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Superallowed nuclear β decay: symmetry breaking, CVC and CKM unitarity
Nuclear Physics A 844 (2010) 138c–142c www.elsevier.com/locate/nuclphysa
Superallowed nuclear β decay: symmetry breaking, CVC and CKM unitarity J. C. Hardya∗ and I. S. Townera a
Cyclotron Institute, Texas A&M University, College Station, TX 77843-3366, U.S.A.
Currently, the most restrictive test of the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is anchored by nuclear beta decay. Precise measurements of the f t-values for superallowed beta transitions between analog 0+ states are used to determine GV , the vector coupling constant; this, in turn, yields Vud , the up-down element of the CabibboKobayashi-Maskawa (CKM) matrix. When the value for Vud obtained from the latest survey of world data is combined with Vus and Vub , the other top-row elements, it leads to the most demanding test available of the unitarity of that matrix. Important ingredients required in the extraction of Vud from experiment are calculated radiative corrections as well as corrections to account for the isospin symmetry breaking that occurs between the analog parent and daughter states of each superallowed transition. The latter are well supported by comparison with experiment. The current value for the unitarity sum is 0.99995(61), a result that is in remarkable agreement with Standard Model expectations. 1. THE DETERMINATION OF Vud Beta decay between nuclear analog states of spin-parity, J π = 0+ , and isospin, T = 1, has a unique simplicity: it is a pure vector transition and is nearly independent of the nuclear structure of the parent and daughter states. The measured strength of such a transition – expressed as an “f t value” – can then be related directly to the vector coupling constant, GV with the intervention of only a few small (∼1%) calculated terms to account for radiative and nuclear-structure-dependent effects. Once GV has been determined in this way, it is only another short step to obtain a value for Vud , the up-down mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, with which it is possible to test the top-row unitarity of that matrix. In dealing with these so-called superallowed decays, it is convenient to combine some of the small correction terms with the measured f t-value and define a “corrected” Ft-value. Thus, we write [1] Ft ≡ f t(1 + δR )(1 + δN S − δC ) =
K , 2G2V (1 + ΔVR )
(1)
∗ The work of J. C. H. was supported by the U. S. Dept. of Energy under Grant DE-FG02-93ER40773 and by the Robert A. Welch Foundation under Grant A-1397. I. S. T. would like to thank the Cyclotron Institute of Texas A & M University for its hospitality during annual two-month visits.
0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.05.024
139c
J.C. Hardy, I.S. Towner / Nuclear Physics A 844 (2010) 138c–142c
3090 3080
10
22
C
Mg
14
O
3090
Ft (s)
ft (s)
3070
Alm
26
3060 3050
38 m 46
K
V
Ar 34 Cl 42Sc
50
34
Mn 62Ga 54 Co
74
Rb
3080 3070
3040 0
10
20
Z of daughter
30
40
3060
0
10
20
30
40
Z of daughter
Figure 1. Results from the 2009 survey [1]. The uncorrected f t values for the twelve best known superallowed decays (left), compared with the same results after application of δR , δC and δN S correction terms. The grey band in the right panel is the average Ft value, including its uncertainty.
where K = 8120.2787(11)×10−10 GeV−4 s; δC is the isospin-symmetry-breaking correction and ΔVR is the transition-independent part of the radiative correction. The terms δR and δN S 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 δC , depends in its evaluation on the details of nuclear structure. From this equation, it can be seen that a measurement of any one superallowed transition establishes an individual value for GV . A measurement of several of them tests the Conserved Vector Current (CVC) hypothesis that GV is not renormalized in the nuclear medium. If indeed GV is constant – i.e. all the Ft-values are the same – then an average value for GV can be determined and Vud obtained from the relation Vud = GV /GF , where GF is the well known [2] weak-interaction constant for purely leptonic muon decay. The f t-value that characterizes any β-transition depends on three measured quantities: the total transition energy, QEC , the half-life, t1/2 , of the parent state and the branching ratio, R, for the particular transition of interest. A new survey of world data on superallowed 0+ → 0+ beta decays has just been published [1], in which the f t values for 13 such transitions have been precisely determined from more than 150 independent measurements of the various input quantities. The evaluated survey results were then used to obtain the f t and corrected Ft values. The results are shown in Fig. 1. From these results we can draw several important conclusions. First, the Ft values are consistent from A=10 to A=74, thus confirming the CVC expectation. Second, this consistency makes it possible to set a tight limit on any contribution from scalar currents: If the scalar current is assumed to be maximally parity violating, like the vector current, then |CS /CV | = 0.0011(14) [1]. Third, with a mutually consistent set of Ft values, no scalar current and the test of CVC passed, one can confidently proceed to determine the value of GV and, from it, Vud . The result is Vud = 0.97425(22), which, when combined with current values for Vus and Vub , yields a CKM unitarity sum of 0.99995(61) [1], in
