CP violation in decays

IFT-2004/30 IPPP/04/85 DCPT/04/170 arXiv:hep-ph/0412253v2 19 Dec 2004 hep-ph/0412253 August 22, 2018 CP violation in Bd0 → τ +τ − de ays Piotr H. Chankowski1 , Jan Kalinowski1,2, Zbigniew Was3 3,4 and Maªgorzata Worek 1 Institute of Theoreti al Physi s, Warsaw University, Ho»a 69, 00-681, Warsaw, Poland 2 IPPP, University of Durham, South Road, Durham DH1 3LE, UK 3 Institute of Nu lear Physi s PAS, Radzikowskiego 152, 31-3420, Cra ow, Poland 4 Institute of Nu lear Physi s, NCSR "Demokritos, 15310 Athens, Gree e Abstra t B 0 (B̄ 0 ) → l+ l− de ays requires a measurement of polariza0 + − 0 + − tion of the nal lepton pair, or a pre ise determination of the B → l l and B̄ → l l Establishing CP violation in rates. We rst argue that if the amplitudes of these de ays are dominated by the s alar and pseudos alar Higgs penguin diagrams, as happens e.g. in supersymmetry with large tan β , the CP asymmetries depend pra ti ally on only one CP violating phase. This phase an be large, of the order of the CKM phase, leading to large CP asymmetries in + − 0 0 the τ τ de ay hannel of Bd (B̄d ) mesons, potentially measurable in BELLE or experiments. Se ondly, we show that the existing mented by its universal interfa e TAUOLA τ -lepton BABAR de ay library supple0 0 + − an e iently be used to sear h for B (B̄ ) → τ τ de ays, and to investigate how the CP asymmetry is ree ted in realisti observables. experimental 1 Introdu tion 40 years after its dis overy in the Kaon system, CP violation has also been rmly established in B -physi s in a series of high statisti s experiments whi h enabled su iently pre ise measurements of the relevant observables. While the reported small dieren e between the time dependent CP asymmetries measured in B → J/ψKS and in B → φKS de ays [1℄, if onrmed, would oni t with the Standard Model (SM) predi tion, there is as yet no onvin ing signal of a ontribution to CP violation from physi s beyond the SM. Su h a signal may eventually be provided by yet more pre ise measurements and joint analysis of CP asymmetries measured in dierent hannels. Equally important is to look for CP violation in hannels in whi h no (or negligibly small) ee ts violating CP are predi ted by the SM. Observation of a non-zero ee t in su h pro esses would be an unambiguous signal of new physi s ontribution to CP violation. From many possible hannels1 we onsider in this letter avour hanging de ays of 0 the neutral B mesons into lepton pairs, Bd,s → l+ l− . This hannel has attra ted a lot of attention sin e it is very sensitive to new physi s whi h ae ts the b-quark Yukawa ouplings [25℄. Approximate and full one-loop al ulations in the supersymmetri extension of the SM with large tan β (the ratio of the va uum expe tation values of the two Higgs doublets) [46℄ showed that in su h a s enario one an expe t truly spe ta ular 0 enhan ement of the rates of the de ays Bd,s → l+ l− . Moreover, new physi s an also lead to observable CP violation in these de ays, whi h is not predi ted by the SM. For example, CP violation ould manifest itself through non-equal leptoni de ay rates of the tagged B 0 (t) and B̄ 0 (t) states (B 0 (t) and B̄ 0 (t) are the states whi h at t = 0 are tagged as B 0 and B̄ 0 , respe tively). If polarization of nal state leptons an be determined, additional information on CP violation ould be + − + − lR ) [or Γ(B 0 (t) → lR lR ) and provided by non-equal Γ(B 0 (t) → lL+ lL− ) and Γ(B̄ 0 (t) → lR + − 0 Γ(B̄ (t) → lL lL )℄ de ay rates [7, 8℄. Theoreti ally leptoni de ays are parti ularly lean as the only nonperturbative quantities, on whi h their rates depend, are the B 0 meson de ay onstants FBd,s . Moreover, FBd,s an el out in suitably dened CP asymmetries. On the other hand it is not lear whether the CP violation in these hannels an be a essible experimentally. First of all these de ays have not yet been seen: the best upper limit on the Bd0 → µ+ µ− bran hing fra tion omes at present from the BABAR experiment [9℄: Br(Bd0 → µ+ µ− ) < 8.3 × 10−8 , (1) Br(Bs0 → µ+ µ− ) < 2.7 × 10−7 (2) whi h improves on previous limits of BELLE (1.6 × 10−7 [10℄) and CDF (1.5 × 10−7 [11℄). The Bs0 → µ− µ+ bran hing fra tion is bounded by [12℄ resulting from the ombination of the CDF (5.8 × 10−7 [11℄) and D0 (4.1 × 10−7 [13℄) (all limits are at 90% CL). The limits (1) and (2) are still about 2-3 orders of magnitude 1 For example, also very small CP violation is predi ted by the SM in the 1 harm se tor. above the orresponding predi tions of the SM: Br(Bd0 → µ+ µ− ) ≈ 1.3 × 10−10 and Br(Bs0 → µ+ µ− ) ≈ 3.6 × 10−9 [1416℄. The main un ertainties of these predi tions ome from the de ay onstants FBs and FBd whi h are known up to a pre ision of ∼ 15% [17℄. 