New CP-violating parameters in cascade decays

Transcription

1 New C-violating parameters in cascade decays A. Amorim a,b, Mário G. Santos a and João. Silva a,c a Centro de Física Nuclear da Universidade de Lisboa arxiv:hep-ph/987364v2 8 Sep 998 Avenida rofessor Gama into, 2, 699 Lisboa Codex, ortugal b epartamento de Física Faculdade de Ciências da Universidade de Lisboa Campo Grande, 7 Lisboa, ortugal c Centro de Física Instituto Superior de Engenharia de Lisboa 9 Lisboa, ortugal February, 28 Abstract We consider decay chains of the type M + f +, where M is a neutral meson that may mix with its antiparticle M, before decaying into the final state f. may be either a heavier neutral meson or a charged meson. We perform a rephasing-invariant analysis of the quantities that show up in such cascade decays. If the decay M + (or the decay M + ) is forbidden, we find the usual λ f parameters describing the interference between the mixing of a neutral meson system and the decay from that system into the final state f. However, when both the M + and M + decays are allowed, we find a new class of rephasing-invariant parameters, ξ i, that measure the interference between the mixing of a neutral meson system and the decay from the initial state into that system. We show that the quantities λ f and ξ i are necessary and sufficient to describe all the interference effects present in the most general cascade decay. We discuss the various cascade decays in turn, highlighting the special features of each one. Introduction The particle-antiparticle neutral meson systems have long been identified as an ideal setting to search for C violation (CV). This effect was discovered in 964 in the

2 neutral kaon system [], and it is believed that this will soon be complemented with measurements of CV in the decays of neutral B mesons. The C violation found in the kaon system and the null results found elsewhere can be accommodated in the Standard Model (SM), through an irremovable complex phase in the Cabibbo-Kobayashi-Maskawa quark mixing matrix [2]. The major goal of the upcoming B-factory experiments is to submit this explanation to stringent tests [3]. In 989, Azimov [4] pointed out that additional tests could be performed by looking for B d J/ψK J/ψ[f] K cascade decays, involving the neutral B d and neutral kaon systems in succession. This idea has been followed by ass and Sarma [5]. Recently, cascade decays have received renewed attention. Kayser and Stodolsky have developed a matrix method to deal with cascade decays [6]; Kayser has shown that one may use B d J/ψK J/ψ[πlν] K decays to get at cos 2β [7]; and Azimov and unietz pointed out that one may use B s J/ψK J/ψ[f] K decays to measure m Bs, even if m Bs turns out to be too large to be determined experimentally from the direct decays of the B s mesons [8]. In all these cases, an initial B meson can only decay to one of the kaon s flavour eigenstates. To leading order in the SM, the decays B d K + X and B s K + X, and the respective C conjugate decays are forbidden. In this article we will consider a more general situation, in which a given meson can decay into both flavour eigenstates of a lighter neutral meson system. This situation is the one relevant for the decays B ± + X ± [f] + X ±, B + X [f] + X, ± K + X ± [f] K + X ±, K + X [f] K + X. () ecays of the first type have been discussed by Meca and Silva [9] in the context of uncovering new physics effects in the system. Here we show that all these decays depend on a new type of C-violating observables, over and above those required for the description of direct decays of neutral mesons. In section 2 we describe all the quantities involved in cascade decays of the type M + X [f] M + X, where and M describe two neutral meson systems, and [f] M describes a set of particles into which M can decay, and which have an invariant mass equal to that of the M M system. This analysis is performed by noting that all physical quantities must remain invariant under a rephasing of the kets in the problem. In section 3 we develop the conditions for C invariance, thus identifying all the C-violating rephasing-invariant quantities involved in cascade decays. Naturally, we obtain the usual C-violating observables. These describe CV in the mixing of the neutral meson systems; CV in the decays; and CV in the interference between the mixing of the neutral meson system and the decay from that system into the final state. However, when both the M + and M + decays are allowed, we find a new class of rephasing invariant quantities that measure the interference between the mixing of the intermediate neutral meson system M, and the decay from the initial state into that system. These quantities are used in section 4 to develop We will take B to stand for both Bd and B s. Moreover, K refers to a generic combination of K and K, and similarly for, B d and B s. 2

3 the formulae describing a generic cascade decay. Section 5 contains a classification of cascade decays and an extended discussion of the special features of each case. We draw our conclusions in section 6. 2 Rephasing-invariant quantities Let us consider a decay of the type M + X [f] M + X, (2) where both and M belong to neutral meson systems. In general, the amplitude for the overall decay chain involves the amplitudes of the primary decays A M M X T, A M M X T, A M M X T, A M M X T, (3) followed by the amplitudes of the secondary decay, A M f f T M, A M f f T M. (4) Since there is mixing in the neutral system, the decay rate will also involve the parameter q /p which arises in the transformation from the flavour eigenstates into the mass eigenstates: H = p + q, L = p q. (5) Here, H and L refer to the heavy and light mass eigenstates, respectively. Similarly, the M M mixing enters in the expressions through the parameter q M /p M. For completeness, we describe in appendix A the time evolution of a generic neutral meson system. In quantum mechanics, all kets may be rephased at will, 2 e iγ, e iγ, M e iγ M M, M e iγ M M, (6) f e iγ f f, f e iγ f f. Under this rephasing, the amplitudes and mixing parameters also change, according to q p e i(γ γ ) q p, q M p M e i(γ M γ M ) q M pm, A M ei(γ γ M ) A M, A M e i(γ γm) A, M A M e i(γ γ M ) A M, A M e i(γ γ M ) A M, (7) A M f e i(γ M γ f ) A M f, A M f e i(γ M γ f) A M f, A M f e i(γ M γ f ) A M f, A M f e i(γ M γ f ) A M f. 2 The rephasing-invariant analysis of cascade decays presented in this section is inspired in the analysis of direct decays performed in Ref. []. 3