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remarkable agreement with Standard Model expectations. This important result does depend to a certain extent on nuclear-structure-dependent corrections. It is crucial then to evaluate their reliability, and we will focus on that issue in what follows. 2. THE NUCLEAR-STRUCTURE-DEPENDENT CORRECTIONS The most important observation, which strongly supports the validity of the nuclearstructure-dependent corrections, is offered by Fig. 1. Since δR is only weakly Z-dependent [3], it is these corrections, δC and δN S , that are responsible for replacing the considerable scatter among the uncorrected f t values with complete uniformity among the corrected Ft values. It is important to note that δC and δN S were not somehow adjusted specifically to achieve this effect. They were instead calculated [3] with well-established shell-model wave functions whose origins were completely independent of the superallowed data. Thus, the consistency of the Ft values is a powerful validation of the calculated corrections used in their derivation. That is not the only support for the structure-dependent corrections however. Following the model used in Ref. [3], the isospin-symmetry-breaking correction, δC , can be expressed as the sum of two terms, δC1 + δC2 . The first of these terms, δC1 , accounts for the difference in configuration mixing between the parent and daughter states, and was calculated in a modest-sized shell-model, which incorporated charge-dependent interactions and used well-established two-body matrix elements based on a wide range of spectroscopic data. Charge-dependence was introduced in three different ways: first the single-particle energies of the proton orbits were shifted relative to those of neutrons according to the known levels in the closed-shell-plus-proton or closed-shell-plus-neutron nuclei; second a two-body Coulomb interaction among the valence protons was added, and its strength adjusted to reproduce the measured b coefficient of the isobaric multiplet mass equation (IMME) for each mass chain; and third, to incorporate the charge dependence of the nuclear force, all the T = 1 proton-neutron matrix elements were increased by about 2% relative to the neutron-neutron matrix elements, the precise amount being determined by requiring agreement with the measured c coefficient of the IMME. The second term, δC2 , is a measure of the mismatch in the radial wave functions between the parent and daughter states. It was evaluated from full-parentage Woods-Saxon wave functions [3] matched both to the known binding energy of the states involved and to the nuclear charge radii as determined from electron scattering. Recently the same calculation was repeated with Hartree-Fock wave functions [1] to set limits on possible systematic model-dependence of the results. The accuracy of these calculations of δC1 and δC2 for each superallowed transition naturally depends on the completeness of the spectroscopic data available in the surrounding region of nuclei. In particular, for transitions between nuclei with 10 ≤ A ≤ 38, where large amounts of good spectroscopic data have been reliably reproduced by the shell model, the calculated correction terms are considered to be reliable as well. However, since the superallowed transitions occur between nuclei with nearly equal numbers of neutrons and protons, the situation deteriorates as A increases and spectroscopic information on such nuclei becomes more meager. As a result, the calculated corrections for
J.C. Hardy, I.S. Towner / Nuclear Physics A 844 (2010) 138c–142c
141c
Table 1 Comparison of calculation with experiment for the branching ratios from superallowed parents to the first excited non-analog 0+ states in their daughters. The excitation energy is given for each daughter state of interest. Parent nucleus 38
Km Sc 46 V 50 Mn 54 Co 62 Ga 42
Ex (0+ Branching ratio (ppm) 1) (keV) Experimental Calculated [3] 3377 1837 2611 3827 2561 2342
< 12 [4] 59 ± 14 [1] 39 ± 4 [5] < 3 [5] 45 ± 6 [5] 53 ± 25 [6]
6±2 22 ± 22 18 ± 15 8±4 65 ± 26 240 ± 80
the cases with 42 ≤ A ≤ 54 are somewhat less reliable than those for the lighter nuclei, and corrections for the A ≥ 62 cases are very much less reliable. All these considerations have been reflected in the uncertainties assigned to the correction terms [1]. So far we have described the way in which the isospin-symmetry-breaking corrections were calculated following a semi-phenomenological approach, with parameters determined so far as possible from experiment. We now turn to two independent tests of the calculations, one experimental, the other theoretical. The experimental one specifically tests the calculation of δC1 . If isospin were an exact symmetry, then the parent 0+ (T =1) state would decay exclusively to its analog state in the daughter nucleus – the superallowed transition – and β transitions to all other 0+ states would be strictly forbidden. But, with isospin symmetry broken, weak transitions (with branching ratios measured in parts per million) can occur to these other 0+ states. Naturally the same calculation that obtains δC1 for a superallowed transition also predicts the branching ratio to other daughter 0+ states and this can be compared to experiment. The experiments are difficult and there are a limited number of daughter 0+ states that are energetically available to β decay, especially among the lighter superallowed emitters, but some data are available and these are presented in Table 1 where they are compared with the theoretical predictions. Clearly the agreement is excellent with the single exception of 62 Ga. As already noted, the shellmodel parameters for nuclei in this mass region are poorly known so a discrepancy of this size should not be surprising. The second independent test of the calculations is a theoretical one that applies to the total isospin-symmetry-breaking correction, δC . Only for the lightest superallowed emitter has it been possible to approach an exact treatment of the calculation rather than the semi-phenomenological model that has been used so far. Caurier et al. [7] have reported a large no-core shell-model calculation for the decay of 10 C but, even though they were able to extend their basis states up to 8¯hω, their calculated δC still had not converged to a stable value. However they used their results together with perturbation theory to estimate that the full value of δC should be about 0.19%. This result agrees completely with the model calculated result of 0.18(2)% [3].
