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Stanford preprint ITP912/91

arXiv:hep-ph/9205240v1 27 May 1992April 19, 2018LBL-31431UCB-PTH-59/91OHSTPY-HEP-T-92-003Stanford preprint ITP912/91Predictions for Neutral K and B Meson Physics ∗Savas DimopoulosDepartment of PhysicsStanford UniversityStanford, CA 94305Lawrence J. HallDepartment of PhysicsUniversity of CaliforniaandTheoretical Physics GroupLawrence Berkeley Laboratory1 Cyclotron RoadBerkeley, California 94720Stuart RabyDepartment of PhysicsThe Ohio State UniversityColumbus, OH 43210AbstractUsing supersymmetric grand unified theories, we have recently in-vented a framework which allows the prediction of three quark masses,two of the parameters of the Kobayashi-Maskawa matrix and tan β, theratio of the two electroweak vacuum expectation values. These predic-tions are used to calculate ǫ and ǫ′ in the kaon system, the mass mixingin the B0d and B0s systems, and the size of CP asymmetries in the decaysof neutral B mesons to explicit final states of given CP.∗This work was supported in part by the Director, Office of Energy Research, Office ofHigh Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Departmentof Energy under Contracts DE-AC03-76SF00098 and DOE-ER-01545-573 and in part by theNational Science Foundation under grants PHY90-21139 and PHY86-12280.

DisclaimerThis document was prepared as an account of work sponsored by the United States Gov-ernment.Neither the United States Government nor any agency thereof, nor The Regentsof the University of California, nor any of their employees, makes any warranty, express orimplied, or assumes any legal liability or responsibility for the accuracy, completeness, or use-fulness of any information, apparatus, product, or process disclosed, or represents that its usewould not infringe privately owned rights. Reference herein to any specific commercial prod-ucts process, or service by its trade name, trademark, manufacturer, or otherwise, does notnecessarily constitute or imply its endorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof, or The Regents of the University of California.

Theviews and opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof of The Regents of the University of Californiaand shall not be used for advertising or product endorsement purposes.Lawrence Berkeley Laboratory is an equal opportunity employer.ii

In previous papers [1] we have invented a predictive framework for quarkand lepton masses and mixings based on the Georgi-Jarlskog ansatz [2] for theform of the mass matrices in supersymmetric grand unified theories [3]. In thispaper we use this scheme to make predictions for parameters of the neutral Kand B meson systems.

In particular we show that the CP asymmetries in neutralB meson decays are large and will provide a powerful test of the scheme. Webegin by reviewing the predictions for quark masses and mixings.The top mass is predicted to be heavymt = 179 GeVmb4.15 GeV mc1.22 GeV .053Vcb2 1.46ηb!

1.84ηc!. (1)where ηi is the QCD enhancement of a quark mass scaled from mt to mi.

Per-turbativity of the top Yukawa coupling also requires that mt <187 GeV. In thispaper the central values of ηi quoted correspond to a complete two loop QCDcalculation with αs(MZ) = .109, whereas in reference 1 an approximate two loopresult was given.The value of αs = .109 comes from a 1 loop analysis of the unification ofgauge couplings which for simplicity ignored threshold corrections at the grandunified and supersymmetry breaking scales [1].

However these threshold correc-tions will be present at some level in all grand unified models [4], and would nothave to be very large for our predicted value of the QCD coupling to range overall values allowed by the LEP data: .115±.008. Whatever the threshold correc-tions are, they must give an acceptable value for the strong coupling.

Hence it isimportant to consider the range of top masses allowed by the LEP range of αs.Larger values of αs lead to larger ηi reducing mt. Increasing αs(MZ) from .109to .123 decreases mt from 179 GeV to 150 GeV.

Alternatively, αs(MZ) = .123allows the top mass to be near the fixed point value of 187 GeV with Vcb = .047.The above numbers refer to the running mass parameter. The pole mass, whichis to be compared with experiment, is 4.5% larger.The particular form of the quark mass matrices leads to an unusual formfor the Kobayashi-Maskawa matrixV =c1 −s1s2e−iφs1 + c1s2e−iφs2s3−c1s2 −s1e−iφc1e−iφ −s1s2s3s1s3−c1s3eiφ(2)where s1 = sin θ1, etc, and we have set c2 = c3 = 1.∗We do not lose anygenerality by choosing the phases of quark fields such that θ1, θ2 and θ3 all liein the first quadrant.

