

 29 May 95

HEPSY 95-01

May 1995Rare

b Decays Stephen Playfer and Sheldon Stone

Physics Department

Syracuse University Syracuse, New York 13244-1130

(To be published in International Journal of Modern Physics Letters)

Abstract Rare b decays provide a unique opportunity to measure Standard Model parameters and probe beyond the Standard Model. We review here the experimental progress made in measuring these decays, and the importance of future measurements, including the possible observation of CP violation.

I. Introduction

The dominant decays of the b quark are charged current couplings via a W \Gamma to a c quark as shown in Figure 1(a). There are also rare decays to a u quark. Observation of these decays has led to measurements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements jVcbj and jVubj [1].

The b quark can make transitions in other ways. B0 \Gamma _B0 mixing, the process where a particle changes into its antiparticle, occurs via a "box" diagram with virtual W bosons and t quarks inside the box [Figure 1(b)]. The box diagram gives rise to large fractions of mixed events: 17% for B0 and 50% for Bs mesons.

Flavor-changing neutral currents lead to the transitions b ! s and b ! d. These can be described in the Standard Model by one-loop diagrams, known as "penguin" diagrams, where a W \Gamma is emitted and reabsorbed [2]. The first such process to be observed was b ! sfl, described by the diagram in Figure 1(c), where the fl can be radiated from any charged particle line. Another process which is important in rare b decays is b ! sg, where g designates a gluon radiated from a quark line [Figure 1(d)]. A third example of such processes is the transition b ! s`+`\Gamma which can occur through the diagrams shown in Figures 1(e) and 1(f). We consider the loop processes shown in Figures 1(c)-(f) to be among the most interesting and important rare b decays.

The decay amplitudes for the diagrams shown in Figure 1 are proportional to the CKM matrix elements present at each vertex. For the loop diagrams there are additional factors of ff if a fl is radiated and ffs if a gluon is radiated, as well as a kinematic factor which is a function of (mq=mW )2. Since the heaviest quark is the top quark, it is usually the amplitude involving the top quark that dominates in decays via loop diagrams.

1



b W(a)

b

d

t,c,u

t,c,u

W- b

d

b

Ws,d

g t,c,u

(c)

b

Ws,d

g t,c,u

(d)

(b)

b

Ws,dt,c,u g,Z ll +

(e)

b t,c,uW- s,dWl l+ -n(f)

c,u

Figure 1: Feynman Diagrams for b Decays B meson decays presently provide the only experimental evidence for penguin decays. Although neutral kaons have long been known to mix, penguin contributions to kaon decay are hard to identify since the s ! d transition leads to the same final states as the s ! u transition [3]. Only the dilepton final states can give direct evidence for penguins in kaon decay. The channels K+ ! ss+* _* and K0 ! ss0e+e\Gamma are the most promising, but the predicted branching ratios are small [4], and thus far they have not been observed.

Loop diagrams in charm decays are suppressed either by CKM matrix elements or by small values of (mq=mW )2, since the heaviest available quark is the b quark. The mixing diagram for D0 decay is proportional to jVcbVubj2 \Theta (mb=mW )2 for bquark exchange and to jVcsVusj2 \Theta (ms=mW )2 for s-quark exchange. The same kind of suppression factors apply in the case of the penguin diagrams. As a result decays such as D ! aefl are expected to be dominated by long distance contributions such as rescattering from the related hadronic decay D ! aeae. In the case of mixing a long distance effect would be D0 ! K+K\Gamma ! _D0. If these long distance effects are not too large, rare charm decays may be sensitive to non-Standard Model effects, since the Standard Model predictions for the loop diagrams are so small [5].

It is expected that CP violation will be significant in rare b decays. Charge conjugation, C, changes a particle to an anti-particle, while parity, P, changes lefthanded particles to right-handed or vice-versa. In 1964 it was found that the combined

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operation, CP, showed an asymmetry in neutral kaon decays [6]. CP violation is a necessary ingredient in explaining why our local position in the Universe consists of matter rather than anti-matter, and thus why we exist. It is of great importance to find out whether or not the Standard Model can quantitatively describe CP violation in the B system.

In the parameterization of Wolfenstein [7], the CKM matrix can be described by four independent parameters *, A, ae and j. The matrix is given in equation (1).

Vij = 0B@

1 \Gamma *2=2 * A*3(ae \Gamma ij)\Gamma

* 1 \Gamma *2=2 A*2 A*3(1 \Gamma ae \Gamma ij) \Gamma A*2 1

1CA

(1)

The * and A parameters have been measured in semileptonic decays of s and b quarks. Although j and ae have not been determined separately, constraints on these parameters are given by measurements of the ffl parameter describing CP violation in K0L decay, and by B0 \Gamma _B0 mixing and semileptonic b ! X`* decays. An analysis of the allowed parameter space is shown in Figure 2 [8]. Overlaid on the figure is a

triangle that results from the requirement V3kV y1k = 0, i.e. that the CKM matrix be unitary. Measurements of CP violation in B decays can, in principle, determine each of the angles ff, fi and fl of this triangle independently.

Figure 2: The CKM triangle overlaid upon constraints in the ae \Gamma j plane, from measured values of Vub=Vcb, B0 \Gamma _B0 mixing and ffl in the K0 system. The allowed region is given by the intersection of the three bands.

In this paper we will review the experimental data on rare b decays and compare it with the Standard Model predictions. Following this we will discuss the sensitivity of the data to extensions of the Standard Model. Finally we discuss the importance of CP violation and the propects for experimental measurements.

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II. B0 \Gamma _B0 Mixing

The transformation of a B0 meson into a _B0 meson can occur via the diagram shown in Figure 1(b). As this is not the main topic of this paper we give only a brief summary here and refer the reader to an excellent review for more details [9]. The variable that is measured by experiments is x j \Delta M=\Gamma , where \Delta M is the mass difference between the light and heavy neutral B mesons. The CKM elements are related to x via

x = G

2 F

6ss2 BBf

2 Bmbo/BjV \Lambda tbVtdj2F m

2 t

M 2W ! jQCD; (2)

where GF is the Fermi constant. The constant BB and the B meson decay constant fB have been calculated theoretically, but the large uncertainties in these calculations limit the ability to extract the CKM element jVtdj from the measurement of x. To determine x experiments either measure the ratio of mixed events to total events integrated over time (ARGUS and CLEO), or they measure the explicit time dependence (ALEPH and OPAL). The extracted x values are shown in Table 1.

Table 1: x = \Delta M=\Gamma Values from B0d mixing measurements

Experiment x

CLEO[10] 0.65\Sigma 0.10 ALEPH[11] 0.76\Sigma 0.12

OPAL[12] 0.73\Sigma 0.14 ARGUS[13] 0.75\Sigma 0.15

AVERAGE 0.71\Sigma 0.06

The band in Figure 2 is derived from equation (2) by assuming BB = 1, and taking an fB range of 160-240 MeV that corresponds to recent theoretical estimates.

The fraction of mixed events is given by

O/ = x

2

2(1 + x2) (3)

The measurements of x correspond to a O/ value of 17% for B0 events. There are also predictions and experimental limits on mixing in the Bs system indicating that the mixing has an almost maximal value of 50% [9].

III. Observation of Radiative Penguin Decays

Figure 1(c) depicts the process b ! sfl, where the photon can be radiated by any charged object in the diagram. This decay is uniquely described in the Standard

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Model by a "penguin" diagram, with corrections from other diagrams, often called "long distance" effects, estimated to be only a few percent (see Section V(A) for a detailed discussion). The inclusive process b ! sfl leads to many exclusive final states where the s quark hadronizes with the spectator quark. Angular momentum conservation forbids the decay B ! Kfl, but it is expected that K \Lambda (892)fl will be a significant fraction of the inclusive rate. The remaining inclusive rate comes from higher mass K\Lambda resonances and non-resonant K(nss) final states. There are large variations among the theoretical predictions for the fraction of b ! sfl that hadronizes as B ! K\Lambda fl.

A) Observation of B ! K\Lambda fl

The first successful search for b ! sfl by the CLEO collaboration was for the exclusive K\Lambda fl final state [14]. This is much easier than trying to measure the inclusive branching ratio for b ! sfl, because the final state is completely kinematically constrained, and the analysis is similar to that used for reconstructing hadronic B meson final states at the \Upsilon (4S) [15]. Neutral clusters in a CsI calorimeter are selected with energies between 2.1 and 2.9 GeV, if they have a shower shape consistent with a single fl, and if they cannot be combined with another fl to form a ss0. The K\Lambda (892) candidates are searched for in three channels: K\Lambda 0 ! K+ss\Gamma , K\Lambda \Gamma ! K\Gamma ss0 and K\Lambda \Gamma ! K0ss\Gamma . If the energy sum of the K\Lambda and the fl is within 75 MeV of the known beam energy Ebeam, then the beam constrained invariant mass

mB = sE2beam \Gamma `\Gamma \Gamma !PK\Lambda + \Gamma !Pfl '

2 (4)

is plotted for each candidate event and an excess is looked for at the known B meson mass.