4 Only those quantities which are invariant under this redefinition may have physical meaning. Clearly, the magnitudes of all the quantities in Eq. 7 satisfy this condition. In addition, we find some rephasing-invariant quantities which arise from the clash between the phases in the mixing and the phases in the decay amplitudes: λ M q p A M A M, λ M q p A M A M, (8) λ M f q M p M A M f A M f, λ M f q M p M A M f A M f, (9) ξ M A M A M p M q M, ξ M A M A M p M q M. () The quantities in Eq. 8 are well known. They describe the interference between the mixing in the system and the decay from that system into the flavour specific meson M or M, respectively. Similarly, the parameters in Eq. 9 describe the interference between the mixing in the M M system and the decay from that system into the final state f and f, respectively. On the contrary, the parameters in Eq. describe a new type of interference: the interference between the mixing in the M M system and the decay into that system (originating from, in the case of ξ M, or from, in the case of ξ M ). We notice that the parameters described thus far are not all independent. In particular λ M ξ M = λ M ξ M. () This means that, of the six phases contained in Eqs. 8, 9, and, only five are independent. This is obvious from Eqs. 7 which contain ten quantities and five rephasings. 3 Conditions for C-invariance In order to study C violation we first consider the consequences of C invariance. If C is a good symmetry, then there exist three phases α, α M and α f, and two C eigenvalues η = ± and η M = ±, such that: and C = e iα, C M = e iα M M, C f = e iα f f, (2) C H = η H, C L = η L, C M H = η M M H, C M L = η M M L. Notice that, since we define the sign of m to be positive, it is the value of η and of η M that must be taken from experiment. For example, we know experimentally that, if the small C violation in the neutral kaon system were absent, then the heavy kaon would coincide with the long-lived kaon, which would be C odd. Therefore, we must take η K = whenever we neglect C violation in the neutral kaon system. For the 4 (3)

5 other neutral meson systems, these parameters must still be determined experimentally. In the meantime, one sometimes uses the SM predictions for these parameters. For instance, the SM prediction for the Bs B s system is that the heavy state should be mostly C odd []. In this case, neglecting the C violation in the Bs Bs system leads to η Bs =. Moreover, the C transformation of the composite intermediate state is C XM = η X e iα M X M. (4) Here η X contains any relevant C transformation of the state X, as well as the parity corresponding to the relative angular momentum between X and M. For example, in the decays B J/ψK, η X = because J/ψ is C even, while the J/ψ and the kaon have a relative L = orbital angular momentum [2]. As a result of Eqs. 2, 3 and 4, the C invariance conditions are q p = η e iα, q M p M = η M e iα M, (5) and A M = η Xe i(α α M ) A M, A M = η X e i(α +α M ) A M, A M f = e i(α M α f ) A M f, A M f = e i(α M+α f ) A M f. (6) Therefore, if C is conserved we have q p = q M pm = A M = A A A M M = M A M f = =. A M f A M f A M f (7) Moreover, the parameters describing interference C violation become related by λ M λ M = λ M f λ M f = ξ M ξ M = λ M ξ M = λ M ξ M = η X η η M. (8) We have used Eq. on the last line. One may use the relations above to develop more complicated conditions for C invariance, such as A M A M f A A M M f = A M A M fa A. (9) M M f A very important special case occurs when f is a C eigenstate. Then η f e iα f = ±, and the conditions for C invariance become A M f = and λ M f = η M η f. (2) A M f 5

6 q p g g - + g + M X M X g M + M M X p q g MM M - q p M M g M - M X f Figure : The most general MX [f] M X cascade decay. We have found the usual types of C violation: q/p probes C violation in the mixing of neutral mesons; A i f A f ī probes C violation in the decays; while arg λ i f + arg λ i f measures C violation arising in the interference between the mixing of the initial neutral meson system, i, and the decay from that system into the final states f and f. If f is a C eigenstate, then the C-violating observable arg λ i f +arg λ i f is related to Imλ i f. In addition, we have discovered a new type of C-violating observables given by arg ξ M + arg ξ M. These observables measure the C violation which arises from the interference between the mixing in the M M system and the decay from the initial state ( and ) into that system. 4 The decay rate for a generic cascade decay 4. ecay amplitudes Let us consider a neutral meson that decays into an intermediate state containing a combination of M and M at time t, measured in the rest-frame of. The resulting combination of M and M will evolve in time and decay into the final state f at time t M, measured in its rest-frame. The decay amplitude for this process is given by A [ t M t M f ] = g+ (t )g+ M (t M) [ A M A M f + A A M M f [ + g+ (t )g M (t qm M) A p M A M f + p ] M A M q M A M f M + g (t )g M (t M) q [ pm A p q M A M f + q ] M A M p M A M f M + g (t )g+ M (t M) q [ A A p + A ] M M A f M M f. (2) This result is easiest to derive with the help of the evolution diagram presented in Fig.. Henceforth we shall not show the explicit time dependence of the g-functions. The decay amplitude may also be written as A [ t M t M f ] = ( A M A M f + A A M M f) [ g + g+ M + χ g+ gm + χ 2 g gm + χ ] 3g gm +, (22) ] 6