We have predicted [1]s1 = .196∗The angle θ3 used in this paper corresponds to θ3 −θ4 used in reference 1.1

s2 = .053χ(3)whereχ =smu/md0.61.22GeVmcηc1.84(4)while the input Vcb determines s3 which must be chosen quite large in view of (1).The ratio ms/md, which we predict to be 25.15, strongly prefers mu/md < .8,while the present value of |Vub/Vcb| = s2 prefers mu/md larger than .4. In allof our predictions the largest uncertainty lies in mu/md, which we will displaythrough the parameter χ.The angle φ is determined by the requirement that |Vus| = sin θccφ = 1χ0.51 (1 ± 0.11) −0.13χ2= 0.38+.21−.14sφ ≃0.92 1.15 −0.15χ2 (1 ± 0.22)!= 0.92−.11+.05(5)In the first expressions the χ dependence is shown explicitly, together with theuncertainty from the measured value of the Cabibbo angle sin θc = 0.221 ±.003.Note that the O(1%) uncertainties in θc become greatly magnified in φ. Forthis reason we keep track of the θc dependence in our predictions.

For cos φthe expression is exact, while for sin φ it is good to better than 1%.The finalnumerical expressions correspond to the limits χ2 = 1∓13 and sin θc = .221±.003,which are used for all numerical predictions in this paper. Notice that cφ isdetermined to be positive and the experimental data on Re ǫ in the kaon systemforces sφ positive.Hence there is no quadrant ambiguity: choosing θ1,2,3 allin the first quadrant means that φ is also in the first quadrant.

The rephaseinvariant measure of CP violation [5] is given in our model byJ = ImVudVtbV ∗ubV ∗td = c1c2c3s1s2s23sφ= 2.6 × 10−5 Vcb.0532f(χ)(6)wheref(χ) = χ 1.15 −0.15χ2 (1 ± 0.22)!= 1−.29+.23. (7)The scheme which leads to these predictions involves mass matrices at the uni-fication scale with seven unknown real parameters.

Six of these are needed todescribe the eigenvalues: mu ≪mc ≪mt and md ≪ms ≪mb, while theseventh is the CP violating phase. Hence a more predictive theory, having fewerthan seven input parameters, must either relate the up mass matrix to that ofthe down, or must have an intrinsic understanding of the family mass hierarchy.Without solving these problems the most predictive possible theory will involve2

seven Yukawa parameters. Such a predictive scheme can only be obtained byrelating the parameters of the lepton mass matrix to those of the down quarkmass matrix.

To our knowledge the only way of doing this while maintainingpredictivity is to use the ansatz invented by Georgi and Jarlskog [2]. The crucialpoint about our scheme for fermion masses is that it is the unique scheme whichincorporates the GUT scale mass relations mb = mτ, ms = mµ/3 and md = 3mewith seven or less Yukawa parameters and completely independent up and downquark matrices.

The factors of three result from there being three quark colors.It is because of this uniqueness that the detailed confrontation of this modelwith experiment is important. If the model is excluded, for example by improv-ing measurements of mt, Vcb or Vub/Vcb, then the whole approach of searchingfor a maximally predictive grand unified scheme may well be incorrect.

Alter-natively it may mean that there is a very predictive scheme, but it involvesrelations between the up and down matrices in an important way. It is im-portant to calculate the observable parameters of the neutral K and B mesonsystems as accurately as possible, so that future experiments and lattice gaugetheory calculations will allow precision tests of this scheme.Our form for the Kobayashi-Maskawa matrix, equation (2), is very unfamil-iar and so we give predictions for the parameters that appear in the Wolfensteinform of the matrix [6].

The Wolfenstein form is an approximate form for thematrix which is unitary only to order λ3, where λ = sin θc = |Vus|. To order λ4we find the matrix can be written as:V (4) =1 −λ22 −λ48λAλ4(α −iβ)−λ1 −λ22 −λ48 −A2λ42Aλ2Aλ3 −Aλ4(α + iβ)−Aλ2 + Aλ421 −A2λ42where A is defined by Vcb and α, β by Vub.

