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Testing the Standard Model and Schemes for Quark Mass

arXiv:hep-ph/9207225v1 9 Jul 1992WIS-92/52/Jun-PHLBL-32563hep-ph/9207225July 1992(T/E)Testing the Standard Model and Schemes for Quark MassMatrices with CP Asymmetries in B DecaysYosef Nir1Physics Department, Weizmann Institute of Science, Rehovot 76100, IsraelUri Sarid2Physics Department, Weizmann Institute of Science, Rehovot 76100, IsraelandTheoretical Physics Group3Lawrence Berkeley Laboratory1 Cyclotron RoadBerkeley, California 94720AbstractThe values of sin 2α and sin 2β, where α and β are angles of the unitaritytriangle, will be readily measured in a B factory (and maybe also in hadroncolliders).We study the standard model constraints in the sin 2α −sin 2βplane. We use the results from recent analyses of fB and τb|Vcb|2 which takeinto account heavy quark symmetry considerations.

We find sin 2β ≥0.15 andmost likely sin 2β >∼0.6, and emphasize the strong correlations between sin 2αand sin 2β. Various schemes for quark mass matrices allow much smaller areasin the sin 2α −sin 2β plane.

We study the schemes of Fritzsch, of Dimopoulos,Hall and Raby, and of Giudice, as well as the “symmetric CKM” idea, and showhow CP asymmetries in B decays will crucially test each of these schemes.1Incumbent of The Ruth E. Recu Career Development Chair. Supported in part by the IsraelCommission for Basic Research, United States–Israel Binational Science Foundation and the MinervaFoundation.

E-mail: ftnir@weizmann.weizmann.ac.il2E-mail: sarid%theorm.hepnet@lbl.gov3Permanent address

DisclaimerThis document was prepared as an account of work sponsored by the United States Government.Neither the United States Government nor any agency thereof, nor The Regents of the University ofCalifornia, nor any of their employees, makes any warranty, express or implied, or assumes any legalliability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus,product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial products process, or service by its trade name, trademark,manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation,or favoring by the United States Government or any agency thereof, or The Regents of the Universityof California. The views and opinions of authors expressed herein do not necessarily state or reflectthose of the United States Government or any agency thereof of The Regents of the University ofCalifornia and shall not be used for advertising or product endorsement purposes.Lawrence Berkeley Laboratory is an equal opportunity employer.ii

CP asymmetries in neutral B decays will provide a unique way to measure theCKM parameters. In a high-luminosity e+e−collider running at the energy of theΥ(4S) resonance (a “B factory”), two of the three angles of the unitarity triangle(see Fig.

1) will be readily measured [1]: the CP asymmetry in e.g. B →π+π−willdetermine sin 2α, while that in e.g.

B →ψKS will determine sin 2β. It may also bepossible to measure sin 2β in a hadron collider, but sin 2α would be difficult due to thelarge background (see, e.g., [2]).

The experimental measurements are expected to behighly accurate and the theoretical calculations are, to a large extent, free of hadronicuncertainties. Furthermore, CP asymmetries in neutral B decays are a powerful probeinto possible sources of CP violation beyond the standard model (SM).

The richnessof available B decay modes would allow one to determine detailed features of the newsources of CP violation if the SM predictions are not borne out. In this work, werefer to both aspects of CP asymmetries in B decays, namely the determination ofthe CKM parameters within the SM, and the testing of extensions of the SM, with aspecial emphasis on the information that can be extracted by measuring two anglesof the unitarity triangle rather than, say, sin 2β alone.In the first part of this work, we investigate in detail the SM predictions for sin 2αand sin 2β.

In particular, we study the correlation between the two quantities andpresent our results in the sin 2α −sin 2β plane. We update previous analyses withemphasis on recent theoretical developments which involve the heavy quark symmetry.In the second part of this work, we show how various schemes for quark massmatrices can be tested through their predictions for sin 2α and sin 2β.