7 where we have used χ A M q M χ 2 q p A +A M p M M f M qm A M f A M A M f +A M A M f p A M q M qm A M f +A M A M p M M f p A M A M f +A M A M f χ 3 q p A M A M f +A M A M f A M A M f +A M A M f = λ M f+ξ M +ξ M λ M f, = λ M ξ +λ M f M +ξ M λ M f, ξ λ = λ M f + M M +ξ M λ M f. (23) Notice that the parameters χ n involve ξ M and ξ M, which describe the new type of interference effects which are the subject of this article. However, χ 3 may always be reinterpreted in a different way. Let us consider the state M into f A M f M + A M f M. (24) This is the M M combination that decays into the final state f at time t M. Indeed, this state is orthogonal to the state M not into f A M f M A M f M, (25) which clearly cannot decay into the final state f because f T M notinto f =. Using Eq. 24, it is easy to show that χ 3 = λ Minto f. (26) Still, in general, χ and χ 2 may not be reinterpreted as parameters λ. One may wonder why these effects did not show up in the analysis presented in [4, 5, 6, 7, 8]. The reason is simple: in those cases a meson can only decay into one of the flavour eigenstates of the intermediate meson system. For example, can decay into M, but not into M. Indeed, taking A M = = A M, we find, in addition to Eq. 26, χ = λ M f and χ 2 = λ MH. (27) The last equality involves M H, the heavy mass eigenstate of the M M system, and holds if and only if p M /q M =. In these cases, we do not need to introduce the parameters ξ. On the contrary, these parameters are required in the discussion of the B + K + [f] K + cascade decays, as found by Meca and Silva [9]. The same parameters would enter the analysis of the decays, B + H,L K +, if only these decays could be disentangled from one another [3]. Unfortunately, they cannot [3]. It should now be clear that the effects described by ξ M and ξ M are mandatory and, when combined with the usual C-violating observables, are also sufficient for a discussion of the B and K decay chains. In the example just discussed, in which A M = = A M, we find that χ 3 = χ χ 2. This relation, although not valid in general, holds in a variety of special cases. Indeed, ( ) ( λ 2 M f ξ χ 3 χ χ 2 = M ξ ) M λ M ( + ξ Mλ M f ) 2. (28) This result is completely general. We conclude that χ 3 χ χ 2 vanishes if any of the following conditions holds: 7

8 . λ 2 M f =, which, if f is a C eigenstate, means that there is C conservation when the M M system decays into the final state f; 2. ξ M ξ M =, meaning that there is C conservation when the system decays into the M M system; 3. λ M =, meaning that the decay M X is forbidden. Analogously, the difference also vanishes if M X is forbidden. If any of these conditions holds, the amplitude for the cascade decay depends only on two independent parameters (for example χ and χ 2 ), in addition to the overall normalization factor. We see that χ 3 χ χ 2 is only different from zero if there are interference effects in every step of the decay chain. 3 We now turn to the C invariance constraints on the χ n parameters. We find that, if there is C conservation, then χ 2 = η X η η M, (29) and χ 3 = χ χ 2. In addition, if f is a C eigenstate, then we find two further conditions for C invariance: χ = η M η f and χ 3 = η X η η f. (3) We now consider the case in which we start with a tagged. The decay amplitude for the cascade chain becomes A [ t M t M f ] = p ( A q M A ) M f + A A M M f [ ] χ3 g+g + M + χ 2 g+g M + χ g g M + g g + M. (3) This result may be obtained from Eq. 22 by interchanging g+ g the result by p /q. and multiplying 4.2 ecay rates Using Eq. 22, we find for the decay MX [f] M X, Γ [ t M t M f ] = N [ g + gm + g+ g M χ 2 g + g M 2 Re { χ g+ M g M 2 Re { χ 3 g+ g + 2Re { χ 2 g + g gm + gm 2 + χ 2 2 g g M } + 2 g 2 Re { χ2 χ 3g M } + 2 g M 2 Re { χ χ 2 g 2 + χ 3 2 g g+ M } + g M } + g } { + 2Re χ χ 3 g+ g gm + gm 2 }], (32) where N A M A M f + A M A M f 2. (33) 3 Informally, we may think of χ 3 χ χ 2 as the all hell breaks loose parameter. 8

9 + + X X ε - i p B + 2 q x τ - i q x τ 2 p + X + X ε - + ( K π ) X + Figure 2: The B + X + (K π + ) X + cascade decay. Similarly, the decay rate for an initial is Γ [ t M t M f ] p /q 2 = N [ χ 3 2 g + g M χ 2 2 g + g M 2 + χ 2 g g M + 2 g 2 { } + Re χ2 χ g 3 gm + gm { Re χ g M + 2 g M 2 { } + Re χ 3 g+ g + 2 g M 2 { Re χ χ 2 g + 2Re { χ χ 3 g + g gm + gm 2 + } + gm } + g } { + 2Re χ 2 g+ g gm + gm g g M + 2 }], (34) One may also use the rates in Eqs. 32 and 34 to describe decays of the type ± MX ± [f] M X ±. To this end, one substitutes +,, g + exp ( Γ +t /2), g, and q /p. 5 Classification of cascade decays To simplify the notation we will suppress the explicit reference to the final stage of the decay chain, [f] M X, unless it is required. Traditionally, one has considered cascade decays of the type B d, B s K, using the known mixing parameters of the K K system as an analyzer for the parameters of the heavier B d B d [4, 5, 6, 7] and B s B s [8] systems. More recently, Meca and Silva [9] have stressed the role of B ± decays as a probe for new physics contributions to the mixing. Clearly, cascade decays of the type ± MX ± [f] M X ± are useful for the second strategy, while cascade decays of the type MX [f] M X could, in principle, be useful for both. We analyze them in turn. 5. Cascade decays from charged mesons ecays of this type include + K K +, + {K, K }π +, s + K π +, and s + {K, K }K +. Since the kaon system is well tested experimentally, these decays are of limited use. This is also true of the pure penguin decays B + K π + and B + K K +. More interesting are the decays B + {, }K +, such as the one represented in Fig. 2. We use 9