The reason that we prefer to work withthe matrix at 0(λ4) is that for us the Vub entry numerically really is 0(λ4). Thuswe have A, α, β = 0(1).

Notice that the order λ4 contributions to Vud, Vcs, Vtsand Vtb can be dropped unless an accuracy of greater than 2 1/2% is required.This means that V (4) is actually the same as V (3), the usual Wolfenstein formwith ρ = λα and η = λβ.Hence we use the usual Wolfenstein parametersA, ρ, η. We findA = 1.09|Vcb|.053ρ = s2cφλ= 0.12(1 −0.25χ2)η = s2sφλ= 0.22f(χ).

(8)The leading dependence on the uncertain quantity mu/md is shown explic-itly, through the parameter χ. The uncertainties in ρ and η coming from sinθc3

are less than 10% and 4% respectively. These are not shown as the Wolfen-stein approximation is itself only good to about 20%.

Notice that we can writeρ + iη = reiφ, wherer = s2λ = 0.24χ. (9)Requiring unitarity for the imaginary part of V to 0(λ5) [6] one deducesJ ≃A2λ6η = 3.1 × 10−5 Vcb.0532f(χ).

(10)This agrees well with the exact result of equations 5, since (10) is expected tohave 0(λ) ∼0(20%) corrections.Beneath the scale of grand unification our effective theory is just that ofthe minimal supersymmetric standard model (MSSM). Hence in the rest of thispaper we wish to give the predictions for ǫ, ǫ′, xd, xs and the CP violating an-gles α, β, γ in the minimal supersymmetric standard model, with the KobayashiMaskawa matrix given by equation 2, and with our predicted values for quarkmasses.In reference 7 it will be shown that in the MSSM the supersymmetric con-tributions to ǫ, xd, xs and to the CP violating angles α, β, and γ in B mesondecay are small.

Here we will simply give a simplified discussion of why thesecontributions are negligable for quark masses and mixings of interest to us. Inthe standard model with a heavy top quark the quantities ǫ, xd and xs are dom-inated by box diagrams with two internal top quarks.

The amplitude of thesestandard model box diagrams can be written asBij = ASMVtiV ∗tj2(11)where i, j = d, s, b label the relevant external mass eigenstate quark flavors ofthe diagram and the dependence on the Kobayashi-Maskawa matrix elementsis shown explicitly. The leading supersymmetric contributions to these threequantities come from box diagrams with internal squarks and gluinos.

In thiscase the flavor changes occur through off-diagonal squark masses: M2ij. Theamplitudes for these box diagrams can be written asB′ij = AMSSMM2ij2(12)where again only the relevant flavor structure has been shown explicitly.

It hasbeen assumed that squarks of flavor i and j are degenerate.In the MSSM the squarks are all taken to be degenerate at the grand unifiedscale.The squark mass matrices evolve according to renormalization groupequations which generate non-degeneracies and flavor-changing entries. Whenall quarks are light a very simple approximation for the flavor-changing entries4

of the mass matrices results [8]. Since we predict a top Yukawa coupling close tounity, this is not good enough for our purposes.

We use the analytic solutions ofthe renormalization group equations valid to one loop order in the top Yukawacoupling, but with other Yukawa couplings neglected [9].This is the sameapproximation used to obtain our quark mass and mixing predictions [1] and issufficient providing tan β is not so large as to make the bottom Yukawa couplinglarge. In this approximation only SU(2) doublet squarks have flavor changingmasses.

We are able to find a very convenient approximation for the inducedflavor changing mass squared matrix elements for the down type doublet squarksM2ijM2 ≃0.4VtiV ∗tj 1 + 3ξ21 + 5.5ξ2! (13)where M is the mass of the (nearly) degenerate squarks and ξ is the ratio of thegluino to squark mass at the grand unified scale.

For values of the top Yukawacoupling consistent with the prediction of equation 1, this approximation is goodto better than a factor of two. The exact result has only slight sensitivity to thetrilinear scalar coupling A of the MSSM, which we have neglected.Comparing the standard model box amplitude (equation 11) to that ofthe MSSM superbox amplitude (equations 12 and 13) it is apparent that thedependence on the Kobayashi-Maskawa matrix elements is identical.

