Effect of fourth generation CP phase on b s transitions

Transcription

1 Eur. Phys. J. C 7, (003) Digital Oject Identifier (DOI) /epjc/s THE EUROPEAN PHYSICAL JOURNAL C Effect of fourth generation CP phase on s transitions A. Arhri 1,, W.-S. Hou 3 1 Max-Planck Institut für Physik, Föhringer Ring 6, München, Germany LPHEA, Physics Department, Faculty of Science-Semlalia, P.O.B. 390, Marrakesh, Morocco 3 Department of Physics, National Taiwan University, Taipei, Taiwan 10764, R.O.C. Received: 0 Novemer 00 / Pulished online: 7 March 003 c Springer-Verlag / Società Italiana di Fisica 003 Astract. We study the effect of a sequential fourth generation on sγ, B s mixing, B K ( ) l + l, X sl + l and φk S, taking into account the presence of a new CP phase in quark mixing. We find that the effects via electromagnetic and strong penguins, such as in sγ or CP-violation in B φk S, are rather mild, ut the impact via oxes or Z penguins is quite significant. Not only can one have a B s mixing much larger than expected, even if it is measured near the present limit, mixing-dependent CP-violation could ecome maximal. For electroweak penguin modes, the fourth generation can provide the enhancement that perhaps may e needed, while the m l + l dependence of the forward ackward asymmetry in B K l + l can further proe for the fourth generation. These effects can e studied in detail at the B factories and the Tevatron. 1 Introduction The source of CP-violation within the standard model lies in the Caio Koayashi Maskawa (CKM) quark mixing matrix, where a unique KM phase emerges with three generations of quarks and leptons. Despite its great success, there are reasons to elieve that the three generation standard model (SM, or SM3) may e incomplete. For example, the generation structure is not understood, while recent oservations of neutrino oscillations point towards an enlarged neutrino sector [1]. A simple enlargement is to add a sequential fourth generation (SM4); with additional mixing elements, one crucial aspect is that the source for CP-violation is no longer unique. The B factories at KEK and SLAC have turned on with a splash, and CP-violation has een oserved in B 0 J/ψK S decay modes which is consistent with SM3. Both experiments have recently oserved the B Kl + l [ 5] decay mode, while the Belle Collaoration has just reported the oservation of the inclusive B X s l + l mode [6]. Though a it on the high side, the results are again not inconsistent with SM3. However, there may e a hint for new physics in the mixing-dependent CPviolation in B 0 φk S decay [7]. At any rate, together with many related loop-induced processes, we have entered an age where the uniqueness of the three generation SM phase can start to e checked. The B Kl + l type of decays are highly suppressed in the SM ecause they occur only via electroweak penguin processes. The first measurement of such processes through sγ was performed some years ago y CLEO [8]. The measured ranching ratio has een used to constrain the Wilson coefficient C7 eff [9]. The measurement of the B (K, K )l + l and X s l + l modes can provide information on C9 eff and C 10, as well as sign information on C7 eff. The contriutions from the fourth generation to rare decays have een extensively studied [10 13], where the measured sγ decay rate has een used [13] to put stringent constraints on the additional CKM matrix elements. However, most of these studies neglect the effect of new CP phases y taking λ t Vt s V t to e real. The sγ constraint then implies that, for fixed t quark mass, λ t can take on only one of two values. In this paper, we study the effect of the fourth generation on s transitions y including the CP phase in Vt s V t. We use the updated measured value of sγ [15] and the lower ound on Bs 0 B s 0 mixing, to put a constraint on fourth generation parameters. The effect of the allowed region of parameter space on s transitions are compared with Belle and BaBar measurements. We further discuss the forward ackward asymmetry of B (K, K )l + l, and indirect CP-violation in B s B s mixings and B 0 φk S decay. We stress that we focus on Vt s V t only, in anticipation of new data. Thus, only one additional new CP phase is introduced. The K + π + ν ν process, B d mixing and its CP phase etc., involve Vt d V t s, Vt d V t respectively, which are quite independent from the CP phase information contained in V t sv t. Strategy The sγ process provides a stringent constraint on new physics, while the non-oservation of B s B s mixing also puts constraints on a fourth generation. We shall discuss these two traditional constraints efore turning to more recent experimental oservations.

