A new critical survey of all world data on superallowed beta decays provides
demanding tests of, and tight constraints on, the weak interaction. In
confirmation of the conserved vector current (CVC) hypothesis, the vector
coupling coupling constant is demonstrated to be constant to better than 3
parts in 10^4, and any induced scalar current is limited to fs \leq 0.0013 in
electron rest-mass units. The possible existence of a fundamental scalar
current is similarly limited to |Cs/Cv| \leq 0.0013. The superallowed data also
determine the CKM matrix element to be Vud = 0.9738(4). With PDG values for Vus
and Vub, the top-row test of CKM unitarity yields Vud^2 + Vus^2 + Vub^2 =
0.9966(14); although, if a recent result on Ke3 decay is used exclusively to
obtain Vus, this sum becomes 0.9999(16). Limits on possible right-hand currents
are given.Comment: 4 pages, one postscript fil

Hardy, J. C.

Towner, I. S.

Physical Review Letters

English

arXiv

Texas A&M University

New Limits on Fundamental Weak-Interaction Parameters from Superallowed β Decay

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:nucl-th/0412050v1  14 Dec 2004New limits on fundamental weak-interaction parameters from superallowed β decayJ.C. Hardy and I.S. Towner∗Cyclotron Institute, Texas A&M University, College Station, Texas 77843(Dated: July 2, 2018)A new critical survey of all world data on superallowed 0+ → 0+ β decays provides demanding testsof, and tight constraints on, the weak interaction. In confirmation of the conserved vector current(CVC) hypothesis, the vector coupling constant, GV, is demonstrated to be constant to better than3 parts in 104, and any induced scalar current is limited to fS ≤ 0.0013 in electron rest-mass units.The possible existence of a fundamental scalar current is similarly limited to |CS/CV | ≤ 0.0013. Thesuperallowed data also determine the up-down element of the Cabibbo-Kobayashi-Maskawa (CKM)matrix to be Vud = 0.9738(4). With Particle Data Group values for Vus and Vub, the top-row testof CKM unitarity yields |Vud|2 + |Vus|2 + |Vub|2 = 0.9966(14); although, if a recent result on Ke3decay is used exclusively to obtain Vus, this sum becomes 0.9999(16). Either unitarity result can beexpressed in terms of the possible existence of right-hand currents.PACS numbers: 23.40.Bw, 12.15.Hh, 12.60.-iBeta decay between nuclear analog states of spin-parity, Jpi = 0+, and isospin, T = 1, has been a subject ofcontinuous and often intense study for five decades. Thestrengths, or ft-values, of such transitions are nearly in-dependent of nuclear-structure ambiguities and dependuniquely on the vector part of the weak interaction.Thus, their measurement has given nuclear physicists ac-cess to clean tests of some of the fundamental precepts ofweak-interaction theory and, over the years, this strongmotivation has led to very high precision being achievedin both the experiments and the theory required to in-terpret them.As befits such an important issue, there have been peri-odic surveys of the relevant world data (see, for example,Refs. [1, 2, 3, 4]). Because the last survey appeared in1990 and a large amount of new data has appeared inthe decade and a half since then, we have just completeda thorough new overview [5] in which we critically sur-veyed all relevant measurements, adjusted original datato take account of the most modern calibration stan-dards, obtained statistically rigorous average results foreach transition, and used updated and consistent calcula-tions to extract weak-interaction parameters from thoseresults. The outcome includes more exacting confirma-tion of the conserved vector current (CVC) hypothesis,a reduced limit on any possible scalar currents, and animproved value for the up-down quark-mixing elementof the Cabibbo-Kobayashi-Maskawa (CKM) matrix, Vud.The latter is an important component of the most de-manding available test of the unitarity of that matrix andcan help constrain or rule out the influence of right-handcurrents.For any superallowed 0+ → 0+ β decay between T = 1analog states, the experimental ft-value can be relatedto the vector coupling constant via an expression thatincludes several small (∼1%) correction terms. It is con-venient to combine some of these terms with the ft-valueand define a “corrected” Ft-value. Thus, we write [6]Ft ≡ ft(1 + δ′R)(1 + δNS − δC) =K2G2V(1 + ∆VR), (1)where K/(h¯c)6 = 2pi3h¯ ln 2/(mec2)5 = 8120.271(12) ×10−10 GeV−4s, GV is the vector coupling constantfor semi-leptonic weak interactions, δC is the isospin-symmetry-breaking correction and ∆VRis the transition-independent part of the radiative correction. The termsδ′R and δNS comprise the transition-dependent part of theradiative correction, the former being a function only ofthe electron’s energy and the Z of the daughter nucleus,while the latter, like δC , depends in its evaluation on thedetails of nuclear structure. From this equation, it canbe seen that each measured transition establishes an in-dividual value for GV and, if the CVC assertion is correctthat GV is not renormalized in the nuclear medium, allsuch values – and all the Ft-values themselves – shouldbe identical within uncertainties, regardless of the spe-cific nuclei involved.The ft-value that characterizes any β-transition de-pends on three measured quantities: the total transitionenergy, QEC , the half-life, t1/2, of the parent state andthe branching ratio, R, for the particular transition ofinterest. The QEC-value is required to determine thestatistical rate function, f , while the half-life and branch-ing ratio combine to yield the partial half-life, t. In ourtreatment [5] of the data, we considered all measurementsformally published before November 2004 and those weknew to be in an advanced state of preparation for pub-lication by that date. We scrutinized all the original ex-perimental reports in detail and, where possible, adjustedthose QEC -values that were based on outdated calibra-tion standards. If corrections to any measurements wereevidently required but insufficient information was pro-vided to make them, the results were rejected. Of thesurviving results, only those with (updated) uncertain-ties that are within a factor of ten of the most precise2TABLE I: Final average ft values, correction terms, and Ftvalues [5] for the twelve best known superallowed transitions.Parent ft δ′R δC − δNS FtNucleus (s) (%) (%) (s)10C 3039.5(47) 1.652(4) 0.540(39) 3073.0(49)14O 3043.3(19) 1.529(8) 0.570(56) 3071.9(26)22Mg 3052.4(72) 1.446(17) 0.505(24) 3080.9(74)26mAl 3036.7(12) 1.458(20) 0.261(24) 3072.9(15)34Cl 3050.5(47) 1.425(32) 0.720(39) 3071.7(19)34Ar 3059(12) 1.394(35) 0.825(44) 3076(12)38mK 3051.1(10) 1.423(39) 0.720(47) 3072.2(21)42Sc 3046.0(15) 1.437(47) 0.460(47) 3075.6(25)46V 3045.5(22) 1.429(54) 0.465(33) 3074.7(30)50Mn 3044.5(15) 1.429(62) 0.547(37) 3071.1(27)54Co 3047.4(15) 1.428(71) 0.639(43) 3071.2(28)74Rb 3083.8(75) 1.49(12) 1.50(41) 3083(15)Average, Ft 3072.7(8)χ2/ν 0.42measurement for each quantity were retained for averag-ing. The statistical procedures we followed in analyzingthe data were those used by the Particle Data Group intheir periodic reviews of particle properties (e.g. [7]).The final average ft values obtained from the survey[5], together with applied correction terms and the re-sulting Ft values, are given in Table I for the twelvesuperallowed transitions that are now known to 0.4%precision or better; all but four are actually known tobetter than 0.1%. They cover a broad range of nuclearmasses from A = 10 to A = 74. As anticipated by CVC(see Eq. 1) the Ft values are statistically consistant withone another, yielding an average value Ft = 3072.7(8) s,with a corresponding chi-square per degree of freedom ofχ2/ν = 0.42. These results have many important out-comes. We will deal with each, individually identified.Nuclear-structure-dependent corrections. As can beseen from Eq. 1, the Ft values differ from their ft-valuecounterparts by the application of the δ′R radiative cor-rection and the nuclear-structure-dependent corrections,δNS− δC . To isolate and illustrate the effect of the lattercorrections, we plot in the top panel of Fig. 1 the valuesof ft(1+δ′R) for the 12 transtions listed in Table I and thecorresponding values of Ft in the bottom panel. Theydiffer only by the inclusion of the structure-dependentcorrections in the latter