Hence theratio of box diagrams is independent of the external flavors i,j B′ijBij!= I (x)100GeVM2 1 + 3ξ21 + 5.5ξ2!2(14)where x = m˜g/M and m˜g is the gluino mass. The monotonic function I resultsfrom the momentum integral of the superbox diagram [8] and varies from I(1)= 1/30 to I(0) = 1/3.

If the squarks are taken light (eg 150 GeV) in an attemptto enhance the superbox amplitude, then x ≥1 to avoid an unacceptably lightgluino, resulting in I ≤1/30. To increase I therefore requires an increase inM, but this rapidly decreases the importance of the superbox diagram.

The ξdependent factor in equation 14 is always less than unity. We conclude that thesupersymmetric contributions to ǫ, xd and xs are unimportant in our scheme.We have shown that, in the MSSM with degenerate squarks, the superboxdiagrams have the same Kobayashi-Maskawa phases as in the standard modelbox diagrams.

This implies that the supersymmetric diagrams do not affect theCP asymmetry parameters α, β and γ in Bo decay, as is well known [10].We have assumed that at the grand unified scale the squark mass matricesare proportional to the unit matrix. In supergravity theories this proportion-ality is expected only at the Planck scale.

In general it is possible that largeinteractions of the quarks with superheavy fields could introduce large flavor5

changing effects from renormalisation group scaling between Planck and grandscales [11]. We assume that this does not happen, as would be the case if theonly large Yukawa coupling in the grand unified theory is that which generatesthe top quark mass.We now proceed to our predictions.

Since the KL −KS mass differencereceives large long distance contributions we do not think it provides a usefultest of our theory. On the other hand, all observed CP violation is described bythe single parameter ǫ, which is reliably calculated from short distance physics.To calculate ǫ precisely we do not use the Wolfenstein form for the KM matrix.Instead we use a manifestly phase invariant formula for ǫ in terms of J [12].

Wefind that the box diagram with internal top quarks dominates. Including a 20%contribution from the diagrams with one top and one charm we find|ǫ| = 7.2 10−3BKmt176GeV2 J2.79.10−5 Vcb.0532 s21s2c.

(15)The parameter BK describes the large uncertainty in the matrix element of afour quark operator between kaon states. We use experiment for |ǫ| and make aprediction for BK:BK = 0.40(1 ± .01 ± .03)4.15GeVmb2 1.22GeVmc2 ηb1.462 ηc1.842f(χ)−1(16)where we have used equations 1 and 6 for mt and J.

The first uncertainty showncomes from the experimental value of |ǫ| = (2.26 ± .02) × 10−3 while the secondcomes from sinθc. This prediction for BK is strikingly successful.

We stress thatmu/md cannot vary too much in our theory: values larger than 0.6 are stronglydisfavored by the fact that ms/md is 25, while the present experimental valuesof Vub/Vcb disfavors mu/md lower than 0.6. Allowing χ2 = 1 ± 1/3, sin θc =0.221±.003 gives the range BK = 0.31−0.57 (for mb = 4.15 GeV and mc = 1.22GeV).

This should be compared with recent lattice results BK = 0.7 ± 0.2 inthe quenched approximation [13].In the standard model, predictions for ǫ′/ǫ are very uncertain because theydepend sensitively on: i) various strong interaction matrix elements, ii) the valueof ΛQCD, iii) the value of the strange quark mass ms and iv) the value of thetop quark mass. We use our central predictions for ΛQCD and ms, and rely onthe 1/N approximation for the QCD matrix elements [14].

Using the analyticexpression given in reference 14 we are able to derive our predictionǫ′ǫ ≃3.9 × 10−4 2.7m−.5t+ 0.5m.1t −2.2m.4tχ.4 0.4BK.8(17)where mt is to be given in units of 179 GeV and is predicted in equation 1, andBK is predicted in equation 16. This small result is not unexpected, given the6

large value of mt [15], and we stress that it is uncertain because we do not knowhow well to trust the 1/N matrix elements.The leading supersymmetric contribution to ǫ′/ǫ comes from a diagramwith an internal gluino and an insertion of the flavor-changing squark mass ofequation 13. We find that this superpenguin amplitude is small compared tothe ordinary penguin:A(superpenguin)A(penguin)≃0.04150GeVM2 1 + 3ξ21 + 5.5ξ2!