2 556 A. Arhri, W.-S. Hou: Effect of fourth generation CP phase on s transitions.1 The sγ constraint With a sequential fourth generation, the Wilson coefficients C 7 and C 8 receive contriutions from the t quark loop, which we will denote as C7,8 new. Because a sequential fourth generation couples in a similar way to the photon and W, the effective Hamiltonian relevant for sγ decay has the following form: H eff = 4G i=8 F [λ t Ci SM (µ)+λ t Ci new (µ)]o i (µ), (1) i=1 with λ f = Vfs V f, and the O i s are just as in SM. The unitarity of the 4 4 CKM matrix leads to λ u + λ c + λ t + λ t = 0. Taking into account that λ u is very small compared to the others, one has [10,11] λ t = λc λ t, () and λ c = V csv c is real y convention. It follows that λ t C SM 7,8 + λ t C new 7,8 = λ c C SM 7,8 + λ t (C new 7,8 C SM 7,8 ), (3) where the first term corresponds to the usual SM contriution. It is clear that, for the m t m t and λ t 0 limits, the λ t (C7,8 new C7,8 SM ) term vanishes, as required y the GIM mechanism. We parameterize λ t r s e iφs, (4) where Φ s is a new CP-violating phase. The ranching ratio of sγ is computed y using formulas given in [16]. Using B(B X c e ν e )=10.4%, λ c =0.04, we find B(B X s γ) in SM. The present world average for B X s γ rate is [15], B(B X s γ)=(3.3 ± 0.4) (5) We will keep the sγ ranching ratio in the 1σ range of (.9 3.7) 10 4 in the presence of a fourth generation. We note that the CDF Collaoration excludes [17] the quark in the mass range of 100 GeV <m < 199 GeV at 95% C.L., if B( Z) = 100% 1. Precise EW data provide a stringent constraint on the fourth generation: the splitting etween t and is severely constrained y m t m m Z [15,18,19]. Assuming that m t >m,we conclude that the t should e heavier than 199 GeV + m Z. In our analysis we will consider only a t heavier than 50 GeV. In Fig. 1 we give a contour plot of B(B X s γ)inthe (Φ s,r s ) plane for m t = 50 and 350 GeV. We see that the CP phase Φ s does make a considerale impact on the range of B(B X s γ), ut one not particularly sensitive to m t. In general, for oth the m t = 50 and 350 GeV cases, r s cannot e much larger than 0.0 if one wishes to keep B(B X s γ) in the range of (.9 3.7) But for Φ s π/, 3π/, when Vt s V t is largely imaginary, r s can take on much larger values; for these cases the t contriution 1 Such a limit can e relaxed [18] if we consider that the decays H and cw may e competitive with Z Fig. 1. Contours for B(B X sγ)( 10 4 ) in the (Φ s,r s) plane, for m t = 50 (350) GeV on the left (right) adds only in quadrature to the SM contriution, hence is much more accommodating. In this sense, our approach is different from the approach of [10,13] where λ t is taken as real, and can take on only one of two values for a fixed m t. Since the chiral structure is the same as in SM, and since a heavy state such as t does not ring in extra asorptive parts, CP asymmetry in sγ remains small. However, the possiility of sizale CP odd t contriutions allowed y sγ, while not easily distinguished in the sγ process itself, can make an impact on CP oservales in other s transitions, as we shall see.. B 0 s B 0 s mixing constraint Because of a strong m t dependence, the fourth generation contriution can easily have an impact on B 0 s B 0 s mixing and its CP phase Φ Bs, and can e accessile soon at the Tevatron Run II. We use the definition of m Bs = M B 1 and M B 1 = M B 1 e iφ B S, where M B 1 is given y [0] M B 1 = G F M W 1π M B s B Bs f B s {λ t ηs 0 (x tw ) (6) + η λ t S 0(x t W )+ η λ t λ t S0 (x tw,x t W )}, with x fw = m f /M W, and S 0 (x) = 4x 11x + x 3 4(1 x) 3 log xx 3 (1 x) 3, (7) { [ S 0 (x, y) 1 1 = xy y x y 3 ] 1 4 (1 y) log y + 1 [ 1 x y (1 x) 3 ] 1 4 (1 x) log x 3 } 1, (8) 4 (1 x)(1 y) where η =0.55 is the QCD correction factor. Taking into account the threshold effect from the quark, we find η = α s (m t ) 6/3 ( αs (m ) α s (m t ) ) 6/1 ( ) 6/19 αs (m t ). α s (m )