10 and [4] ǫ ǫ A K π + A K π + A B + K + A B + K + V cd V us.5, (35) V cs V ud V ub V cs a 2.9, (36) V cb V us where a 2 /a.26 accounts for the fact that the B + K + decay is colorsuppressed, while the B + K + decay is not. Meca and Silva have pointed out that these decays may be used to get at x m /Γ [9]. This is easy to see from Fig. 2, where we have taken y =, and used the small x approximation, g + and g i/2 x τ, with τ Γ t. If m =, then there are no mixed decay paths and we only have two contributions to the decay amplitude. In suitable units, one is of order ǫ and the other of order ǫ. On the other hand, if m, then there are two further decay paths; one is of order x and the other of order ǫ ǫx. Now, values of x 2 are easy to obtain in many models of new physics [5]. In that case, the last term may be dropped, but the term linear in x corrects the unmixed ǫ and ǫ terms by about %. This has a corresponding impact on the Gronau-London-Wyler [6, 7] and Atwood-unietz-Soni [4] methods to get at the CKM phase γ with the B + {, }K + decays [9]. The decay rates for chains like the one in Fig. 2 are easy to calculate. Using Eq. 32 we find, where Γ [ B + t B t f ] = e Γ B + t B AB + A f + A B A + 2 f [ g 2 + 2Re { χ g+ g } + χ 2 g 2 ], (37) + a χ λ f + ξ B +. (38) + λ f ξ B + Taking y = and the small x approximation, the term dependent on g + g is proportional to Imχ x. It contributes to the rate due to two independent interference effects: one related to Imλ f, the other related to Imξ B + (see Eq. 4 bellow). This is easiest to understand with the aid of the parametrizations A K π + = A, A K π + = ǫ A e i, A B + K + = B, A B + K + = ǫ eiγ B e i B, (39) where A, and B contain form factors, strong phases and magnitudes of CKM matrix elements. and B are the differences of strong phases that exist between the amplitudes involving and the corresponding amplitudes involving. In addition, we take q /p = e 2iθ, thus allowing for the fact that the new physics effects might also bring with them new phases into the mixing. Therefore, λ K π + = ǫ ei(2θ + ) and ξ B + K + = ǫ e i(γ+2θ + B ). (4)

11 ε K K - (X l ν ) K l + B + K + i p - q x τ 2 Figure 3: The B + K + (X l + ν l ) K + cascade decay. The term linear in x becomes proportional to Im χ Im λ K π + ( ξb + K + 2) + Im ξ B + K + ( λ K π + 2) ǫ 2 [ǫ sin (2θ + ) ǫ sin (γ + 2θ + B )], (4) where we have neglected higher order terms in ǫ and ǫ. On the one hand, there is a contribution proportional to ǫx which arises from the Imλ f piece in χ. This involves θ, which vanishes in the SM, and the difference of strong phases, which is difficult to estimate but could also be small. Obviously, these two effects are the same that appear in the direct decay K π + ; they were studied in this context by [8], and [9, 2], respectively. On the other hand, there is a contribution proportional to ǫx which arises from the Imξ B + piece in χ. This is guaranteed large even in the SM, for it involves the large CKM phase γ. Therefore, these decays may be used to probe values of x 2. This case is discussed in detail in reference [9]. ecays like B + {, }π + and B + {, }ρ + might also be useful in the search for x. These decay chains have much larger branching ratios than B + {, }K +. Indeed, BR[B + ρ + ] = (.34 ±.8) 2, BR[B + π + ] = (5.3 ±.5) 3 [2], while BR[B + K + ]/BR[B + π + ] =.55 ±.5 [22]. This means that the overall normalization factor B 2 in Eq. 37 is roughly 8 (46) times larger in B + π + (B + ρ + ) decays than it is in B + K + decays. But, in these cases, ǫ is further suppressed by a factor of about.5 with respect to its value in the B + K + decay chains. This suppresses the ǫ ǫ and ǫx interference terms in the decay rate, meaning that these interference effects come into all the decay chains at roughly the same level. On the contrary, the term proportional to ǫx is the same for all the decay chains. Therefore, it should be much easier to detect in B + π + and B + ρ + decays than it is in the original B + K + decays proposed by Meca and Silva [9]. We recall that the ǫx term involves Imλ f which also shows up in direct f decays. Hence, it is only detectable if the new physics also produces large C-violating effects, θ, in the system [8], or if the strong phase difference,, is considerable [9, 2]. The crucial role played by the ξ i parameters in the B + {, }K + [f] K + cascade decays is seen most vividly by taking f = X l + ν l. This case is illustrated in Fig. 3. Since the decay X l + ν l is forbidden, we have λ f =, χ = ξ B + and, thus, the interference is determined solely by ξ B +. Indeed, it is clear from

12 Fig. 3 that the only interference effect present is that arising from the clash between the mixing and the decay from the B + into that system. Notice the crucial role played by the fact that there are two decay paths from B + into the system; B + and B +. Both must be present in order for the decay chain to display this new type of interference. Finally, there are B + s + and B + + decays. However, there are no B + s + and B + + decays. As a result, in Fig. 2 there is only one decay path from the initial B + into the intermediate state and, thus, the effects of that decay factor out in the overall decay rates. Everything works as if we had started from an initial meson. 5.2 Cascade decays B K Let us consider decay chains of the type B d K X [f] K X, where X may be J/ψ, φ, ω, ρ, π, etc. In this case, we have A B d K = and A B d K = in the SM. Therefore, the parameters needed for Eqs. 32 and 34 become N = A B d K A K f 2, χ = λ K f, χ 2 = q B d p K A B d K = λ Bd K p Bd q K A S, B d K χ 3 = χ χ 2. (42) We had already anticipated these results in Eq. 27, where we mentioned that the last equality on the third line holds if q K /p K =. From the experimental measurement of C violation in the K K system, we know that this equality holds to the level of 3. The situation is very similar for the decay chains of the type B s K X [f] K X. Here, A B s K = and A B s K = in the SM. Hence, N = A B s K A K f 2, χ = /λ K f, χ 2 = q B s q K A B s K = λ Bs K p Bs p K A S, B s K χ 3 = χ χ 2. (43) The articles [4, 5, 6, 7] discuss the first case, with X = J/ψ; while reference [8] describes the second case, also for X = J/ψ. As we have stressed before, in these cases, the fact that there is only one decay path from the B into the K K system implies that all the interference effects (and any C violation therein) may be written in terms of the classic λ parameters. To find the decay rates, we could substitute Eq. 42 or Eq. 43 into Eq. 32. However, we know that the lifetimes of the mass eigenstates of the K K system are very 2

