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UG–FT–69/96 hep-ph/9703410 March 1997 CP violation from new quarks in the chiral limit F. del Aguila, J. A. Aguilar–Saavedra Departamento de F´ısica Teo´rica y del Cosmos, Universidad de Granada 9 18071 Granada, Spain 9 9 1 G. C. Branco n Departamento de Fisica and CFIF/UTL, Instituto Superior T´ecnico a 1096 Lisboa, Portugal J 4 Abstract 2 We characterize CP violation in the SU(2) U(1) model due to an extra vector- 3 × like quark or sequential family, giving special emphasis to the chiral limit m =0. v u,d,s 0 In this limit, CP is conserved in the three generation Standard Model (SM), thus 1 implying that all CP violation is due to the two new CP violating phases whose 4 effects may manifest either at high energy in processes involving the new quark or as 3 deviations from SM unitarity equalities among imaginary parts of invariant quartets 0 (or, equivalently, areas of unitarity triangles). In our analysis we use an invariant 7 9 formulation, independent of the choice of weak quark basis or the phase convention / in the generalized Cabibbo-Kobayashi-Maskawamatrix. We identify the three weak- h p basis invariants, as well as the three imaginary parts of quartets B1−3 which, in the - chirallimit,givethestrengthofCPviolationbeyondtheSM.Wefindthatforanextra p vector-like quark B 10−4, whereas for an extra sequential family B 10−2. i i e | |≤ | |≤ h PACS: 11.30.Er, 12.15.Ff, 12.60.-i, 14.80.-j : v i X 1 Introduction r a In the Standard Model (SM) CP violation is parametrized by one CP violating phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1]. Although this phase accounts for the observed CP violation in the K0-K¯0 system [2], there is no deep understanding of CP violation. Furthermore, it has been established that the amount of CP violation present in the SM is not sufficient to generate the baryon asymmetry of the universe [3]. This provides motivation for looking for new sources of CP violation which can lead to deviationsfromtheSMpredictionsforCPasymmetriesinB0 decaysand/ortonewsignals of CP violation observable at high energy, in future colliders. In the SM the CKM matrix V is a 3 3 unitary matrix whose matrix elements CKM × V are strongly constrained by unitarity which implies, for example, that the imaginary ij 1 parts of all invariant quartets, Im V V∗V V∗ (i = k, j = l), are equal up to a sign. In ij kj kl il 6 6 particular, one has T Im V V∗V V∗ +Im V V∗V V∗ = 0, 1 ud cd cb ub us cs cb ub ≡ T Im V V∗V V∗ +ImV V∗V V∗ = 0, 2 ud td tb ub us ts tb ub ≡ T Im V V∗V V∗ +Im V V∗V V∗ = 0. (1) 3 cd td tb cb cs ts tb cb ≡ In the SM one may have CP violation in the limit m = 0, but degeneracy of two quarks u,d of the same charge does imply CP invariance. Hence in the chiral limit m = 0, with u,d,s d and s quark masses degenerate, CP violation can only originate in physics beyond the SM. These CP properties of the SM are summarized through a necessary and sufficient condition for CP invariance, expressed in terms of a weak quark basis invariant [4, 5] 1 I det[M M†,M M†] = tr [M M†,M M†]3 ≡ u u d d 3 u u d d = 2i(m2 m2)(m2 m2)(m2 m2)(m2 m2) t c t u c u b s − − − − − (m2 m2)(m2 m2)Im V V∗V V∗ = 0, (2) b d s d ud cd cs us × − − where M are the up and down quark mass matrices and m the mass of the quark i. u,d i Although for three generations the two invariants of Eq. (2) are proportional to each other, tr [M M†,M M†]3 = 0 has the advantage of being a nontrivial necessary condition u u d d for CP invariance in the SM, for an arbitrary number of generations [5]. Within the three generation SM, CP violation in the B system is not suppressed by the factor (m m ) due to the fact that one is able to distinguish B from B in the s d d s − initial state and kaons from pions in the final state. This flavour identification is crucial in