3 A. Arhri, W.-S. Hou: Effect of fourth generation CP phase on s transitions 557 Fig.. Contours for m Bs (ps 1 ) in the (Φ s,r s) plane, for m t = 50(350) GeV on the left (right), with the results of Fig. 1 overlayed as dashed lines for comparison Fig. 3. m Bs (left) and sin Φ Bs (right) versus Φ s for m t = 50 GeV and r s =0.00 (short dash), 0.01 (long dash) and 0.0 (solid) Given the splitting etween t and, it turns out that η is also close to 0.55 for 50 GeV <m t < 350 GeV, which is not surprising. We shall take η = η. In our calculation we use B Bs fb s = (60 MeV) and m t (m t ) = 168 GeV, which gives m SM B s 17.0ps 1 with vanishing sin Φ SM B s. For illustration, in Fig. we give a contour plot of m Bs in the (Φ s,r s ) plane, for m t = 50 and 350 GeV. The result of Fig. 1 is overlayed for comparison. We see that m Bs can reach 90 ps 1 for r s 0.0 and Φ s π, ut for heavier m t, B(B X s γ) will reach eyond and ecome less likely. For Φ s π/, 3π/, where sγ is more tolerant, m Bs can get greatly enhanced. On the other hand, the experimental lower ound of m Bs > 14.9ps 1 implies that the region where 0 r s 0.03 and cos Φ s > 0 is disfavored, which rules out a region allowed y sγ. We now see that the allowed parameter space is larger for the m t = 50 GeV case, i.e. for heavier m t, r s can only take on smaller values. This is ecause the m t dependence for m Bs is rather strong, ut is much weaker in case of sγ. We stress that the Φ s π/, 3π/ cases are much more forgiving ecause the CP phase for s transitions is practically real in SM3, hence a purely imaginary fourth generation contriution adds only in quadrature. We thus turn to the study of the possile impact of a fourth generation, especially through its new CP phase, on other s transitions. 3 CP-violation in B meson mixing Here we mean Bs 0 B s 0 mixing. A fourth generation can affect Bd 0 B d 0 mixing and its CP phase through Vt d V t. Since there is no apparent deviation from SM3 [15] expectations, we ignore it. While an enhanced m Bs can easily evade present ounds, of particular interest is whether and how a fourth generation could affect the mixing-dependent CP-violating oservale sin Φ Bs, which is analogous to the recently measured CP phase in the Bd 0 B d 0 mixing amplitude. We plot m Bs and sin Φ Bs versus Φ s in Figs. 3 and 4, for m t = 50 and 350 GeV, respectively, for several values of r s. In general, the interference etween SM3 and fourth generation effects is constructive for cos Φ s < 0, and can Fig. 4. Same as Fig. 3 for m t = 350 GeV and r s = 0.00 (short dash), (long dash) and (solid) enhance m Bs up to ps 1 for large r s 0.0, as we have seen already in Fig.. Conversely, destructive interference occurs for cos Φ s > 0, ut this would give m Bs < m SM B s, hence is disfavored y the present lower ound. We are thus more interested in the constructive interference scenario, or when Φ s π/, 3π/, where the sγ and m Bs constraints are more forgiving. We illustrate the m t = 50 GeV case with a larger range of r s values. This not only allows for a greatly enhanced m Bs, it makes sin Φ Bs very interesting. We see from Fig. 3 that the sin Φ Bs structure is rich: for r s > 0.01, sin Φ Bs can reach ±100%. This can e traced to the dependence of M 1 on Φ s. We see from (6) that the λ t, λ t, and λ t -independent terms imply that M 1 takes the following form: M 1 = M 1 e iφ Bs r s e iφs A + r s e iφs B + C, (9) where A and B are explicit functions of m t and m t and C is the usual SM3 contriution. For large enough r s, the rse iφs term dominates, and one has e iφs modulation. But as r s decreases, the quadratic term ecomes unimportant, and our term is only suject to the e iφs modulation arising from the interference term. Thus, for r s as small as 0.00 (compare λ c = VcsV c = 0.04), m Bs is close to the SM3 value, ut sin Φ Bs can already reach 5% near Φ s = π/, 3π/. For r s =0.01, Φ s is restricted y the m Bs lower ound to the range of 70 Φ s 300, where sin Φ Bs can take on extremal values of 100% near the oundaries. This is very interesting, for if m Bs is measured soon near the present ounds, we may find the associated CP-violating phase to e maximal! For r s =0.0, the allowed range for Φ s is slightly larger than (roughly 50 Φ s 310 ) the r s =0.01 case. The maximal sin Φ Bs now occurs close to Φ s = π/, 3π/ since the fourth generation effect overwhelms the SM3 top quark effect. In all these cases, sin Φ Bs vanishes for Φ s = π when one has maximal constructive interference in m Bs.