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3. CONCLUSIONS Evidently, the calculated nuclear-structure-dependent corrections have been based on a wide range of nuclear data and have passed several independent tests. Moreover they lead to a consistent set of corrected Ft values for the superallowed transitions, the ultimate independent test. We can therefore be confident in the value of Vud derived from superallowed nuclear β decays and in the unitarity sum to which it contributes. REFERENCES 1. J. C. Hardy and I. S. Towner, Phys. Rev. C 79 (2009) 055502. 2. C. Amsler et. al. [Particle Data Group], Phys. Lett. B 667 (2008) 1 and 2009 partial update for the 2010 edition. 3. I. S. Towner and J. C. Hardy, Phys. Rev. C 77 (2008) 025501. 4. K. G. Leach et. al., Phys. Rev. Lett. 100 192504. 5. E. Hagberg et. al., Phys. Rev. Lett. 73 396. 6. P. Finlay et. al., Phys. Rev. C 78 025502. 7. E. Caurier et. al., Phys. Rev. C 66 024314.
Superallowed nuclear β decay: symmetry breaking, CVC and CKM unitarity J. C. Hardya∗ and I. S. Townera a
Cyclotron Institute, Texas A&M University, College Station, TX 77843-3366, U.S.A.
Currently, the most restrictive test of the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is anchored by nuclear beta decay. Precise measurements of the f t-values for superallowed beta transitions between analog 0+ states are used to determine GV , the vector coupling constant; this, in turn, yields Vud , the up-down element of the CabibboKobayashi-Maskawa (CKM) matrix. When the value for Vud obtained from the latest survey of world data is combined with Vus and Vub , the other top-row elements, it leads to the most demanding test available of the unitarity of that matrix. Important ingredients required in the extraction of Vud from experiment are calculated radiative corrections as well as corrections to account for the isospin symmetry breaking that occurs between the analog parent and daughter states of each superallowed transition. The latter are well supported by comparison with experiment. The current value for the unitarity sum is 0.99995(61), a result that is in remarkable agreement with Standard Model expectations. 1. THE DETERMINATION OF Vud Beta decay between nuclear analog states of spin-parity, J π = 0+ , and isospin, T = 1, has a unique simplicity: it is a pure vector transition and is nearly independent of the nuclear structure of the parent and daughter states. The measured strength of such a transition – expressed as an “f t value” – can then be related directly to the vector coupling constant, GV with the intervention of only a few small (∼1%) calculated terms to account for radiative and nuclear-structure-dependent effects. Once GV has been determined in this way, it is only another short step to obtain a value for Vud , the up-down mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, with which it is possible to test the top-row unitarity of that matrix. In dealing with these so-called superallowed decays, it is convenient to combine some of the small correction terms with the measured f t-value and define a “corrected” Ft-value. Thus, we write [1] Ft ≡ f t(1 + δR )(1 + δN S − δC ) =
K , 2G2V (1 + ΔVR )
(1)
∗ The work of J. C. H. was supported by the U. S. Dept. of Energy under Grant DE-FG02-93ER40773 and by the Robert A. Welch Foundation under Grant A-1397. I. S. T. would like to thank the Cyclotron Institute of Texas A & M University for its hospitality during annual two-month visits.
