standard Model is flawed, or approximate, and that some “new physics” must be injected. To date there is no evidence for this. Nevertheless unitarity tests continue to be pursued with ever-increasing precision since they provide limits on any possible new physics, and these limits themselves yield interesting conclusions.
There are two types of tests for CKM unitarity:
Test the orthogonality of rows and columns:
It is difficult to achieve high precision with this type of test since six elements are involved, from all three quark generations. Though it is currently a very active field of research, the results cannot yet match the precision achieved by normalization tests. We will not pursue such tests any further here.
Test the normalization of rows and columns:
For the first row or first column, this normalization test is made much easier by the realization that the contribution from the third-generation member is negligibly small, so only two matrix elements must be determined precisely. To date, the best precision is obtained in the top-row test
In this article, we will concentrate on the top-row normalization test beginning with a discussion of the determination of from nuclear beta decay.
2 Measurements of superallowed β decay
For the past decade at least, the most precise value for the top left element of the CKM matrix, , has been obtained from measurements of β-decay between nuclear states with () = (0+, 1). This type of “superallowed” transition offers a powerful simplicity: Because the axial current cannot contribute to transitions between spin-0 states, only the vector current is involved. Consequently, the measured strength of the transition – its “ft value” – relates directly to the fundamental vector coupling constant GV; so, according to the Conserved Vector Current (CVC) hypothesis, the measured ft values should be the same for all such transitions regardless of their specific host nuclei.
In practice, the expression for the ft value of a superallowed transition must incorporate several small correction terms, which account for radiative and isospin symmetry-breaking effects, both to be discussed in more detail in the next section. It is convenient to combine some of these terms with the ft value and define a “corrected” value, which we write 
where GeV−4s, is the isospin-symmetry-breaking correction and is the transition-independent part of the radiative correction. The terms and comprise the transition-dependent part of the radiative correction, the former being a function only of the electron's energy and the Z of the daughter nucleus, while the latter, like , depends in its evaluation on the details of the nuclear structure of the parent and daughter states. All these correction terms are small, of order 1% or less, with uncertainties at least an order of magnitude smaller than that, so equation (5) provides an experimental method for determining GV to better than a part in a thousand. This result can then yield a value for with the same precision since = , where is the well known  weak-interaction constant for purely leptonic muon decay.
Experimentally, the ft value that characterizes a superallowed transition – or any β transition for that matter – is determined from three measured quantities: the total transition energy, , the half-life, , of the parent state, and the branching ratio, R, for the particular transition of interest. The -value is required to determine the phase-space integral, f, while the half-life and branching ratio combine to yield the partial half-life, t. In a 2009 survey of world data , the ft values of thirteen superallowed transitions in nuclei ranging from 10C to 74Rb were obtained with high precision. Although some more measurements have been published since that time, the overall outcome is unchanged.
The 2009 survey included the results from more than 150 independent measurements with comparable precision of the three properties, , and R, for all thirteen superallowed transitions. Thus, on average, each quantity has been independently determined four times. This is a very robust data set. An overview of the results is presented in Fig. 1, where the three measured properties for each transition appear in the top three panels, and the corresponding ft and values are in the bottom two panels. They are plotted against the Z values of the daughter nuclei. The ft and values also appear numerically in Table 1. It is striking to see in the figure that the three experimental quantities change over orders of magnitude in the span from Z = 5 to Z = 36, yet the ft values change by less than 3%, and the final corrected values are all consistent with a single constant having precision. This, in itself, is convincing proof that CVC is satisfied and that GV can be reliably extracted from the data.
As will be explained in the next section, the correction terms applied to obtain are based on criteria that are completely independent of these results so, in addition to confirming CVC, the results in Fig. 1 strongly support the validity of these correction terms since their presence in the values completely removes the vestigial transition-to-transition variations evident in the ft values.
Table 1. Experimentally determined ft values and corresponding derived values. From ref. 
