The magnitudes of the CKM matrix elements $V_{td}$, $V_{ts}$, and $V_{tb}$ are determined for the first time without any assumptions of unitarity. The implications for the unitarity of the CKM matrix as a whole are discussed

Swain, J D

Taylor, L

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First unitarity-independent determination of the CKM matrix elementsVtd, Vts, and Vtb and the implications for unitarityJohn Swain and Lucas TaylorDepartment of Physics, Northeastern University, Boston, Massachusetts 02115, USA(December 19, 1997)The magnitudes of the CKM matrix elements Vtd , Vts , andVtb are determined for the rst time without any assumptionsof unitarity. The implications for the unitarity of the CKMmatrix as a whole are discussed.12.15.Ff, 12.15.Lk, 13.38.Dg, 14.65.HaI. INTRODUCTIONThe relationship between weak and mass eigenstates ofquarks, assuming there are three generations, is describedby the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1]V =0@ Vud Vus VubVcd Vcs VcbVtd Vts Vtb1A : (1)Unless the CKM matrix is assumed to be unitary the bot-tom row elements, Vtd , Vts , and Vtb , are undetermined.There are, however, a number of measured quantitieswhich are sensitive to combinations of these elements.We use our recent determination of jVtb j from elec-troweak corrections to Z decays [2] to \unlock" the bot-tom row of the CKM matrix. From a combined t to datafrom the LEP, SLC, CESR, Tevatron, and other experi-ments we determine all the CKM elements, independentof unitarity assumptions, and proceed to test unitarity ofthe full CKM matrix for the rst time.II. CONSTRAINTS ON THE CKM ELEMENTSWe use experimental results and the correspondingtheoretical predictions to construct a chisquare which isthen minimised to obtain all the CKM matrix elements.The following sections describe individual chisquare con-straints which are ultimately summed to form the globalchisquare.A. Constraint on jVtb jIn a previous paper [2] we described a new method forthe determination of jVtb j from electroweak loop correc-tions, in particular to the process Z ! bb. We dene achisquare 2tb(mZ ;mt ;mH ; s; ; Vtb) for the agreementof the theory with nineteen experimental measurements.Full details of the input parameter values, errors, andcorrelations are given in Ref. [2].B. Constraint on jV tbVtd jThe product jV tbVtd j is constrained by measurementsof B0d−B0d mixing. Therefore, using these measurementsand our constraint on jVtb j enables us to extract jVtd j.The mixing is expressed in terms of the mass dierencemd of the CP eigenstates [3]md =G2F62jV tbVtd j2m2WmBd(BBdf2BdBS0 (xt) ; (2)wherepBBdfBd = (201  42) MeV [4] comes from acombined analysis of lattice calculations, which are com-patible with predictions of QCD sum rules, and theQCD factor is B = 0:55  0:01 [5]. The functionS0(xt = m2t =m2W ) is given by [3]S0(xt) =xt4"1 +3− 9xt(xt − 1)2 +6x2t lnxt(xt − 1)3#; (3)where mt is the running top mass [5] as required for con-sistency with B. It is related to the pole mass mt asmeasured by the Tevatron, according tomt = mt1−43s(mt ) mt − 8 GeV: (4)Given the measurement of md = 0:472 0:018 ps−1 [6]we dene the chisquare constraint on jV tbVtd j2tdtb =(md(mt ;mW )− 0:472)2(0:418md (mt ;mW ))2 + (0:018)2(5)where the theoretical prediction in the numerator de-pends on mt and mW . We vary mt explicitly in thet; mW is varied implicitly since it is a function of theother electroweak parameters which vary explicitly [2].The two terms in the denominator correspond respec-tively to the theoretical and experimental uncertainties,the former being dominated by the 20% relative error onpBBdfBd .1C. Constraint on jV tsVtb j=jVcb jThe rate for the process b! sγ, which involves a loopwith a virtual top quark, is proportional to jV tsVtb j2. Ourconstraint on jVtb j therefore enables us to determine jVts j.The theoretical prediction in the Standard Model hasbeen calculated to be [7]BR(b! sγ)th: = (3:28 0:33) 10−4jV tsVtb j=jVcb j0:9762: (6)The element Vcb enters through the normalisation to themeasurement