13 different. Therefore, it is more appropriate to write our expressions in terms of γ S and γ L, which are the lifetimes of the short-lived (K S ) and long-lived (K L ) mass eigenstates, respectively. Since in both cases χ 3 = χ χ 2, Eq. 22 may be rewritten as A [ B t B K t K f ] e iµk H t K ( + χ ) ( g B + + χ 2 g B ) + e iµ K L t K ( χ ) ( g B + χ 2 g B ). (44) efining e iφ ( + χ )( χ ) ( + χ )( χ ), (45) the decay rate becomes 4 Γ [ B t B K t K f ] /N = e γ St K χ 2 g B + χ 2 g B 2 + e γ Lt K + χ 2 g B + + χ 2 g B 2 [ ( 2e γ S +γ L g t 2 K B χ + χ cos ( m K t K φ) 2 χ2 g B 2) +2 sin ( m K t K φ)im ( χ 2 g B + gb )]. (46) Now, using Eqs. 58 and recalling that χ is either equal to λ f or to /λ f, we notice that χ A KS f, + χ A KL f. (47) Therefore, the first term in Eq. 46 describes the decays in which the final state f has come from K S. Taking f = π + π, this is the dominant term for times t K < /γ S. Likewise, the second term refers to the decays in which the final state f has come from K L. At late times, t K /γ S, the cascade chain is dominated by these decays since by then the K S component will have decayed away. The last terms describe the interference between the path going through K S and that going through K L. These will become important at intermediate times [4, 7]. Let us consider decays of the type B d KJ/ψ. In the SM, these decays get tree level and penguin contributions which, to an excellent approximation, share a common weak phase. Moreover, in the SM one has Γ Bd and q B /p B. As a result χ 2 2 =, Reχ 2 = cos 2β, and Imχ 2 = sin 2β. Also, since Γ B, we get from Eq. 65 that Re(g+ B gb ). This is the origin of the standard result4 : the pure K S (and the pure K L ) term of Eq. 46 only measures sin 2β, leaving a fourfold ambiguity in the determination of β. Kayser has pointed out that one can use the K S K L interference on the last line of Eq. 46 to get at cos 2β, thus reducing the ambiguity [7]. 4 It should be stressed that the classic B d J/ψK S experiments involve, in fact, cascade decays, since the kaon will ultimately be detected through its decay into π + π. However, since the K L is mostly C odd, the detected π + π is very unlikely to have come from a K L (χ η K = in Eqs. 3 and 46). Moreover, the experiments will select events with low t K [23]. We see from Eq. 46 that, under these circumstances, the rate is overwhelmingly dominated by the first term, which corresponds to the pure B d J/ψK S decay [23]. + 3

14 One may also consider those decay chains where the primary decay is B d K ρ, B d K π, etc. If these primary decays are dominated by the top-mediated penguin diagram, then χ 2 measures again the weak phase β. However, although suppressed by V ub V us /(V tb V ts ) and by color, the tree level diagram comes in with a different weak phase. The resulting hadronic uncertainties imply that these decay chains cannot compete with B d KJ/ψ in the determination of the CKM phase β. We now turn to decays of the type B s K J/ψ. The considerations made about the B d KJ/ψ case also apply here, with one exception. Here, the SM tree level and penguin contributions have a large weak phase difference, which turns out to be equal to β. In fact, the decay amplitudes may be parametrized as 5 A B s K J/ψ = V cbv cd T + V tbv td λ 3 T + λ 3 e iβ, A B s K J/ψ = V cbv cd T + V tbv td λ3 T + λ 3 e +iβ. (48) The symbols T and stand for the tree level and penguin diagram contributions, with the CKM factor taken out; they include form-factors, strong phases, the gauge couplings (weak and strong, respectively), and also includes the usual one-loop prefactor. Since the penguin diagram is not CKM suppressed with respect to the tree level diagram, χ 2 = λ Bs J/ψK S may differ from one by a few percent. Azimov and unietz have pointed out that one may use these decay chains to get at the elusive m Bs mass difference [8]. It is well known that m Bs is difficult to determine from time-dependent measurements because of the vertexing limits. These preclude measurements of x s much larger than about 2. The situation is also bleak if we integrate the decays over t B because the effects of x s show up in the rate into a flavour-specific final state as /( + x 2 ). This should be compared to what one gets by integrating Eq. 46 over t B. As shown in the appendix, the result may be gotten in a straightforward way with the substitutions g+ B 2 G B +, gb 2 G B, and gb + gb GB +. Now, Azimov and unietz point out that, upon integration over t B, the last term of Eq. 46 contains a term proportional to Reχ 2 ImG B +. And, the function ImGB + shown in Eq. 67 involves x/(+x 2 ) rather than /(+x 2 ) [8]. This increases the sensitivity to x s considerably. Moreover, this term involves Reχ 2 which is nonzero even in the limit of C conservation [8]. To appreciate this effect, we take x s, y s, and we assume that there is C conservation. In that case, there will be no phase differences in Eqs. 48 and Eq. 29 reads χ 2 = η X η Bs η K =. Under these assumptions, we find 4Γ Bs q K 2 Γ [ B A B s K 2 s t B K t K f ] dt B = e γ St K A KS f 2 ( y s ) + e γ Lt K A KL f 2 ( + y s ) 2e γ S +γ L 2 t K A KS fa KL f sin ( m K t K φ) x s. (49) Therefore, as pointed out by Azimov and unietz, the interference term is sensitive to /x s [8]. 5 In these expressions, we have used the standard phase convention for the CKM matrix elements, and we have ignored the C transformation phases of B s and K. However, to be consistent, we have kept the η X = factor obtained in getting from the first to the second line [2]. 4

