CKM unitarity normalization tests, present and future

Authors


Corresponding author E-mail: hardy@comp.tamu.edu

Abstract

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+inline image 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.

1 Introduction

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 [1], 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 [2], 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):

display math(1)

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:
    display math(2)
    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:
    display math(3)
    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
    display math(4)

In this article, we will concentrate on the top-row normalization test beginning with a discussion of the determination of inline image 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, inline image, has been obtained from measurements of β-decay between nuclear states with (inline image) = (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” inline image value, which we write [3]

display math(5)

where inline image GeV−4s, inline image is the isospin-symmetry-breaking correction and inline image is the transition-independent part of the radiative correction. The terms inline image and inline image 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 inline image, 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 inline image with the same precision since inline image = inline image, where inline image is the well known [4] 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, inline image, the half-life, inline image, of the parent state, and the branching ratio, R, for the particular transition of interest. The inline image-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 [3], 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, inline image, inline image 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 inline image values are in the bottom two panels. They are plotted against the Z values of the daughter nuclei. The ft and inline image 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 inline image values are all consistent with a single constant having inline image precision. This, in itself, is convincing proof that CVC is satisfied and that GV can be reliably extracted from the data.

Figure 1.

Results from the recent survey [3] of 13 precisely measured superallowed β transitions between (inline image) = (0+, 1) states. The parents of these transitions, from left to right, are 10C, 14O, 22Mg, 26Alm, 34Cl, 34Ar, 38Km, 42Sc, 46V, 50Mn, 54Co, 62Ga and 74Rb. The top three panels present the average of measured inline image, log inline image and log R values for each transition. The bottom two panels give the corresponding ft and inline image values. The shaded horizontal line in the bottom panel represents the overall average inline image value for all transitions. All error bars are shown: in the cases where none are visible, they are smaller than the data point.

As will be explained in the next section, the correction terms applied to obtain inline image 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 inline image values completely removes the vestigial transition-to-transition variations evident in the ft values.

Table 1. Experimentally determined ft values and corresponding derived inline image values. From ref. [3]
Nucleusft (s)inline image (s)
10C3041.7(43)3076.7(46)
14O3042.3(27)3071.5(33)
22Mg3052.0(72)3078.0(74)
26Alm3036.9(9)3072.4(14)
34Cl3049.4(12)3070.6(21)
34Ar3052.7(82)3069.6(85)
38Km3051.9(10)3072.5(24)
42Sc3047.7(14)3072.4(27)
46V3049.5(9)3073.3(27)
50Mn3048.4(12)3070.9(28)
54Co3050.8(15)3069.9(33)
62Ga3074.1(15)3071.5(72)
74Rb3084.9(78)3078(13)

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:

display math(6)

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 inline image-box and inline image-box diagrams. At low energies there is an infrared divergence in the bremsstrahlung process that is cancelled by a corresponding divergence in the inline image-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):

display math(7)

The short-distance correction, inline image, includes the inline image-box and the high-energy part of the inline image-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 [5], is

display math(8)

The long-distance correction, inline image, includes the bremsstrahlung and low-energy part of the inline image-box diagram and requires a model calculation of the hadronic structure. Also included in inline image are contributions of order inline image and inline image in which the electron in the inline image-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 inline image-vertex and inline image-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 inline image-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 inline image into two pieces:

display math(9)

where inline image depends only trivially on the nucleus involved, while inline image requires a nuclear-structure calculation.

Further details on the calculation of inline image and inline image are given in ref. [6]. In Table 2 we give current values of inline image and inline image for the neutron and 13 superallowed transitions of interest.

Table 2. Calculated radiative corrections inline image and inline image in percent units. From ref. [6]
Nucleusinline image (%)inline image (%)
Neutron1.4902(2)0.0
10C1.679(4)−0.345(35)
14O1.543(8)−0.245(50)
22Mg1.466(17)−0.225(20)
26Alm1.478(20)0.005(20)
34Cl1.443(32)−0.085(15)
34Ar1.412(35)−0.180(15)
38Km1.440(39)−0.100(15)
42Sc1.453(47)0.035(20)
46V1.445(54)−0.035(10)
50Mn1.445(62)−0.040(10)
54Co1.443(71)−0.035(10)
62Ga1.459(87)−0.045(20)
74Rb1.50(12)−0.075(30)

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

display math(10)

where inline image is the symmetry-limit value of the Fermi matrix element squared, and inline image is the correction to it.