of BR(B ! Xcee) = (10:4  0:4)% [8]which reduces the otherwise large error due to the b-quark mass uncertainty. CLEO has measured BR(b !sγ) = (2:32  0:57  0:35)  10−4 [9], and ALEPHhas presented a preliminary measurement of BR(b !sγ) = (3:29  0:71  0:68)  10−4 [10]. We aver-age these two results, assuming a common uncertaintyof 0:3  10−4 for the theoretical modelling, to obtainBR(b ! sγ)ex: = (2:60  0:59)  10−4. The chisquareconstraint on jV tsVtb j=jVcb j is then2tstb=cb =(BR(b! sγ)th: − BR(b! sγ)ex:)22th: + 2ex:; (7)where th: and ex: are the theoretical and experimentalerrors on BR(b ! sγ) respectively. We neglect the ex-plicit dependence on mt and s but we do include their3% contribution to the theoretical uncertainty of 10%.D. Constraints on other CKM elementsThe most precise constraints on the element jVud j arefrom nuclear beta decay. We use the recent measurementof jVud j from the Chalk River Laboratories [11] to denethe chisquare constraining jVud j2ud = (jVud j − 0:9740)2=(0:0005)2: (8)We use combined results from measurements of Ke3 andhyperon decays [8] to construct the jVus j chisquare2us = (jVus j − 0:2205)2=(0:0018)2: (9)We use the results of experiments measuring the produc-tion of charm from (anti-)neutrino interactions with dvalence quarks [8] to construct the jVcd j chisquare2cd = (jVcd j − 0:224)2=(0:016)2: (10)Measurements of the De3 decay D ! Ke+ [8] are usedto construct the jVcs j chisquare2cs = (jVcs j − 1:01)2=(0:18)2; (11)where the error is dominated by uncertainties in thehadronic form factors.The existence of b ! u transitions, which depend onjVub j, is well established from measurements of the end-point of the charged lepton energy spectrum in b !X‘− decays. Such analyses yield the ratio jVub j=jVcb j [8]from which we derive the chisquare2ub=cb = (jVub j=jVcb j − 0:08)2=(0:02)2: (12)Recently CLEO reported the rst measurements ofBR(B0 ! −‘+= (1:8 0:4 0:3 0:2)  10−4 andBR(B0 ! −‘+=(2:5 0:4+0:5−0:7  0:5 10−4 wherethe errors correspond respectively to the statistical, sys-tematic, and model-dependent uncertainties [12]. Theseresults yield the chisquare2ub = (jVub j − 0:0033)2=(0:0008)2: (13)The element jVcb j is determined both from the averageb-lifetime and from B ! D‘ decays in the limit of zerorecoil. The resulting chisquare for jVcb j is [5]2cb = (jVcb j − 0:040)2=(0:003)2: (14)III. UNITARITY-FREE FITWe t for all the CKM matrix elements without as-suming unitarity of the CKM matrix by minimising thechisquare, dened as2 = 2ud (Vud ) + 2us(Vus ) + 2cd(Vcd ) + 2cs(Vcs)+2ub=cb(Vub ; Vcb) + 2ub(Vub) + 2cb(Vcb)+2tdtb(Vtb ; Vtd ;mt ;mW )+2tstb=cb(Vts ; Vtb ; Vcb)+2tb(mZ ;mt ;mH ; s; ; Vtb); (15)where for completeness we have included all the depen-dences of the individual chisquare denitions. The 14parameters which vary in the t are mZ , mt , log10(mH ),s(mZ), , and the magnitudes of the nine CKM ele-ments. The results of the t are shown in the secondcolumn of table I, where the errors include the eectsof experimental and theoretical uncertainties and all cor-relations. The chisquare per degree of freedom for theunitarity-free t is 15:1=(28− 14) = 15:1=14. The proba-bility to obtain a value greater than this is 37% indicatingthat the data are in good agreement with the underlyingmodel.The bottom row CKM elements, jVtd j, jVts j, and jVtb jare determined for the rst time, independent of any uni-tarity assumptions.2IV. UNITARITY-CONSTRAINED FITWe also t the data imposing CKM unitarity as a con-straint. The four parameters which dene a three dimen-sional unitary matrix can be uniquely determined fromthe moduli of the elements. We choose the parametrisa-tion of the CKM matrix advocated by the Particle DataGroup [8] which enforces exact unitarityVud = c12c13 (16)Vus = s12c13 (17)Vub = s13e−i13 (18)Vcd = −s12c23 − c12s23s13ei13 (19)Vcs = c12c23 − s12s23s13ei13 (20)Vcb = s23c13 (21)Vtd = s12s23 − c12c23s13ei13 (22)Vts = −c12s23 − s12c23s13ei13 (23)Vtb = c23c13 (24)where cij = cos ij and sij = sin ij for three angles, 12,13, and 23, which lie in the rst