order to detect CP violation effects arising from gauge interactions [6, 7, 8, 9]. At high energy colliders, the natural asymptotic states are no longer hadronic states, but quark jets [10]. Now, at high energies, it will be very hard, if not impossible, to identify the flavour of light quark jets (d, s, u). In this limit, CP violation can only originate in physics beyondtheSM.Indeedmany simpleSMextensions incorporatenewsourcesof CP violation which may be observable at future colliders in the chiral limit [11]. In this paper we concentrate on the case of extra quarks, with special emphasis on vector-like quarks, i. e. quarks whose left-handed and right-handed components are in the same type of multiplet. Vector-like quarks naturally arise in various extensions of the SM, for example in grand-unifiedtheories based on E , as well as inother superstringinspiredextensions of 6 the SM. In most cases, these vector-like quarks are isosinglets and their mass could be of the order of the electroweak scale [2]. A new sequential quark family is also allowed [16], although precision electroweak data put an upper limit on their square mass difference [2]. We will show in this paper that with the adition of an extra quark, either sequential 2 or vector-like, there is CP violation even in the limit where m = 0. Therefore, in u,d,s,c these simple extensions of the SM, CP violation effects can beseen even if only the flavour of heavy quarks (b, t, b′) is identified. In both cases, for an isosinglet quark and for a sequential extra family, CP violation mediated by gauge bosons is parametrized by a 4 4 × unitarymatrixV defineduptoquarkmasseigenstate phaseredefinitions. For anewdown (up) vector-like quark the charged couplings are described by the CKM matrix V , the CKM first 3 rows (columns) of V. But these 3 rows (columns) V completely fix V. The CKM neutral couplings are a function of the V matrix elements and are not independent. In CKM thecaseof anisosinglet quark, theneutralcouplingsarenolonger diagonal, i. e. thereare flavour changing neutral currents (FCNC). The3 3block of the CKMmatrix connecting × standard quarks is no longer unitary either, but deviations from unitarity are naturally suppressedbypowersofm/M, wheremisastandardquarkmassandM denotes themass of the isosinglet quark. The strength of FCNC among standard quarks is also suppressed by powers of m/M, since FCNC are proportional to deviations from 3 3 unitarity in × V . For a new sequential family V = V and the neutral couplings are diagonal and CKM CKM real. It turns out that in both cases there are three CP violating phases in V [18]. CKM In our analysis we will adopt the following strategy: First, we identify a set of weak- basis invariants which completely specify the properties of the model considered in the sense that if any one of the invariants is nonzero there is CP violation, while if all the invariants of the set vanish there is CP invariance. These weak-basis invariants are phys- ically meaningful quantities, and they are the analog of the invariant in Eq. (2) for the class of models we are considering. They can be expressed in terms of quark masses and the imaginary parts of various invariant products of CKM matrix elements. We then study the chiral limit m = 0, starting with the simpler case m = 0. Both are u,d,s u,d,s,c especially interesting since in these cases the three generation SM conserves CP, thus im- plying that in the chiral limit CP violation arises exclusively from physics beyond the SM. The chiral limit is not only physically natural for studying CP violation beyond the SM but phenomenologically relevant at high energy, where the light fermion masses are negligible. The above mentioned weak-basis invariants are especially usefulin the analysis of the chiral limit, allowing one to readily identify which Im