and, obviously, these correctionsact very well to remove the considerable “scatter” thatis apparent in the top panel and is effectively absent inthe bottom one. It is important to note that, with thesingle exception of 74Rb, the calculations of δNS and δC[8] are based on well-established shell-model wave func-tions, and were further tuned to measured binding ener-gies, charge radii and coefficients of the isobaric multipletmass equation. Their origins are completely independentof the superallowed decay data. Thus the consistency ofthe corrected Ft values appearing in the bottom panelof the figure is a powerful validation of the calculatedcorrections used in their derivation.CVC(1) – vector coupling constant. The CVC hypoth-esis – that the weak vector current is just an isospin rota-tion of a conserved electromagnetic vector current – hasseveral direct implications. The first is that GV is indeedconstant, independent of the nuclear medium. The factthat the Ft values, listed in Table I and plotted in thebottom panel of Fig. 1, form a consistent set over a widerange of nuclei, 10 ≤ A ≤ 74, verifies this expectation10C14O22Mg26 mAl34Cl34Ar38 mK42Sc46V50Mn54Co74Rb308030903100311031303120ft(1+)(s)dR’310030903080307010 30203060Z of daughtert(s)3140FIG. 1: In the top panel are plotted the experimental ft-values corrected only for δ′R, those radiative effects that areindependent of nuclear structure. In the bottom panel, thecorresponding Ft values are given; they differ from the toppanel simply by inclusion of the nuclear-structure-dependentcorrections, δNS and δC . (See Eq. 1.) Note that the in-creased uncertainty for 74Rb reflects the lack of experimentalconstraints on the nuclear shell model in this mass region.The horizontal grey band in the bottom panel indicates theaverage Ft value with its uncertainty. The curved lines rep-resent the approximate loci the Ft values would follow if aninduced scalar current existed with values of fS = ±0.002 inelectron rest-mass units or a fundamental scalar current withCS/CV = ±0.002.3of CVC at the level of 3 × 10−4, which is the fractionaluncertainty we obtain for Ft. This is a 30% improve-ment over the best previous value [4] – also obtainedfrom superallowed beta decay – and can be attributed toimprovements in the experimental data.CVC(2) – induced scalar current. A second implica-tion of the CVC hypothesis is that the “induced” scalarterm in the vector part of the weak interaction – as dis-tinct from any possible “fundamental” scalar current –must be zero. An independent argument [9], that therebe no second-class currents in the hadronic weak inter-action, also requires it to vanish. The induced scalarterm is characterized by the coupling constant fS (usingthe notation of Behrens and Bu¨hring [10]), the presenceof which would affect the calculation of the statistical-rate function, f , via a term in the shape-correction func-tion that is inversely proportional to the positron energy.Since the total superallowed-transition decay energy in-creases with Z, an induced scalar term would thereforehave its greatest effect on the Ft values at low Z, in-troducing curvature in that region. The bottom panel ofFig. 1 shows the loci of Ft values that would be expectedif fS = ±0.002. Obviously, the Ft values do not exhibitany such curvature and, from a least-square fit to thedata, we obtain, fS = −0.00005(130) or, expressed as alimit, |fS| ≤ 0.0013, in electron rest-mass units.Fundamental scalar current. Although the weak inter-action is normally described by an equal mix of vector-and axial-vector interactions, which maximizes parity vi-olation, there is no a priori reason to rule out otherforms of fundamental couplings, notably scalar and ten-sor interactions. It turns out that a “fundamental”scalar current would have the same effect on the Ft val-ues as would an “induced” scalar term: it would intro-duce curvature in the loci of values at low Z as illus-trated in Fig. 1. The superallowed data in this caseyield the result CS/CV = −0.00005(130), or a limit of|CS/CV | ≤ 0.0013, as expressed in the conventional no-tation of Jackson, Trieiman and Wyld [11]. The cor-responding result for the