quadrant. The phaseparameter 13 lies in the range 0 < 13 < 2. The 9parameters which vary in the unitarity-constrained t aremZ , mt , log10(mH ), s(mZ), , s12, s13, s23, and cos 13.The results of the t are shown in table I together withthe corresponding values of the CKM elements.The changes in mt , mH , and s, compared to theunitarity-free t, are due to their correlation with jVtb j,as discussed in Ref. [2]. Since only terms of the formcos 13 appear in this analysis, there is a two-fold ambi-guity between the regions 0 < 13 < , as favoured by CPnon-conservation in the kaon system, and  < 13 < 2.Unfortunately, the tted value of cos 13 = −0:26+0:82−0:74has rather large errors; in fact, the lower error corre-sponds to the physical limit of cos 13 = −1.The chisquare per degree of freedom for the unitarity-constrained t is 25:5=(28− 9) = 25:5=19. The probabil-ity to obtain a value larger than this is 14% which indi-cates reasonable consistency of the data and the model,albeit with less probability than the unitarity-free t.V. TESTS OF CKM UNITARITYUnitarity implies V V y = V yV = U , where U is theunit matrix, and hence the normality conditionsi  jVi1j2 + jVi2j2 + jVi3j2 = 1; and (25)j  jV1j j2 + jV2j j2 + jV3j j2 = 1; (26)for i = 1; 2; 3 rows and j = 1; 2; 3 columns. From only themagnitudes of the nine CKM elements it is not possibleto test orthogonality of pairs of dierent rows or pairs ofdierent columns. Allowing for correlations, the resultsof the six normality tests are1  jVud j2 + jVus j2 + jVub j2 = 0:997 0:001; (27)2  jVcd j2 + jVcs j2 + jVcb j2 = 1:105 0:367; (28)3  jVtd j2 + jVts j2 + jVtb j2 = 0:623 0:315; (29)1  jVud j2 + jVcd j2 + jVtd j2 = 0:999 0:007; (30)2  jVus j2 + jVcs j2 + jVts j2 = 1:104 0:367; (31)3  jVub j2 + jVcb j2 + jVtb j2 = 0:629 0:316: (32)The respective similarities between the three row con-straints and the three column constraints reflects the rel-atively small values of the o-diagonal elements. Only1 is signicantly dierent from unity, being 2:1 standarddeviations low. Given that there are six, albeit somewhatcorrelated, conditions we do not attach great signicanceto this although its future evolution is of interest.The trend is for the i and j to be somewhat lowerthan the expectations of a unitary matrix, as would bethe case if there were more than three generations. Totest this possibility with a single number we dene thequantityCKM =p(123)(123) (33)which should be unity for a unitary matrix. We ndCKM = 0:69  0:43 where all correlations have beentaken into account. This is lower than unity but onlyby 0:7 standard deviations, indicating that there is nocompelling evidence for a deviation from unitarity.These results constitute the rst tests of unitarity ofthe complete CKM matrix. In particular, 3, 1, 2, 3,and CKM are measured for the rst time.VI. OUTLOOKThe error on jVtd j is limited by the large theoreti-cal uncertainty on (BBdf2Bd), which is improving onlyslowly with time. In future jVtd j will also be deter-mined from the process K+ ! +, which is sensi-tive to the product jV tdVts j. Recently, the E787 exper-iment observed one candidate event with an estimatedbackground of (0:08 0:03) events, from which they de-termined BR(K+ ! + ) = (4:2+9:7−3:5)  10−10 [13].From this they derive, assuming unitarity, the constraint0:006 < jVtd j < 0:060, which is in agreement with ourmore precise unitarity-independent result. Ultimately,the measurement of BR(K+ ! + ) will yield preciseand theoretically well understood constraints on jV tdVts j.The uncertainty on jVts j from the process b ! sγ isstatistics limited, so some improvement can be expectedfrom CLEO and the B factory experiments. The param-eter ms describing Bs − Bs mixing is also sensitive tojVts j. The latest experimental limit is ms > 10:2 ps−1at the 95% C.L. [6]. If ms were to be measured, itsinterpretation in terms of jVts j would suer from the3large errors on (BBsf2Bs). The ratio md=ms is, how-ever, considerably less sensitive to theoretical uncertain-ties and its measurement would yield an important con-straint on the ratio jVtd=Vts j2. Similarly, a future mea-surement of the ratio BR(B ! (=!)