0 If the de ays Bd,s → µ+ µ− o ur at the rates as predi ted by the SM, their dete tion will be ome possible only at the LHC. New physi s (like supersymmetry) an in rease signi antly their rates to a level that they an be observed at BABAR, BELLE or Tevatron in near future. However, as we will show, in this ase the ratio of time integrated leptoni de ay rates of the B 0 (t) and B̄ 0 (t) states to µ+ µ− is unlikely to deviate appre iably from unity. Polarization measurement also seems very di ult in the ase of the µ+ µ− hannel [18℄. In the τ + τ − hannel the situation is quite dierent: large CP violating ee ts an be expe ted, and τ polarization measurement is possible. So far this hannel has not been mu h explored experimentally on a ount of di ulties with identifying τ 0 leptons. As a result, pra ti ally no experimental limits on Br(Bd,s → τ + τ − ) exist despite the fa t that the orresponding rates are expe ted to be a fa tor m2τ /m2µ ∼ 300 larger than the ones for de ays into µ+ µ− nal states. The purpose of this paper is twofold. Firstly, we point out that in realisti s enarios of large tan β MSSM, where the rates of B 0 (B̄ 0 ) → l+ l− de ays are signi antly in reased to a level measurable at the running BABAR and BELLE experiments, the CP asymmetry in the Bd0 (B̄d0 ) → τ + τ − hannel an be quite large and potentially measurable. Se ondly, we identify realisti experimental observables in whi h the CP asymmetry an be dete ted. In addition to the ratio of time integrated leptoni Bd0 (t) and B̄d0 (t) already mentioned, we onsider two omplementary observables: the π ± energies from τ → πν de ays, and the a oplanarity angle between the de ay planes of the ρ mesons whi h originate from τ → ρν . The former is sensitive to the longitudinal, while the latter to the transverse polarizations of τ 's oming from Bd0 and B̄d0 de ays. We show how the existing TAUOLA pa kage [1921℄ and its universal interfa e [2224℄ an be used to sear h for these de ays and for the CP violation, demonstrating that the ne essary tools for full Monte Carlo simulations are reliable and ready for use in the Bd0 (B̄d0 ) → τ + τ − physi s. 2 General formulae The ee tive Lagrangian des ribing Bd0 (B̄d0 ) → l+ l− de ays an su in tly be written as 0 0 Leff = Bs,d ψ̄l (bs,d + as,d γ 5 )ψl + B̄s,d ψ̄l (b̄s,d + ās,d γ 5 )ψl . (3) To simplify the notation the subs ripts d and s referring to non-strange and strange B 0 mesons, unless expli itly written, will be omitted. Hermiti ity (CPT invarian e) implies b̄ = b∗ , ā = −a∗ . (4) The amplitudes of B 0 de ays into two heli ity eigenstates read 0 AL ≡ hlL+ lL− |Bs,d i = MB (a + b β) , + − 0 AR ≡ hlR lR |Bs,d i = MB (a − b β) , 2 (5) where β = (1 − 4m2l /MB2 )1/2 . Similar formulae with a and b repla ed by ā and b̄, respe tively, give the amplitudes ĀL and ĀR for the orresponding B̄ 0 de ays. Evidently, if both oe ients a and b are simultaneously nonzero, Γ(B 0 → lL+ lL− ) 6= Γ(B 0 → lR+ lR− ) but this is not yet a signal of CP violation. CP is violated, for example, if (6) (7) + − Γ(B 0 → lL+ lL− ) 6= Γ(B̄ 0 → lR lR ) , 0 + − 0 + − Γ(B → lR lR ) 6= Γ(B̄ → lL lL ) , be ause the initial and nal states on both sides transform into ea h other under CP [7℄. As there are no strong phases involved, this an o ur only through the mixing of the 0 0 (t)) whi h at B 0 and B̄ 0 mesons. In the standard formalism [25℄ the state Bphys (t) (B̄phys 0 0 t = 0 is a pure B (B̄ ) evolves in time a ording to q 0 |Bphys (t)i = g+ (t)|B 0 i + g− (t)|B̄ 0 i , p p 0 0 |B̄phys (t)i = g+ (t)|B̄ i + g− (t)|B 0 i . q Negle ting the dieren e of the de ay widths of the two B 0 mass eigenstates one nds Γ ∆M t, 2 ∆M i sin t, 2 g+ (t) = e−iMB t− 2 t cos Γ g− (t) = e−iMB t− 2 t (8) where ∆M ≡ MBH0 − MBL0 ≪ MB ≡ (MBH0 + MBL0 )/2. The ratio p/q ( al ulable in the SM or its extensions) is given by p = q s ∗ ∗ H12 M12 1 Γ12 ≈ 1 − Im H12 |M12 | 2 M12   (9) , with H12 ≡ M12 + 2i Γ12 = hB 0 |Heff |B̄ 0 i, et . [25℄. The probability amplitude that the state, whi h initially was a B 0 , de ays at time t into left-handed leptons is therefore given by q 0 hlL+ lL− |Bphys (t)i = g+ (t) AL + g− (t) ĀL , p (10) and the rates of B 0 → lL+ lL− and B̄ 0 → lR+ lR− de ays are proportional to   2   2 q g− (t) 0 |hlL+ lL− |Bphys (t)i|2 = |g+ (t)|2 |AL |2 1 +  p g+ (t) p g− (t) + − 0 |hlR lR |B̄phys (t)i|2 = |g+ (t)|2 |ĀR |2 1 + q g+ (t) ĀL AL 2 AR ĀR 2  ! q g− (t) ĀL  + 2Re , p g+ (t) AL   ! p g− (t) AR  + 2Re . q g+ (t) ĀR  The matrix elements for B 0 → lR+ lR− and B̄ 0 → lL+ lL− an be obtained from the ones above by inter hanging L ↔ R. 3 Sin e |AL | = |ĀR |, |AR | = |ĀL | and, as follows from (4) and (5), AR ĀL = AL ĀR  ∗ , (11) one sees that the CP is violated if either q 6= 1 p  Im (λL ) 6= Im λ−1 R or  , where λL ≡ q ĀL , p AL and λR ≡ q ĀR . p AR (12) The simplest quantities measuring the amount of CP violation are the asymmetries onstru ted out of time integrated polarized de ay rates A1CP (t1 , t2 ) A2CP (t1 , t2 ) ≡ R t2 ≡ R t2 t R t12 t1 t R t12 t1 + − 0 0 dt Γ(Bphys (t) → lL+ lL− ) − tt12 dt Γ(B̄phys (t) → lR lR ) R t2 + − + − , 0 0 dt Γ(Bphys (t) → lL lL ) + t1 dt Γ(B̄phys (t) → lR lR ) R (13) + − 0 0 dt Γ(Bphys (t) → lR lR ) − tt12 dt Γ(B̄phys (t) → lL+ lL− ) . R + − 0 0 dt Γ(Bphys (t) → lR lR ) + tt12 dt Γ(B̄phys (t) → lL+ lL− ) R (14) and the ratio of integrated unpolarized de ay rates R t2 Rl (t1 , t2 ) ≡ Rtt12 t1 The time interval (t1 , t2 ) 0 dt Γ(Bphys (t) → l+ l− ) . 