Fierz interference constant isbF = +0.0001(26). These results represent a factor of30 reduction in the central values compared to our pre-viously published result [6] with the standard deviationbeing essentially unchanged. These are by far the moststringent limits on |CS/CV | or |bF | ever obtained fromnuclear beta decay.The Vud element of the CKM matrix. With a mutu-ally consistent set of Ft values, we can now insert theiraverage value [16], Ft, into Eq. 1 and determine the vec-tor coupling constant GV using the value ∆VR= 2.40(8)%calculated for the transition-independent radiation cor-rection by Marciano and Sirlin [12]. The value of GV it-self is of little interest but, when combined with GF, theweak interaction constant for the purely leptonic muondecay, it yields a value for the element Vud of the CKMmatrix: Vud = GV/GF. Taking the Particle Data Group(PDG) value [7] of GF/(h¯c)3 = 1.16639(1)×10−5 GeV−2,we obtain |Vud| = 0.9738(4). Compared to our previ-ously recommended value [6], this result differs by twounits in the last digit quoted and has a reduced uncer-tainty. It also agrees with, but is considerably more pre-cise than, the result obtained using the same statisticalprocedures from the world data for neutron decay [6]:viz. |Vud| = 0.9745(16). Note that, by more than an or-der of magnitude, Vud is the most precisely determinedelement of the CKM matrix.CKM unitarity – top row. The CKM matrix trans-forms the basis of quark mass eigenstates to that of thequark weak-interaction eigenstates, and it must thereforebe unitary in order that the basis remains orthonormal.An important test of the standard model is whether theexperimentally determined matrix elements indeed sat-isfy a unitarity condition. Currently, the sum of thesquares of the top-row elements, which should equalone, constitutes the most demanding such test avail-able. With the value just obtained for |Vud| and thePDG’s recommended values [7] of |Vus| = 0.2200(26) and|Vub| = 0.00367(47), this test yields|Vud|2 + |Vus|2 + |Vub|2 = 0.9966± 0.0014, (2)which fails unitarity by 2.4 standard deviations. A recentmeasurement of the K+ → pi0e+νe (K+e3) branching ratiofrom the Brookhaven E865 experiment [14] obtains Vus =0.2272± 0.0030. If this value alone were adopted for Vusrather than the PDG average of many experiments, thesum in Eq. 2 would equal 0.9999(16) and unitarity wouldbe fully satisfied.CKM unitarity – first column. Though for now it is aless demanding test, the first column of the CKM matrixcan also be used to test unitarity. With our value for Vudand the PDG values for Vcd and Vtd, the sum of squaresof these three elements is 0.9985(54), in agreement withunitarity. The larger uncertainty is entirely due to Vcd.Right-hand currents. If parity violation were not max-imal, then right-hand currents should be included inthe analysis that yields a value for Vud. Thus, we canset an upper limit on the possible role of right-handcurrents by attributing any apparent non-unitarity ofthe CKM matrix entirely to that source. We expressthe result in terms of the right-left and left-left cou-pling constants, aRL and aLL, defined by Herczeg [15].If we accept the unitarity test in Eq. 2, then we findReaLR/aLL = −0.00176(74). Within the context of themanifest left-right symmetric model, this result corre-sponds to a mixing angle of ζ = 0.00176(74). If, in-stead, we adopt the E865 value for Vus, the result be-comes ReaLR/aLL = −0.00007(84).The accumulated world data on superallowed 0+ → 0+β decay comprises the results of over one hundred mea-surements of comparable precision [5]. Virtually all theimportant experimental parameters used as input to theft-value determinations have been measured in at least4two, and often four or five, independent experiments.Obviously, just another measurement will not have muchimpact on the precision of the weak-interaction parame-ters quoted here. Nevertheless, it is still possible for wellselected experiments with existing or currently foreseentechniques to make real improvements. For example, thevalidation of the nuclear-structure-dependent correctionterms, δNS and δC , exemplified by the comparison ofthe two panels in Fig. 