3 Theoretical corrections to superallowed β decay
3.1 Radiative corrections
In a beta-decay half-life experiment, the rate measured includes not only the bare decay but also radiative decay processes, in which for example the beta particle emits a bremsstrahlung photon. Since it is just the half-life for the bare beta-decay process that is required for the ft value, the measured result needs to be amended with a calculated radiative correction, RC:
In such a calculation it is necessary to consider not just bremsstrahlung but also the exchange of virtual photons and Z-bosons. To order α, the fine-structure constant, the principal graphs are one-photon bremsstrahlung, the -box and -box diagrams. At low energies there is an infrared divergence in the bremsstrahlung process that is cancelled by a corresponding divergence in the -box graph. For these reasons it is convenient to separate the contributions from these graphs into contributions at low energies (long distances) from those at high energies (short distances):
The short-distance correction, , includes the -box and the high-energy part of the -box diagrams and is evaluated by ignoring the hadronic structure and using free-quark Lagrangians. This contribution therefore is universal, being independent of which nucleus is involved in the beta decay. The current value, from Marciano and Sirlin , is
The long-distance correction, , includes the bremsstrahlung and low-energy part of the -box diagram and requires a model calculation of the hadronic structure. Also included in are contributions of order and in which the electron in the -box and bremsstrahlung diagrams is allowed to interact with the Coulomb field of the nucleus. If in the evaluation of the box graphs, it is assumed that the -vertex and -vertex are both with the same nucleon N, then the box graph, like the bremsstrahlung graph, becomes proportional to the Fermi matrix element. This means these contributions are independent of nuclear structure, depending only trivially on the nucleus involved through its total charge Z and the energy of the electron emitted.
By contrast, in the -box graph, if the γ- and W-interactions are with different nucleons in the nucleus, then the evaluation involves two-nucleon operators, necessitating a nuclear-structure calculation. It is convenient therefore to separate into two pieces:
where depends only trivially on the nucleus involved, while requires a nuclear-structure calculation.
Further details on the calculation of and are given in ref. . In Table 2 we give current values of and for the neutron and 13 superallowed transitions of interest.
Table 2. Calculated radiative corrections and in percent units. From ref. 
3.2 Isospin-symmetry breaking
A central tenet of the Standard Model is the Conserved Vector Current (CVC) hypothesis. It asserts that the coupling constant governing the strength of a vector weak interaction is not renormalized by the hadronic environment. Thus, irrespective of which nucleus is under study with superallowed beta decay, the coupling constant GV should remain the same. This tenet, however, is only true in the limit of isospin symmetry. In nuclei, the Coulomb interaction among protons, for example, breaks this symmetry. Thus to be able to test, or use, the constancy of GV it is necessary first to have an estimate of the degree of isospin-symmetry breaking that occurs in the nuclei under study. The symmetry breaking is characterized as a reduction in the square of the Fermi matrix element
where is the symmetry-limit value of the Fermi matrix element squared, and is the correction to it.
Over the years there have been many computations of by various authors using a diverse set of nuclear models [3, 6-14]. We have selected five sets from these works to display in Table 3. Two of the sets use the nuclear shell model to account for configuration mixing, and a mean field – taken to be either a phenomenological Woods-Saxon potential (SM-WS) or a Hartree-Fock computed potential (SM-HF) – to account for the radial mismatch of proton and neutron single-particle wave functions caused by Coulomb polarization. These results by Towner and Hardy [3, 6] are considered to be semi-phenomenological in that a number of isospin-specific nuclear properties have been fitted in their derivation: the different proton and neutron separation energies in the parents and daughters respectively, and the b- and c-coefficients in the isobaric multiplet mass equation (IMME) for each T=1 multiplet that includes a parent and daughter state.
Alternative models do not lend themselves to local phenomenological constraints. One, by Sagawa et al.  and improved upon by Liang et al. , treats the even-even nucleus of the parent-daughter pair as a core, and the analog odd-odd nucleus as a particle-hole excitation built on that core. The particle-hole calculation is carried out in the charge-exchange random-phase approximation (RPA). The more recent work  replaces zero-range interactions with finite-range meson-exchange potentials and a relativistic rather than nonrelativistic treatment (RHF-RPA) is used. In a variation of this approach, density-dependent meson-nucleon vertices were introduced in a Hartree (only) computation with nonlocal interactions (RH-RPA).
Most recently, Satula et al.  used an isospin- and angular-momentum-projected density functional theory (DFT). This method accounts for spontaneous symmetry breaking, configuration mixing and the long-range Coulomb polarization effects.
Table 3. Five recent sets of calculations from model approaches labelled DFT, RHF-RPA, RH-RPA, SM-HF and SM-WS (see text). Also given is the chi square per degree of freedom, , from the confidence test proposed in Ref. 