0 dt Γ(B̄phys (t) → l+ l− ) an be hosen a (15) ording to the experimental onvenien e. If the number of tagged events is not very large, or there is a large un ertainty in experimental determination of the de ay time t, one and A2CP ≡ A2CP (0, ∞) and the ratio obtain [7, 26℄ A1CP = A2CP = 1 2 x 2  an exploit the asymmetries Rl ≡ Rl (0, ∞)  for whi h it is straightforward to  −1 |λ2L | − |λ−2 R | − x Im λL − λR   A1CP ≡ A1CP (0, ∞)   −1 2 + x2 + 12 x2 |λ2L | + |λ−2 R | − x Im λL + λR 1 2 x 2    −1 |λ2R | − |λ−2 L | − x Im λR − λL      −1 2 + x2 + 12 x2 |λ2R | + |λ−2 L | − x Im λR + λL  , (16) , (17) and Rl = (|AL |2 + |AR |2 ) (1 + 21 x2 + 21 x2 | pq |2 ) − x {|AL |2 Im(λL ) + |AR |2 Im(λR )} n −1 2 (|AL |2 + |AR |2 ) (1 + 21 x2 + 12 x2 | pq |2 ) − x |AL |2 Im(λ−1 R ) + |AR | Im(λL ) 4 o , (18) −1 where x ≡ ∆M/Γ. If |q/p| = 1, the relation (11) implies |λL | = |λ−1 R |. Moreover, λL + λR is then real, and the formulae (16), (17) and (18) simplify to A1CP = Rl = −2 x Im λL , 2 + x2 + x2 |λL |2 A2CP = −2 x ImλR , 2 + x2 + x2 |λR |2 (|AL |2 + |AR |2 ) (1 + x2 ) − x {|AL |2 Im(λL ) + |AR|2 Im(λR )} . (|AL |2 + |AR |2 ) (1 + x2 ) + x {|AL |2 Im(λL ) + |AR |2 Im(λR )} (19) The asymmetries A1CP , A2CP , as fun tions of λL,R , are bounded from above by [7℄ A1,2 CP ≤ √ 1 . 2 + x2 (20) Sin e xs > 20.6 for the Bs0 -B̄s0 system, the CP asymmetries in the leptoni Bs0 (B̄s0 ) de ays an rea h at best ∼ 4.5%. In ontrast, for the Bd0 -B̄d0 system, for whi h xd = 0.771 ± 0.012 they an be as large as ∼ 60%. This is fortunate, sin e Bd0 B̄d0 are opiously produ ed at BABAR and BELLE in a relatively lean environment ( ompared to the B 0 produ tion at hadron olliders). For this reason we will onsider only the CP asymmetries in the Bd0 (B̄d0 ) → l+ l− de ays. The quantities (13) and (14) depend on asymmetries of B 0 and B̄ 0 de ays into longitudinally polarized leptons. In the ase of the τ + τ − de ay mode they are best identied by measuring the π ± energy spe tra from τ → πν de ays [27℄. The density matrix formalism outlined in Se tion 4 allows to onstru t also observables sensitive to transverse polarization of the nal state τ 's [28℄. These observables will prove omplementary sin e in some s enarios the signal of CP violation an learly be visible in the latter observables while hidden in the former (as Rl an be expressed in terms of polarized de ay rates, Rl is then equal 1). 3 Supersymmetry s enario If the Cabibbo-Kobayashi-Maskawa (CKM) matrix is the only sour e of both avour and ∗ CP violation (as in the SM), then q/p ≈ −Vtb∗ Vtd(s) /Vtb Vtd(s) implying |q/p| ≈ 1. Moreover, ∗ ∗ AL , AR ∝ Vtb Vtd(s) , ĀL , ĀR ∝ Vtb Vtd(s) , so that λL and λR are almost real. The CP asymmetries in the B 0 → l+ l− de ay are then negligible.2 In models of new physi s one an have Im(λL ) 6= Im(λ−1 R ) and/or |q/p| 6= 1, but if the predi ted rates of these de ays are still of the same order as the SM predi ts, the dete tion of the µ+ µ− de ay mode will be ome possible only at the LHC where most probably the measurement of the muon polarization will not be feasible. Dete ting CP violation in this mode ould be possible then only by measuring the ratio Rµ . The dete tion of the Bd0 → τ + τ − mode o urring at the SM model level (Br ∼ 5 × 10−8) is possible at BELLE and BABAR 3 but the number of 2 In 3A O(10−2 ) [7℄ as a result of a small departure of |q/p| from 1. 8 0 0 tion, among 5 × 10 events of B B̄ d d pair produ tion olle ted by these 0 0 + − already be some 50 events of B (B̄ ) de ays into τ τ . d d the SM they are estimated to be ording to the SM predi experiments so far there should 5 re onstru ted events might be too small to dete t any CP violation. On the other hand, 0 → τ + τ − events will be larger, the identi ation of the at LHC where the number of Bd,s τ de ay mode will probably be quite di ult and the CP asymmetry hard to dete t. Mu h more promising situation an o ur in the supersymmetri s enario with a large ratio of the va uum expe tation values of the two Higgs doublets, vu /vd ≡ tan β ∼ 40÷50. 0 The oe ients a and b (and ā and b̄) in (3), and hen e, the amplitudes of the Bd,s → l+ l− de ays an then re eive important ontributions from the Higgs penguin diagrams with s- hannel H 0 and A0 Higgs boson ex hanges [4, 5, 8℄. If the mass s ale of the Higgs parti les H 0 and A0 (for vu /vd ≫ 1 MH ≈ MA ) is not too high, of order ∼< 500 GeV, the de ay amplitudes an be dominated totally by these diagrams easily saturating the experimental limits (1), (2). This an happen even if the supersymmetri parti le masses are quite large, say in the TeV range [5, 29℄. As illustrative examples of new physi s we onsider here two dierent supersymmetri s enarios: minimal (MFV) and non-minimal (NMFV) avour violating, both with large ratio of VEVs. MFV S enario: In this ase no additional avour violation in the sfermion mass matri es is assumed (i.e. the gluino-quark-squark verti es onserve avour) but additional CP violation phases are introdu ed by omplex parameters µ and At (the higgsino mass term and left-right top squark mixing, respe tively). The relevant ee tive avour violating ouplings of the two heavy neutral Higgs bosons to the down-type quarks4 an be written as H 0 ,A0 bd Leff md At 0 mb A∗t (H 0 + iA0 )PL + (H − iA0 )PR d =C b̄ MW µ MW µ∗ " # ∗ m m A A d b t t +C Vtd∗ Vtb d¯ (H 0 + iA0 )PL + (H 0 − iA0 )PR b , MW µ MW µ∗ Vtb∗ Vtd # " where PL,R = (1 ∓ γ 5 )/2 are hiral proje tors and the oe ient C is given by C= H2 g 3 m2t tan2 β κ 2 4 MW 16π 2 with a dimensionless fun tion of higgsino and stop masses H2 of order O(1). The fa tor κ ∝ tan β summarises some renements in the al ulation (resummation of tan β enhan ed terms, for details see [6, 29℄). It is the fa tor tan2 β whi h makes these ouplings so important. Combining these ouplings with the H 0 and A0 ouplings to l+ l− and using M2 M2 i i h0|d̄PL b|B̄d0 i = + FBd B , h0|b̄PL d|Bd0 i = − FBd B , 2 mb 2 mb 2 2 i M ¯ R b|B̄ 0 i = − i FB MB , h0|b̄PR d|Bd0 i = + FBd B , h0|dP d 2 mb 2 d mb 4 The ee tive avour violating oupling of the lightest CP even Higgs h0 to the down-type quarks is negligible. 