1, can be improved by the addi-tion of new transitions selected from amongst those withlarge calculated corrections. If the ft-values measuredfor cases with large calculated corrections also turn intocorrected Ft values that are consistent with the others,then this must verify the calculations’ reliability for theexisting cases, which have smaller corrections. In fact,the cases of 34Ar and 74Rb, which have only recentlybeen measured, were chosen for this very reason and,although their precision does not yet equal that of theothers, they do indicate that the corrections so far areliving up to expectations. The precision of these newmeasurements can certainly be improved, and other newcases with large calculated corrections [8], such as 18Ne,30S and 62Ga, would be valuable additions.Another area of potential improvement is in the limitset on scalar currents. The bottom panel of Fig. 1 clearlyillustrates the sensitivity of the low-Z cases to the pos-sible presence of scalar currents. Reduced uncertainties,particularly on the decays of 10C and 14O, could thusreduce the scalar-current limit significantly.Finally, it is important to point out that the biggestcontribution to Vud uncertainty – 90% of it, in fact –comes from the calculation of the transition-independentpart of the radiative correction [12], ∆VR= 2.40(8)%. Notonly is the uncertainty quoted for ∆VRthe principal limita-tion on the precision with which Vud can be determinedfrom nuclear superallowed β decay, but it will have asimilar limiting effect on the determination of Vud fromneutron and pion decays as well. No significant improve-ment in the precision of the Vud contribution to the CKMunitarity test can come from any source without a moredefinitive calculation of ∆VR.The work of JCH was supported by the U. S. Dept.of Energy under Grant DE-FG03-93ER40773 and by theRobert A. Welch Foundation. IST would like to thankthe Cyclotron Institute of Texas A&M University for itshospitality during several two-month visits.∗ Present address: Department of Physics, Queen’s Uni-versity, Kingston, Ontario K7L 3N6, Canada[1] I.S. Towner and J.C. Hardy, Nucl. Phys. A205, 33(1973).[2] J.C. Hardy and I.S. Towner, Nucl. Phys. A254, 221(1975).[3] V.T. Koslowsky, E. Hagberg, J.C. Hardy, H. Schmeing,R.E. Azuma and I.S. Towner, in Proc. 7th Int. Conf. onatomic masses and fundamental constants, Darmstadt-Seeheim, ed. O. Klepper (T.H. Darmstadt, 1984) p. 572[4] J.C. Hardy, I.S. Towner, V.T. Koslowsky, E. Hagbergand H. Schmeing, Nucl. Phys. A509, 429 (1990).[5] J.C. Hardy and I.S. Towner, submitted to Phys. Rev. C.[6] I.S. Towner and J.C. Hardy, J. Phys. G: Nucl. Part. Phys.29, 197 (2003).[7] S. Eidelman et al., Phys. Lett. B592, 1 (2004).[8] I.S. Towner and J.C. Hardy, Phys. Rev. C 66 035501(2002).[9] S. Weinberg, Phys. Rev. 112, 1375 (1958).[10] H. Behrens and W. Bu¨hring, Electron Radial Wave Func-tions and Nuclear Beta-decay (Clarendon Press, Oxford,1982).[11] J.D. Jackson, S.B. Treiman and H.W. Wyld Jr., Phys.Rev. 106, 517 (1957).[12] W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 56, 22(1986); A. Sirlin, in Precision Tests of the Standard Elec-troweak Model, edited by P. Langacker (World-Scientific,Singapore, 1994).[13] W.E. Ormand and B.A. Brown, Phys. Rev. C 52, 2455(1995); W.E. Ormand and B.A. Brown, Phys. Rev. Lett.62, 866 (1989); W.E. Ormand and B.A. Brown, Nucl.Phys. A440, 274 (1985).[14] A. Sher et al., Phys. Rev. Lett. 91, 261802 (2003).[15] P. Herczeg, Phys. Rev. D 34, 3449 (1986); P. Herczeg,Prog. Part. Nucl. Phys., 46, 413 (2001).[16] For this purpose we use just the nine most accurate Ftvalues. Further, we also make a small adjustment [5] toaccount for possible systematic uncertainties in the calcu-lated values of δC by averaging our values given in Table Iwith those of Ormand and Brown [13] and increasing theerror assigned to Ft. We use Ft = 3073.5(12).

New Limits on Fundamental Weak-Interaction Parameters from Superallowed �� Decay

New limit on fundamental weak-interaction parameters from superallowed
  beta decay

New limit on fundamental weak-interaction parameters from superallowed beta decay

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