γ)=BR(B ! Kγ)would constrain jVtd=Vts j2.Given that LEP has nished running on the Z, the un-certainty on jVtb j from our method of tting electroweakdata is unlikely to go below approximately 20% [2]. Thesingle top quark production rate at hadron colliders is,however, also sensitive to jVtb j. The estimated uncer-tainty at the end of the Tevatron Run II is jVtb j=jVtb j =12% − 19%, where the range reflects the uncertainty ofthe gluon structure functions [14].VII. SUMMARYThe elements jVtd j, jVts j, and jVtb j, and hence the fullCKM matrix, are determined for the rst time indepen-dent of any unitarity assumptions:jVtd j = 0:0113+0:0060−0:0029; (34)jVts j = 0:045+0:022−0:010; (35)jVtb j = 0:77+0:18−0:24: (36)We test unitarity and determine that all six CKM nor-mality conditions are consistent with unitarity. Four ofthese tests are performed for the rst time.ACKNOWLEDGEMENTSWe thank the National Science Foundation for nan-cial support and the Department of Physics, UniversidadNacional de La Plata for their generous hospitality.[1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963), M.Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652(1973).[2] J. Swain, and L. Taylor, hep-ph/9712420 (unpub-lished).[3] J. Rosner, in B Decays (revised 2nd edition), edited byS. Stone, World Scientic, Singapore, (1994).[4] J.M. Flynn and C.T. Sachrajda, hep-lat/9710057(1997). To appear in \Heavy Flavours II", World Sci-entic, Singapore, Eds. A.J. Buras and M. Lindner.[5] A.J. Buras and R. Fleischer, hep-ph/9704376 (1997).To appear in \Heavy Flavours II", World Scientic, Sin-gapore, Eds. A.J. Buras and M. Lindner.[6] M. Jimack, Invited talk at the International EurophysicsConference on High Energy Physics, Jerusalem, 1997,(unpublished).[7] K. Chetyrkin, M. Misiak and M. Mu¨nz, Phys. Lett.B400, 206 (1997).[8] R. M. Barnett et al., Phys. Rev. D54, 1 (1996).[9] M.S. Alam et al., Phys. Rev. Lett. 74, 2885 (1995).[10] P. Kluit, Invited talk at the International EurophysicsConference on High Energy Physics, Jerusalem, 1997,(unpublished).[11] E. Hagberg, J.C. Hardy, V.T. Koslowsky, G. Savard, andI.S. Towner, Invited talk at the NNDF 96, Osaka, 2-5Sep, (1996), nucl-ex/9609002 (unpublished).[12] J.P. Alexander et al., CLEO CONF 96-16 (unpub-lished); we do not use the updated result [CLEOCONF 97-30 (unpublished)] of BR(B0 ! −‘+=(2:0  0:3 0:3)  10−4 since the uncertainty from themodel dependence has yet to be evaluated.[13] S. Adler, et al., Phys. Rev. Lett. 79, 2204 (1997).[14] A.P. Heinson, A.S. Belyaev and E.E. Boos, Phys. Rev.D56, 3114 (1997).TABLE I. Results of the unitarity-free and unitar-ity-constrained ts described in the text. Values marked bya dagger (y) are not explicitly free in the t but are derivedfrom the other parameters in the same column.Free t Constrained tmZ (GeV) 91:187  0:002 91:187  0:002mt (GeV) 174:2  5:7 172:4  5:3log10(mH=GeV) 2:15+0:30−0:38 2:03+0:30−0:37s(mZ) 0:1171  0:0025 0:1188  0:0021−1(mZ) 128:913  0:092 128:904  0:092jVud j 0:9740  0:0005 0:9748  0:0003 yjVus j 0:2205  0:0018 0:2230  0:0014 yjVub j 0:00325  0:00058 0:0032  0:0006 yjVcd j 0:224  0:016 0:2228  0:0014 yjVcs j 1:01  0:18 0:9741  0:0003 yjVcb j 0:0401  0:0029 0:0397  0:0029 yjVtd j 0:0113+0:0060−0:0029 0:0093  0:0018 yjVts j 0:045+0:022−0:010 0:0387  0:0029 yjVtb j 0:77+0:18−0:24 0:9992  0:0001 ys12 − 0:2230  0:0014s13 − 0:0032  0:0006s23 − 0:0396  0:0030cos 13 − −0:26+0:82−0:744

First unitarity-independent determination of the CKM matrix elements $V_{td}$ , $V_{ts}$ , and $V_{tb}$ and the implications for unitarity

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First unitarity-independent determination of the CKM matrix elements VtdV_{td}Vtd​, VtsV_{ts}Vts​, and VtbV_{tb}Vtb​ and the implications for unitarity

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First unitarity-independent determination of the CKM matrix elements $V_{td}$ , $V_{ts}$ , and $V_{tb}$ and the implications for unitarity