6 one arrives at a = −ā∗ = C ′ Vtb∗ Vtd b = b̄∗ = C ′ Vtb∗ Vtd mb A∗t md At ≈ C ′ Vtb∗ Vtd + MW µ MW µ∗ ! mb A∗t md At − ≈ C ′ Vtb∗ Vtd MW µ MW µ∗ MB2 MA2 MB2 MH2 ! mb MB2 A∗t , MW MA2 µ mb MB2 A∗t , MW MH2 µ with C′ = − g 4 m2t ml FB H2 tan3 β κ 2 16 MW MW mb 16π 2 In this ase the amplitude of the Bd0 -B̄d0 mixing is not modied (in ontrast to the one for Bs0 -B̄s0 mixing) [29℄, so that |q/p| ≈ 1 holds true as in the SM. One then nds λL ≈ e−2iδCP 1−β , 1+β λR ≈ e−2iδCP 1+β , 1−β (21) where the ee tive CP violating phase is given as δCP = arg(µ∗ A∗t ). The time integrated asymmetries then read A1CP = −0.09 × sin(2δCP ) , A2CP = −0.35 × sin(2δCP ) . (22) Sin e the sparti les giving rise to substantial Higgs penguin ontributions to a and b (ā and b̄) an be quite heavy, even order O(1) phase of µAt needs not produ e una eptable ele tri dipole moments. NMFV S enario: In this ase squark mass matri es violate avour and the orre tions generating the avour hanging ouplings of the neutral Higgs bosons H 0 ad A0 are dominated by gluino loops [5℄. The relevant ee tive Lagrangian is then 0 ,A LH eff with 0 bd h i = D b̄ αb (H 0 + iA0 )PL + αd∗ (H 0 − iA0 )PR d +D d¯ αd (H 0 + iA0 )PL + αb∗ (H 0 − iA0 )PR b , h i (23)  m∗g̃  md db mb db αd = δ + δ µ MW LL MW RR  m∗g̃  mb bd md bd δ + δ , αb = µ MW LL MW RR and 4 |µ|2 H3 D = gs2g tan2 β 2 κ, 3 Mq̃ 16π 2 where H3 is another dimensionless fun tion of order O(1) of gluino and sbottom masses bd and κ again denotes dominant higher order ontributions [30℄. The mass insertions δLL 7 bd and δRR et . are the ratios of the diagonal entries of the down-type squark mass squared matri es to the average squark mass squared Mq̃2 . One then nds mb MB2 m∗g̃ bd mg̃ db∗ a = −ā = D (αb + ≈D , δLL + ∗ δRR 2 MW MA µ µ ! 2 m∗g̃ bd mg̃ db∗ ∗ ′ ∗ ′ mb MB δ − ∗ δRR , b = b̄ = D (αb − αd ) ≈ D MW MH2 µ LL µ ∗ ′ αd∗ ) ! ′ with |µ|2 H3 FB ml 1 tan3 β 2 κ . D ′ = − gs2 g 2 2 3 MW Mq̃ 16π mb bd bd The insertions δLL and δRR are not very tightly onstrained. Typi ally the bound of bd order |δLL(RR) | < 0.2 × (Mq̃ /1 TeV) arising from ∆MBd is quoted [31℄. This estimate, however, does not take into a ount the ontribution of the so- alled double-penguin diagrams [30, 3234℄, whi h an signi antly ae t the Bd0 -B̄d0 mixing amplitude. This ontribution arises from the H 0 and A0 ex hanges between two ee tive (1-loop generated) verti es (23) of whi h one vertex annihilates a right- hiral and the other a left- hiral bd db∗ bd bd quark5 and is therefore proportional to αb αd∗ ∝ δLL δRR = δLL δRR (with the small dquark mass negle ted). Sin e this produ t is onstrained mu h stronger [30℄, to avoid a potential oni t with the value of ∆MBd and the time dependent CP asymmetry bd bd bd bd aJ/ψKS (t) measured in B → J/ψKS de ay, we assume that either δRR ≫ δLL or δRR ≪ δLL . This leads to a ≈ ±b and, as in the previous s enario, to |q/p| ≈ 1. The asymmetries then read A1CP = −0.35 × sin(2δCP ) , A2CP = −0.09 × sin(2δCP ) , where now bd δCP = arg(Vtb∗ Vtd ) − arg(mg̃ µδLL(RR) ). (24) It is interesting to note here that the CP asymmetries an be nonzero even if the phase of the CKM matrix remains the only sour e of CP violation (i.e. all supersymmetri parameters are real). Moreover, sin e |arg(Vtb∗ Vtd )| is of order 1, the total phase violating CP needs not be small. 0 In both s enarios, in whi h the Bd,s → l+ l− amplitudes are dominated by the ex hange of H 0 and A0 Higgs bosons whose ee tive avour violating ouplings to bd (or bs) dier only by a fa tor i, one gets 5 Contributions of H0 and A0 a≈b or a ≈ −b , ex hanges between verti es annihilating same mixing amplitude are proportional to 2 1/MH − 1/MA2 ≈ 0 8 [32℄. hirality quarks to the (up to ∼< 15%). For a = b the fa tors λL and λR dened in Eq. (12) simplify to: λL = − q a∗ 1 − β , p a 1+β λR = − q a∗ 1 + β p a 1−β (25) and all CP-sensitive quantities depend on one ee tive phase whi h an be taken as 1 δCP = − arg(λL ) . 