15 B d q p B B B g + B g - B B d d ε ε X X X - i p 2 q x τ i q - 2 p x τ X ε - + ( K π ) X Figure 4: The B d X (K π + ) X cascade decay. We now look at the cascade decays with B s K π, B s K ρ, etc. If these decays are dominated by the tree-level diagram, then the phase of χ 2 measures the CKM phase γ. However, the penguin diagram comes in with a different weak phase and, moreover, the tree level diagram is suppressed by color and by V ub V ud /(V tb V td ). The tree level and the penguin diagrams could come into χ 2 in commensurate proportions, with correspondingly large hadronic uncertainties. As a result, these decays may be of limited use. Of course, if we were to assume that there is C conservation, then the phase of χ 2 would become well defined. But, in this case, such an assumption is not warranted for it would just hide the large theoretical errors that do exist in the determination of the phase of χ Cascade decays B Let us consider decays of the type Bd {, }π, B d {, }ρ, etc. Here, the decay Bd X is suppressed with respect to the decay Bd X by ǫ = V ub V cd /(V cb V ud ). One might be tempted to conclude that everything follows as for the B + {, }X + decay chains, and, in particular, that one might also use these decays to get at x. This is not the case, because of the Bd B d mixing. In order to understand this, one should compare Fig. 2 and Fig. 4. The crucial point about Fig. 2 is that the two paths which exist in the absence of mixing are both suppressed: one by ǫ, the other by ǫ. This is what makes the x corrections important; they are to be compared with ǫ and ǫ, and not with [9]. On the contrary, in the B d X decays of Fig. 4 there are four paths in the absence of mixing. Two of the paths correspond to the unmixed Bd B d time evolution. One path is suppressed by ǫ, the other by ǫ. These paths are analogous to the ones in the B + X + decay chain. However, in Fig. 4 there are also two paths corresponding to the mixed evolution of Bd into Bd. One of these is suppressed by ǫ ǫ, but the other, given schematically by B d B d X X (K π + ) X, (5) is not suppressed at all. Therefore, in B d X decays, the x corrections are to be compared with, not with ǫ or ǫ. For x 2, this correction amounts to about %, and may be safely neglected. There are, however, two caveats to the analysis presented. First, if one is using the B d X decays to look for precision measurements, such as effects proportional to ǫ 5

16 or ǫ, then one must indeed take the x corrections into account. Second, things would change if one could select events with t B / m Bd. These times are too short for the Bd to mix into B d and, for them, the B d decay chain mimics that of a B+. We now turn to decays of the type B s X, where X is a meson whose quark content contains a s s piece. Examples include φ, η, etc. The analysis of these systems is exactly the same as for the previous case, except that the suppression of ǫ is substituted here by a rather mild suppression given by V ub V cs /(V cb V us ) = R b. We have used V ub = R b λ 3, with R b ranging from /3 to /2 to account for the fact that V ub is in fact closer to λ 3 /2 or even λ 3 /3. ecays of this type have been discussed by Gronau and London [6]. As before, the presence of mixing (here the Bs B s mixing) means that the cascade decays based upon B s φ are much less affected by a large value of x than the cascade decays started by B + X The no-oscillation requirement While we may obtain values of x 2 in many models of new physics, the value of x in the SM is much smaller than that. One might wonder what happens to the B cascade decays if we keep with the SM and take x y. In this case there is no oscillation in the system: g+ = and g =. The decay amplitude in Eq. 22 becomes A [ B t B t [f] ] = AB into f [ g B + + λ B into f g B ]. (5) In writing this expression we have used Eq. 26 and noticed that Eq. 24 implies A B into f into f B = A B A f + A B A f. (52) Therefore, in the absence of mixing, the decay chain B X [f] X reduces to the direct decay of an initial B into the final state into f X. This is as expected. Indeed, we know that into f is the combination of and that decays into the final state [f] at time t. But, since there is no oscillation, that is also the state that one must overlap with the state obtained from the decay of the B meson into the system (at time t = ). Typically, f is taken to be a flavour specific final state, which picks up either or. In fact, we see from Eq. 24 that, in this case, into f, or into f. Alternatively, f is taken to be a C eigenstate, f cp, with eigenvalue e iα f = ηf = ±. In this context, it is usually assumed that there is no C violation in the decays of neutral mesons into the final state f cp. We can use Eqs. 6 and 24 to show that, under these conditions, C into fcp = η f into fcp, as expected. This no-oscillation requirement is the one behind the B {,, into fcp }X triangle construction in the Gronau-London-Wyler method to measure the CKM phase γ [6, 7]. 5.4 Cascade decays B into f K into f [f] K. Strictly speaking, in the B K decay chains we must follow three time dependences separately; those of the evolution in the B B,, and K K systems. 6

17 However, if we continue to assume that there is no evolution in the system, as we have done in Eq. 5, then these decay chains reduce to B into f K into f [f K ]. (53) This is effectively a B KX [f] K X decay chain of the type described by Eqs. 44 and 46. There are two cases to be considered. For B d K decay chains we need the amplitude of Bd K, which is proportional to Vub V cs, and the amplitude of Bd K, which is proportional to VcbV us. These two amplitudes correspond to color-suppressed diagrams and differ by about R b. They come into the amplitude of Bd into f K in the proportions given by Eq. 52, which depend on the final state f into which the combination decays. We find, A B d into f K A B d into f K = A Bd K A f + A B d K A f λ 3 T A f + R b λ 3 e iγ TA f, = A B A d K f + A B A d K f R b λ 3 e iγ TA f + λ 3 T A f, (54) where T and T contain the relevant form factors and strong phases. Gronau and London [6] have noticed that the two amplitudes involved in these decays have roughly the same magnitude. Using this fact they proposed a triangle construction based on the comparison between A B, A d K B d K, A Bd into fcpk, and the magnitudes of the C conjugated decays. This allows the extraction of the CKM phase γ, up to discrete ambiguities. Gronau and London [6] also noted that the interference Cviolating terms in these decays allow the extraction of the unusual combination of CKM phases 2β + γ. The two amplitudes involved in B s K decays are those of Bs K, proportional to Vub V cd, and Bs K, proportional to Vcb V ud. The former is CKM suppressed with respect to the latter by roughly a factor of a few times 2. This implies that triangles similar to those in the Gronau London construction [6] would be extremely narrow, thus precluding a good measurement of γ in this case. In fact, using Eq. 52 we get A B s into f K A B s into f K = A B s K A f + A B s K A f λ 2 T A f R b λ 4 e iγ TA f, = A B s K A f + A B s K A f R b λ 4 e iγ TA f + λ 2 T A f. (55) However, this strong hierarchy has its advantages too. Let us consider f = f cp and assume that there is no C violation in the decays of neutral mesons into the final state f cp. In this case Eq. 2 holds: A M f = A M f. By comparing Eqs. 55 with Eqs. 48 we conclude that the proposal to measure m Bs put forth by Azimov and unietz [8], can be replicated by using the cascade decay B s into f K into f [f K ] 7