The five sets of values in Table 3 show a wide variation. It would be useful if some yardstick were available to distinguish the quality of one set relative to another. Towner and Hardy  proposed such a test using the premise that the CVC hypothesis is valid. The requirement is that a calculated set of values should produce a statistically consistent set of values, the average of which we can write as . Then Eq. (5) can be written for each individual transition in the set as
For any set of corrections to be acceptable, the calculated value of for each superallowed transition must satisfy this equation where ft is the measured result for that transition and has the same value for all of them. Thus to test a set of values for n superallowed transitions, one can treat as a single adjustable parameter and use it to bring the n results for the right hand side of Eq. (11), which are based predominantly on experiment, into the best possible agreement with the corresponding n calculated values of . The normalized χ2, minimized by this process, then provides a figure of merit for that set of calculations, The χ2 for each fit, expressed as , where is the number of degrees of freedom, is given in the last row of Table 3.
The most obvious outcome of this analysis is that the model, SM-WS, has a χ2 smaller by a factor of five than the other four cases cited. For this reason the SM-WS values are used in the determination of the CKM matrix element, . However, the other cases can be used for establishing an error assignment on this analysis.
4 Results for
The consistency of the values (and consequently GV) obtained from the 13 measured superallowed 0+ β transitions (see Sec. 2) is a crucially important result. Without consistency, there is no coupling “constant” and there can be no justification for extracting a value of . With consistency demonstrated, however, can be reliably obtained from the following equation :
where GeV−2  and is the average measured value. Taking = 3071.81(83) s from the 2009 survey , we arrive at = 0.97425(22), a value with 0.02% precision.
This result is certainly the most precise current determination of , but superallowed 0+ β decay is not the only experimental approach to . Neutron decay, nuclear mirror decays, and pion beta decay have all been used for this purpose. Though these other methods cannot as yet compete with 0+ decays for precision, they can lend useful confirmation to the more precise result.
Neutron β decay has one important advantage to offer: Its analysis does not require the application of corrections for isospin-symmetry breaking () or for nuclear-structure-dependent radiative effects (). But it also has big disadvantages: neutrons are notoriously difficult to confine; and their decay is not restricted to the vector weak interaction, as are the 0+ decays. The axial-vector current is also present so, in addition to its ft value, some other parameter, usually the β asymmetry from the decay of polarized neutrons, must be measured to determine the ratio of the vector to axial-vector contributions. These experimental challenges have so far limited the achievable precision and, indeed, have at times led to completely inconsistent results. Even as of 2012, when the world average value for the neutron lifetime is τ = 880.4(10) s [4, 17] and for is −1.2731(20) [4, 18, 19], both have normalized χ2 values well above unity, 2.7 for the former and 4.1 for the latter, which have caused their uncertainties to be enlarged. Together, these results yield a value for of 0.97520(140), which agrees with the superallowed result but has an uncertainty more than six times larger.
The nuclear mirror decays, like neutron decay, also require an additional correlation measurement to sort the weak vector contribution from the axial-vector one. However, unlike neutron decay, they provide no simplification in avoiding the nuclear-structure-dependent corrections, and . A recent analysis of the five measured mirror transitions , which incorporated and corrections, yielded = 0.97190(170), a result that is only slightly less precise that the neutron result and is consistent within 1.2 combined standard deviations with the value obtained from the superallowed 0+ decays.
Finally, pion β decay should, in principle, be the best way to determine . It is a pure vector decay between two spin-zero members of an isospin triplet and is therefore analogous to the superallowed decays, but is uncontaminated by nuclear-structure uncertainties. Unfortunately, in practice the branching ratio is very small (∼10−8) and difficult to measure with sufficient precision. The most recent, and best, measurement is by the PIBETA group , whose result corresponds to a value for of 0.97420(260) .
The results from all four types of measurement are plotted in Fig. 2. Obviously they are consistent with one another but, because the nuclear 0+ value is so much more precise, it dominates the average. Furthermore, it is the most secure result, its value having changed little in the past two decades except for a steady (and significant) improvement in its uncertainty (see Fig. 6 in Ref. ). Thus we continue our practice  of adopting the 0+ result as the recommended value, namely
while noting that the results from the other measurement techniques are entirely consistent with this value.