2 (26) The immediate onsequen e of a = b, with |q/p| lose to 1, is that for the µ+ µ− nal state the parameters |λL | and |λR | assume values ∼ 4 × 10−4 and ∼ 2.5 × 103 , respe tively, sin e in this ase β = (1−4m2µ /MB2 )1/2 is almost 1. As a result, the predi ted asymmetries are very small: |A1CP | ∼< 2 × 10−4 , |A2CP | ∼< 10−3 . The same is true if a and b are somewhat split. In ontrast, for the τ + τ − nal states, for whi h β = (1 − 4m2τ /MB2 )1/2 diers substantially from 1, we have |λL| ∼ 0.15, |λR | ∼ 6.7 and the maximal possible values of the asymmetries are (A1CP )max = 9% (A2CP )max = 35% . and (27) The omparison of magnitudes of possible CP violating ee ts in the ratio (18) for µ µ and τ + τ − de ay modes is shown in Fig. 1, where Rµ and Rτ are plotted as fun tions of b/a for four dierent values of the phase δCP (keeping arg(a) =arg(b) and |p/q| = 1). The plots show, that the ratios Rl approa h unity for a ≈ ±bβ . (Vanishing of Rl for a ≈ ±bβ follows also from the formula (19) if one takes into a ount that for a → b ± β λR ∼ 1/(a ∓ bβ) whereas |AR |2 ∼ |a ∓ bβ|2 .) Therefore, for a ≈ ±b the deviation from unity of Rµ is tiny while for Rτ it an be quite substantial. + − The asymmetries A1,2 CP (13), (14) and the ratio Rl (15) are the observables in whi h the signal of CP violation (i.e. a non-zero phase δCP ) vanishes for a ≈ ±bβ . As the spin density matrix formalism will show, transverse polarization of the nal state leptons is free from this problem. However, observables sensitive to the transverse polarization an experimentally be onstru ted only for τ 's whi h de ay into hadrons. Thus, for a ≈ ±b + − only the τ τ hannel provides an interesting opportunity to look for CP asymmetry in the leptoni Bd0 (B̄d0 ) de ays. The oe ients a and b (ā and b̄) are onstrained by the experimental limit Eq. (1), whi h in the ase a ≈ ±b gives < 4.9 × 10−9 . |a| ≈ |b| ∼ (28) < 2 × 10−9 . |a| ≈ |b| ∼ (29) In the minimal avour violation s enario, the parameters a and b are also onstrained indire tly6 by the limit (2). Taking (mτ /mµ )(|Vtd |/|Vts|) ∼ 4 this onstraint is satised if 6 The limit imposed by the Bs0 -B̄s0 mass dieren e [29,32℄ an be avoided on a ount of a dierent tan β and MA,H dependen e as ompared to the the Higgs penguin diagram ontributions to Bd0 (B̄d0 ) → l+ l− de ays [16℄. 9 Figure 1: The ratios Rµ and Rτ as fun tions of b/a for the phase δCP = − 12 arg(λL ) = −1 (solid line), 0.3 (dashed), 0.5 (dotted) and 0.75 (dash-dotted). Rl (−δCP ) = Rl (δCP ). 0.1 In our simulations in Se tion 5 we onservatively set |a| = |b| ∼< 10−9 and treat both s enarios simultaneously, as all what matters are the values of |a| = |b| and the single CP violating phase δCP . In the MFV ase δCP = arg(µ∗ A∗t ) while in the NMFV ase it is given by Eq. (24); in both ases the phase an be of order 1. Finally let us noti e that the SM as well as supersymmetri box and Z 0 penguin ontributions spoil the exa t equality a = ±b. In addition, a nite dieren e of A0 and H 0 masses also splits these two oe ients. It is therefore likely that |a| and |b| and the phases of a and b dier from ea h other by some 10÷15% even for |a| and |b| saturating the bounds (1), (2). Nevertheless, our simplied analysis will demonstrate that in supersymmetry there are good reasons to expe t substantial CP violation in Bd0 (B̄d0 ) → τ + τ − de ays. Therefore in the following se tions we present tools whi h an be used to sear h for these de ays and look for CP violation. 4 Spin density matrix formalism To see how the CP asymmetry in Bd0 (B̄d0 ) → τ + τ − de ays are ree ted in realisti observables we use the TAUOLA τ -lepton de ay library whi h allows us to simulate fully the ee ts of τ polarization. The exhaustive des ription of the method and numeri al algorithm is given in papers [19℄ and [22℄. The input to the TAUOLA Universal Interfa e [24℄ is the spin density matrix of the τ + τ − system resulting from the de ay of a neutral parti le. In this se tion we olle t the ne essary formulae for this matrix for a τ + τ − pair originating 10 from B 0 (B̄ 0 ). The time dependent B 0 -meson mixing an easily be dealt with by introdu ing time dependent ee tive fa tors aeff , beff , āeff and b̄eff dened as: q q g− (t) = a g+ (t) − a∗ g− (t) , p p q ∗ q beff (t) = b g+ (t) + b̄ g− (t) = b g+ (t) + b g− (t) , p p p p āeff (t) = ā g+ (t) + a g− (t) = −a∗ g+ (t) + a g− (t) , q q p p b̄eff (t) = b̄ g+ (t) + b g− (t) = b∗ g+ (t) + b g− (t) , q q aeff (t) = a g+ (t) + ā so that the instantaneous B 0 widths into left- or right-handed τ 's read MB β |aeff (t) + β beff (t)|2 , 16π MB 0 + − Γ(Bphys (t) → τR τR ) = β |aeff (t) − β beff (t)|2 , 16π (30) 0 Γ(Bphys (t) → τL+ τL− ) = (31) and those for B̄ 0 are given by similar formulae with aeff (t), beff (t) repla ed by āeff (t), b̄eff (t). CP is violated be ause in general āeff (t) 6= −a∗eff (t), and b̄eff (t) 6= b∗eff (t). The spin weight for the τ + τ − pair originating from B 0 de ay at time t is given by wt = 1 T (s1 , p1 , s2 , p2 ) , 4 T (0, p1, 0, p2 ) (32) where i h T (s1 , p1 , s2 , p2 ) = Tr P (s1 ) (6 p1 + ml ) (beff + aeff γ 5 )P (s2) (6 p2 − ml ) (b∗eff − a∗eff γ 5 ) , with P (s) ≡ 21 (1 + γ 5 6 s), and p1 , s1 (p2 , s2 ) are the momentum and spin four-ve tors of the τ − (τ + ) lepton, respe tively. The τ -lepton spin four-ve tors sa in the B 0 (B̄ 0 ) rest frame are related to the spin three-ve tors ~σa in the τa -lepton rest frames as follows7 MB β σ1z , 2ml MB z sza = σ , 2ml a s02 = − s01 = MB β σ2z , 2ml x,y sx,y a = σa . (33) Combining (32) with the τ -lepton de ay matrix elements, the τ -lepton rest frame spin ve tors ~σ1 and ~σ2 get repla