We analyzethe Dimopoulos-Hall-Raby (DHR) scheme [3], the Giudice scheme [4], the Fritzschscheme [5], and the idea that the CKM matrix is symmetric in the absolute values ofits entries [6] (including the two-angle parametrization of Kielanowski [7]). Each ofthese schemes allows a range for the asymmetries which is much smaller than in theSM and thus may be clearly excluded when the asymmetries are measured.Various bounds on the CKM parameters are usually presented as constraints onthe form of the unitarity triangle (for a review see [8, 9, 2] and references therein).However, the quantities directly measurable via CP violation in a B-factory are sin 2αand sin 2β, so we will present our constraints in terms of these observables.

The time-dependent CP asymmetry in the decay of a B or ¯B into some final CP-eigenstate fis given byΓ(B0(t) →f) −Γ(¯B0(t) →f)Γ(B0(t) →f) + Γ(¯B0(t) →f) = −Imλ(f) sin ∆M t ,(1)where ∆M ≡M(BHeavy) −M(BLight), B0(t) (¯B0(t)) is a state which starts out as theflavor eigenstate B0 (¯B0) at a time t = 0, and λ(f) is a complex number with (almostexactly) unit magnitude. Then, within the SM (and in all schemes considered in this1

work),Imλ(π+π−) = sin 2α,Imλ(ψKS) = sin 2β(2)(where we took into account the fact that ψKS is a CP-odd state). Thus, our figuresin the sin 2α −sin 2β plane simply present the allowed range in the Imλ(π+π−) −Imλ(ψKS) plane.

This gives an important advantage to our method: the presentationin the Imλ(π+π−)−Imλ(ψKS) plane allows a direct comparison of the SM predictions(or the experimental results) with models of new physics where the asymmetries arenot related to angles of the unitarity triangle.We use the following relations to transform from the (ρ, η) coordinates of the freevertex A of the unitarity triangle to (sin 2α, sin 2β):sin 2α=2η[η2 + ρ(ρ −1)][η2 + (1 −ρ)2][η2 + ρ2],sin 2β=2η(1 −ρ)η2 + (1 −ρ)2. (3)Note that these coordinate transformations are highly nonlinear; hence the predictionsin the sin 2α −sin 2β plane will be very different from the more familiar constraintsin the ρ −η plane.

Furthermore, since (3) are not (uniquely) invertible, we may notsimply map the regions in the ρ −η plane allowed by each of the various constraintsinto corresponding regions in the sin 2α −sin 2β plane, and then assume that theoverlap in the latter is allowed. To see this, note that a single point in the overlapregion in the sin 2α −sin 2β plane may correspond to two different points in the ρ−ηplane.

If each of these two points is allowed by one constraint but forbidden by theother, then the original point in the sin 2α −sin 2β plane is in fact forbidden thoughit is in the overlap of two regions allowed by the individual constraints. We thereforeform the overlap in the ρ−η plane first, and then map this overall-allowed region intosin 2α−sin 2β coordinates.

Finally, even in the ρ−η plane the overlap of two allowedregions may not all be allowed: a given point in the overlap may meet the variousconstraints only by using different values of some parameter which enters into bothconstraints. But this correlation is unimportant in practice, since the uncertaintiesin the parameters which enter into more than one constraint never dominate bothconstraints.We now analyze the SM predictions for sin 2α and sin 2β, updating previous anal-yses of constraints on the CKM parameters.

The most significant update is in theconstraint from B−¯B mixing, which determines the length of one side of the unitaritytriangle:(1 −ρ)2 + η2 =(1.3 × 107 GeV) xd(BBf 2B)ytf2(yt)(τB|Vcb|2)|Vcd|2ηB(4)2

where ηB = 0.85 is a QCD correction, yt = (mt/MW)2 and f2(x) = 1 −34x(1 + x)(1 −x)−2[1 + 2x(1 −x2)−1 ln(x)]. Recently, both lattice and QCD sum-rule calculations ofthe fB decay constant were made which rely on heavy quark symmetry considerations.Results from the two techniques now converge to a consistent range and, we believe,should be preferred over previous, more model-dependent, calculations.

We use theresult of ref. [10] from QCD sum-rules, which is consistent with lattice calculations(see [11] and references therein),fB = 190 ± 50 MeV.

(5)Since the BB factor is expected to be close to unity, we simply take BB = 1 andneglect the uncertainty in BB relative to that in fB (or, equivalently, absorb it intothe uncertainty in (5)). Heavy quark symmetry considerations have also been appliedto find the combination |Vcb|2τB.