The difference in shape between jetlike continuum events and spherical B _B events is exploited by making cuts on several event shape variables to suppress the continuum background. The most useful variables are the angle of the thrust axis of the rest of the event relative to the candidate thrust axis (cos`T ), the second Fox-Wolfram moment (R2) [16], and the sum of the momenta in a 90o cone perpendicular to the candidate axis (s?) [14, 17]. There is a significant background due to initial state radiation (ISR). To suppress this background the events are transformed to the rest system of the e+e\Gamma following the emission of the photon. In this primed frame the variables cos`0T and R02 are recalculated.

In 1:4f b\Gamma 1 of \Upsilon (4S) data there are eight K\Lambda 0fl and five K\Lambda \Gamma fl candidates within 6 MeV of MB. The continuum background level is one event in each of K\Lambda 0 ! K+ss\Gamma and K\Lambda \Gamma ! K\Gamma ss0, and zero in K\Lambda \Gamma ! K0ss\Gamma where there are two candidates. This is a clear signal for the decay B ! K\Lambda fl (Figure 3). The yields of B0 ! K\Lambda 0fl and B\Gamma ! K\Lambda \Gamma fl are consistent. If the relative fractions of B\Gamma and B0 produced at the \Upsilon (4S) are assumed to be equal, the average branching ratio is (4:5 \Sigma 1:5 \Sigma 0:9) \Theta 10\Gamma 5.

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Figure 3: Beam constrained mass distribution in GeV for the B ! K\Lambda fl candidates, dark shaded K\Lambda 0 ! K+ss\Gamma , light shaded K\Lambda \Gamma ! K\Gamma ss0, unshaded K\Lambda \Gamma ! K0ss\Gamma

B) Measurement of Inclusive b ! sfl

Recently, the CLEO collaboration has also made the first measurement of the inclusive b ! sfl branching ratio [18]. The signature for the inclusive process is a photon with energy between 2.2 and 2.7 GeV. This region contains 75-90% of the signal according to calculations that include the smearing due to the Fermi motion of the quarks in the B meson, and the motion of a B meson produced at the \Upsilon (4S).

There are large backgrounds to the inclusive signal from continuum jets (e+e\Gamma ! q _q) and initial state radiation (ISR). These backgrounds are suppressed by two methods: a shape variable analysis using a neural network, and a B reconstruction analysis. After these cuts have been made, the remaining continuum background is subtracted using scaled off-resonance data. There are also small backgrounds from other B decays, mostly consisting of photons from ss0 and j decays that survive a ss0(j) mass cut because the other photon was not found. As a first approximation these are taken from a Monte Carlo simulation, but then a correction is made for any differences that are observed between the ss0 and j spectra measured in data, and those predicted by the Monte Carlo. This takes into account any omissions in the Monte Carlo (e.g. b ! sg).

The neural network analysis uses a set of eight variables defining the event shape. The variables R2, s?, R02 and cos`0T are as defined in the previous section. In addition the energies in 20o and 30o cones parallel and antiparallel to the fl direction are used. The energies in the "away" cones relative to the fl are found to be particularily useful in discriminating against both q _q and ISR backgrounds. However since the eight variables are highly correlated, and none of them has clear discriminating power

6

compared to the others, they are combined into a joint variable, r, which tends to +1 for signal, and -1 for continuum background. A neural net is used for this purpose since it is the best method of taking into account the correlations between the shape variables.

The B reconstruction analysis combines the high energy photon with a candidate Xs system, where Xs contains either a Ks ! ss+ss\Gamma or a charged track consistent with a kaon, and an additional 1-4 pions, of which one may be a ss0. To be accepted the reconstructed decay candidate must satisfy a thrust axis cut, jcos`T j ! 0:7, and a O/2 cut on the combined \Delta E and mB information. If there is more than one candidate per event, the one with the smallest O/2 is selected. The reconstruction ambiguities usually have the same high energy photon, but different Xs systems. This is not important if the method is used only to suppress continuum background, and no attempt has been made to obtain the corrected Xs mass distribution in the CLEO analysis. Figure 4 shows the apparent Xs mass distribution, with a fit indicating the presence of a large component from K\Lambda (892). With larger data samples it will be possible to study the Xs mass distribution and obtain additional information about the exclusive decay modes that contribute.



1851194-00420 15 10

5

0 -5 -100.6 0.8 1.0 1.2 1.4 1.6 1.8

M (Xs) (GeV)

Eve nts / 0 .1 G

eV

Figure 4: Apparent XS mass distribution from the B reconstruction analysis. The solid curve is a fit to the expected distribution from a spectator model. The dashed curve shows the non-K*(892) component of the fit.

The two methods for suppressing continuum are complementary. The neural net has high efficiency (32%) but modest background suppression, whereas the B reconstruction method has low efficiency (9%), but suppresses the background by an additional factor of 14. According to Monte Carlo studies they should be equally sensitive and only slightly correlated with each other. Figure 5 shows the photon en7



1851194 2000

1000

0 150

50

0 2.0 2.5 3.0 3.5 4.0 4.5 5.0

( a ) ( b ) flEr^ (GeV) Eve nts / 0 .1 G

eV



1851194-003

( c )

( d )

200 150

100

50

0

25

0

-251.8 2.0 2.2 2.4 2.6 2.8

Er^ (GeV)

Eve nts / 0 .1 G

eV

Figure 5: Photon energy spectra from the neural net analysis, (a) & (b), and from the B reconstruction analysis, (c) & (d). In (a) & (c) the on resonance data are the solid lines, the scaled off resonance data are the dashed lines, and the sum of backgrounds from off resonance data and b ! c Monte Carlo are shown as the square points with error bars. In (b) & (d) the backgrounds have been subtracted to show the net signal for b ! sfl. The solid lines are fits of the signal shape using a spectator model prediction.

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ergy spectra from the two analyses. In Figures 5(b) and (d) the signal shape is taken from a spectator model prediction [19]. There is a small excess above the kinematical endpoint in Figure 5(b) that is attributed to a statistical fluctuation in the continuum background. The measured branching ratios are B(b ! sfl) = (1:88 \Sigma 0:74) \Theta 10\Gamma 4 from the event-shape analysis and B(b ! sfl) = (2:75 \Sigma 0:67) \Theta 10\Gamma 4 from the B reconstruction analysis. The average result, after taking into account the small correlations between the two analyses, is B(b ! sfl) = (2:32 \Sigma 0:57 \Sigma 0:35) \Theta 10\Gamma 4 , where the first error is statistical, and the second systematic. Details of the contributions to the systematic error can be found in [20].

IV. Theory of Radiative Penguin Decays A) Standard Model Prediction for b ! sfl

The partial decay width for b ! sfl is given by [21, 22]:

\Gamma (b ! sfl) = ffG

2 F m5b

128ss4 jV

\Lambda tsVtbC

eff7 (_)j2 (5)

Since the quark mass, mb, is not well known, the m5b dependence is removed by normalizing to the decay rate for b ! c`*:

\Gamma (b ! sfl) \Gamma (b ! c`*) = j

V \Lambda tsVtbj2j

Vcbj2

ff 6ssg(mc=mb) jC

eff7 (_)j2 (6)

where the factor g(mc=mb) corrects for phase space. In these expressions Ceff7 (_) is an effective coefficient of the electromagnetic loop operator:

O7 = e8ss2 mb_sffoe_* (1 + fl5)bffF_* (7) The value of C7 can be calculated perturbatively at the mass scale _ = MW . The explicit expression for C7(MW ) as a function of (m2t =M 2W ) can be found in [23]. The evolution from MW down to a mass scale _ = mb introduces large QCD corrections. These are calculated using an operator product expansion based on an effective Hamiltonian: H

eff (b ! sfl) = \Gamma 2p2GF V \Lambda tsVtb

8X

i=1

Ci(_)Oi(_) (8)

Renormalization of the coefficients, Ci, and operator mixing, lead to a value of Ceff7 (_) significantly larger than C7(MW ) [23]. This increases the predicted rate for b ! sfl by a factor of 2-3.

Evidently the prediction for the rate is very sensitive to the QCD corrections. The leading log calculation is uncertain to about 25%, primarily because it is unclear at which renormalization scale, _, the effective coefficient, Ceff7 (_), resulting from

9

the operator product expansion, should be evaluated. Values between _ = 12mb and _ = 2mb have been suggested. A next-to-leading order calculation requires the evaluation of additional two-loop diagrams, as well as some three-loop diagrams. It is hoped that these calculations can be done, since they are expected to reduce the uncertainty in the Standard Model prediction to about 10%.

B) Comparison between b ! sfl Experiment and Theory

The leading log prediction for B(b ! sfl) is (2:8 \Sigma 0:8) \Theta 10\Gamma 4 [22, 23]. If the next-to-leading order terms that have been calculated are included they tend to reduce the prediction to about 1:9 \Theta 10\Gamma 4 [24]. Both these predictions are in excellent agreement with the experimental result of (2:3\Sigma 0:6\Sigma 0:4)\Theta 10\Gamma 4. Since the theoretical uncertainties are dominated by the choice of the renormalization scale, _, it is difficult to obtain useful constraints on other Standard Model parameters such as mt and Vts. The combined CDF and D0 measurement of mt = (180 \Sigma 12) GeV [25] is well within the range required for consistency with b ! sfl. Ali et al. [26] have set bounds on Vts:

0:62 ! jVtsjjV

cbj ! 1:10 (9)

This ratio is expected to be one if the CKM matrix is unitary.