ed by the polarimetri ve tors ~h1 and ~h2 whi h are determined solely by the dynami of the onsidered τ de ay pro ess. 7 In the rest frame of B 0 /B̄ 0 the z -axis is aligned with τ − momentum, exa tly as in TAUOLA Universal Interfa e. However, here as a parti le number 1 the τ − lepton is taken (not τ + ), resulting in the transposition of the matrix Rµν in the TAUOLA Universal Interfa e. 11 The spin weight for the omplete event (B → τ τ → de ay produ ts) an be written in the form   X X X X 1 R0j hj2  . Ri0 hi1 + Rij hi1 hj2 + W T = 1 + 4 j=x,y,z i=x,y,z i=x,y,z j=x,y,z (34) Expanding (32) and omparing with (34) we nd R00 = +1, Rzz = −1, Rx0 = Ry0 = R0x = R0y = 0, Rxz = Ryz = Rzx = Rzy = 0, 2Re(aeff b∗eff ) β , R0z = −Rz0 = |beff |2 β 2 + |aeff |2 2Im(aeff b∗eff ) β |beff |2 β 2 − |aeff |2 Rxy = −Ryx = − , R = R = , xx yy |beff |2 β 2 + |aeff |2 |beff |2 β 2 + |aeff |2 (35) Eq. (34) with Rµν repla ed by R̄µν omputed as above but with āeff (t) and b̄eff (t) repla ing aeff (t) and beff (t), respe tively, gives the spin weight for the events from B̄ 0 de ays. CP violating ee ts in the B 0 → τ + τ − and B̄ 0 → τ + τ − de ays are absent if R0z = R̄z0 , Rxy = R̄yx and Rxx = R̄xx . For simulations of the time integrated measurements, the time averaged matrix hRµν i has to be used 8 hRµν i ≡ R 0 dt Γ(Bphys (t) → τ + τ − ) Rµν (t) R 0 dt Γ(Bphys (t) → τ + τ − ) and hR̄µν i given by a similar formula. The asymmetry (13) is then given by A1CP = where Γint = Z (1 − hRz0 i)Γint − (1 + hR̄z0 i)Γ̄int , (1 − hRz0 i)Γint + (1 + hR̄z0 i)Γ̄int 0 dt Γ(Bphys (t) → τ + τ − ), Γ̄int = Z (36) 0 dt Γ(B̄phys (t) → τ + τ − ). A2CP (t1 , t2 ) dened in (14) is given by (36) reversing the signs in the bra kets. It is also easy to he k that for |q/p| = 1 and a = ±b one has hRz0 iΓint = −hR̄z0 iΓ̄int (i.e. Re(aeff b∗eff ) = −Re(āeff b̄∗eff )). Then the two fa tors: hRz0 iΓint and hR̄z0 iΓ̄int an el out in the numerator of (36) and of the similar formula for A2CP (t1 , t2 ). As a result, nonzero asymmetries A1,2 CP are possible only if Γint 6= Γ̄int , whi h in view of the equalities |aeff (t)| = |b̄eff (t)|, |beff (t)| = |āeff(t) | requires β signi antly dierent from 1. This again onrms our observation made in Se tion 2 that for a ≈ ±b the asymmetry in the µ+µ− hannel is suppressed. 8 Simulations for time non-integrated measurements, with time-dependent de ays expli itly generated are also possible. 12 Sin e in the limit a = ±bβ the two integrated rates Γint and Γ̄int are equal one gets hRz0 i = −hR̄z0 i, whi h means that even for δCP 6= 0 there an be no CP violating ee ts in the observables sensitive to the longitudinal polarization of τ 's (nor in Rτ ). In ontrast, it is easy to he k by using the expli it analyti al expressions, that in this limit the elements xx and xy of these matri es need not satisfy hRxy i = −hR̄xy i and hRxx i = hR̄xx i. Hen e, the observables sensitive to transverse τ polarization an reveal CP violation even if the observables introdu ed in Se tion 2 fail to signal it. 5 Results of the Monte Carlo simulations As B 0 → τ + τ − de ays have not yet been seen it might seem premature to study dierential distributions in this hannel, in luding τ polarization. Nevertheless, our analysis an serve as a good starting point for the future experimental work if indeed a umulated experimental samples would turn to be large enough. By providing a Monte Carlo tool useful in al ulating e.g. dete tor responses, our study may also be onsidered as ben hmarks for the simulations to be used in setting the upper limit for B 0 → τ − τ + bran hing ratios. There are some similarities between this study and the one aiming at assessing measurability of the Higgs boson parity at future Linear Colliders [28℄. The dieren e is that now the oe ients a and b (and also ā and b̄) in Eq. (3) an be omplex, while in [28℄ they were taken to be purely imaginary and real, respe tively. As a result, in the formula Eq. (34) terms linear in the polarimetri ve tor omponents also appear. Therefore we have extended the TAUOLA Universal Interfa e to in lude su h a possibility as well. The algorithm for simulating B 0 de ay into τ leptons is nearly identi al to the one for H 0 de ay. Changes ne essary for implementation in the Universal Interfa e of the TAUOLA Monte Carlo library are limited to the repla ement of Higgs identier with the one for Bd0 and B̄d0 mesons, and the values of the spin density matrix with the ones omputed in the pre eding se tion. The formulae given in the previous se tion are general. However, motivated by the two supersymmetri s enarios dis ussed in Se tion 3, in our numeri al study we will show rst the results obtained in the limit a = ±b. Then we will dis uss ee ts of a small departure from this relation. We will limit ourselves to two observables, whi h are known to provide valuable and omplementary information on the spin state of de aying τ lepton pairs. π ± energies: As the rst observable we take the π + and π − energy spe tra in the de ay hannels τ + → π + ν̄τ (or τ − → π − ντ ). Sin e they ree t the longitudinal polarization of the individual τ ± leptons, the spe tra are sensitive to Rz0 and R̄0z