4 558 A. Arhri, W.-S. Hou: Effect of fourth generation CP phase on s transitions It should e noted that, for purely imaginary e iφs, r s could e consideraly larger than 0.0, leading to a very large m Bs and maximal sin Φ Bs. The measurements of such cases, however, are more difficult. For the heavier m t = 350 GeV scenario of Fig. 4, sγ puts a slightly tighter constraint, hence we have illustrated the Φ s dependence of m Bs and sin Φ Bs for r s only up to For this parameter space, the rse iφs term in (9) is sudominant, and one largely has e iφs modulation. The case is similar to the low r s discussion of Fig. 3. Once again, for Φ s π/, 3π/, r s could e much larger than the illustrated range, and sin Φ Bs would ecome maximal. 4 Electroweak penguin B (K, K )l + l A recent highlight from B factories is the emerging electroweak penguins. Based on 30 f 1 data, the first oservation of B Kµ + µ was reported y the Belle experiment in 001, giving B(B Kµ + µ )=( ) 10 6 []. This was updated [3] in 00 to ( ± 0.08) 10 6 with twice the data at 60 f 1. Comining with the B Ke + e mode, one finds B(B Kl + l )=( ± 0.06) 10 6, (10) at 5.4σ significance. These results have een confirmed [5] y the BaBar Collaoration with a dataset of 78 f 1, giving B(B Kl + l ) = ( ) 10 6, consistent with an earlier value of ( ) 10 6 at 56.4 f 1 ; the pulished BaBar upper limit in 001, however, gives [4] B(B Kl + l ) < The latest upper limits for B(B K l + l ) are for Belle and, for BaBar, ,or( ± 0.8) 10 6 at.8σ significance. Given the volatility, including the muon efficiency prolem of BaBar, we shall not comine the aove values. However, while B(B Kl + l ) ( ) 10 6 is not inconsistent with the SM predictions [1], it is somewhat on the high side, as we shall see. Similarly, the new Belle oservation [6] of inclusive B X s l + l decay, B(B X s l + l )=(6.1 ± ) 10 6, (11) at 5.4σ significance, is also a it on the high side. We apply the same approach we have introduced for sγ. Using the notations and formulas from [1], the effective Hamiltonian for sl + l is given y H eff = 4G i=10 F [λ t Ci SM (µ)+λ t Ci new (µ)]o i (µ), (1) i=1 where a similar decomposition as in (3) can e made. We use the Wilson coefficients Ci SM calculated in naive dimensional regularization []. The analytic expressions for all Wilson coefficients can e found in [3]. For simplicity, we will not take into account long-distance effects from real c c intermediate states. Fig. 5. Non-resonant ranching ratios of B Kl + l and B K l + l, where solid, dashed and long dashed curves are for (r s,m t ) = (0.0, 50 GeV), (0.01, 350 GeV), (0.0, 350 GeV), respectively. The horizontal and is the SM expectation at NLO using LCSR form factors, with the central line standing for the central value. The present experimental ranges are ( ) 10 6 and ( ) 10 6, respectively We work with the MS ottom mass evaluated at µ = 5 GeV. At the NLO, { m (µ) =m pole 1 4 ( α s (µ) 1 3 )} 3 π log(mpole /µ), where m pole =4.8GeV is the quark pole mass. For the other parameters, we take 1.3 GeV as the charm quark mass, m t (m t ) = 168 GeV, m K = GeV, m K = GeV, m B =5.8 GeV, λ c = VcsV c =0.04 and sin θ W =0.3. To compute the B (K, K )l + l rates, we need the B K and B K transition form factors, where we take the values from QCD sum rules on the light-cone (LCSR) [1]. We thus otain the non-resonant ranching ratios of B(B Kµ + µ )= and B(B K µ + µ )= (compare [1]) in SM for central values of the form factors. We note that the more recent NNLO calculations give results that are 40% smaller [4], and could suggest the need for new physics, such as the fourth generation. For B X s l + l, with the set of parameters chosen aove, and again without including the long-distance contriution, we find that the non-resonant ranching ratio within SM3 is of order , where we integrate ŝ =(p