We again believe that the new results, in which onlythe corrections to the heavy quark limit are model-dependent, should replace previouscalculations which were completely model dependent. We take the analysis of ref.

[12]with updated input data [13]:|Vcb| (τb/1.3 ps)1/2 = 0.040 ± 0.005 . (6)For the mixing parameter xd, we use [14]xd = 0.67 ± 0.11 .

(7)Finally, we use |Vcd| = |Vus| = 0.221 ± 0.002.Our second constraint comes from the endpoint of the lepton spectrum in charm-less semileptonic B decays. We adopt the range quoted by the Particle Data Group[15]:|Vub/Vcb| = 0.10 ± 0.03 .

(8)This determines the length of the other side of the unitarity triangle:ρ2 + η2 =VubVcbVcd2. (9)The third constraint comes from the CP-violating ǫ parameter in the K0 system:ρ ="1 + (η3f3(yc, yt) −η1)ycη2ytf2(yt)|Vcb|2#−1η"2.5 × 10−5|ǫ|η2ytf2(yt)|Vcb|4BK|Vcd|2#(10)where η1 = 0.7, η2 = 0.6 and η3 = 0.4 are QCD corrections [16], yc = (mc/MW)2 andf3(x, y) = ln(y/x) −34y(1 −y)−1[1 + y(1 −y)−1 ln(y)].

The uncertainties here lie inthe value of the BK parameter, estimated to beBK = 2/3 ± 1/3,(11)3

and in the range for |Vcb|. Using [15] τB = 1.29 ± 0.05 ps, we deduce from (6):|Vcb| = 0.040 ± 0.007.

(12)We further use |ǫ| = (2.26 ± 0.02) × 10−3 and [17] mc(mc) = 1.27 ± 0.05 GeV.Since the xd and ǫ constraints depend on mt, we have carried out our analysis forvarious mt values within the range 90 GeV ≤mt ≤185 GeV. We present our results inFig.

2 in two ways. First, the thin black curves encompass all values of (sin 2α, sin 2β)which satisfy all three constraints using values of the input parameters within their1 −σ ranges (or within the theoretically favored ranges for the parameters BK andfB).

That is, the SM can accommodate a B-factory result anywhere within thesecurves without stretching any input parameter beyond its 1 −σ range. We will referto these regions as the “allowed” areas of the SM.

(A somewhat similar plot of sin 2αversus ρ appears in [11]). Second (and similarly to [18]), in order to get a sense ofthe expected value of (sin 2α, sin 2β) given our current knowledge of the various inputparameters, we generated numerous sample values for these parameters based on aGaussian distribution for |Vcd|, τB|Vcb|2, |Vub/Vcb|, τB, xd, mc and |ǫ|, and a uniformdistribution (= 0 outside of the “1 −σ” range) for fB.

For each sample set we usedthe constraints (4) and (9) to determine ρ and η, and then rejected those sets whichdid not meet the constraint (10) for 1/3 ≤BK ≤1. We binned the sets which passedin the sin 2α−sin 2β plane, and thus obtained their probability distribution.

We showin Fig. 2 the resulting 68% and 90% probability contours in dark gray and light gray,respectively.

Since we do not know the true origin of the CKM parameters and thus donot know the true probability distribution from which the experimental inputs result,and since the theoretical restrictions on fB and BK cannot be posed statistically, wecan only interpret these probability contours as an indication of likely outcomes forB-factory results based on the SM. For example, the “tail” of the allowed areas whichextends towards small values of (sin 2α, sin 2β) requires many of the parameters to bestretched to their 1−σ bounds and so seems unlikely and lies outside both probabilitycontours.Similarly to previous analyses (see, e.g., [2, 11, 19, 21, 20, 22, 23]), we find thatsin 2α can have any value in the full range from −1 to 1, while sin 2β is always positiveand has a lower boundsin 2β ≥0.15 .

(13)Furthermore, sin 2α is likely to be positive if the top mass is near its present lowerbound, and most importantly the favored values for sin 2β are above 0.5. We alsofind that the bounds on the two quantities are correlated (as also noted in [21]).