Table 2: Predictions for the ratio of B ! K\Lambda fl to b ! sfl. Author(s) Reference Method B ! K\Lambda fl Fraction Altomari [27] Spectator Quark Model 4.5% Deshpande & Trampetic [28] Relativistic Quark Model 6 - 14% Aliev et al [29] QCD Sum Rules 39% Ali & Greub [19] Spectator Quark Model (13\Sigma 3)% O'Donnell & Tung [30] Heavy Quark Symmetry 10% Ball [31] QCD Sum Rules (20\Sigma 6)% Atwood & Soni [32] Bound State Resonances 1.6 - 2.5% Bernard, Hsieh & Soni [33] Lattice QCD (6.0\Sigma 1.2\Sigma 3.4)% UKQCD collaboration [34] Lattice QCD 15 - 35%

The fraction of the inclusive b ! sfl rate hadronizing as B ! K \Lambda fl depends on the B ! K\Lambda form factor. This has been calculated by many authors using either QCD sum rules, Lattice QCD, or Heavy Quark Effective Theory (HQET). Table 2 summarizes these predictions for the ratio of B ! K\Lambda fl to b ! sfl. It can be seen that the predictions range from a few percent to 40%. The data suggest a value of (21 \Sigma 7)% for this ratio, which is not accurate enough to limit the range of acceptable form factor models. It has been suggested by Isgur [35] that the discrepancies between

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the models could be resolved by using the measured D ! K\Lambda form factors as a basis for calculating all heavy to light quark form factors. Larger data samples will make it possible to distinguish between the predictions in Table 2 and improve our understanding of the B ! K\Lambda form factor. Until accurate predictions are available for the exclusive measurements, they are not as useful as the inclusive measurement for constraining the Standard Model or new physics.

C) Effect of Extensions of the Standard Model on b ! sfl

The measurement of b ! sfl has inspired a large number of theoretical investigations of extensions of the Standard Model that could lead to significant changes in the predicted rate for b ! sfl. These studies use the upper and lower limits (95% C.L.):

1:0 \Theta 10\Gamma 4 ! B(b ! sfl) ! 4:2 \Theta 10\Gamma 4 (10) to constrain the allowed parameter space of the Standard Model extension being considered. Among the most widely discussed models are Higgs doublets, Supersymmetry, anomalous W W fl couplings, and anomalous top quark couplings. We give a brief summary of these cases below. For investigations into other theoretical ideas such as leptoquarks, a fourth generation and left-right symmetric models the reader is referred to the review article by Hewett [36].

In two-Higgs doublet models there is a charged Higgs that can be inserted into the loop instead of the W boson. There are two models for the couplings of the Higgs doublets to the quarks, depending on how the fermion masses are generated. In both cases the free parameters are the charged Higgs mass, MH+ , and the ratio of the doublet vacuum expectation values, tanfi. With Model I couplings, the b ! sfl rate is enhanced at low tanfi, suppressed for values of tanfi between 0.5 and 1.0, and is rather insensitive to large tanfi. Model II couplings always enhance the b ! sfl rate. In this case the experimental upper limit requires MH+ to be at least 240 GeV even for large values of tanfi [37].

Supersymmetry introduces many additional particles that can appear inside the loop. In the limit of exact supersymmetry these additional contributions cancel the Standard Model contribution, and b ! sfl does not occur at all. In supersymmetric models there are charged Higgs bosons with Model II type couplings that enhance the rate for b ! sfl. Contributions in which down type squarks and either neutralinos or gluinos are inserted into the loop are usually found to be negligible. However, there are significant contributions when an up type squark and a chargino are inserted in the loop. There are several recent analyses of the size and sign of the chargino contributions relative to the Standard Model and charged Higgs contributions. It appears that there are some regions of the parameter space where the supersymmetric model predicts a rate comparable to or below the Standard Model, even for small values of MH+. This requires a small stop quark mass, a large value of tanfi, and a higgsino mass parameter _ ! 0 [38].

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D) Constraints on Anomalous Couplings

The existence of anomalous couplings at the W W fl vertex can be constrained by tree-level processes such as e+e\Gamma ! W +W \Gamma and p_p ! W fl, and by loop diagrams in processes such as b ! sfl [39]. The anomalous couplings are described by two parameters, * and \Delta ^, which are zero in the Standard Model, but can acquire nonzero values in some extensions of the Standard Model. They modify the value of C7(MW ), and hence the predicted rate for b ! sfl:

C7(MW ) = C7(MW )SM + A1\Delta ^ + A2* (11) The coefficients A1 and A2 are functions of (m2t =M 2W ). Since A1 is larger than A2 for mt = 180 GeV, b ! sfl is three times more sensitivity to \Delta ^ than to *. Figure 6



1851194-0052.0 1.0

0 -1.0

-2.0-5 -4 -3 -2 -1 0 1 2 3 4 5

y"

1851194-0052.0 1.0

0 -1.0

-2.0-5 -4 -3 -2 -1 0 1 2 3 4 5

\Delta u*

Figure 6: Limits on anomalous W W fl couplings. The shaded regions are allowed by the b ! sfl measurement. The region between the shaded regions is excluded by the lower limit, the outer unshaded regions by the upper limit. The ellipse shows the limit obtained at the Tevatron by the D0 experiment (CDF has a similar limit).

shows the bounds that can be set on * and \Delta ^ from existing data. The limits from b ! sfl are complementary to the limits obtained at the Tevatron [40]. Large positive and negative values of \Delta ^ and * are excluded by the upper limit on b ! sfl. In the region around \Delta ^ = \Gamma 1 there is a complicated interference between the three terms in equation (11). This leads to the exclusion of a narrow band by the lower limit on b ! sfl.

Anomalous top quark couplings have also been considered [41]. The first possibility is that there are anomalous ttfl couplings in analogy to the W W fl case considered above. Once again this would modify C7 through two additional parameters. There is also the possibility of anomalous gluon couplings to the top quark that would modify C8, but the constraints on these couplings from b ! sfl are found to be rather weak.

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Finally there is the interesting point that b ! sfl probes the V-A structure of the tbW and tsW couplings [42].

V. Searches for Other Radiative Penguin Decays A) The Decay B ! aefl

It was suggested by Ali [43] that the ratio of CKM elements jVtdj=jVtsj could be extracted from a measurement of:

B(B\Gamma ! ae\Gamma fl)B

(B\Gamma ! K\Lambda \Gamma fl) = B

(B0 ! ae0fl) + (B0 ! !fl)B

(B0 ! K\Lambda 0fl) = j

Vtdj2j

Vtsj2 ,\Omega (12)

where \Omega corrects for phase space, and , corrects for SU(3) symmetry breaking.



b W(c)

g (a)

W(b)

u

ud uu

} }

r g

b }

}

u u

d r g Wb }

}

c c

s,d K*,q

(d)

g W-b }r

u

d

or

-d u

Figure 7: Non-penguin contributions to radiative decays. (a) Color suppressed diagram for B ! K\Lambda (ae)fl with a cc intermediate state such as a . (b) Color suppressed diagram for B0 ! ae0fl with a uu intermediate state such as a ae. (c) Tree level diagram for B\Gamma ! ae\Gamma fl with a uu intermediate state such as a ae. (d) Annihilation diagram for B\Gamma ! ae\Gamma fl where the fl can be radiated from any of the lines.

Equation (12) is only valid if contributions other than the top-quark loop can be neglected in both decay modes. According to Soni [44] there are significant differences between the long-distance contributions to b ! sfl and b ! dfl. Examples of such additional diagrams are shown in Figure 7. A recent estimate of the long distance contributions from virtual and ae mesons is ! 10% for both b ! sfl and b ! dfl [45]. However, there is one contribution from an annihilation diagram (Figure 7(d)), that is predicted to be significant by Eilam et al [46], and may be as much as 60% of the top-quark loop. Note that this annihilation diagram includes the contribution from Figure 7(c) via rearrangement of the quark lines. Since this annihilation diagram only applies to B\Gamma decays it is expected that \Gamma (B\Gamma ! ae\Gamma fl) is different from \Gamma (B0 ! ae0fl).

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Deshpande et al. [45] also discuss contributions from the u and c quark loops to b ! dfl. These contributions are larger than in b ! sfl and may be as much as 20% of the top-quark loop. This would again complicate the extraction of Vtd from Equation (12).

CLEO has made a preliminary search for B\Gamma ! ae\Gamma fl, B0 ! ae0fl and B0 ! !fl [47]. A data sample of 2:0f b\Gamma 1 at the \Upsilon (4S) results in upper limits between 1.0 and 2.5\Theta 10\Gamma 5 for the three modes. This corresponds to a limit on the ratio in equation (12) of 0.34 at 90% confidence level. The search is beginning to be background limited. In !fl the background is primarily from the continuum, whereas in ae\Gamma fl and particularily ae0fl there is significant feeddown from misidentified K\Lambda fl events. Future detectors with better particle identification will be able to suppress this feeddown [48], but the continuum background may still be a problem.

B) Searches for b ! s`+`\Gamma

The process b ! s`+`\Gamma occurs through a loop diagram with a virtual fl or Z boson (Figure 1(e)), or through a box diagram containing two W bosons (Figure 1(f)). In addition the hadronic decays B ! (

0)K(\Lambda ) contribute to the related exclusive decays

B ! K(\Lambda )`+`\Gamma through the secondary decays (

0) ! `+`\Gamma . A full understanding of

b ! s`+`\Gamma has to include both the short distance contributions from the loop and box diagrams, and the long distance contributions from the decays, and the interference between them [26, 49].