The top row CKM matrix element is best determined from kaon semi-leptonic decays, , of both charged and neutral kaons, and from the purely leptonic decay of the kaon, the most important mode being . Other determinations from hyperon decays and hadronic tau decays do not have the precision at the present time to challenge the results from kaon decays and therefore will not be considered here.
The experimental inputs for the determination of are the rates – lifetimes and branching ratios – and the form factors required for the phase space integration. The best current value, presented at the CIPANP12 workshop  is
Here is the semi-leptonic decay form factor at zero-momentum transfer. Its value is close to unity. In fact, the CVC hypothesis in the exact SU(3) symmetry limit establishes its value to be exactly one. But SU(3) symmetry is broken to some extent and a theoretical calculation is required to estimate the departure of from unity. Today, lattice QCD calculations are used for this purpose, replacing the former semi-analytic approaches based on chiral perturbation theory. Two groups working on the classification and averaging of results from lattice QCD [23, 24] have joined their efforts, forming the Flavor Lattice Average Group (FLAG-2) to provide recommended values for these form factors . For their recommended value, after symmetrizing the uncertainty, is , yielding
An independent determination of can be obtained from the purely leptonic decay of the kaon. Further, if it is considered as a ratio with the leptonic decay of the pion, , the hadronic uncertainties can be minimized and the result yields the ratio of the CKM matrix elements . In the analysis of the FlaviaNet group , the current result is
The most recent FLAG-2 average of the four published determinations of the decay constants in lattice QCD is . More significant is a new calculation of a correction to the ratio of experimentally determined decay rates, for isospin-symmetry breaking  that nearly doubles the formerly-used correction. The resulting value for reported at the CIPANP12 workshop  is
Thus, we now have three pieces of data – from nuclear decays, Eq. (13), from decays, Eq. (15), and the ratio from decays, Eq. (17) – from which to determine two parameters and . An unconstrained fit does not change the input value of and yields for our final selection
The third element of the top row of the CKM matrix, , is very small and hardly impacts on the unitarity test at all. Its value from the 2012 PDG compliation  is
6 CKM unitarity tests
We can test the unitarity of the CKM matrix by evaluating the sum of squares of the measured elements in any one of the rows or columns of the matrix, and then comparing the result with unity. We have set the stage for testing the top row elements by individually examining the results for , and . Now, combining the results from Eqs. (13), (18) and (19), we obtain the result
a result that shows unitarity to be fully satisfied to a precision of 0.06%. Only and contribute perceptibly to the uncertainty and their contributions to it are almost equal. This may seem surprising since is known to much higher relative precision than , but it follows from the fact that contributes 95% to the unitarity sum itself.
No other row or column approaches this precision on a unitarity test. The first column comes closest, with = 1.0021(51) , but this is a factor of ten less precise than the top-row sum. The corresponding sums for the second row and second column are another order of magnitude less precise. Without question the top row sum provides the most demanding test of CKM unitarity, is its dominant contributor, and superallowed 0+ β decay is effectively the sole experimental source for the value of .
7 Outcome from unitarity tests
The unitarity sum established in Eq. (20) can be used to set limits on new physics beyond the Standard Model in two ways: (i) directly, via a new semi-leptonic interaction (e.g. right-hand currents) and (ii) indirectly, via loop-graph contributions to the radiative correction (e.g. extra Z-bosons). One example of each type is discussed in some detail in Sect. 9 of Ref. . Here we just update the results given there for the current unitarity sum.
First, for right-hand currents, we write
Here is the amplitude for a semi-leptonic weak interaction with left-hand couplings in the lepton sector and right-hand couplings in the hadron sector. Clearly, the result in Eq. (21) is consistent with no right-hand currents.
Second, for extra Z-bosons, we write
yielding the one-standard deviation limits on Δ of
Marciano and Sirlin  relate Δ to the mass of the extra Z boson via
Here is the mass of the extra Z-boson contributing to the radiative correction and the W-boson mass. Noting that the correction Δ is negative, we obtain from the lower limit in Eq. (23)
The confidence level is 68%. This limit is slightly weaker than the limit of 460 GeV given in  as the lower limit on Δ in Eq. (23) has been increased slightly through a more recent revision in the value (see Section ' and '). Impressive though this limit remains, somewhat higher limits have been obtained in direct searches at proton and electron colliders, as described in the survey of Erler and Langacker  in the 2012 PDG listings.