18 q - i 2 p x τ λ 2 λ 2 K K π π g Ḵ K π - + (X l ν ) l π K Figure 5: The cascade decay Kπ [X l + ν l ] K π. instead. We recall that Eqs. 48 imply that χ 2 may differ from by about /T. Here this problem is mitigated by the fact that the second amplitude in the first Eq. 55 is suppressed by Cascade decays K Let us now consider decays of the type {K, K }X [f] K X. These decays do not look promising. We discuss them briefly, for the sake of completeness. The decays involve the CKM matrix elements of the first two families. In the standard phase convention for the CKM matrix, these matrix elements are very approximately real. Moreover, since δ K p K 2 q K 2 is known experimentally to be very small, the only rephasing-invariant quantities in {K, K }X [f] K X decays which have non-negligible weak phases are those involving q /p = e 2iθ, and this only if the new physics effects turn out to make θ large. Since the large weak phases (β and γ) are not involved, and since the kaon system is well tested experimentally, these decays can only be used to obtain the mixing parameters in the system. However, due to the fact that the mixing of the meson is very small (we continue to take x 2 as our reference value) this mixing can only be relevant if the amplitudes of the other paths available to get to the final state are suppressed in some way (as they are in the decay chain shown in Figs. 2 and 3). This has two important consequences. First, one of the amplitudes, K X or K X, must be suppressed relatively to the other. The decays K π and K π are of this kind, since A K π λ2 A K π. (56) Second, the final state [f] K cannot be a C eigenstate, since one of the K [f] K decay amplitudes must also be suppressed relatively to the other (as in Fig. 2). This second requirement is clearly satisfied by the semileptonic decays of the neutral kaons, shown in Fig. 5. In that case A K f =. However, the large mixing in the K K system will make it possible, even then, to get to the final state (X l + ν l ) with no suppression at all. This effect is similar to the one described in section 5.3 for the mixing of the B d. So, to measure the mixing parameters in the system with this kind of decay, we will also need to select events with very small t K. The idea is that, for very short times t K the neutral kaons have not yet mixed appreciably, and one might probe effects which are due mainly to 8

19 the mixing in the system. But, by looking at Fig. 5 we recognize that this requires a time cut of order t K x / m K or t K x / Γ K for the x effect to be dominant. Experimentally, Γ K γ S 2 m K. Therefore, we would need to look at times of order t K x τ S, where τ S s is the lifetime of K S! This result can be checked directly in the decay rate. Eq. 57 shows the expression for the decay rate, integrated over t and expanded only up to the terms linear in x K τ K and y K τ K. Using Eq. 32, we find 6 e τ K Γ [ t K t K f ] dt + Imχ 3 x [ + y K τ K Im(χ χ 3 ) x ] 2 Imχ x 2 2 Reχ [ + x K τ K Imχ + Re(χ χ 3 ) x ] 2 Reχ 2x 2 We have expanded in x and taken y =. If we take the (mathematical) limit τ K =, then we get a decay of the type already studied by Wolfenstein [8], with a term proportional to Imχ 3 x. However, by developing Eq. 57 up to order τ 2 K and taking χ 3 = χ χ 2, we find that the terms proportional to x are only dominant up to times of order t K x Reχ 2 / m K, or t K x Imχ 2 / Γ K. (Notice that Imχ 2 can only be different from zero if the mixing phase is also different from zero [8], or if there is some strong phase involved [9, 2].) This confirms the result that we had guessed just by looking at Fig. 5. We conclude that, for times t K larger than about x τ S, the effect of x will be at most of order %. (57) 6 Conclusions We have developed the formalism needed to study the most general cascade decays of the type M + f +, which involve two neutral meson systems, and M, in succession. The resulting decay rates exhibit the usual sources of C violation: CV in the mixing of neutral meson systems, probed by q/p ; CV present directly in the decays, detected by A i f A ī f ; and the CV in the interference between the mixing in the initial neutral meson system, i, and the decay from that system into the final state f, measured by arg λ i f + arg λ i f (which is proportional to Imλ i f when f is a C eigenstate). But, when both the M + and M + decays are allowed, we find a new class of rephasing-invariant parameters, ξ i, that measure the interference between the mixing in the M M system and the decay from the initial state ( and ) into that system. We have applied this formalism to a variety of cases. The main results are the following. The proposal by Meca and Silva [9] to detect new physics in x through the decays B + {, }K + [f] K + may be extended to the B + {, }π + 6 It is instructive to derive this formula in a different way. Neglecting the weak phases of the first two families, χ 3 = χ χ 2 and the {K, K }X [f] K X decay rate may be written in a form similar to that of the B K decay rate shown in Eq. 46. It is straightforward to use this to rederive Eq