At low dilepton masses the dominant contribution from the virtual fl can be directly related to b ! sfl. There are also sharp peaks from the contributions at m and m0 which can be directly related to the measurements of the exclusive hadronic decays. At high dilepton masses the Z and box contributions are expected to be important, as are possible additional contributions from other heavy mass particles. The interference between the various diagrams can be studied by measuring the shape of the dilepton mass spectrum, and by measuring the lepton-pair asymmetry.

The high dilepton mass range has been studied at hadron colliders where there is a good signature for dimuon pairs. The first search for events with dimuon masses between 3.9 and 4.4 GeV was performed by the UA1 experiment [50]. They found upper limits of 5:0 \Theta 10\Gamma 5 for the inclusive process b ! s_+_\Gamma , and 2:3 \Theta 10\Gamma 5 for the exclusive channel B0 ! K\Lambda 0_+_\Gamma . Both these limits should be interpreted as referring only to the short distance contributions from the loop and box diagrams, since there is an extrapolation to the remainder of the phase space under the assumption that the long distance contributions are negligible above m0. Recently the CDF collaboration has presented preliminary results from the Tevatron collider. They search over the dimuon mass ranges 3.2-3.5 and 3.8-4.4 GeV, and again extrapolate to the full dimuon mass range to get upper limits on the short distance contribution of 3:5 \Theta 10\Gamma 5 and 5:3\Theta 10\Gamma 5 for the exclusive channels B0 ! K\Lambda 0_+_\Gamma and B\Gamma ! K\Gamma _+_\Gamma , respectively [51] .

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In contrast to the hadron collider experiments CLEO has searched for all dilepton masses except for the ranges 2.9-3.2 and 3.5-3.8 GeV where the (

0) contributions

dominate [52]. The analysis uses standard methods to reconstruct exclusive B meson decays from a candidate K or K\Lambda meson and a pair of identified leptons. Some typical plots of the beam-constrained mass distributions are shown in Figure 8. In an \Upsilon (4S)

Figure 8: Beam constrained mass distributions from CLEO for B ! K(\Lambda )`+`\Gamma : (a) K+e+e\Gamma (b) K+_+_\Gamma (c) K\Lambda 0e+e\Gamma (d) K\Lambda 0_+_\Gamma

data sample of 2.0 f b\Gamma 1 the background is less than one event in the signal region for each of the exclusive channels. The residual background is half from the continuum and half from B _B events where both B mesons decay semileptonically. Table 2 summarizes the preliminary upper limits from CLEO for the exclusive channels B ! K(\Lambda )_+_\Gamma and B ! K(\Lambda )e+e\Gamma . The rate for the decays to electron pairs is predicted to be larger than that to muon pairs due to the contribution from low mass pairs below the dimuon mass threshold. In some cases the limits from CLEO are close to the theoretical expectations. In the future significant increases in statistics at both hadron colliders and at \Upsilon (4S) machines are expected to lead to the observation of b ! s`+`\Gamma . Eventually there should be enough statistics to measure the dilepton mass distribution, and other kinematic variables characterizing the three-body final state. Of particular interest is the forward-backward asymmetry of the lepton pair, since this is expected to be large in the Standard Model, and may be rather sensitive to non-Standard Model physics [26, 54].

15

Table 3: Results of b ! s`+`\Gamma searches at CLEO. B Decay Candidate Detection 90% C.L. Standard Model Mode Events Efficiency Upper Limit Prediction[53] K+e+e\Gamma 2 24.4% 12:0 \Theta 10\Gamma 6 0:6 \Theta 10\Gamma 6 K+_+_\Gamma 0 15.1% 9:0 \Theta 10\Gamma 6 0:6 \Theta 10\Gamma 6 K\Lambda 0e+e\Gamma 0 9.8% 16:0 \Theta 10\Gamma 6 5:6 \Theta 10\Gamma 6 K\Lambda 0_+_\Gamma 0 5.0% 31:0 \Theta 10\Gamma 6 2:9 \Theta 10\Gamma 6

VI. Rare Hadronic Decays

Rare hadronic B decays are described by a combination of a b ! u spectator diagram (Figure 1(a)), and a gluonic penguin diagram (Figure 1(d)). Decay modes such as ss+ss\Gamma and ss\Sigma ae\Upsilon are expected to be described mainly by the spectator diagram with a ss\Gamma or ae\Gamma being produced by the W \Gamma . A small contribution from a b ! d penguin diagram is also expected in these modes. Decay modes such as K \Gamma ss+ and K\Lambda \Gamma ss+ are expected to result mainly from a b ! s penguin diagram, with a small contribution coming from the Cabibbo-suppressed spectator diagram where the W \Gamma produces a K(\Lambda )\Gamma . There are also a few decay modes, such as K0ss0 and K(\Lambda )OE that are described only by a gluonic penguin diagram, and a few modes such as ss\Gamma ss0 that are described only by a spectator diagram. In modes where both penguin and spectator diagrams are significant direct CP violation can occur, as will be discussed in Section VII.

To establish the relative importance of the penguin and spectator amplitudes it is necessary to study a large number of decay modes. The first evidence for hadronic B meson decays to final states without charmed mesons came from the CLEO observation of a signal in the sum of the two decay modes B0 ! K+ss\Gamma and B0 ! ss+ss\Gamma [55]. We will refer to this sum as B0 ! h+ss\Gamma . Since this publication the CLEO data sample has increased by almost a factor of two, and a number of other charmless hadronic decay modes have been studied. We also note that there are a few candidate events for B0 ! h+ss\Gamma or Bs ! h+K\Gamma from the DELPHI and ALEPH experiments at LEP [56].

A) Decays to Kss and ssss Final States

The signature for a B0 decay to two charged tracks is a particularily simple one. At the \Upsilon (4S) the B is almost at rest, and the tracks are back-to-back with momenta of about 2.6 GeV. CLEO observed a signal for such events in 1.4 f b\Gamma 1 of \Upsilon (4S) data [55]. Here we discuss new results from a larger data sample of 2.4 f b\Gamma 1 [57].

The same two kinematical variables are used as in the B ! K\Lambda fl analysis (Section

16

III(A)), i.e. the energy sum of the two tracks relative to the beam energy (\Delta E), which has an r.m.s. resolution of 25 MeV, and the beam constrained invariant mass (mB), which has an r.m.s. resolution of 2.6 MeV. Particle identification uses dE=dx information from the main tracking chamber. Separation between the K+ss\Gamma and ss+ss\Gamma hypotheses comes from the dE=dx information (1.8oe, where oe is the rms resolution), and from the difference in \Delta E (1.7oe). The overall separation of 2.5oe is rather marginal, and is expected to be much better in future detectors [48].

The background is due to continuum production of two light quark jets. From studies of off-resonance data samples it is known that a cut on the thrust axis, cos`T , discussed in Section III(A), is most effective against this background. Requiring cos`T ! 0:7 removes 95% of the background and only 35% of the signal. There is some additional discrimination from the energy distribution of the rest of the event, the direction defined by the axis of the two tracks, and the direction of the B meson. This information is combined into one variable (F ), using a linear Fisher discriminant technique [58].

The final signal yields are obtained from a likelihood fit to the four variables, \Delta E, mB, F and dE=dx, using an event sample containing the signal region and a large sideband in \Delta E and mB from which the background is determined. In the first version of this fit all three signal hypotheses are allowed, ss+ss\Gamma , K+ss\Gamma and K+K\Gamma . It is found that the best fit has zero yield for a K+K\Gamma signal. This is expected since this decay mode cannot occur via a penguin or spectator diagram. After setting an upper limit of 4:0 \Theta 10\Gamma 6 (90% C.L.) on B0 ! K+K\Gamma , a second fit is done in which only the first two signal hypotheses are included. The results of this fit projected onto the mB and \Delta E axes, are shown as the solid and dotted lines in Figure 9. The event histograms result from an event-counting analysis that will be described in section VI(B).

The statistical significance of the signal yield in Figure 9 is determined from the probability that the fitted background fluctuates up to the combined yield of signal plus background. Although this is determined by Poisson statistics for these small event samples, it is conventional to quote the probability in the equivalent number of oe of a Gaussian distribution. With this definition, the combined significance of the two signal modes is quoted as 5oe, with each individual mode having a significance of about 2.5oe. These results are interpreted as an observation of the sum of the two decays, but not yet as a significant result for either of the individual channels.

The combined branching ratio for B0 ! K+ss\Gamma and B0 ! ss+ss\Gamma is measured to be (1:8 \Sigma 0:6 \Sigma 0:2) \Theta 10\Gamma 5. The signal yields and the upper limits on the individual branching fractions are given in Table 4. These results are consistent with the theoretical predictions given in the last column of Table 4.

CLEO has made a similar analysis of the decay modes B+ ! h+ss0. Here the continuum background is larger and the K=ss separation is weaker, since the presence of a ss0 leads to a \Delta E resolution of 50 MeV. There are also results for the decay modes B0 ! ss0ss0, B+ ! K0ss+ and B0 ! K0ss0. In these three cases only one

17

Figure 9: Projections of the B0 ! h+ss\Gamma candidates onto the mB and \Delta E variables. The lines show the result of the likelihood fit. In the upper plot the solid line is the fitted background, and the dotted line is the fitted signal. In the lower plot the lower solid line is the background, the dashed line is the fitted B0 ! ss+ss\Gamma signal the dotted line is the fitted B0 ! K+ss\Gamma signal and the upper curve is the sum of all three contributions. Shaded events are identitifed as K+ss\Gamma , unshaded as ss+ss\Gamma .