V V∗V V∗ continue being ij kj kl il physically meaningful and nonvanishing when taking these degenerate mass limits. In a 4 4 unitary matrix 9 of these imaginary products are independent, and all of them can × be made to vanish if CP is conserved. In the chiral limit m = 0, there are two CP u,d,s violating phases and three independent imaginary products physically relevant. If m is c also neglected compared to m , m = 0, there is one CP violating phase left and one t u,d,s,c independent imaginary product physically significant. These imaginary products B1 ≡ Im VcbV4∗bV4b′Vc∗b′ = T1−T3, B2 ≡ Im VtbV4∗bV4b′Vt∗b′ = T2+T3, 3 B3 Im VcbVt∗bVtb′Vc∗b′ = T3, (3) ≡ which survive in the chiral limit and which involve mixings in the heavy quark sector (t, b, c and the new quark(s) ), can also be expressed in terms of the rephasing invariants T defined in Eqs. (1). (We will use for the subindex of the fourth row ‘4’ and ‘t′’ when i referring to the vector-like and sequential cases, respectively. When referring to both we will also use ‘4’.) These invariants T only involve mixings among standard quarks i and they vanish in the SM. Thus, the effects of physics beyond the SM may be seen measuring B at high energy in processes involving new quarks or at low energy through i the nonvanishing of T . We shall show that the present bounds on B are 10−2 and 10−4 i i | | for a fourth family and a new vector-like quark, respectively. In our analysis, we will only take into account CP violating effects arising from gauge interactions (in some specific models with isosinglet quarks the Higgs sector is more involved than in the SM, leading to new CP violating contributions from scalar interactions) and we will neglect the Higgs contributions to CP violation. It may be worth to emphasize that the invariant formulation of CP violation requires an educated use of symbolic programs [20]. However to go beyond the simplest cases is difficultforasexplainedintheAppendixthenumberofinvariantsneededtogetacomplete set grows very rapidly with the number of phases. We study the simplest cases of a new vector-like quark or an extra sequential family, deriving limits for observables involving known particles as final states. If there exist more vector-like or sequential quarks, larger CP violating effects than the ones studied here are possible but in observables involving several of these new quarks. This paper is organized as follows. In Section 2 we set our notation and for the case of an extra down (up) quark isosinglet we propose a complete set of weak-basis invariant conditions which are necessary and sufficient for CP invariance. We also give the explicit expressions of the invariants in terms of quark masses and imaginary parts of invariant products. The proof that these invariant conditions form a complete set is given in the Appendix. The corresponding set of invariants for the case of a sequential family was discussed in Ref. [19]. In Section 3 we use these invariants to study the simplest case of two degenerate (massless) up and down quark masses, m = 0, and the chiral u,d,s,c limit, m = 0. In Section 4 we discuss the corresponding geometrical description of u,d,s CP violation with triangles and quadrangles and the bounds on the CP violating effects of these new fermions commenting on the prospects to measure them (at large colliders). Section 5 is devoted to our conclusions. 