8 Future prospects
The CKM unitarity sum tests a key premise of the Standard Model so the limits of its uncertainty set boundaries on the scope of possible new physics that may exist beyond the Standard Model. The opportunity to expose new physics, or to further limit the possible scope of new physics, together provide ample motivation to push for still tighter uncertainties on the unitarity sum. What are the prospects for achieving that goal?
As of 2012, after a concentrated effort, the measurements on decays have arrived at the value for in Eq. (14), which has a relative uncertainty of about . However, the lattice QCD estimate for the form factor, , has a much larger uncertainty, so it dominates the uncertainty on . While little improvement in the experimental result is anticipated in the near future , Van de Water  predicts that the uncertainty on the lattice calculation will be decreased to 0.2% in 2014 and to 0.1% by 2020. Thus, we can anticipate an improvement in the uncertainty of by a factor of 2 in the near future and perhaps by a bit more in five or ten years.
The error budget for the determination of is similarly dominated by uncertainties in the theoretical correction terms. Of the total 0.023% uncertainty on as derived from the superallowed 0+ decays, 0.018% is contributed by the uncertainty associated with the radiative correction and 0.010% comes from the nuclear-structure-dependent corrections, . Only 0.008% can be considered experimental in origin. This error budget is illustrated in the lower left panel of Fig. 2.
It is also instructive to observe that the uncertainty is the same size in the middle two lower panels of Fig. 2 as well. Thus, even if the experimental uncertainties can be reduced dramatically for neutron decay or for the mirror decays, the overall uncertainty on will be limited to about the same value by . For pion decay, in the bottom right panel, the uncertainty in is less than for the other types of measurement, but there is no apparent path to experimental error bars that are reduced by the factor of 20 needed for the result to have an impact on .
Any improvement in is a problem for theory and, unfortunately, we are not aware of any attempts currently underway to reduce its uncertainty.
In the meantime some small improvement in can be obtained from a reduction in the uncertainty associated with the nuclear-structure-dependent corrections, . As already discussed in Sec. 3.2, there is considerable activity in this area on the theoretical side. However, experiment can also play an important role too: Every set of nuclear-structure-dependent corrections can be tested by being applied to the uncorrected experimental ft values to obtain a set of values, which can then be evaluated for the consistency required by CVC (see Eq. (11) and Table 3). The more precisely the ft values have been measured, the more discriminating this test can be; and if new superallowed transitions with larger predicted nuclear corrections can be measured, the test will be improved still more.
Measurements with reduced uncertainties are regularly being reported for the thirteen previously measured transitions, which are listed in Tables 2 and 3. Additionally, it is to be anticipated that at least three new superallowed 0+ transitions – from 26Si, 34Ar and 38Ca – which all have relatively large predicted nuclear corrections, will be fully characterized within two or three years. Together, these improvements and additions should serve to reduce the overall uncertainty attributed to by 15%.
Considering anticipated improvements to both and , the uncertainty on the unitarity sum will likely be dropped from ±0.00060 to ±0.00040 in the next few years. If the uncertainty on could also be reduced by, say, a factor of two, then the unitarity uncertainty would become even less, ±0.00030. This reduction would of course improve the limits given in Section 'Outcome from unitarity tests': for example, with the limits , the limit on the extra Z boson mass would become 650 GeV, which is comparable to the limits from direct searches.
This is likely the extent of improvements for the foreseeable future.
This work was supported by the U.S. Department of Energy under Grant No. DE-FG03-93ER40773 and by the Robert A. Welch Foundation under Grant No. A-1397.
John Hardy received his Ph.D. degree in nuclear physics from McGill University (Montréal) in 1965. He was a post-doctoral fellow at Oxford University and a Miller Fellow at the University of California in Berkeley before joining the Chalk River Nuclear Labs of Atomic Energy of Canada in 1970. In 1997, he joined the Texas A&M University where he is now a Distinguished Professor. In 2006, he shared the Bonner Prize with Ian Towner from the American Physical Society.
Ian Towner is a retired professor of physics from the Queen's University, Canada and an adjunct professor of physics at the Texas A&M University. Together with John Hardy, he won the American Physical Society's Bonner Prize in 2006.