Inparticular, we note that:4

• The magnitude of at least one of the two asymmetries is always larger than 0.2,and probably larger than 0.6.• If sin 2β ≤0.4, then sin 2α must be positive—in fact, above 0.2.Once the top mass is measured firmer predictions will of course be possible, based onone of the graphs in Fig. 2.Various estimates may be made of the allowed ranges for the input parameters.In particular, there is no single obvious way to evaluate theoretical uncertainties.Furthermore, future improvement in both experimental measurements and theoreticalanalyses would certainly strengthen the constraints.

Thus, it is useful to understandthe sensitivity of our analysis to the various uncertainties.To this end we havedisplayed in Fig. 3 how the allowed regions of the SM depend on the choice of inputparameters, for a representative top mass of 130 GeV.

For Figs. 3a, 3b and 3c wehave allowed somewhat larger ranges for 0.05 ≤|Vub/Vcb| ≤0.15 and 100 MeV ≤fB ≤300 MeV.

All other ranges are kept as before. The 5 solid lines of Fig.

3a correspond,from bottom to top, to the constraint (9) when |Vub/Vcb| increases from 0.05 to 0.15.The 12 solid lines of Fig. 3b correspond, from left to right, to the constraint (4) whenthe values of fB and τB|Vcb|2 decrease within their respective ranges.

The 6 solid linesof Fig. 3c correspond, from left to right, to the constraint (10) when |Vcb|2 and BKdecrease within their respective ranges.

(Note that each solid line in these figuresmust meet all three constraints. For Fig.

3b this disallows the lower end of the rangefor fB and τB|Vcb|2, while for Fig. 3c it is the lower end of the range for |Vcb|2 and BKthat is not allowed.) One can then read offthe approximate allowed region for a morerestricted choice of input parameter ranges.

For completeness we have also plottedin Fig. 3d the allowed region obtained by accepting the range 0.15 ≤|Vub/Vcb| ≤0.20suggested by Isgur et al.

[24], while keeping all other parameters as in the rest ofFig. 3.

In this case it is likely that sin 2β is very close to unity, or else (and this isunlikely) sin 2α ∼sin 2β and they can both be as small as roughly 0.1 if |Vub/Vcb| andBK are as large as possible and fB is as small as possible.We next turn to the testing of various schemes for quark mass matrices. We usethe following ranges for quark masses at 1 GeV [17]:mc = 1.36 ± 0.05 GeV,mb = 5.6 ± 0.4 GeV,(14)and for mass ratios:mdms= 0.051 ± 0.004,mumc= 0.0038 ± 0.0012,msmb= 0.030 ± 0.011.

(15)In the remainder of our analysis we allow only 1 −σ ranges for all inputs, since webelieve that if any of these schemes need to be stretched beyond their 1−σ predictions5

then their motivation is largely lost. These 1 −σ ranges should only be viewed as thefavored values within the schemes; one should not rule out any scheme simply on thebasis that the experimental results do not quite fall within the 1 −σ predictions weobtain.

In Fig. 4 we display these predictions of the four schemes for the same samplevalues of mt as in Fig.

2. Only the symmetric CKM ansatz admits a sufficientlylarge range of mt to be included in more than one graph.

For reference we have alsoindicated, in gray, the 1 −σ allowed areas of the SM.We first discuss the Fritzsch scheme [5],Mu =0au0au0bu0bucu,Md =0adeiφ10ade−iφ10bdeiφ20bde−iφ2cd. (16)It fits ten parameters (6 masses, 3 mixing angles and a CP-violating phase) with eightparameters and therefore makes two predictions.

It is now nearly excluded [25]. Themain difficulty lies in the relation|Vcb| =smsmb−e−iφ2smcmt ,(17)which can only be fulfilled if the top quark is close to the experimental lower bound:mt ∼90 GeV.

(18)If the top quark is indeed this light, then the next crucial test for the Fritzsch schemewould be its predictions for CP asymmetries in B0 decays. The allowed range for(sin 2α, sin 2β) is shown as the black wedge in Fig.

4a. We find0.10 ≤sin 2α ≤0.67;0.56 ≤sin 2β ≤0.60.