18

Figure 10: Beam constrained mass distributions for rare hadronic decays to pseudoscalar mesons. The arrows indicate the signal region.

Table 4: Results of CLEO II Searches for Rare Hadronic B Decays to Two Pseudoscalar Mesons.

B Decay Signal Yield Fitted B.R. (90% C.L.) Predictions Mode (Likelihood Fit) Background \Theta 10\Gamma 5 \Theta 10\Gamma 5 [22, 59]

ss+ss\Gamma 9.4+4:9\Gamma 4:1 ! 2.0 1.0-2.6

5.8\Sigma 0.3 K+ss\Gamma 7.9+4:5\Gamma 3:6 ! 1.7 1.0-2.0

ss+ss0 5.0+4:2\Gamma 3:2 ! 1.7 0.6-2.1

12.6\Sigma 0.5 K+ss0 4.9+3:6\Gamma 2:8 ! 1.4 0.3-1.3

ss0ss0 1.2+1:7\Gamma 0:9 2.1\Sigma 0.2 ! 0.9 0.03-0.10 K0ss+ 5.2+3:5\Gamma 2:8 1.6\Sigma 0.1 ! 4.8 1.1-1.2 K0ss0 2.3+2:2\Gamma 1:5 0.7\Sigma 0.1 ! 4.0 0.5-0.8

19

signal hypothesis is assumed and no dE=dx information is used in the likelihood fit. The mass distributions for these modes are shown in Figure 10, and the results of the likelihood fits are summarized in Table 4. It is found that no individual mode has a significance greater than 3oe, although most of them are fitted with a small positive yield. With the exception of ss0ss0 the other modes in Table 4 are expected to be observed with branching fractions comparable to or just below the h+ss\Gamma channels, so it is likely that many of these modes will be observed in the near future.

B) Decays to Vector and Pseudoscalar Mesons

In this section we discuss searches for the decays B ! ssae, B ! Kae, B ! K\Lambda ss and B ! K(\Lambda )OE. The final states for these decays contain three or more particles, up to two of which may be ss0s, and one of which may be a Ks. The ae, K\Lambda and OE final states are selected by a cut of one natural width about the resonance mass. The dE=dx information from the main tracking chamber is used to select the most probable decay mode in cases where this is ambiguous.

In contrast to the previous section, CLEO has used a simple event-counting analysis to search for these modes rather than a full likelihood fit. Cuts are made on cos`T , F and dE=dx. In addition for decays to a vector and a pseudoscalar meson the decay helicity angle, `H, is defined as the angle in the vector meson rest frame between the direction of the B meson and one of the decay products of the vector meson. Since signal events have a cos2`H distribution, a jcos`H j ? 0:5 cut can be used to suppress continuum background. After all these cuts have been made, the event yield in the signal region in the mB \Gamma \Delta E plane is compared to the yield expected from an extrapolation of a large two-dimensional sideband region.

Table 5 summarizes the results from 2.4 f b\Gamma 1 of \Upsilon (4S) data. In most cases there are few events in the signal region. In the K0ae and K(\Lambda )OE channels there are also few events in the sideband region, and we do not quote an estimated background number because of the difficulty in extrapolating the yield from such small statistics. There are no significant signals in any of the decay modes in Table 5, although ss\Sigma ae\Upsilon and K\Lambda +ss\Gamma do have more events in the signal region than expected purely from background. The upper limits on B0 ! ss\Sigma ae\Upsilon and B+ ! K+OE are close to the theoretical predictions.

VII. CP violation in Rare Decays

In the Standard Model CP violation arises from a complex phase in the CKM matrix, which relates the mass eigenstates to the weak eigenstates. This is an inevitable consequence of having three families of quarks. In general, if we have two interfering amplitudes, we can write each of them as a product of a strong decay amplitude and a weak decay amplitude

A = asei`sawei`w

20

Table 5: Results of CLEO II Searches for Rare Hadronic B Decays to Final States with Vector Mesons.

B Decay Event Yield Estimated B.R.(90% C.L.) Prediction Mode (Signal Region) Background \Theta 10\Gamma 5 \Theta 10\Gamma 5 [22, 59]

ss\Sigma ae\Upsilon 7 2.9\Sigma 0.7 ! 8.8 1.9-8.8 ss0ae0 1 1.8\Sigma 0.6 ! 2.4 0.07-0.23 ss+ae0 4 2.3\Sigma 0.3 ! 4.3 0.0-1.4 ss0ae+ 8 5.5\Sigma 1.2 ! 7.7 1.5-3.9

K\Lambda +ss\Gamma 3 0.7\Sigma 0.2 ! 7.2 0.1-1.9 K\Lambda 0ss0 0 1.1\Sigma 0.3 ! 2.8 0.3-0.5 K\Lambda +ss0 4 1.9\Sigma 0.7 ! 9.9 0.1-0.9 K\Lambda 0ss+ 2 1.0\Sigma 0.6 ! 4.1 0.6-0.9

K+ae\Gamma 2 2.0\Sigma 0.4 ! 3.5 0.00-0.20 K0ae0 0 ! 3.9 0.01-0.04 K+ae0 1 3.8\Sigma 0.2 ! 1.9 0.01-0.06 K0ae+ 0 ! 4.8 0.00-0.03

K0OE 1 ! 8.8 0.1-1.3 K\Lambda 0OE 2 ! 4.3 0.0-3.1 K+OE 0 ! 1.2 0.1-1.5 K\Lambda +OE 1 ! 7.0 0.0-3.1

B = bseiffisbweiffiw : (13) Applying the CP operators to these amplitudes results in

A = asei`sawe\Gamma i`wB

= bseiffisbwe\Gamma iffiw: (14)

Note that the weak phase has changed sign, while the strong phase has not. The rate difference, which may exhibit CP violation is

\Gamma \Gamma \Gamma = jA + Bj2 \Gamma jA + Bj2 = 2asawbsbwsin(ffis \Gamma `s)sin(ffiw \Gamma `w): (15) If two distinct weak decays processes are possible which go via CKM elements with a phase difference,then sin(ffiw \Gamma `w) 6= 0. Guaranteeing a strong phase shift, however,

21

is not possible. In fact, the theory of strong decays lacks sufficient power to be able to accurately predict the magnitude of such phase differences, or their sign relative to the weak phase.

A) CP violation in B\Sigma Decays to Two Pseudoscalars

Direct CP violation can occur in charged B meson decays due to interference between any two diagrams with different weak and strong phases. In decays to two pseudoscalar mesons the penguin and tree diagrams can give rise to integral rate asymmetries such as:

\Delta (Kss0) = \Gamma (B

\Gamma ! K\Gamma ss0) \Gamma \Gamma (B+ ! K+ss0)

\Gamma (B\Gamma ! K\Gamma ss0) + \Gamma (B+ ! K+ss0) (16) that are manifestly CP violating.

There have been several suggestions for measurements of decay rates of B\Sigma mesons that could be used to determine the CKM phase sinfl [60, 61]. Although the discussion of the derivation of sinfl is complicated, we would like to present the arguments for comparing the rates of B\Sigma decays to two pseudoscalar mesons in order to extract sinfl [60], since it is likely that future experiments will measure these decay rates [48].

The final state K\Gamma ss0 has an amplitude (Ts) from the tree diagram in Figure 11(a) if the W \Gamma materializes as a K\Gamma , an amplitude (Cs) from the "color suppressed" diagram 11(b), and an amplitude (Ps) from the penguin diagram in Figure 11(c). For the final state ss\Gamma ss0 there are analogous tree (T ) and color suppressed (C) amplitudes where the s quark in these diagrams is replaced by a d quark. However there is no analogous penguin amplitude because the gluon can form d _d as well as u_u and these amplitudes cancel. This is the same as the statement that the I= 32 ss+ss0 final state cannot be made with a \Delta I= 12 penguin amplitude. For the K0ss\Gamma final state there is only one contribution from the penguin diagram shown in Figure 11(d).

Assuming SU(3) symmetry, the strange amplitudes are related to the non-strange amplitudes by: T

sT = CsC = r = Vus

Vud

fK

fss = 0:28 (17) The amplitudes can be summarized as

A(B\Gamma ! ss\Gamma ss0) = \Gamma 1p2 (T + C) (18) A(B\Gamma ! K0ss\Gamma ) = Ps (19) A(B\Gamma ! K\Gamma ss0) = \Gamma 1p2 (Ts + Cs + Ps) (20)

These amplitudes can be related byp

2A(B\Gamma ! K\Gamma ss0) + A(B\Gamma ! K0ss\Gamma ) = rp2A(B\Gamma ! ss\Gamma ss0): (21)

22



b W(a)

b

Ws g

t

(c)

u

s} K u

uu} po

u uuu} K} po

b W(b)

us} K u

uu} po

b

Ws g

t

(d)

u dud}K} p

o

Figure 11: Amplitudes contributing to B\Gamma decays to two pseudoscalars. (a) tree diagram for K\Gamma ss0 (b) color-suppressed diagram for K\Gamma ss0 (c) penguin diagram for K\Gamma ss0 (d) penguin diagram for K0ss\Gamma . The ss\Gamma ss0 final state goes through (a) and (b), with the W \Gamma ! _ud rather than _us.