0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.05.024
139c
J.C. Hardy, I.S. Towner / Nuclear Physics A 844 (2010) 138c–142c
3090 3080
10
22
C
Mg
14
O
3090
Ft (s)
ft (s)
3070
Alm
26
3060 3050
38 m 46
K
V
Ar 34 Cl 42Sc
50
34
Mn 62Ga 54 Co
74
Rb
3080 3070
3040 0
10
20
Z of daughter
30
40
3060
0
10
20
30
40
Z of daughter
Figure 1. Results from the 2009 survey [1]. The uncorrected f t values for the twelve best known superallowed decays (left), compared with the same results after application of δR , δC and δN S correction terms. The grey band in the right panel is the average Ft value, including its uncertainty.
where K = 8120.2787(11)×10−10 GeV−4 s; δC is the isospin-symmetry-breaking correction and ΔVR is the transition-independent part of the radiative correction. The terms δR and δN S 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 δC , depends in its evaluation on the details of nuclear structure. From this equation, it can be seen that a measurement of any one superallowed transition establishes an individual value for GV . A measurement of several of them tests the Conserved Vector Current (CVC) hypothesis that GV is not renormalized in the nuclear medium. If indeed GV is constant – i.e. all the Ft-values are the same – then an average value for GV can be determined and Vud obtained from the relation Vud = GV /GF , where GF is the well known [2] weak-interaction constant for purely leptonic muon decay. The f t-value that characterizes any β-transition depends on three measured quantities: the total transition energy, QEC , the half-life, t1/2 , of the parent state and the branching ratio, R, for the particular transition of interest. A new survey of world data on superallowed 0+ → 0+ beta decays has just been published [1], in which the f t values for 13 such transitions have been precisely determined from more than 150 independent measurements of the various input quantities. The evaluated survey results were then used to obtain the f t and corrected Ft values. The results are shown in Fig. 1. From these results we can draw several important conclusions. First, the Ft values are consistent from A=10 to A=74, thus confirming the CVC expectation. Second, this consistency makes it possible to set a tight limit on any contribution from scalar currents: If the scalar current is assumed to be maximally parity violating, like the vector current, then |CS /CV | = 0.0011(14) [1]. Third, with a mutually consistent set of Ft values, no scalar current and the test of CVC passed, one can confidently proceed to determine the value of GV and, from it, Vud . The result is Vud = 0.97425(22), which, when combined with current values for Vus and Vub , yields a CKM unitarity sum of 0.99995(61) [1], in
140c
J.C. Hardy, I.S. Towner / Nuclear Physics A 844 (2010) 138c–142c
remarkable agreement with Standard Model expectations. This important result does depend to a certain extent on nuclear-structure-dependent corrections. It is crucial then to evaluate their reliability, and we will focus on that issue in what follows. 2. THE NUCLEAR-STRUCTURE-DEPENDENT CORRECTIONS The most important observation, which strongly supports the validity of the nuclearstructure-dependent corrections, is offered by Fig. 1. Since δR is only weakly Z-dependent [3], it is these corrections, δC and δN S , that are responsible for replacing the considerable scatter among the uncorrected f t values with complete uniformity among the corrected Ft values. It is important to note that δC and δN S were not somehow adjusted specifically to achieve this effect. They were instead calculated [3] with well-established shell-model wave functions whose origins were completely independent of the superallowed data. Thus, the consistency of the Ft values is a powerful validation of the calculated corrections used in their derivation. That is not the only support for the structure-dependent corrections however. Following the model used in Ref. [3], the isospin-symmetry-breaking correction, δC , can be expressed as the sum of two terms, δC1 + δC2 . The first of these terms, δC1 , accounts for the difference in configuration mixing between the parent and daughter states, and was calculated in a modest-sized shell-model, which incorporated charge-dependent interactions and used well-established two-body matrix elements based on a wide range of spectroscopic data. Charge-dependence was introduced in three different ways: first the single-particle energies of the proton orbits were shifted relative to those of neutrons according to the known levels in the closed-shell-plus-proton or closed-shell-plus-neutron nuclei; second a two-body Coulomb interaction among the valence protons was added, and its strength adjusted to reproduce the measured b coefficient of the isobaric multiplet mass equation (IMME) for each mass chain; and third, to incorporate the charge dependence of the nuclear force, all the T = 1 proton-neutron matrix elements were increased by about 2% relative to the neutron-neutron matrix elements, the precise amount being determined by requiring agreement with the measured c coefficient of the IMME. The second term, δC2 , is a measure of the mismatch in the radial wave functions between the parent and daughter states. It was evaluated from full-parentage Woods-Saxon wave functions [3] matched both to the known binding energy of the states involved and to the nuclear charge radii as determined from electron scattering. Recently the same calculation was repeated with Hartree-Fock wave functions [1] to set limits on possible systematic model-dependence of the results. The accuracy of these calculations of δC1 and δC2 for each superallowed transition naturally depends on the completeness of the spectroscopic data available in the surrounding region of nuclei. In particular, for transitions between nuclei with 10 ≤ A ≤ 38, where large amounts of good spectroscopic data have been reliably reproduced by the shell model, the calculated correction terms are considered to be reliable as well. However, since the superallowed transitions occur between nuclei with nearly equal numbers of neutrons and protons, the situation deteriorates as A increases and spectroscopic information on such nuclei becomes more meager. As a result, the calculated corrections for
J.C. Hardy, I.S. Towner / Nuclear Physics A 844 (2010) 138c–142c
141c
Table 1 Comparison of calculation with experiment for the branching ratios from superallowed parents to the first excited non-analog 0+ states in their daughters. The excitation energy is given for each daughter state of interest. Parent nucleus 38
Km Sc 46 V 50 Mn 54 Co 62 Ga 42
Ex (0+ Branching ratio (ppm) 1) (keV) Experimental Calculated [3] 3377 1837 2611 3827 2561 2342
< 12 [4] 59 ± 14 [1] 39 ± 4 [5] < 3 [5] 45 ± 6 [5] 53 ± 25 [6]
6±2 22 ± 22 18 ± 15 8±4 65 ± 26 240 ± 80
the cases with 42 ≤ A ≤ 54 are somewhat less reliable than those for the lighter nuclei, and corrections for the A ≥ 62 cases are very much less reliable. All these considerations have been reflected in the uncertainties assigned to the correction terms [1]. So far we have described the way in which the isospin-symmetry-breaking corrections were calculated following a semi-phenomenological approach, with parameters determined so far as possible from experiment. We now turn to two independent tests of the calculations, one experimental, the other theoretical. The experimental one specifically tests the calculation of δC1 . If isospin were an exact symmetry, then the parent 0+ (T =1) state would decay exclusively to its analog state in the daughter nucleus – the superallowed transition – and β transitions to all other 0+ states would be strictly forbidden. But, with isospin symmetry broken, weak transitions (with branching ratios measured in parts per million) can occur to these other 0+ states. Naturally the same calculation that obtains δC1 for a superallowed transition also predicts the branching ratio to other daughter 0+ states and this can be compared to experiment. The experiments are difficult and there are a limited number of daughter 0+ states that are energetically available to β decay, especially among the lighter superallowed emitters, but some data are available and these are presented in Table 1 where they are compared with the theoretical predictions. Clearly the agreement is excellent with the single exception of 62 Ga. As already noted, the shellmodel parameters for nuclei in this mass region are poorly known so a discrepancy of this size should not be surprising. The second independent test of the calculations is a theoretical one that applies to the total isospin-symmetry-breaking correction, δC . Only for the lightest superallowed emitter has it been possible to approach an exact treatment of the calculation rather than the semi-phenomenological model that has been used so far. Caurier et al. [7] have reported a large no-core shell-model calculation for the decay of 10 C but, even though they were able to extend their basis states up to 8¯hω, their calculated δC still had not converged to a stable value. However they used their results together with perturbation theory to estimate that the full value of δC should be about 0.19%. This result agrees completely with the model calculated result of 0.18(2)% [3].
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J.C. Hardy, I.S. Towner / Nuclear Physics A 844 (2010) 138c–142c
3. CONCLUSIONS Evidently, the calculated nuclear-structure-dependent corrections have been based on a wide range of nuclear data and have passed several independent tests. Moreover they lead to a consistent set of corrected Ft values for the superallowed transitions, the ultimate independent test. We can therefore be confident in the value of Vud derived from superallowed nuclear β decays and in the unitarity sum to which it contributes. REFERENCES 1. J. C. Hardy and I. S. Towner, Phys. Rev. C 79 (2009) 055502. 2. C. Amsler et. al. [Particle Data Group], Phys. Lett. B 667 (2008) 1 and 2009 partial update for the 2010 edition. 3. I. S. Towner and J. C. Hardy, Phys. Rev. C 77 (2008) 025501. 4. K. G. Leach et. al., Phys. Rev. Lett. 100 192504. 5. E. Hagberg et. al., Phys. Rev. Lett. 73 396. 6. P. Finlay et. al., Phys. Rev. C 78 025502. 7. E. Caurier et. al., Phys. Rev. C 66 024314.