l + + p l ) /m over the range [4m l /m, 1]. The strong m t dependence [11] makes the electroweak penguin a particularly good proe for the presence of a fourth generation. Fourth generation contriutions to B (K, K )l + l have een studied in [11,5]. These studies, however, took λ t to e real, which would then e constrained y sγ to two specific λ t values for a given m t [13] (i.e. for Φ s = 0 and π). As we pointed out, allowing for a CP phase in λ t, for fixed m t a large region in the (Φ s,r s ) plane is actually still allowed. In Fig. 5 we plot the non-resonant ranching ratio for B Kl + l and B K l + l versus CP phase Φ s for several values of r s and m t. The horizontal ands give the SM3 range taking into account form factor uncertainties. The pattern is similar to m Bs : for cos Φ s > 0wehave destructive interference, and the ranching fraction is elow SM; for cos Φ s < 0 we have constructive interference and the ranching ratio is largest for Φ s = π. We see that the non-resonant B(B Kl + l ) can reach approximately to 3 times the SM value depend-

5 A. Arhri, W.-S. Hou: Effect of fourth generation CP phase on s transitions 559 Fig. 6. Contour plot for B X sl + l in (Φ s,r s) plane for m t = 50 (350) GeV on the left (right). The present experimental result is (6.1 ± ) 10 6 a Fig. 7a,. Normalized forward ackward asymmetry contours for B K l + l, for Φ s =0,ina r s versus s/m B for m t = 350 GeV and m t versus s/m B for r s =0.0 ing on m t, r s, and Φ s. As stated, such an enhancement may ecome really called for if the NNLO result [4] of B(B Kl + l )=(0.35 ± 0.1) 10 6 is orne out. In any case, a possile factor of 3 enhancement over SM is still within the experimental range of (10), ut the case of r s 0.0 with m t = 350 GeV may have too large a ranching ratio for Φ s π, hence V t sv t and m t should not e simultaneously too large. Thus, the electroweak penguin modes have similar power as the m Bs ound, which was illustrated in Fig., in proing a fourth generation at present. But it may improve rapidly once the experiments converge. We note that there may e some troule with the B(B K l + l )/ B(B Kl + l ) ratio etween theory and experiment. This may e either an experimental prolem (fluctuation), or a prolem with form factors. For this reason, the inclusive B X s l + l may e more interesting, since it suffers less from hadronic uncertainties. We illustrate y a contour plot in Fig. 6 the non-resonant contriution to B(B X s l + l ) 10 6 in the (Φ s,r s ) plane, for m t = 50 and 350 GeV. As one can see, in oth cases, cos Φ s > 0 reduces B X s l + l to values less than and is less favored, while for π/ <Φ s < 3π/ the ranching ratio is enhanced and grows with r s. The ehavior is similar to B s mixing. We note that the effect of the fourth generation on B X s l + l has een studied recently in [14]. The constraints we otain on r s from B X s l + l are consistent with [14]. The dilepton invariant mass spectrum does not contain further information eyond the rates themselves. With proper rescaling, the shape is the same as in SM. Let us now discuss the forward ackward asymmetry (FBA) of B K l + l. The usual definition is [1] da FB dŝ = û(ŝ) 0 dû d Γ 0 dûdŝ + dû d Γ û(ŝ) dûdŝ, (13) where û(ŝ) = (1 4ˆm l /ŝ)λ and λ = λ(1, ˆm K, ŝ) = 1+ ˆm 4 K +ŝ ŝ ˆm K (1 + ŝ) with ˆm = m /MB. From the experimental point of view the normalized FBA (ĀFB) is more useful and is defined y dāfb dŝ = da FB dŝ / dγ dŝ. (14) a Fig. 8a,. Normalized B K l + l forward ackward asymmetry contours for Φ s versus s/m B, for a m t = 50 GeV, r s =0.04, and m t = 350 GeV, r s =0.0 Note that the FBA for B Kl + l vanishes. In the SM, the FBA for B K l + l is positive for ŝ<0.1 with values going up to 0.1, vanishes at ŝ 0 =0.1 and turns negative for 0.7 > ŝ>0.1 with FBA values going down to 0.4. This reflects the