Amplitude triangles can be constructed from this relationship and its complex conjugate for B+ decays (Figure 12).

Since A(B\Gamma ! K0ss\Gamma ) involves only one penguin diagram, it is equal to A(B+ ! K0ss+). On the other hand,

rp2A(B\Gamma ! ss\Gamma ss0) = aT eiffiT e\Gamma ifl (22)

rp2A(B+ ! ss+ss0) = aT eiffiT eifl; (23) where fl =arg(V \Lambda ubVus). Note, that the rates of these two processes are equal since they involve a single weak phase and a single strong phase; however, there is a difference in phase of 2fl between them. In the case of K\Gamma ss0 and K+ss0 the penguin and tree contributions interfere and there is a net weak phase shift of arg(VubV \Lambda usV \Lambda tbVts). The strong phase shift, ffi = ffiT \Gamma ffiP, is also important in determining the actual rate asymmetry, \Delta (Kss0), which has been estimated to be a few percent by several authors [62]. CP violation would be explicit if a measured rate difference between K\Gamma ss0 and K+ss0 existed, but this requires a strong phase shift as well as the weak phase shift. It has been argued that by constructing amplitude triangles as shown in Figure 12, the angle fl can be determined with a twofold ambiguity. Note that if ffi is zero fl can be derived unambiguously even though there is no explicit CP violation in B\Sigma ! K\Sigma ss0 [60].

This procedure is only valid if there are no additional contributions other than those discussed above. For example the c and u quark penguin loops could generate significant asymmetries with different phases from the t quark penguin. There

23



A(K p ) = A(K p )-o +o

O"2A(K

p )o

O"2A(K p )

- +

2g

rO"2A(p p )o + o

rO"2A

(pp

)o Figure 12: Amplitude triangles relating B\Sigma ! K\Sigma ss0, B\Sigma ! ss\Sigma ss0 and B\Sigma ! K0ss\Sigma . are several experimental checks that other diagrams can be neglected. The decays B\Gamma ! K0ss\Gamma and B+ ! K0ss+ should have equal rates, as should B\Gamma ! ss\Gamma ss0 and B+ ! ss+ss0. If non-equal rates were observed in these decays this would also be an observation of direct CP violation, but it would not be possible to extract the angle fl from the triangle relation any more. Another check comes from B0 ! K+K\Gamma , since this cannot be produced by the tree and penguin diagrams of Figure 11. It is important that this final state not be observed at a small fraction of the other decays.

Deshpande and He [63] have pointed out that there is another class of penguin diagrams where the gluon in Figure 11(c) and (d) is replaced by a fl, Z, or a box diagram. Gluonic penguins couple equally to u_u and d _d, whereas the fl and Z penguins couple differently to u and d quarks. These electroweak penguin contributions are expected to be small in B\Sigma ! ss\Sigma ss0, but they could be as large as the tree level contributions in B\Sigma ! K\Sigma ss0. If this is the case, the method we have described above cannot be used to extract fl without information from other decay modes.

There have been two suggestions on how to extract sinfl allowing for possible electroweak penguin contributions. Gronau et al. suggest forming an amplitude quadrangle including the additional decay mode Bs ! jss0 [64]. However, measuring this rare Bs decay appears to be extremely difficult. Deshpande and He suggest in a recent preprint the construction of additional amplitude triangles using the decays B\Gamma ! K\Gamma j(

0) and B+ ! K+j(0) [65]. The octet part of the j=j0 system is defined as

j8: A

(K\Gamma j8) = A(K\Gamma j) cos ` + A(K\Gamma j0) sin `; (24)

where ` is the j \Gamma j0 mixing angle of about 20o [6]. Two new amplitude triangles can be constructed: p

2A(K\Gamma ss0) \Gamma 2A( _K0ss\Gamma ) = p6A(K\Gamma j8) (25)p

2A(K+ss0) \Gamma 2A(K0ss+) = p6A(K+j8): (26)

24

a b 1 2

A 2A

3

4 4

Figure 13: Amplitude triangles and two solutions a and b for the magnitude of sinfl. The amplitude A represents A(K0ss\Gamma ). The amplitude 2A forms a triangle with the amplitudes p2A(K\Gamma ss0) (line 1) and p6A(K\Gamma j8) (line 2). This triangle determines the amplitude B (line 3). The dotted lines show the equivalent construction for the B+ decays, leading to the amplitude B (line 4). There is a two fold ambiguity in the angle between 3 and 4 due to the possibility of flipping the B+ triangle with respect to the B\Gamma triangle. This leads to the two results for B \Gamma B shown as a and b.

It is then convenient to construct the amplitude combinations

B = p2A(K\Gamma ss0) \Gamma A( _K0ss\Gamma ) (27) B = p2A(K+ss0) \Gamma A(K0ss+): (28) The complete amplitude construction is shown in Figure 13. The solid lines show the amplitude triangle for the B\Gamma decays (equation (25)), while the dotted lines show the B+ amplitudes (equation (26)). From these triangles B (equation (27)), andB

(equation (28)), are constructed. The difference B \Gamma B (equation (29)), has two possible solutions a and b, which are related to the amplitude for B\Gamma ! ss\Gamma ss0 and to sinfl by: B \Gamma B

= \Gamma i2p2eiffififififi VusV

ud fifififi fififiA(ss

\Gamma ss0)fififi sin fl; (29)

where ffi is a strong phase shift. From the measured difference from the triangle construction a or b, and the measured rate for B\Gamma ! ss\Gamma ss0 the angle fl can be determined with a twofold ambiguity.

The calculation of Deshpande and He assumes that the couplings of kaons, pions and etas are related by SU(3), and that the decay amplitudes can be factorized. In

25

equation (29) the amplitude A(ss\Gamma ss0) should be multiplied by a factor (fK=fss), and in equations (25) and (26) the amplitude A(K\Gamma j8) by a factor (fss=fj). Although there is some uncertainty in these theoretical assumptions, the success of factorization in explaining B decays to exclusive final states with a D\Lambda + and a light hadron has been encouraging [66].

The angle fl can also be determined using measured rates in charged B decays to D0K final states. The method proposed by Gronau and Wyler [61] uses the three related decay modes B\Gamma ! D0K\Gamma , B\Gamma ! _D0K\Gamma , B\Gamma ! DCP K\Gamma , where DCP indicates that the D0 decays into a CP eigenstate, and the corresponding modes for B+. The decay B\Gamma ! D0K\Gamma is a Cabibbo suppressed version of B\Gamma ! D0ss\Gamma , while the decay B\Gamma ! _D0K\Gamma is a color suppressed b ! u transition where the virtual W \Gamma transforms itself into a cs pair. Interference is possible between these two decays modes if the D0 decays into a CP eigenstate. Examples of such final states include K+K\Gamma , Ksss0, and Ksj. To simplify the discussion only states that are in specific angular momentum configurations are used so that their CP is defined as +1 or -1; these states are usually denoted as D01 and D02. We have

D01 = 1p2 hD0 + _D0i ; and D02 = 1p2 hD0 \Gamma _D0i : The amplitudes for the three B\Gamma decays modes are related by:p

2A\Gamma 1 (B\Gamma ! D01K\Gamma ) = A(B\Gamma ! _D0K\Gamma ) + A(B\Gamma ! D0K\Gamma ): Denoting the hadronic phase as ffi, givesp

2A\Gamma 1 (B\Gamma ! D01K\Gamma ) = jAjei(OEs+ffi) + Aeiffi: The decays to D01 need not be equal for B+ and B\Gamma , and an asymmetry in them is a manifest demonstration of CP violation. A triangle construction serves to determine sinfl. For more details see [61, 67].

B) CP violation in B0 ! ss+ss\Gamma due to mixing

The final state ss+ss\Gamma is one of the simplest in B decay, since there are only two pseudoscalar particles in the final state, and there is no spin or angular momentum to consider. This final state can be reached from either a B0 or a _B0, and the s-wave production of the two spinless particles means that this is a CP eigenstate. The two interfering amplitudes necessary for CP violation are provided by the direct B0 decay, and the indirect decay following _B0 \Gamma B0 mixing. When mixing provides the second amplitude for a decay to a CP eigenstate, the strong phase shift disappears from the equation relating the measured CP asymmetry to the CKM angles.

To measure CP violation using mixing we need to make use of the correlated production of B0 _B0 pairs. This is done by tagging the number of ss+ss\Gamma events

26

produced with the other B decaying as a B0, as opposed to a _B0. An example of a suitable tag is lepton flavor. These numbers are time dependent functions of T = t \Gamma t0 (in units of mean lifetime), where t is the decay time of the ss+ss\Gamma and t0 is the decay time of the other B:

R(T ) / e\Gamma jT j(1 \Gamma sin \Delta m\Gamma T sin 2ff) (for ss+ss\Gamma ; B0); (30)

R(T ) / e\Gamma jT j(1 + sin \Delta m\Gamma T sin 2ff) (for ss+ss\Gamma ; _B0): (31) It can be shown that R(T ) = R(\Gamma T ): The CP asymmetry is

AT (ss+ss\Gamma ) = R(T ) \Gamma R(T )R(T ) + R(T ) = sin \Delta m\Gamma T sin 2ff (32) We can measure a time independent asymmetry by integrating over t\Gamma t0. However, if the C parity of the intial B0 _B0 pair is -1, as is the case in \Upsilon (4S) decay, the time integrated rate is zero. This is not the case in hadron colliders, or if the initial state is B0\Lambda _B0.