5 −4mt180GeV2!−1(18)for the case of degenerate squarks and gluino of mass M. This is partly becausethe loop integral is numerically smaller, but is also because the ordinary penguindiagram is enhanced by an order of magnitude by a large ln mt factor. Eventhough these supersymmetric contributions are negligable, the uncertainties inthe QCD matrix elements still imply that ǫ′/ǫ cannot be considered a precisiontest of our scheme.The dominant standard model contribution to B0B0 mass mixing arisesfrom the box diagram with internal top quarks.

We find that for the B0d:xd = ∆mΓ= .25 √BfB150 MeV!2 mtGeV2|Vtd|2. (19)Using equation (1) for mt, Vtd = s1s3 and the experimental value for xd of .67±.10we predict√BfB = 134MeV4.15GeVmb 1.22GeVmc ηb1.46 ηc1.84 |Vcb|.053.

(20)We note that if xd, mt and Vtd are allowed to range over their experimentallyallowed values, the prediction of the box diagram (equation 19) implies that√BfB will have to range over an order of magnitude. It is therefore a non-trivialsuccess for our theory that it gives a prediction for√BfB which is close to thequoted values for fB√B.

Our prediction should be compared with recent latticeresults: fB = 205±40 MeV and√BfB = 220±40 MeV [16]. Our predictions formt (1), BK (16) and fB (20) all depend on ηi which depend on αs.

The numbersquoted are for the fairly low value of αs = .109. Threshold corrections at thegrand unified scale could increase this, easily resulting in a 20% increase in ηbηc.This not only reduces the top mass, but gives improved agreement with latticecalculations for both BK and fB.

At any rate, once the top mass is measuredour predictions for BK and fB will be sharpened considerably. Our results forBK and fB agree with those obtained reference 17.The standard model box diagram relates the mass mixing in B0s to that in7

B0d by:xsxd= |Vts|2|Vtd|2 Bsf 2BsBdf 2Bd!= 25 Bsf 2BsBdf 2Bd! (21)where we used our result for the ratio of Kobayashi-Maskawa factors: c21/s21 =25 with negligible uncertainty.

If the ratio of B meson decay constants could beaccurately calculated, and if large values of xs (say 15 to 25) could be measured,then (21) provides a precision test of our theory.Finally we consider CP asymmetries which result when B0 and B0 candecay to the same CP eigenstate f [18]. Unitarity of the KM matrix implies the1st and 3rd columns are orthogonal: VudV ∗ub + VcdV ∗cb + VtdV ∗tb = 0.

This can berepresented as a triangle since the sum of three vectors is zero. Labelling theangles opposite these three vectors as β, α, and γ respectively, one finds thatthe CP asymmetries are proportional to sin 2β (for Bd →ψKs, etc), sin 2α (forBd →π+π−etc.) or sin 2γ (for Bs →φKs, etc).In the approximation that c2 = c3 = 1 and that s1s2 ≪1, we calculate sin2α, sin 2β and sin 2γ to an accuracy of (1 +0(λ3)), ie to 1% accuracy:sin 2α = −2cφsφsin 2β = 2c1s1s2sφs2c(1 + c1s2cφs1)sin 2γ = 2cφsφs21s2c(1 + c1s2cφs1).

(22)It is interesting to note that s3 does not appear anywhere in these results. Thisis because all lengths of the unitarity triangle are simply proportional to s3.This is similar to the well known result that in the Wolfenstein approximationall lengths of the triangle are proportional to A.

For us this lack of sensitivityto s3 is an essentially exact result. Given that we know s1 precisely, and thatφ is extracted from s1, s2 and the Cabibbo angle sc, the only uncertainties innumerically evaluating α, β and γ come from experimental uncertainties in sin θcand in the dependence of s2 on mu/md, mc and ηc via χ shown in equations 3 and4.

We calculate sin 2α, sin 2β and sin 2γ in terms of χ for sinθc = .221±.003. Theresults are shown in the Figure.