interference etween the sγ versus sz mediated processes. The presence of a fourth generation affects the latter more strongly and hence can change the pattern. We are interested in the parameter space where the shape of FBA changes completely from that of SM. In Fig. 7, we give contour plots for normalized FBA, for Φ s =0,inthe(ŝ, r s ) plane for m t = 350 GeV (left) and in the (ŝ, m t ) plane for r s =0.0 (right). We see that FBA can vanish not only for ŝ =0.1 ut also in the full range of 0.7 > ŝ>0.1. In the left plot this happens for rs and m t = 350 GeV while in the right plot this seems to happen for m 0 t 335 GeV and r s =0.0. From the left contour, note that for r s <rs, 0 the FBA is negative for ŝ>0.1 exactly as in the SM, ut for r s >rs, 0 the FBA remains positive even for ŝ>0.1. The same oservation can e made for ŝ>0.1 in the right contour: the FBA is negative for m t <m 0 t ut positive for m t >m 0 t, and FBA can reach +0.4 or more. In Fig. 8, we illustrate the effect of the CP phase Φ s for normalized FBA in the (ŝ, Φ s ) plane for m t = 50 GeV, r s =0.04 (left), and for m t = 350 GeV, r s =0.0 (right). From the left plot, one can see that FBA has a constant sign (negative) for large ŝ>0.1 and can ecome larger than 0.3 for a large CP phase and large ŝ. In the right

6 560 A. Arhri, W.-S. Hou: Effect of fourth generation CP phase on s transitions plot the FBA can flip sign for Φ s 0.5 and for values of ŝ> CP-violation in charmless B decays Although it is too early to draw conclusions [6], oth Belle and BaBar presented [7] some hints for mixingdependent CP violation in the charmless B 0 φk S decay mode, with the two experiments agreeing in sign. It is intriguing that the sign is opposite to that of B 0 J/ψK S! We do not advocate that new physics is already called for, ut illustrate the impact of a fourth generation. It is known that the CP asymmetry in oth B 0 φk S and J/ψK S should measure the same quantity sin φ 1 in SM, where the uncertainty is estimated to e less than O(λ ) [8]. The effective Hamiltonian for the B = 1 transition can e written as [7] H eff = G F {λ u (C 1 O u 1 + C O u )+λ c (C 1 O c 1 + C O c ) } i=10 [λ t Ci SM (µ)+λ t Ci new (µ)]o i (µ). (15) i=3 For the SM part, we have used the next-to-leading order formulas given in [7] for the Wilson coefficients, while for the fourth generation effect in Ci new, we have used the leading order analytic formulas given in [9]. Following the notation of [7], in the factorization approach, the amplitude of oth B K φ and B 0 K 0 φ is proportional to V t Vts(a 3 +a 4 +a 5 1/(a 7 +a 9 +a 10 )). The exact form of the factorizale hadronic matrix element is irrelevant for us since we are interested only in CP asymmetry oservales in SM4, and the ratio of the ranching ratios in SM3 and SM4. The coefficients a i are defined y a i = Ci eff +1/N C Ci±1 eff for odd, even i. The CP asymmetry sin(φ φks ) is defined y sin(φ φks )= ImΛ 1+ Λ, (16) where Λ =e iφ1 (A/A) with A eing the CP-conjugate amplitude of A. We define also the following ratio: R φks = A SM4(B 0 φk S ) A SM3 (B 0 φk S ). (17) In our analysis we use the following set of parameters: µ =.5GeV, m t = 168 GeV, m =4.88 GeV, k = m /, α =1/18 and s W =0.3 and α s(m W )= In SM3 (i.e. without a fourth generation) we find that A/A =0.979, which is close to 1, and which means that sin Φ φks = sin φ1 = sin Φ J/ψKS. In our numerical evaluation we take φ 1 close to 4 which reproduces the world average of sin φ 1 =0.734 coming from B 0 J/ψK S. IN SM4, the ratio A/A has an additional CP phase, which we denote θ, that arises from our CP phase Φ s. a Fig. 9. a Ratio R φks and sin(φ φks ) as a function of the CP phase Φ s, where solid, dashed and long dashed curves are for r s =0.01, 0.0 and 0.04, respectively, with m t = 400 GeV So if we use the definition of Λ defined