Over the years there have been many computations of inline image 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: inline image 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. [9] and improved upon by Liang et al. [10], 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 [10] 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. [14] 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 inline image 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, inline image, from the confidence test proposed in Ref. [15]
 inline image
NucleusDFTRHF-RPARH-RPASM-HFSM-WS
10C0.462(65)0.0820.1500.225(36)0.175(18)
14O0.480(48)0.1140.1970.310(36)0.330(25)
22Mg0.432(49)  0.260(56)0.380(22)
26Alm0.307(62)0.1390.1980.440(51)0.310(18)
34Cl 0.2340.3070.695(56)0.650(46)
34Ar1.08(42)0.2680.3760.540(61)0.665(56)
38Km 0.2780.3710.745(63)0.655(59)
42Sc0.70(32)0.3330.4480.640(56)0.665(56)
46V0.375(96)  0.600(63)0.620(63)
50Mn0.39(13)  0.620(59)0.655(54)
54Co0.51(20)0.3190.3930.685(63)0.770(67)
62Ga   1.21(17)1.48(21)
74Rb0.90(22)1.0881.2581.42(17)1.63(31)
inline image1.92.72.12.20.4

The five sets of inline image 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 [15] proposed such a test using the premise that the CVC hypothesis is valid. The requirement is that a calculated set of inline image values should produce a statistically consistent set of inline image values, the average of which we can write as inline image. Then Eq. (5) can be written for each individual transition in the set as

display math(11)

For any set of corrections to be acceptable, the calculated value of inline image for each superallowed transition must satisfy this equation where ft is the measured result for that transition and inline image has the same value for all of them. Thus to test a set of inline image values for n superallowed transitions, one can treat inline image 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 inline image. 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 inline image, where inline image 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 inline image values are used in the determination of the CKM matrix element, inline image. However, the other cases can be used for establishing an error assignment on this analysis.

4 Results for inline image

The consistency of the inline image values (and consequently GV) obtained from the 13 measured superallowed 0+inline image 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 inline image. With consistency demonstrated, however, inline image can be reliably obtained from the following equation [16]:

display math(12)

where inline image GeV−2 [4] and inline image is the average measured inline image value. Taking inline image = 3071.81(83) s from the 2009 survey [3], we arrive at inline image = 0.97425(22), a value with 0.02% precision.

This result is certainly the most precise current determination of inline image, but superallowed 0+inline image β decay is not the only experimental approach to inline image. Neutron decay, nuclear inline image mirror decays, and pion beta decay have all been used for this purpose. Though these other methods cannot as yet compete with 0+inline image 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 (inline image) or for nuclear-structure-dependent radiative effects (inline image). 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+inline image 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 inline image 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 inline image of 0.97520(140), which agrees with the superallowed result but has an uncertainty more than six times larger.

The nuclear inline image 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, inline image and inline image. A recent analysis of the five measured mirror transitions [20], which incorporated inline image and inline image corrections, yielded inline image = 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+inline image decays.

Finally, pion β decay should, in principle, be the best way to determine inline image. 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 [21], whose result corresponds to a value for inline image of 0.97420(260) [16].

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+inline image 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. [3]). Thus we continue our practice [16] of adopting the 0+inline image result as the recommended value, namely

display math(13)

while noting that the results from the other measurement techniques are entirely consistent with this value.

Figure 2.