as an be inferred from the expression (34), i.e. they are sensitive to Re(aeff b∗eff ) as follows from (35). The CP violation is ree ted in the dieren e between the energy spe trum of π − (π + ) originating from B 0 (B̄ 0 ) and the energy spe trum of π + (π − ) originating from B 0 (B̄ 0 ). This observable 13 was exploited before, for example in the Z 0 /γ ∗ → τ + τ − study (for a review of the method and its extensions, see e.g. Ref. [35℄). We will show energy spe tra in the rest frame of the B 0 (B̄ 0 ) meson assuming that the re onstru tion of the event kinemati s in the BELLE and BABAR experiments is su iently good for that purpose, that is, that the momenta of τ de ay produ ts in the rest frame of B 0 (B̄ 0 ) an be re onstru ted with the pre ision of a fra tion of the τ mass. A oplanarity angle ϕ∗ : As the se ond observable we use the a oplanarity angle ϕ∗ between two planes spanned by the momenta of de ay produ ts of ρ± → π ± π 0 oming from de ays of both τ leptons into ρντ [28℄. This quantity is sensitive to orrelations between transverse omponents of τ -lepton spins (i.e. to the elements Rxx and Rxy whi h in turn probe Im(aeff b∗eff ), as an be seen from (35)). For the denition of the a oplanarity the orientation of de ay planes and pion momenta has to be properly taken into a ount. The a oplanarity angle ϕ∗ is dened with the help of two ve tors n± normal to the planes determined by the momenta of pions whi h originate from ρ± de ays: n± = pπ± × pπ0 . ·n− If cos ξ = |nn++||n then −| ∗ ϕ = ( ξ 2π − ξ for sgn(pπ− · n+ ) < 0 for sgn(pπ− · n+ ) > 0 (37) making the full range of the variable 0 < ϕ∗ < 2π of physi al interest. Note that under CP, ϕ∗ → 2π − ϕ∗ sin e the ondition sgn(pπ− · n+ ) is always evaluated from the orientation of π − momentum with respe t to the ve tor n+ . In addition, we also have to sort events depending whether y1 y2 > 0 or y1 y2 < 0, where y1 = Eπ+ − Eπ0 , Eπ+ + Eπ0 y2 = Eπ− − Eπ0 , Eπ− + Eπ0 (38) sin e otherwise the spin orrelations are washed out, as explained in Ref. [36℄. The best would be to use in (38) the energies of π ± and π 0 's in the rest frames of the orresponding τ ± leptons, but they are not dire tly measurable. Sin e in the B 0 (B̄ 0 ) rest frame the τ leptons are only mildly relativisti , the dieren e of pion energies in this frame and respe tive rest frames of τ ± should not be very important. The a oplanarity distribution is then evaluated in the rest frame of the ρ+ ρ− pair, but with the energies of π ± and π 0 's in (38) taken in the rest frame of the B 0 (B̄ 0 ). The CP violation is ree ted in the dieren e between the distributions of the a oplanarity angle ϕ∗ measured in B 0 de ays and the angle 2π − ϕ∗ measured in B̄ 0 de ays for the same signs of y1 y2 . Fig. 2 shows the pion energy spe tra and the a oplanarity distributions assuming |q/p| = 1, a = b = 10−9 and the CP violating phase δCP = 0.7. For all plots the same number of 5 × 105 τ + τ − events from Bd0 and B̄d0 de ays has been generated with TAUOLA, although for the parameters hosen the ratio Rτ = 1.32, see Fig. 1. In the upper left panel the energy spe tra of π − from B 0 de ays (thi k line with the slope proportional to hRz0 i [37℄) and of π + from B̄ 0 (thin line; slope ∝ hR̄0z i) are shown, while in the lower left panel shown are the spe tra of π + from B 0 de ays (thi k line; slope ∝ hR0z i) and of π − from B̄ 0 (thin line; slope ∝ hR̄z0 i). The harder π − energy spe trum from Bd0 de ays 14 N Nevents events 3:0
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103 1:00 1:0 2:0 3:0 ' E =MB  Results for the CP violating phase δCP = 0.7 and a = b. Left panels: Single π ± energy spe tra in B 0 (B̄ 0 ) → τ + τ − , τ ± → π ± ντ (ν̄τ ). In the upper (lower) panel the thi k line orresponds to the energy spe trum of π − (of π + ) from B 0 de ays and the thin line to the energy spe trum of π + (of π − ) from B̄ 0 . Spe tra are plotted in the rest frame of B 0 (B̄ 0 ). Right panels: a oplanarity distributions of the ρ+ ρ− de ay produ ts in B 0 (B̄ 0 ) → τ + τ − , τ ± → ρ± ντ (ν̄τ ), ρ± → π ± π 0 . The thi k lines orrespond to the a oplanarity angle ϕ∗ measured in B 0 de ays and the thin ones are for the angle 2π − ϕ∗ measured in B̄ 0 de ays. The a oplanarity angles are dened in the rest frame of the ρ+ ρ− pair. Events in the upper (lower) panel have y1 y2 > 0 (y1y2 < 0). Figure 2: π + from B̄d0 (i.e. larger slope of the thi k line) in the upper left panel indi ates + − + − that Br(Bd0 → τR τR ) > Br(B̄d0 → τL τL ), whi h is a lear signal of CP violation. In the a oplanarity plots (right panels) thi k lines orrespond to the distribution of ϕ∗ measured in B 0 de ays, and the thin lines to the distribution of 2π − ϕ∗ measured in B̄ 0 de ays; in the upper right panel y1 y2 > 0, and y1 y2 < 0 in the lower right one. The shapes of the than thi k and thin lines are des ribed by the formulae NB (ϕ∗ ) = const − sgn(y1 y2 )AR cos(ϕ∗ − δR ) 15 for B0 → τ +τ − NB̄ (ϕ∗ ) = const − sgn(y1 y2 )AR̄ cos(2π − ϕ∗ + δR̄ ) for B̄ 0 → τ + τ − 2 2 1/2 where AR = (Rxx + Rxy ) , sin δR = Rxy /AR , cos δR = Rxx /AR and AR̄ and δR̄ are given by analogous formulae with Rij repla ed by the R̄ij . Dierent shapes of thi k and thin lines seen in the right panels of Fig. 2 again indi ate CP violation. In both energy and a oplanarity plots the CP violation is learly seen and should be measurable even for small statisti s. Note also that if upper and lower plots are ombined (i.e. no sorting a ording to the pion harge or sign of y1 y2 is made), all CP asymmetries are lost. Sin e the lower plots are simple ree tions of the upper ones, in the following only plots orresponding to the upper panels of Fig. 2 are shown. N Nevents events 3:0