aove, Λ =e A A iφ1 A =e i(φ1+θ) A, (18) where we have used A/A =e iθ A/A. We have studied numerically the effect of the CP phase Φ s, r s and m t on oth sin(φ φks ) and R φks and found that for r s < 0.05 and m t 450 GeV the effect is rather small. In Fig. 9 we illustrate the effect of Φ s on R φks (left) and sin(φ φks ) (right) for m t = 400 GeV and several values for r s.forr s =0.0, B 0 φk S in SM4 can e enhanced (reduced) only y aout % for cos Φ s > 0 (< 0). For large r s =0.04, the effect can only reach aout 5%. As one can see in the right plot, the effect of the CP phase Φ s on sin(φ φks ) is also not impressive. Both in SM3 and SM4, we find numerically that a 4 a 9 / for all Φ s, while a 3 a 7 / and a 5 a 10 / in the region where cos Φ s is not close to 1 (Φ s 0 and π). It can e seen that sin(φ φks ) is enhanced for 0 <Φ s <π and reduced for π<φ s < π. Forr s =0.0 (0.04) the enhancement (suppression) is aout +15% ( 30%). The suppression of sin(φ φks ) is insufficient to drive it negative, even for rather sizale r s. For sign change, one would need Φ s 3π/ with an unacceptaly large r s. For example, for m t = 400 GeV and Φ s =3π/, sin Φ φks turns negative for r s > 0.11, which is much larger than V c. 6 Discussion and conclusion A particularly interesting aspect of the existence of a fourth sequential generation is the presence of new CPviolating phases. Restricting ourselves to s transitions, esides the strength of V t sv t r s, one has a unique new CP phase, arg V t sv t Φ s. Existing work in the literature has not emphasized the Φ s 0,π situation, hence has overconstrained the possile impact of the fourth generation. In the present work, we have tried to cover as much ground as possile, and we find a rather enriched phenomenology for the s transitions, for oth CP-violating or CP-independent oservales. The sγ process is not very sensitive to the presence of a fourth generation, since the structure is rather similar to SM, and it is not possile to generate a large CP asymmetry ecause of the chiral structure. It is therefore especially accommodating for Φ s = π/, 3π/, when the fourth generation contriution adds only in quadrature in

7 A. Arhri, W.-S. Hou: Effect of fourth generation CP phase on s transitions 561 rate. These are the most interesting situations, since CPviolating effects elsewhere are the largest. For example, B s mixing and the CP phase sin Φ Bs are very interesting proes of fourth generation effects. While some part of the parameter space allowed y sγ (cos Φ s 0) is ruled out, m Bs can e much larger than SM3 expectations. Any value for sin Φ Bs etween 1 and +1 is in principle possile. What may e interesting is that, even for r s rather small, hence m Bs just aove the present ound, sin Φ Bs could still e sizale. Thus, Tevatron Run II data are eagerly awaited. The electroweak penguin modes B K ( ) l + l and X s l + l have recently een oserved, and the rates tend to e a little on the high side, especially when compared to more recent NNLO results. Again, the est case seems to e for Φ s π/, 3π/, when fourth generation effects add mildly in quadrature. Although the dilepton spectrum, and roughly speaking the direct CP asymmetries as well (analogous to sγ), are not much affected, the forward ackward asymmetry, A FB, can e drastically affected. This is ecause the γ effect is far less sensitive to fourth generation effects than the Z effect, so the s = m l + l /m B dependence of A FB is a sensitive proe of the interference etween the two terms. Thus, while we need the Belle and BaBar measured rates to converge etter, in the long run the forward ackward asymmetry may provide a very interesting proe of the fourth generation. Finally, we studied the mixing-dependent CP-violation in the B φk S charmless decay, since some hint for physics eyond SM appeared