20 and B + {, }ρ + decays, which have much larger branching ratios. As we saw in Eqs. 4, there are two interference effects that probe x. If we neglect the strong phase differences, one is proportional to a possible C-violating phase in the mixing due to new physics. This effect might be easier to detect with the new decays proposed here than it was in the original Meca and Silva proposal. The other interference effect exists even in the absence of new C-violating phases in mixing, and it involves the CKM phase γ. When the branching ratios and suppression factors are taken into account, this effect comes into the new decays at roughly the same level as it does in B + {, }K + decays. Next, we showed that the mixing in the B d B d system implies that the same x effects are much less important in the B d {, }π and B d {, }ρ (unless one could apply a very stringent time cut on the B d decays). The same applies to B d {, }φ decays. Similarly, {K, K }π decays cannot be used to get a handle on x unless one were able to perform a very stringent time cut on t K. On the other hand, if we neglect the mixing, any B K [f] [f] K decay chain (which would normally depend on three time variables, t B, t, and t K ) may be analyzed as a decay chain of the type B into f K into f [f] K. These chains can also be studied with the formalism developed in this article. The most interesting case is B s into fcp K. In principle, this could be used to get at m Bs in the same way as proposed by Azimov and unietz with the B s J/ψK J/ψ[f] K decay chains. We thank C. Meca for discussions on related subjects, and L. Lavoura for this and for reading and criticizing the manuscript. We are indebted to B. Kayser for the seminars on cascade decays that he gave in Lisbon, and which have inspired us to study this subject. This work was partially supported by the ortuguese FCT under contract raxis/2/2./fis/223/94 and by the grant CERN//FAE/4/97. The work of J.. S. is partially supported by the grant CERN/S/FAE/64/97. The work of M. G. Santos is partially supported by the Fundação da Universidade de Lisboa, through CFNUL. A Time dependent functions involved in the decays of neutral meson systems Assuming CT invariance, the mass eigenstates of the M M system are related to the flavour eigenstates by with p M 2 + q M 2 = and M H = p M M + q M M M L = p M M q M M (58) q M = m i Γ 2 R2 = (59) p M 2R 2 R 2 2

21 where m = m H m L (H-heavy, L-light) is positive by definition, Γ = Γ H Γ L, and R 2 is the off-diagonal matrix element in the effective time evolution in the M M space. Consider a M (M ) meson which is created at time t M = and denote by M (t M ) (M (t M )) the state that it evolves into after a time t M, measured in the rest frame of the meson M. We find M (t M ) = g + (t M ) M + q M p M g (t M ) M, M (t M ) = p M q M g (t M ) M + g + (t M ) M, (6) where g ± (t) 2 ( e iµ H t ± e iµ Lt ), (6) µ H m H iγ H /2, and µ L m L iγ L /2. Similar expressions hold for the system. The following formulas are useful: g ± (t) 2 = 4 = e Γt 2 [ e Γ H t + e Γ Lt ± 2e Γt cos( m t) ] (62) [ cosh Γt 2 ± cos ( mt) ], (63) g + (t)g (t) = 4 Integrating over time, we obtain G ± G = e Γt 2 where x m/γ and y = Γ/(2Γ). [ e Γ H t e Γ Lt 2ie Γt sin( m t) ] (64) g ± (t) 2 dt = 2Γ [ sinh Γt 2 + i sin ( mt) ]. (65) g + (t)g (t)dt = 2Γ ( y ± ), (66) 2 + x 2 ( y y + ix ), (67) 2 + x 2 References [] J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, hys. Rev. Lett. 3, 38 (964). [2] N. Cabibbo, hys. Rev. Lett., 53 (963); M. Kobayashi and T. Maskawa, rog. Theor. hys. 49, 652 (973). 2

22 [3] A. B. Carter and A. I. Sanda, hys. Rev. Lett. 45, 952 (98); ibid hys. Rev. 23, 567 (98); I. I. Bigi and A. I. Sanda, Nucl. hys. B93, 85 (98); ibid B28, 4 (987). [4] Ya. I. Azimov, JET Letters 5, 447 (989); hys. Rev. 42, 375 (99). [5] G. V. ass and K. V. L. Sarma, Int. J. Mod. hys. A7, 68 (992); (E) ibid A8, 83 (993). [6] B. Kayser and L. Stodolsky, Max-lanck Institute preprint number MI-HT-96-2, in hep-ph/ [7] B. Kayser, published in the proceedings Les Arcs 997, Electroweak interactions and unified theories, pages Also in hep-ph/ [8] Ya. I. Azimov and I. unietz, hys. Lett. B 395, 334 (997). [9] C. C. Meca and J.. Silva, CFNUL report number CFNUL/98-3, in hepph/ To appear in hys. Rev. Lett. (998). [] G. C. Branco, W. Grimus and L. Lavoura, hys. Lett. B 372, 3 (996). [] I. unietz, hys. Rev. 52, 348 (995). [2] We thank Ya. I. Azimov and I. unietz for stressing the importance of this point, in connection with their article in [8]. [3] Z.-Z. Xing, hys. Lett. B 37, 3 (996). [4]. Atwood, I. unietz, and A. Soni, hys. Rev. Lett. 78, 3257 (997). [5] For a review see, for example, Y. Nir, Nuovo Cim. 9A, 99 (996). Generically, these models keep y Γ /(2Γ ) exceedingly small, and, thus, we will drop it from our analysis. [6] M. Gronau and. London, hys. Lett. B 253, 483 (99). [7] M. Gronau and. Wyler, hys. Lett. B 265, 72 (99). [8] L. Wolfenstein, hys. Rev. Lett. 75, 246 (995). See also T. Liu, Harvard University preprint number HUT-94/E2, in roceedings of the Charm 2 Workshop, FERMILAB-Conf-94/9, page 375, 994. [9] G. Blaylock, A. Seiden, and Y. Nir, hys. Lett. B 355, 555 (995). [2] T. E. Browder and S. akvasa, hys. Lett. B 383, 475 (996). [2] C. Caso et al., European hysical Journal C3, (998), and also the URL: [22] M. Athanas et al., CLEO Collaboration, CLEO preprint number CLEO [23] B. Kayser, private communication. 22