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103 1:00 1:0 2:0 3:0 ' E =MB 4:0 5:0 6:0  Figure 3: As in the upper panels of gure 2 but for δCP = 0.3 and a = b. Fig. 3 shows the orresponding distributions for the CP violating phase δCP = 0.3 keeping still a = b and |q/p| = 1. It is apparent that with de reasing |δCP | the signal of CP violation deteriorates (espe ially in the pion spe tra) and the possibility of distinguishing pion spe tra and a oplanarity distributions from B 0 and B̄ 0 , and hen e the CP violation, would require in reasingly large statisti s whi h may not be attainable at BELLE and BABAR without major upgrades. As we dis ussed, the relation a = ±b is only approximate. Therefore, in Figure 4 (5) we show the π ± spe tra and a oplanarities for a greater (smaller) than b, but keeping as previously their phases equal (and assuming |q/p| = 1). We take b = 0.8 a in Fig. 4 and a = 0.8 b in Fig. 5. In both gures the single CP violating phase δCP = 0.7. Fig 4 shows that for b = 0.8 a with the same value of δCP the CP violating ee ts in π energy spe tra are enhan ed, while in the a oplanarities they are only slightly ae ted ompared to the ase a = b, .f. the upper right panel of Fig. 2. ± On the other hand, for a approa hing bβ (for B → τ + τ − de ays β ≈ 0.74) the ee ts of CP violation in the π ± energy spe tra disappear. This is learly seen by omparing the 16 right panel of Fig. 5 with the upper left one of Fig. 2. This agrees with our observations following Eqs. (25) and (26) and with the dis ussion in Se tion 3. In ontrast, the a oplanarities shown in the right panel of Fig. 5 learly indi ate the CP violation even for a ≈ bβ onrming our dis ussion in Se tion 4. This learly demonstrates the omplementarity of the energy and a oplanarity distributions as a means to dete t CP violation. N Nevents events 3:0
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103 1:00 1:0 2:0 3:0 ' E =MB 4:0 5:0 6:0  Figure 5: As in the upper panels of gure 2 but for δCP = 0.7 and a = 0.8 b. 6 Con lusions In this letter we have investigated possible signals of CP violation in the de ays B 0 (B̄ 0 ) → τ + τ − . We have developed the ne essary formalism and numeri al tools allowing to apply 17 the TAUOLA τ -lepton de ay library together with its universal interfa e to simulate fully the ee ts of the polarization of τ + and τ − originating from su h de ays. We have argued that in the interesting new physi s s enario of supersymmetry with tan β ∼ 40 ÷ 50, in whi h the rates of Bd0 (B̄d0 ) → τ + τ − de ays are enhan ed and ould be dete table in the SLAC and KEK B -fa tories, the dependen e of the CP asymmetries on the model parameters simplies. Moreover, the CP violating phase needs not be small. In the non-minimal avour violation ase it an be of the same order as the phase of the Vtd element of the CKM matrix. Therefore the CP asymmetries an be quite large as opposed to the B 0 (B̄ 0 ) → µ+ µ− de ays in whi h they are kinemati ally suppressed. By using Monte-Carlo simulations we have investigated the possible ee ts of CP violation in two realisti experimental observables and demonstrated that they might be dete table if the CP violating phase is reasonably large, i.e. O(1). Sin e the de ays B 0 (B̄ 0 ) → τ + τ − have not been dis overed yet, we have not dis ussed the statisti s requirements nor attempted at in luding in our analysis any systemati or dete tor un ertainties. It is lear that on e these de ays are dis overed, other τ de ay hannels than the ones investigated here an be analysed jointly to give additional information on the polarization of τ 's. Our numeri al tools are prepared for that. The tools an also be applied to determine the upper limits on the bran hing fra tion of the B 0 (B̄ 0 ) → τ + τ − de ays by the BABAR and BELLE ollaborations. As a nal remark, we point that our analysis an be taken over to Higgs boson produ tion at linear olliders with its subsequent de ay to τ pairs, whi h as yet has not been simulated in onne tion with the omplex s alar and pseudos alar ouplings. A knowledgements The authors would like to thank P. Ball, A. Martin, M. Misiak, H. Paªka and J. Stirling for useful dis ussions. P.H.Ch. thanks the CERN Theory Group for hospitality during the initial stage of this resear h. Work supported by the Polish State Committee for S ienti Resear h Grants 2 P03B 040 24 for years 2003-2005 (P.H.Ch. and J.K.), 1 P03 091 27 for years 2004-2006 (Z.W.), 1 P03B 009 27 for years 2004-2005 (M.W.), and by the EC Contra ts HPRN-CT-200000148 (P.H.Ch.) and HPRN-CT-2000-00149 (J.K). 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