The solid line is for sin θc = .221 while the long(short) dashed lines are for sinθc = .224(.218). Present experiments allow verywide ranges of α, β, γ : −1 < sin 2α, sin 2γ < 1 and .1 < sin 2β < 1[19] so thatour predictions are in a sufficiently narrow range that measurements of theseCP asymmetries will provide a precision test of our model.

Our predictions arevery positive for experimentalists: sin 2β is not near its lower bound, and for thetwo most experimentally challenging cases, sin 2α and sin 2γ, the asymmetriesare close to being maximal.8

How well can our model be tested with an asymmetric B factory operat-ing at the Υ(4S) with luminosity of 3.1033cm−2s−1 [20]? We assume a totalintegrated luminosity of 1041 cm−2, and find, using the numbers in [19], thatfor decay to a final state of branching ratio B, the quantity sin 2α (or sin 2β orsin 2γ) will be measured with an error bar ±δ:δ = .05sB4.10−5s1041cm−2R Ldt(23)where B = (4, 3, 2)10−5 for Bd →ψKs, Bd →π+π−, Bs →ρKs relevant formeasuring sin 2β, sin 2α and sin 2γ respectively.

Measuring all three quantitiesto ±.05 will provide a spectacular precision test of our model. The values ofsin 2α, sin 2β and sin 2γ must be fit by a single value of mu/md which will bedetermined at the ±0.1 level.We stress two important features of our predictions for these CP asymmetryparameters.

Firstly, as in the standard model, they are relatively insensitive tounknown QCD matrix elements. Secondly, they test the Georgi-Jarlskog ansatzin a deep way.For example the only dependence on the renormalization ofgauge couplings beneath the grand scale comes from uncertainties in ηc.

Theseuncertainties could be removed completely by taking the strange quark mass asinput. Taking ms = 180 ± 60 MeV only leads to a 15% uncertainty in χ.In this paper we have made accurate predictions for parameters in theneutral K and B systems, in the belief that the scheme of reference (1) will bedecisively tested in the future.

It is worth stressing that the predictions for BKand√BfB are close to central quoted theoretical values, and thus are alreadystrikingly successful. We have followed the consequences of the only frameworkincorporating the Georgi-Jarlskog mechanism which uses the minimal numberof Yukawa couplings and has independent up and down quark mass matrices:there is absolutely no guarantee that ǫ or xd will be predicted correctly.

Considerfor example the case when ǫ is dominated by the top quark contribution whichis proportional to m2tJBKRe(VtdV ∗tsVusV ∗ud).Even though a theory may givesuccessful predictions for mt(100 −200GeV ), Vtd(.003 −.018) and Vts(.030 −.054) it is not guaranteed that the prediction for ǫ will be anywhere close toexperiment. The quantity m2tRe(VtdV ∗tsVusV ∗ud) has a spread of a factor of 50,and the quantity J which is proportional to s1s2s23sφ could vary over a very widerange.

In particular recall that the phase φ is determined by the requirement that|Vus| = .221±.003. We think that it is extremely non-trivial that the predictionof our theory for m2tRe(VtdV ∗tsVusV ∗ud)J is such that the central prediction for BKis 0.4.The essential results of this paper are given in equation (16) for BK (fromǫ), equation (20) for√BfB (from xd), equation (21) for xs and in the Figure9

for sin 2α, 2β, 2γ. The prediction for ǫ′/ǫ in equation (17) is less important asit involves uncertainties from the matrix elements.

Once the top quark mass isaccurately known, the range of predicted values for BK and√BfB will narrow.The largest uncertainly in BK comes from mu/md. CP asymmetries in decaysof neutral B mesons offer the hope of a precision test of our theory which is freeof strong interaction uncertainties.

An asymmetric B factory operating at theΥ (4S) with an integrated luminosity of 1041 cm−2 can determine sin 2α, β, γ toan accuracy of ±0.05, and this will lead to a determination of mu/md to within±0.1.AcknowledgementsLJH acknowledges partial support from the NSF Presidential Young Inves-tigator Program, thanks Vernon Barger and Gian Giudice for discussions aboutQCD corrections and thanks Uri Sarid for many helpful conversations.References1. S. Dimopoulos, L.J.

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