Measuring CP violation in the ss+ss\Gamma decay determines sin2ff, where

ff = arg(VudV \Lambda ub=VtdV \Lambda tb): (33) However, there is a problem due to the presence of a decay amplitude related to the penguin diagram shown in Figure 1(d). A ss+ss\Gamma final state is produced when the t quark in the loop couples to a d quark rather than an s quark. Although this is suppressed it could turn out to have a significant effect on the CP asymmetry measurement. Gronau and London [68] have shown how this contribution can be isolated by measuring the rates for the ss+ss0 and ss0ss0 processes. Equations (30) and (31) are modified as follows:

R(T ) / e\Gamma jT ji 1 + j,j

2

2 +

1 \Gamma j,j2

2 cos

\Delta m

\Gamma T + Im, sin

\Delta m

\Gamma jT jj (34)

R(T ) / e\Gamma jT ji 1 + j,j

2

2 \Gamma

1 \Gamma j,j2

2 cos

\Delta m

\Gamma T \Gamma Im, sin

\Delta m

\Gamma jT jj; (35) where , is a parameter related to ff. If only the direct and the mixing amplitude are present j,j = 1, and we are left with equations (30) and (31). We can look for the presence of the cosine term experimentally by forming a new time dependent asymmetry:

AjT j(ss+ss\Gamma ) = R(jT j) + R(\Gamma jT j) \Gamma R(jT j) \Gamma R(\Gamma jT j)R(jT j) + R(\Gamma jT j) + R(jT j) + R(\Gamma jT j) = 1 \Gamma j,j

2

1 + j,j2 cos

\Delta m

\Gamma jT j: (36)

27

The asymmetry AjT j leads to a non-vanishing asymmetry, \Delta , in the time integrated rates, even at the \Upsilon (4S):

\Delta (ss+ss\Gamma ) = 1 \Gamma j,j

2

1 + j,j2

1 1 + ( \Delta m\Gamma )2 (37)

If it turns out that the ss0ss0 rate is comparable to ss+ss\Gamma , the penguin amplitude is important but it should be possible to measure it. If it is small enough to be difficult to measure, then the penguin amplitude is likely to be unimportant and ff comes out simply.

Another method for extracting the size of the penguin-tree interference is to compare the integral rates for B0 ! ss+ss\Gamma and B0 ! K+ss\Gamma with those for _B0 ! ss\Gamma ss+ and _B0 ! K\Gamma ss+ [69]. A recent preprint by Deshpande and He [70] shows that the assumption of SU(3) symmetry leads to the following simple relationship between the time integrated rate asymmetries for these decays:

\Delta (ss+ss\Gamma ) ss \Gamma f

2 ss

f 2K \Delta (K

+ss\Gamma ) (38)

and that this result can be used to correct the measurements of the time-dependent CP asymmetry in ss+ss\Gamma for the effect of the penguin amplitude.

C) CP violation in radiative penguin decays

A measurement of the ratio of b ! dfl to b ! sfl determines jVtdj=jVtsj if the amplitudes are described by the t quark loop. As discussed in Section V(A) there may be other amplitudes that are significant in b ! dfl. If this is the case then these additional amplitudes can give rise to direct CP violation in these decays.

One source of CP violation is the presence of three different loop diagrams involving the u, c or t quarks. Gluon exchange provides the necessary strong phase shifts between these loop diagrams. Naively, one would expect that the u and c diagrams would be highly suppressed due to the relatively small quark masses. However both Deshpande et al. [45] and Soares [71] find significant u and c loop contributions in b ! dfl. Soares has explictly calculated the amount of CP violation from these sources and finds asymmetries of (2 \Gamma 8) \Theta 10\Gamma 3 for b ! sfl and a significantly higher value of (2 \Gamma 30) \Theta 10\Gamma 2 for b ! dfl. For the exclusive decay modes Greub et al. [72] estimate that the CP asymmetries are 1% for B ! K\Lambda (890)fl and 15% for B ! aefl.

There are other diagrams that can provide a source of CP violation. Of particular interest is the suggestion that non Standard Model contributions to b ! sfl can lead to large CP asymmetries, since the asymmetry predicted by the Standard Model is rather small. As an example of such possibilities we mention the paper by Wolfenstein and Wu [73] that calculates the expected level of CP violation in a two Higgs doublet model. Here the charged Higgs is present in the loop instead of the W \Gamma , and there

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is an arbitrary phase factor associated with the two Higgs doublets. Depending on this phase the asymmetry in b ! sfl could be anywhere in the range 0-10%. If a CP asymmetry larger than 1% were observed in b ! sfl, it would provide strong evidence for physics beyond the Standard Model.

VIII. Conclusions

Loop diagrams were first discovered in the mixing amplitude for neutral kaon decays. Mixing has also been observed in the neutral B mesons [Figure 1(b)]. CLEO has established the existence of the radiative penguin decay b ! sfl [Figure 1(c)], by measuring the exclusive branching fraction for B ! K\Lambda fl to be (4:5\Sigma 1:5\Sigma 0:9)\Theta 10\Gamma 5, and the inclusive branching fraction for b ! sfl to be (2:3 \Sigma 0:6 \Sigma 0:4) \Theta 10\Gamma 4. These values are in agreement with the Standard Model prediction, and set bounds on the parameters of some extensions of the Standard Model. Other penguin decays such as B ! aefl and b ! s`+`\Gamma have been searched for but have not yet been seen.

There is also a CLEO measurement of (1:8 \Sigma 0:6 \Sigma 0:2) \Theta 10\Gamma 5 for the branching fraction of the sum of the decays B0 ! K+ss\Gamma and B0 ! ss+ss\Gamma . While the K+ss\Gamma final state is thought to occur primarily through the gluonic penguin diagram [Figure 1(d)], the ss+ss\Gamma final state occurs primarily via a b ! u tree level transition [Figure 1(a)]. The data favor equal branching ratios for the two decay modes, but only exclude either one being zero at a level of significance equivalent to about 2:5oe.

Decays of B mesons to two pseudoscalar mesons are particularly important for the study of CP violation. The ss+ss\Gamma final state can be used to measure the angle ff in the CKM triangle. Corrections for a penguin contribution to B0 ! ss+ss\Gamma can be made by making additional measurements of B+ ! ss+ss0 and B0 ! ss0ss0 and using isospin, or by measuring the rate asymmetry between B0 ! K+ss\Gamma and _B0 ! K\Gamma ss+ and using SU(3) symmetry. It is likely that the CKM angle fl will be measured using charged B decays to K\Sigma ss0, ss\Sigma ss0, K0ss\Sigma and K\Sigma j(

0), or using charged B decays to

D0K\Sigma final states.

We are looking forward to the measurement of additional rare b decays such as B ! aefl and b ! s`+`\Gamma . The Standard Model is already constrained by the existing measurements of B0 mixing and b ! sfl. If deviations from the Standard Model are found in other radiative penguin decays, in CP asymmetry measurements, or in K or D meson decays, then possible extensions of the Standard Model must both explain the observed deviations and be consistent with the other measurements.

There are planned improvements to the CESR/CLEO symmetric B-factory, and asymmetric B factories are under construction at KEK and SLAC [48], all with projected luminosities about ten times higher than that currently achieved at CESR/CLEO. In addition hadron collider B experiments are being pursued (HERA-B, Tevatron, LHC)[74]. We hope that these efforts will lead to the observation of CP violation in the B system, and that some evidence for non-Standard Model effects will be found.

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IX. Acknowledgements

We thank the National Science Foundation for support. We acknowledge informative conversations with A. Ali, G. Burdman, N. Deshpande, I. Dunietz, X. He, J. Hewett, N. Isgur, J. Rosner, A. Soni, L. Wolfenstein and our colleagues in the CLEO collaboration.

We apologize to all of those who do not appear in the above list, or who are omitted in the references below, but who have contributed significantly to the study of rare b decays. It has been a constant struggle to finish this paper given the large number of theoretical preprints that appear every week, but it is a great joy to see so much activity in this field.

References

[1] S. Stone, "Semileptonic B Decays", in "B Decays 2nd Edition" by S. Stone,

World Scientific, Singapore (1994).

[2] A. I. Vainshtein, V. I. Zakharov & M. A. Shifman, JETP Lett., 22, 55 (1975);

J. Ellis, M. K. Gaillard & D. V. Nanopoulos, Nucl. Phys. B100, 313 (1975); M. Bander, D. Silverman & A. Soni, Phys. Rev. Lett. 43, 242 (1979).

[3] The radiative decay K ! ssfl is forbidden by angular momentum conservation.

The allowed decay K ! ssssfl can be produced by final state radiation from the dominant ssss decay mode. Gluonic penguin diagrams are also unobservable since they lead to the ssss decay mode.

[4] A. J. Buras, "Rare Decays, CP Violation and QCD," MPI-PhT/95-17 (1995). [5] L. Wolfenstein, Phys. Lett. B 164, 170 (1985); A. Le Yaouanc et al., "Mixing

and CP Violation in D Mesons," LPTHE-Orsay 95/15 (1995); G. Burdman et al., "Radiative Weak Decays of Charmed Mesons," Fermilab-Pub 94/412 (1995).