4 2 Characterization of CP violation for extra quarks Let us consider the SM with N standard families plus n down quark isosinglets (the case d of n up quark isosinglets is similar). In the weak eigenstate basis the gauge couplings to u quarks and the mass terms are g = u¯(d)γµd(d) W†+h.c. Lgauge −√2 L L µ g h(cid:16) (cid:17) i u¯(d)γµu(d) d¯(d)γµd(d) 2s2 Jµ Z eJµ A , (4) −2c L L − L L − W EM µ− EM µ W (cid:16) (cid:17) = u¯(d)M u(s)+d¯(d)M d(s)+d¯(s)m d(s) +h.c., (5) Lmass − L u R L d R L d R (cid:16) (cid:17) (d) (d) (s) (s) (s) where u , d are N SU(2) doublets, d are n SU(2) singlets and u and d are L L L L d L R R N and N +n SU(2) singlets, respectively, and Jµ = 2u¯γµu 1d¯γµd. Hence, the up d L EM 3 − 3 and down quark mass matrices are M d = M , = , (6) Mu u Md md ! with M , M and m submatrices of dimension N N, N (N +n ) and n (N + u d d d d × × × n ), respectively. The weak quark basis can be transformed by unitary matrices without d changing the physics. Under these unitary transformations q(d) U q(d) , d(s) Udd(s) , q(s) Uqq(s), (7) L → L L L → L L R → R R with q = u,d, the mass matrices transform as M U M Uq† , m Udm Ud†, (8) q → L q R d → L d R whereas the gauge couplings remain unchanged. Then the mass matrices are defined up to the unitary transformations in Eq. (8). Asetofphysicalparameterscanbedefinedusingthemasseigenstatebasis. Weassume without loss of generality = diagonal and = V V†, with V unitary and u u d d d M D M D D diagonal (remember that we can always assume and hermitian with nonnegative u d q M M eigenvalues by choosing U appropriately). We define the N (N +n ) matrix V as R × d CKM the firstN rows of the (N+n ) (N+n ) unitary matrix V, and the (N+n ) (N+n ) d d d d × × † matrix X V V . Then the Lagrangian in Eqs. (4,5) reads in the quark mass ≡ CKM CKM eigenstate basis (where no superscripts are needed) g = u¯ γµV d W†+h.c Lgauge −√2 L CKM L µ g (cid:16) (cid:17) u¯ γµu d¯ γµXd 2s2 Jµ Z eJµ A , (9) −2c L L− L L− W EM µ− EM µ W = u¯ (cid:16) u +d¯ d +h.c.. (cid:17) (10) mass L u R L d R L − D D (cid:0) (cid:1) 5 In this basis one can make the counting of CP violating phases. This is equal to the number of phases in V minus the number of independentphase field redefinitions [14], CKM N(N 1) n = N(N +n ) − (2N +n 1) CP d d − 2 − − 1 = (N 1)n + (N 1)(N 2). (11) d − 2 − − CP is conserved if V can be made real. This is the case if all n phases vanish. CKM CP In the SM, N = 3, n = 0, there is only 1 CP violating phase. For one extra quark, d N = 3, n = 1, there are already 3 CP violating phases, the same as for an extra family, d N = 4, n = 0. The number of physical parameters, and in particular of CP violating d phases, grows rapidly with the addition of more quarks. We will stick to these cases which incorporate many of the new features of the addition of new quark fields. For an extra down quark isosinglet, V consists of the first 3 rows of a 4 4 unitary matrix which CKM × can be parametrized as (we explicit it for later use) [18] c1 s1c3 s1s3c5 s1s3s5 −s1c2 c1c2c3+s2s3c6eiδ1 c1c2s3c5−s2c3c5c6eiδ1 c1c2s3s5−s2c3s5c6eiδ1  +s2s5s6ei(δ1+δ3) −s2c5s6ei(δ1+δ3)  −s1s2c4 c1s2c3c4−c2s3c4c6eiδ1 c1s2s3c4c5+c2c3c4c5c6eiδ1 c1s2s3c4s5+c2c3c4s5c6eiδ1  −s3s4s6eıδ2 −c2c4s5s6ei(δ1+δ3) +c2c4c5s6ei(δ1+δ3)  V =  +c3s4c5s6eiδ2 +c3s4s5s6eiδ2  .  +s4s5c6ei(δ2+δ3) −s4c5c6ei(δ2+δ3)   −s1s2s4 c1s2c3s4−c2s3s4c6eiδ1 c1s2s3s4c5+c2c3s4c5c6eiδ1 c1s2s3s4s5+c2c3s4s5c6eiδ1   +s3c4s6eiδ2 −c−2sc43sc54sc65esi6(eδ1iδ+2δ3) +c−2sc43cc54ss65esi6(eδ1iδ+2δ3)   −c4s5c6ei(δ2+δ3) +c4c5c6ei(δ2+δ3)   (12) Note that the first 3 rows completely fix V. On the other hand, V is the CKM matrix for 4 families. In the limit where the new quark does not mix with the standard quarks (i. e. s = s = s = 0), the 3 3 block of V becomes just the standard CKM matrix 4 5 6 × with only one CP violating phase δ . The CP properties of the model with one isosinglet 1 quark can be most conveniently studied by using weak-basis invariants. In the Appendix we present the general treatment and provide the proof that the vanishing of the following set of invariants is necessary and sufficient to have CP invariance: † I = Im tr H H h h , 1 u d d d I = Im tr H2H h h†, 2 u d d d I = Im tr (H3H h h† H2H H h h†), 3 u d d d− u d u d d I = Im tr H H2h h†, 4 u d d d I = Im