(19)We turn next to the scheme of Giudice [4], which requires the charged fermionmass matrices to have the following form at the GUT scale:Mu =00b0b0b0a,Md =0feiφ0fe−iφd2d02dc,Mℓ=0f0f−3d2d02dc. (20)This scheme fits the quark and lepton mass matrices with six parameters and thereforemakes seven predictions.

Among them we findmt ∼125 −155 GeV,|Vcb| ∼0.048,0.07 ≤|Vub/Vcb| ≤0.084130 GeVmt. (21)6

Note that our allowed range for mt is smaller than in ref. [4], due to our strongerbounds on |Vub/Vcb|.

(This range is very sensitive to the bottom quark mass, andthus could be enlarged by adopting more conservative estimates of the uncertainty inmb.) It is not unlikely that this scheme would survive the various measurements untila B-factory starts running.

Then it allows only a narrow band in the sin 2α −sin 2βplane, as shown in Fig. 4b.

The overall constraint is−0.98 ≤sin 2α ≤+1.0;0.2 ≤sin 2β ≤0.7 . (22)However, for low sin 2β values, there is a strong correlation between the two asym-metries.

In particular, if sin 2β <∼0.45, then sin 2α ≥0.65.The scheme by DHR [3] requires that, at the GUT scale, charged fermion massmatrices are of the following form:Mu =0c0c0b0ba,Md =0feiφ0fe−iφe000d,Mℓ=0f0f−3e000d. (23)It has seven parameters and therefore six predictions, among which we find (c.f.

[26])mt ∼185 GeV,|Vcb| ∼0.047 ,|Vub/Vcb| ∼0.065. (24)(Note that the latter prediction, which is at the top of the 1−σ range for this scheme,is just below our allowed range.

We therefore predict a very narrow range of the DHRparameter χ which accounts for much of the uncertainty in this scheme: χ2 ≃4/3. )Thus, future measurements of mt, or theoretical improvement in determining |Vcb|or |Vub/Vcb|, may easily exclude the DHR scheme.

If it survives these tests, then itwould provide very powerful predictions for CP asymmetries in B0 decays. Only avery narrow range in the sin 2α −sin 2β plane is allowed, as shown in Fig.

4d. Theoverall constraint is−0.58 ≤sin 2α ≤−0.33;0.51 ≤sin 2β ≤0.60 .

(25)Once again the values of the two asymmetries are correlated, providing an evenstronger test than implied by (25).Our last example is the symmetric ansatz [6] for the CKM matrix,|Vij| = |Vji|. (26)The theoretical motivation for this ansatz is more obscure than for the previousans¨atze.

In particular, it is still to be demonstrated that the constraints (26) can resultfrom some symmetry of the lagrangian [27]. This ansatz leads to (c.f.

[28, 29, 30, 31])mt >∼160 GeV,|Vub/Vcb| ≥|Vcd|/2 ≃0.11 . (27)7

(This bound on mt is lower than in some previous analyses due to our higher allowedrange of fB, as already remarked in [18].) CP asymmetries in B0 decays would beextremely powerful in testing (26).The correlation between sin 2α and sin 2β isstrongest here, as (26) leads toρ = 1/2 =⇒sin 2α = −2 sin 2β cos 2β.

(28)For a fixed mt value, (28) leads to an allowed curve in the sin 2α −sin 2β plane, asshown in Figs. 4c and 4d.

For the overall bounds we find−1.0 ≤sin 2α ≤−0.76 ;0.68 ≤sin 2β ≤0.91 . (29)The two-angle parametrization of the CKM matrix proposed by Kielanowski [7] is aspecial case of this ansatz, in which η ≃1/(2√3) (to within a few percent).

Conse-quently (c.f. [29, 30]) sin 2α = −√3/2 = −sin 2β, as indicated by the small filledcircle in Figs.

4c and 4d.Before concluding, let us mention a discussion of the structure of quark massmatrices by Bjorken [32]. His assumptions lead to a prediction for the angle γ ofthe unitarity triangle, γ ≈π/2.

For the asymmetries discussed here, this impliessin 2α = sin 2β, which coincides with the predictions of the superweak scenario. Adiscussion of the experimental prospects of excluding such a relation can be found inrefs.