[6] L. Montanet et al., "Review of Particle Properties", (Particle Data Group), Phys.

Rev. D 50, 1173 (1994) (see page 1543).

[7] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1984). [8] S. Stone, "Fundamental Constants from b and c Decay," HEPSY 94-5, to appear in Proceedings of "Particle Strings and Cosmology," meeting, Syracuse, NY (1994), and in Proceedings of DPF94 Meeting, Albuquerque, NM (1994).

[9] H. Schr"oder, "B _B Mixing," in B Decays 2nd Edition, ed. by S. Stone, World

Scientific, Singapore (1994).

[10] J. Bartelt et al., (CLEO) Phys. Rev. Lett. 71, 1680 (1993).

30

[11] D. Buskulic et al., (ALEPH) Phys. Lett. B 322, 441 (1994). [12] R. Akers et al., (OPAL) Zeit. Phys. C 60, 199 (1993). [13] H. Albrecht et al., (ARGUS) Zeit. Phys. C 55, 357 (1992). [14] R. Ammar et al., (CLEO) Phys. Rev. Lett. 71, 674 (1993). [15] T. Browder, K. Honscheid & S. Playfer, "A Review of Hadronic and Rare B

Decays", in "B Decays" 2nd ed. (World Scientific, 1994).

[16] G. Fox and S. Wolfram, Phys. Rev. Lett. 41, 1581 (1978). [17] M. Artuso, "Experimental Facilities for b-quark Physics", in "B Decays" 2nd

ed. (World Scientific, 1994).

[18] M.S. Alam et al., (CLEO) Phys. Rev. Lett. 74, 2885 (1995). [19] A. Ali & C. Greub, Phys. Lett. B259, 182 (1991). [20] J.A. Ernst, Ph.D. Thesis, Univ. of Rochester (1995). [21] S. Bertolini, F. Borzumati & A. Masiero, Phys. Rev. Lett. 59, 180 (1987); R.

Grigjanis et al., Phys. Lett. B 213, 355 (1988); B. Grinstein, R. Springer & M. Wise, Nucl. Phys. B339, 269 (1990).

[22] N. G. Deshpande, "Theory of Penguins in B Decay", in "B Decays 2nd Edition"

by S. Stone, World Scientific, Singapore (1994).

[23] A. J. Buras, M. Misiak, M. M"unz & S. Pokorski, Nucl. Phys. B424, 374 (1994). [24] M. Ciuchini et al., Phys. Lett. B334, 137 (1994). [25] F. Abe et al., (CDF) Phys. Rev. Lett. 74, 2626 (1995); S. Abachi et al., (D0)

Phys. Rev. Lett. 74, 2632 (1995).

[26] A. Ali, G. Giudice & T. Mannel, Preprint CERN-TH.7346 (1994). [27] T. Altomari, Phys. Rev. D37, 677 (1988). [28] N. Deshpande & J. Trampetic, Mod. Phys. Lett. A4, 2095 (1989). [29] T. M. Aliev et al., Phys. Lett. B237, 569 (1990). [30] J. O'Donnell & H. Tung, Phys. Rev. D48, 2145 (1993). [31] P. Ball, "The Decay B ! K\Lambda fl from QCD Sum Rules," TUM-T31-43/93 (1993). [32] D. Atwood & A. Soni, Z. Phys. C64, 241 (1994).

31

[33] C. Bernard, P. Hsieh & A. Soni, Phys. Rev. Lett. 72, 1402 (1994). [34] D. R. Burford et al., preprint FERMILAB-PUB-95/023-T (1995). [35] N. Isgur, private communication. [36] J. Hewett, "Top Ten Models Constrained by b ! sfl," SLAC-PUB-6521 (1994). [37] J. Hewett, Phys. Rev. Lett. 70, 1045 (1993), V. Barger, M. Berger & R. Phillips,

Phys. Rev. Lett. 70, 1368 (1993).

[38] S. Bertolini et al., Nucl. Phys. B353, 591 (1991); R. Barbieri & G. Giudice, Phys.

Lett. B309, 86 (1993); Y. Okada, Phys. Lett. B315, 119 (1993); R. Garisto & J. Ng, Phys. Lett. B315, 372 (1993); F. Borzumati, Z. Phys. C63, 291 (1994); Y. Okada, preprint KEK-TH-428 (1995).

[39] S.P. Chia, Phys. Lett. B240, 465 (1990); K.A. Peterson, Phys. Lett. B282, 207

(1992); T.G. Rizzo, Phys. Lett. B315, 471 (1993); X.G. He & B. Mckellar, Phys. Lett. B320, 165 (1994).

[40] J. Ellison, (D0) Proc. of DPF Meeting, Albuquerque, NM (1994); F. Abe et al.,

(CDF) Phys. Rev. Lett. 74, 1936 (1995).

[41] J. Hewett & T.G. Rizzo, Phys. Rev. D49, 319 (1994). [42] K. Fujiyama & A. Yamada, Phys. Rev. D49, 5890 (1994). [43] A. Ali, V. Braun and H. Simma, Z. Phys. C63, 437 (1994). [44] D. Atwood, B. Blok & A. Soni, Preprint SLAC-PUB-6635 (1994), [45] N. Deshpande, X. He & J. Trampetic, Preprint OITS-564-REV (1994); J. M.

Soares, "The contribution of the J/psi resonance to the radiative B decays," TRI-PP-95-6 (1995).

[46] G. Eilam, A. Ioannissian & R. R. Mendel, "Long Distance Effects and CP Violation in B+ ! ae+fl," TECHNION-PH-95-4 (1995).

[47] M. Athanas et al., CLEO-CONF 94-2, submission to ICHEP94 conference, Glasgow (1994).

[48] CLEO III Detector: Design & Physics Goals, CLEO preprint CLNS 94/1277

(1994); SLAC BABAR Collaboration: Technical Design Report (1995); KEK BELLE Collaboration: Letter of Intent (1994).

[49] A. Buras & M. M"unz, Preprint MPI-PhT/94-096 (1994). [50] Albajar et al., (UA1) Phys. Lett. B 262, 163 (1991).

32

[51] C. Anway-Wiese, contribution to Conference on Vector Boson Self-Interactions,

UCLA (1995).

[52] R. Balest et al., CLEO-CONF 94-4, submission to ICHEP94 conference, Glasgow

(1994).

[53] A. Ali, C. Greub & T. Mannel, "Rare B Decays in the Standard Model," DESY93-016 (1993).

[54] G. Burdman, Preprint FERMILAB-Pub-95/113-T (1995). [55] M. Battle et al., (CLEO) Phys. Rev. Lett. 71, 3922 (1993). [56] P. Abreu et al., (DELPHI) "Search for Charmless Beauty Decays with the DELPHI Detector at LEP", Contribution GLS0163 to IHEP Conference, Glasgow (1994); G. Taylor, K. Zachariadou and M. H. Schune, (ALEPH) "Observation of Charmless B Decays", Contribution GLS0583 to IHEP Conference, Glasgow (1994).

[57] S. Playfer, "Rare b Decays", plenary talk at the spring meeting of the American

Physical Society, Washington D.C. (1995).

[58] F. K. W"urthwein, Ph.D. Thesis, Cornell University (1995). [59] A. Deandrea et al., Phys. Lett. B 318, 549 (1993); A. Deandrea et al., Phys.

Lett. B 320, 170 (1994); L.L. Chau et al., Phys. Rev. D 43, 2178 (1991).

[60] M. Gronau, J. L. Rosner & D. London, Phys. Rev. Lett. 73, 21 (1994). [61] M. Gronau & D. Wyler, Phys. Lett. B 265, 172 (1991). [62] J. M. Gerard & W. S. Hou, Phys. Rev. D 43, 2909 (1991); H. Simma, G. Eilam

& D. Wyler, Nucl. Phys. B352, 367 (1991).

[63] N. G. Deshpande & X.G. He, Phys. Rev. Lett. 74, 26 (1995). [64] M. Gronau et al., "Electroweak Penguins and Two-Body B Decays,"

TECHNION-PH-95-11 (1995).

[65] N. Desphande and X.G. He, "A Method for Determining CP Violating Phase

fl," OITS-576 (1995).

[66] M. S. Alam et al., (CLEO), Phys. Rev. D 50, 43 (1994). [67] S. Stone, Nucl. Instr. & Meth., A333, 15 (1993). [68] M. Gronau & D. London, Phys. Rev. Lett. 65, 3381 (1990).

33

[69] J. P. Silva & L Wolfenstein, Phys. Rev. D 49, 1151 (1994). [70] N. G. Deshpande & X.G. He, "CP asymmetry Relations Between B0 ! ssss and

B0 ! ssK Rates", OITS-566 (1994).

[71] J. M. Soares, Nucl. Phys. B367, 575 (1991). [72] C. Greub, D. Wyler & H. Simma, Nucl. Phys. B434, 39 (1995). [73] L. Wolfenstein & Y. L. Wu, Preprint CMU-HEP 94-24 (1994). [74] "Proceedings of the Second International Workshop on B Physics at Hadron

Machines", ed. P. Schlein, Nucl. Instr. & Meth., A351, 1 (1994).

34