tr H2H2h h†, 5 u d d d 6 I = Im tr (H3H2h h† H2H2H h h†), 6 u d d d− u d u d d I = Im tr H2H H H2, (13) 7 u d u d with H = M M†, H = M M†, h = M m† (see Eq. (6) ). These invariants, which u u u d d d d d d obviously do not depend on the choice of weak quark basis (see Eq. (8) ), can be written in the quark mass eigenstate basis (in the equations below Greek indices run from 1 to 4, Latin indices run from 1 to 3 and a sum over all indices is implicit; m is the mass of the i up quark i and n the mass of the down quark α) α I = m2n2n4 Im V X V∗ , 1 i α β iα αβ iβ I = m4n2n4 Im V X V∗ , 2 i α β iα αβ iβ I = m6n2n4 Im V X V∗ 3 i α β iα αβ iβ m4m2n2n4 Im V V∗ V V∗ i j α β iα jα jβ iβ − +m4m2n2n2n2 Im V V∗V X V∗ , i j α β ρ iρ jρ jα αβ iβ I = m2n2n4n2 Im V X X V∗ , 4 i α β ρ iρ ρα αβ iβ I = m4n2n4n2 Im V X X V∗ , 5 i α β ρ iρ ρα αβ iβ I = m6n2n4n2 Im V X X V∗ 6 i α β ρ iρ ρα αβ iβ m4m2n2n2n4 Im V X V∗V V∗, i j α β ρ iα αβ jβ jρ iρ − I = m4m2n2n2n2 Im V V∗V X V∗ . (14) 7 i j α β ρ iρ jρ jα αβ iβ Notice that X = V∗V , which implies X = X† and X X = X (note however that αβ iα iβ αβ βγ αγ in Eqs. (14) the sums include also mass factors). The imaginary parts involve invariant quartets and invariant sextets, which can be reduced also to products of moduli squared of V elements times imaginary parts of quartets. In the chiral limits there only appear ij imaginary parts of invariant quartets. For the case of four sequential families, the cor- responding set of necessary and sufficient conditions for CP invariance consists of eight invariants which have been given in Ref. [19]. In both cases these invariant conditions completely characterize theCP properties of themodel. If any of theinvariants is nonvan- ishing,thereisCPviolation andthevanishingoftheinvariantsimpliesCPinvariance. The description of the CP properties of a model through invariants is especially useful when consideringlimitingcases wheresomeofthequarkmassescan beconsidered asdegenerate (massless). Theinvariant approach clearly identifies which ones of theIm V V∗V V∗ can ij kj kl il benonvanishingin thevarious limiting cases one considers. Thedifficulttask is, of course, finding a complete set of invariants, but once this is accomplished, the invariant approach is avery convenient methodtodescribeCPviolation. InthenextSection wewillillustrate the usefulness of these invariant conditions, by considering some appropriate chiral limits. 7 3 CP violation in the chiral limit In the chiral limit, m = 0, CP is conserved within the SM. Hence all CP violating u,d,s effects are dueto newphysics. Sizeable CPviolation at high energy is expected to have its origin beyondtheSM.Letusdiscussinturnthesimplestlimitofm m 0, m = 0 t c u,d,s ≫ ∼ and the chiral limit m = 0. We will consider the cases of an extra isosinglet quark and u,d,s of a fourth sequential family. 3.1 m = 0 limit u,d,s,c In this limit there is only one CP violating phase for an extra down quark isosinglet b′ or for a fourth family b′, t′. The best way to study this limit is to substitute m = 0 in u,d,s,c the complete set of invariants characterizing CP. The 7 invariants in Eqs. (13) reduce to I1 = m2tm2b′m2b(m2b′ m2b)Im VtbXbb′Vt∗b′, − I = m2I , 2 t 1 I = m4I , 3 t 1 I4 = (m2b′Xb′b′ +m2bXbb)I1, I = m2I , 5 t 4 I = m4I , 6 t 4 I = 0. (15) 7 It is clear that CP is conserved if and only if ImVtbXbb′Vt∗b′ = 0. Obviously, we have made theassumptionthatb, b′ arenondegenerate, withmb′ > mb. HenceallCPviolating effects are proportional to this imaginary productwhich gives the size of CP violation. Similarly, for 4 families the corresponding 8 invariants [19] reduce to (we use a prime to distinguish them) I1′ = −m2t′m2t(m2t′ −m2t)m2b‘m2b(m2b′ −m2b)Im VtbVt∗′bVt′b′Vt∗b′, I2′ = (m2t′ +m2t)I1′ , I3′ = (m4t′ +m4t)I1′ , I4′ = (m6t′ +m6t)I1′ , I5′ = (m2b′ +m2b)I1′ , I6′ = (m2b′ +m2b)I2′ , I7′ = (m4b′ +m4b)I1′ , I8′ = (m6b′ +m6b)I1′ . (16) 8 In this case CP conservation reduces to requiring Im VtbVt∗′bVt′b′Vt∗b′ = 0, with the implicit assumptionmb′ > mbandmt′ > mt. Notonlythesamecommentsasforanextraisosinglet apply but the observable for SM final states is the same. Thus using unitarity B2 = ImVtbV4∗bV4b′Vt∗b′ = −Im VtbXbb′Vt∗b′ =−Im VtbVi∗bVib′Vt∗b′. (17) Itisclear thatB measuresthestrengthof CPviolation inhighenergyprocessesinvolving 2 the new quark. On the other hand due to the unitarity constraints, B is actually related 2 to the invariants T defined in Eqs. (1), which only depend on standard quark mixings. i Indeed, one obtains B2 = Im VtbVi∗bVib′Vt∗b′ = Im VtbVu∗bVub′Vt∗b′ Im VtbVc∗bVcb′Vt∗b′ − − − = Im V V∗V V∗ +ImV V∗V V∗ tb ub ud td tb ub us ts +Im V V∗V V∗ +Im V V∗V V∗ = T +T , (18) tb cb cd td tb cb cs ts 2 3 with T = 0 in the three generation SM as emphasized in the Introduction. 2,3 All this can also be proven using the CKM matrix, although the physics is less trans- parent. If m = 0, m = 0, the general 4 4 unitary matrix in Eq. (12) (up to quark u,c d,s × field phase redefinitions) can be written c s c s s 0 1 1 3 1 3 s c c c c +s s c eiδ1 c c s s c c eiδ1 s s eiδ1 V =  − 1 2 1 2 3 2 3 6 1 2 3− 2 3 6 − 2 6  , (19) s s c s c c s c eiδ1 c s s +c c c eiδ1 c s eiδ1 1 2 1 2 3 2 3 6 1 2 3 2 3 6 2 6  − −   0 s3s6 c3s6 c6   −    where we have used the freedom to make unitary transformations in the (u,c) and (d,s) spaces. This freedom results of course from the fact that (u,c) and (d,s) are degenerate in the limit we are considering. As expected, in this limit there is only one CP violating phase and the strength of CP violation is given by B2 = ImVtbV4∗bV4b′Vt∗b′ =c1c2s2c3s3c6s26sinδ1. (20) Unitarity allows to recover Eqs. (17,18). In this particular parametrization B is in fact 2 equal to T for T = 0 as defined in Eqs. (1). 3 1,2 3.2 The chiral limit, m = 0 u,d,s In this limit there is no CP violation in the SM, but for one extra quark isosinglet or a fourth sequential family there are two new CP violating phases which remain physical. In 9 the case of the model with an extra down quark isosinglet, CP conservation is equivalent to the vanishing of I in Eqs. (13), because in this limit 1−3 I = m2I +m2I , 1 t t c c I = m4I +m4I , 2 t t c c I3 = m6tIt+m6cIc+m2tm2c(m2t m2c)m2b′m2b(m2b′ m2b)Im VcbVt∗bVtb′Vc∗b′ +I7, − − I4 = (m2b′Xb′b′ +m2bXbb)I1, I5 = (m2b′Xb′b′ +m2bXbb)I2, I6 = (m2b′Xb′b′ +m2bXbb)(m6tIt+m6cIc) m2tm2c(m2t m2c)m2b′m2b[m2b′m2b(Xb′b′ Xbb)Im VcbVt∗bVtb′Vc∗b′ − − − +(m4b′ Vcb′ 2 m4b Vcb 2)Im VtbXbb′Vt∗b′ | | − | | (m4b′ Vtb′ 2 m4b Vtb 2)Im VcbXbb′Vc∗b′], − | | − | | I7 = m2tm2c(m2t m2c)m2b′m2b[(m2b′Xb′b′ m2bXbb)Im VcbVt∗bVtb′Vc∗b′ − − − +(m2b′ Vcb′ 2 m2b Vcb 2)Im VtbXbb′Vt∗b′ | | − | | (m2b′ Vtb′ 2 m2b Vtb 2)Im VcbXbb′Vc∗b′], (21) − | | − | | with Ic = m2cm2b′m2b(m2b′ m2b)Im VcbXbb′Vc∗b′, − It = m2tm2b′m2b(m2b′ m2b)Im VtbXbb′Vt∗b′. (22) − There are only three independent imaginary products, Im VtbXbb′Vt∗b′, Im VcbXbb′Vc∗b′ and Im VcbVt∗bVtb′Vc∗b′, entering in I1−7. Their vanishing guarantees CP conservation. The first one is the only one which survives for m = m as proven in the previous Subsection. c u Analogously, in the case of a fourth family CP invariance is equivalent to the vanishing of I′ in Ref. [19]. In this case the complete set of 8 invariants reduces to 1−3 I1′ = Ic′t+Ic′t′ +It′t′, I2′ = (m2t +m2c)Ic′t+(m2t′ +m2c)Ic′t′ +(m2t′ +m2t)It′t′, I3′ = (m4t +m4c)Ic′t+(m4t′ +m4c)Ic′t′ +(m4t′ +m4t)It′t′, I4′ = (m6t +m6c)Ic′t+(m6t′ +m6c)Ic′t′ +(m6t′ +m6t)It′t′, m2t′m2tm2c(m2t m2c)(m2t′ m2c)(m2t′ m2t)m2b′m2b − − − − [(Vt′b′ 2m2b′ Vt′b 2m2b)Im VcbVt∗bVtb′Vc∗b′ × | | −| | (Vtb′ 2m2b′ Vtb 2m2b)Im VcbVt∗′bVt′b′Vc∗b′ − | | −| | +(Vcb′ 2m2b′ Vcb 2m2b)Im VtbVt∗′bVt′b′Vt∗b′], | | −| | I5′ = (m2b′ +m2b)I1′ , 10

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