The four values for inline image discussed in the text are shown in the top panel, the grey band being the average value. The four panels at the bottom of the figure show the error budgets for each result. The three contributors to the uncertainties – experiment, radiative correction and nuclear correction – are separately identified.

inline image and inline image

The top row CKM matrix element inline image is best determined from kaon semi-leptonic decays, inline image, of both charged and neutral kaons, and from the purely leptonic decay of the kaon, the most important mode being inline image. 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 inline image 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 [22] is

display math(14)

Here inline image 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 inline image 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 [25]. For inline image their recommended value, after symmetrizing the uncertainty, is inline image, yielding

display math(15)

An independent determination of inline image 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, inline image, the hadronic uncertainties can be minimized and the result yields the ratio of the CKM matrix elements inline image. In the analysis of the FlaviaNet group [26], the current result is

display math(16)

The most recent FLAG-2 average of the four published determinations of the decay constants inline image in lattice QCD is inline image [25]. More significant is a new calculation of a correction to the ratio of experimentally determined decay rates, inline image for isospin-symmetry breaking [27] that nearly doubles the formerly-used correction. The resulting value for inline image reported at the CIPANP12 workshop [22] is

display math(17)

Thus, we now have three pieces of data – inline image from nuclear decays, Eq. (13), inline image from inline image decays, Eq. (15), and the ratio inline image from inline image decays, Eq. (17) – from which to determine two parameters inline image and inline image. An unconstrained fit does not change the input value of inline image and yields for inline image our final selection

display math(18)

The third element of the top row of the CKM matrix, inline image, is very small and hardly impacts on the unitarity test at all. Its value from the 2012 PDG compliation [4] is

display math(19)

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 inline image, inline image and inline image. Now, combining the results from Eqs. (13), (18) and (19), we obtain the result

display math(20)

a result that shows unitarity to be fully satisfied to a precision of 0.06%. Only inline image and inline image contribute perceptibly to the uncertainty and their contributions to it are almost equal. This may seem surprising since inline image is known to much higher relative precision than inline image, but it follows from the fact that inline image 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 inline image = 1.0021(51) [4], 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, inline image is its dominant contributor, and superallowed 0+inline image β decay is effectively the sole experimental source for the value of inline image.

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. [16]. Here we just update the results given there for the current unitarity sum.

First, for right-hand currents, we write

display math(21)

Here inline image 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

display math(22)

yielding the one-standard deviation limits on Δ of

display math(23)

Marciano and Sirlin [28] relate Δ to the mass of the extra Z boson via

display math(24)

Here inline image is the mass of the extra Z-boson contributing to the radiative correction and inline image the W-boson mass. Noting that the correction Δ is negative, we obtain from the lower limit in Eq. (23)

display math(25)

and

display math(26)

The confidence level is 68%. This limit is slightly weaker than the limit of 460 GeV given in [15] as the lower limit on Δ in Eq. (23) has been increased slightly through a more recent revision in the inline image value (see Section 'inline image and inline image'). 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 [29] 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 inline image decays have arrived at the value for inline image in Eq. (14), which has a relative uncertainty of about inline image. However, the lattice QCD estimate for the form factor, inline image, has a much larger inline image uncertainty, so it dominates the uncertainty on inline image. While little improvement in the experimental result is anticipated in the near future [22], Van de Water [30] 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 inline image 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 inline image is similarly dominated by uncertainties in the theoretical correction terms. Of the total 0.023% uncertainty on inline image as derived from the superallowed 0+inline image decays, 0.018% is contributed by the uncertainty associated with the radiative correction inline image and 0.010% comes from the nuclear-structure-dependent corrections, inline image. 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 inline image 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 inline image will be limited to about the same value by inline image. For pion decay, in the bottom right panel, the uncertainty in inline image 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 inline image.

Any improvement in inline image 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 inline image can be obtained from a reduction in the uncertainty associated with the nuclear-structure-dependent corrections, inline image. 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 inline image 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+inline image 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 inline image by 15%.

Considering anticipated improvements to both inline image and inline image, 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 inline image 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 inline image, the limit on the extra Z boson mass would become inline image 650 GeV, which is comparable to the limits from direct searches.

This is likely the extent of improvements for the foreseeable future.

Acknowledgements

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.

Biographies

  • Image of creator

    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.

  • Image of creator

    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.