[33, 34].To summarize, we have examined the predictions of the SM and of various quarkmass matrix schemes for sin 2α and sin 2β or, equivalently, for the CP asymmetriesin B →ππ and B →ψKS. Our main results are presented in Figs.

2 and 4. We havedisplayed them in the sin 2α −sin 2β plane to facilitate direct comparison with futureexperiments or non-standard models, and to show the importance of the correlationbetween the predictions for sin 2α and for sin 2β.

(This correlation was also used in[21, 35]). The predictions are quite encouraging for experimenters:• Recent improvements in theoretical calculations lead to a lower bound on theasymmetry in B →ψKS of order 0.15, somewhat higher than previous analyses.• If the asymmetry in B →ψKS is close to its lower bound, than it is highlycorrelated with the asymmetry in B →ππ and at least one of the two is largerthan 0.2.• For the asymmetries to both be small, many parameters have to assume valuesclose to their 1 −σ bounds, which is improbable.

It is more likely that at leastone of the asymmetries is larger than 0.6.8

• Various schemes for quark mass matrices allow a much smaller range for thetwo asymmetries than does the SM. Therefore, they would be stringently testedwhen the asymmetries are measured.US acknowledges partial support from the Albert Einstein Center for TheoreticalPhysics at the Weizmann Institute, and thanks the members of the particle theorygroup at the Weizmann Institute for their kind hospitality and Lawrence Hall forilluminating discussions.References[1] A.B.

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Figure CaptionsFig. 1: The unitarity condition V ∗ubVud+V ∗cbVcd+V ∗tbVtd = 0 represented as a triangle inthe complex plane.

The sides have been divided by |VcbVcd| so that the verticesmay be placed at (0, 0), (1, 0) and (ρ, η). The angles α and β are measuredcounterclockwise as shown.Fig.

2: The SM predictions in the sin 2α(horizontal)−sin 2β(vertical) plane, for fourdifferent top quark masses: (a) 90 GeV, (b) 130 GeV, (c) 160 GeV and (d) 185GeV. The regions allowed by the 1 −σ ranges for all parameters described inthe text are outlined by the thin black lines.

The 68% probability contoursgenerated as described in the text are shown as thick dark-gray lines, while the90% contours are indicated by thinner light-gray lines.Fig. 3: The dependence of the allowed regions in the sin 2α −sin 2β plane on theinput parameter ranges, for a representative value of mt = 130 GeV.

The largestregion allowed by all three constraints is outlined by the dashed lines. For thisfigure we have allowed the wider range 100 MeV ≤fB ≤300 MeV but keptall other ranges as before, with the exception of |Vub/Vcb|: in 3a, b and c weallow the wider range 0.05 ≤|Vub/Vcb| ≤0.15, while in 3d we adopt the higherrange of ref.

[24], 0.15 ≤|Vub/Vcb| ≤0.20. The 5 solid lines in 3a correspond,from bottom to top, to the constraint (9) when |Vub/Vcb| increases from 0.05 to0.15.

The 12 solid lines of 3b correspond, from left to right, to the constraint(4) when its right-hand side increases from 0.29 to 2.71. The 6 solid lines of3c correspond, from left to right, to the constraint (10) when its first bracketedexpression increases from 1.34 to 1.40 and the second increases from 0.24 to0.96.Fig.

4: The 1 −σ allowed regions predicted by various mass matrix schemes in thesin 2α−sin 2β plane, for the same sample values of mt as in Fig. 2.

The allowedregions within the SM are outlined for reference in light gray. In 4a the valueof mt = 90 GeV is consistent only with the Fritzsch ansatz, which predicts thevalues within the thin black wedge.A top mass of mt = 130 GeV in 4b iscompatible only with the scheme of Giudice, which allows the region withinthe band outlined in black.

A symmetric CKM matrix is consistent with a topmass of 160 GeV (4c) and 185 GeV (4d); its predictions lie along the shortblack curve, while the special case of Kielanowski is shown as the small filledcircle in each of these figures. The DHR scheme predicts the heavy top massmt = 185 GeV of 4d, and allows only the tiny region shown in black.11

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출처: arXiv:9207.225 • 원문 보기