here recently. We found the effect of a fourth generation to e insignificant here. This is ecause the strong penguin, even more so than sγ, depends very mildly on the fourth generation, while the harder C 9 term, analogous to the electroweak penguin contriution to B K ( ) ll, is sudominant. Thus, even with its new source of CP-violation, it is unlikely for the fourth generation to change the sign of sin Φ φks with respect to sin Φ J/ψKS in the B d system. The impact of the fourth generation is most prominent in B s mixing and electroweak penguin s transitions, ut mild in electromagnetic and strong penguins. We have found that the fourth generation, even with V t sv t not far elow V cs V c 0.04, could e lurking with a purely imaginary KM phase. Acknowledgements. A. Arhri is supported y the Alexander von Humoldt Foundation. The work of WSH is supported in part y grant NSC M-00-07, the MOE CosPA Project, and the BCP Topical Program of NCTS. References 1. P.H. Frampton, P.Q. Hung, M. Sher, Phys. Rept. 330, 63 (000); J.I. Silva-Marcos, hep-ph/ K. Ae et al. [BELLE Collaoration], Phys. Rev. Lett. 88, (00) 3. K. Ae et al. [BELLE Collaoration], BELLE-CONF- 041, contriuted to 31st International Conference on High Energy Physics (ICHEP 00), Amsterdam, The Netherlands, 4 31 July 00 (see talk y S. Nishida) 4. B. Auert et al. [BABAR Collaoration], Phys. Rev. Lett. 88, (00) 5. B. Auert et al. [BABAR Collaoration], hep-ex/00708, contriuted to ICHEP J. Kaneko et al. [Belle Collaoration], hep-ex/00809, sumitted to Phys. Rev. Lett. 7. Plenary talk y M. Yamauchi (Belle Collaoration) at ICHEP 00; B. Auert et al. [BABAR Collaoration], hep-ex/007070, contriuted to ICHEP 00 (see talk y J. Richman) 8. M.S. Alam et al. [CLEO Collaoration], Phys. Rev. Lett. 74, 885 (1995) 9. M. Ciuchini, G. Degrassi, P. Gamino, G.F. Giudice, Nucl. Phys. B 57, 1 (1998); F.M. Borzumati, C. Greu, Phys. Rev. D 59, (1999) 10. W.S. Hou, A. Soni, H. Steger, Phys. Lett. B 19, 441 (1987) 11. W.S. Hou, R.S. Willey, A. Soni, Phys. Rev. Lett. 58, 1608 (1987) [Erratum-iid. 60, 337 (1987)] 1. T. Hattori, T. Hasuike, S. Wakaizumi, Phys. Rev. D 60, (1999); T.M. Aliev, D.A. Demir, N.K. Pak, Phys. Lett. B 389, 83 (1996); Y. Dincer, Phys. Lett. B 505, 89 (001) and references therin 13. C.S. Huang, W.J. Huo, Y.L. Wu, Mod. Phys. Lett. A 14, 453 (1999); C.S. Huang, W.J. Huo, Y.L. Wu, Phys. Rev. D 64, (001) 14. T. Yanir, JHEP 006, 044 (00) 15. K. Hagiwara et al. [Particle Data Group Collaoration], Phys. Rev. D 66, (00) 16. A.J. Buras et al., Nucl. Phys. B 44, 374 (1994); C.K. Chua, X.G. He, W.S. Hou, Phys. Rev. D 60, (1999) 17. T. Affolder et al. [CDF Collaoration], Phys. Rev. Lett. 84, 835 (000) 18. A. Arhri, W.S. Hou, Phys. Rev. D 64, (001) 19. P.H. Frampton, P.Q. Hung, M. Sher, Phys. Rept. 330, 63 (000); H.J. He, N. Polonsky, S. f. Su, Phys. Rev. D 64, (001); V.A. Novikov, L.B. Okun, A.N. Rozanov, M.I. Vysotsky, Phys. Lett. B 59, 111 (00) 0. T. Inami, C.S. Lim, Prog. Theor. Phys. 65, 97 (1981) [Erratum-iid. 65, 177 (1981)] 1. A. Ali, P. Ball, L.T. Handoko, G. Hiller, Phys. Rev. D 61, (000). A.J. Buras, M. Misiak, M. Munz, S. Pokorski, Nucl. Phys. B 44, 374 (1994) 3. G. Buchalla, A.J. Buras, M.E. Lautenacher, Rev. Mod. Phys. 68, 115 (1996); A.J. Buras, M. Munz, Phys. Rev. D 5, 186 (1995) 4. A. Ali, E. Lunghi, C. Greu, G. Hiller, Phys. Rev. D 66, (00) 5. T.M. Aliev, A. Ozpineci, M. Savci, Nucl. Phys. B 585, 75 (000) 6. Y. Nir, hep-ph/008080, plenary talk presented at ICHEP A. Ali, G. Kramer, C.D. Lu, Phys. Rev. D 58, (1998) 8. Y. Grossman, M.P. Worah, Phys. Lett. B 395, 41 (1997); Y. Grossman, G. Isidori, M.P. Worah, Phys. Rev. D 58, (1998) 9. M. Ciuchini, E. Franco, G. Martinelli, L. Reina, Nucl. Phys. B 415, 403 (1994); A.J. Buras, M. Jamin, M.E. Lautenacher, P.H. Weisz, Nucl. Phys. B 370, 69 (199) [Addendum-iid. B 375, 501 (199)]