Angular distribution of polarised Λ b baryons decaying to Λ ℓ + ℓ −
EEOS-2017-02
Angular distribution of polarised Λ b baryons decaying to Λ(cid:96) + (cid:96) − Thomas Blake a and Michal Kreps a , a Department of Physics, University of Warwick, Coventry CV4 7AL, UKE-Mail: [email protected] , [email protected] . Abstract
Rare b → s(cid:96) + (cid:96) − flavour-changing-neutral-current processes provide important testsof the Standard Model of particle physics. Angular observables in exclusive b → s(cid:96) + (cid:96) − processes can be particularly powerful as they allow hadronic uncertaintiesto be controlled. Amongst the exclusive processes that have been studied by exper-iments, the decay Λ b → Λ(cid:96) + (cid:96) − is unique in that the Λ b baryon can be producedpolarised. In this paper, we derive an expression for the angular distribution of the Λ b → Λ(cid:96) + (cid:96) − decay for the case where the Λ b baryon is produced polarised. Thisextends the number of angular observables in this decay from 10 to 34. StandardModel expectations for the new observables are provided and the sensitivity of theobservables is explored under a variety of new physics models. At low-hadronicrecoil, four of the new observables have a new short distance dependence that is ab-sent in the unpolarised case. The remaining observables depend on the same shortdistance contributions as the unpolarised observables, but with different dependenceon hadronic form-factors. These relations provide possibilities for novel tests of theSM that could be carried out with the data that will become available at the LHCor a future e + e − collider. a r X i v : . [ h e p - ph ] D ec Introduction
Rare b → s(cid:96) + (cid:96) − have been studied extensively by experiments at the B-factories aswell as experiments at the Tevatron and Large Hadron Collider (LHC). Amongst the b → s(cid:96) + (cid:96) − processes that have been studied, the decay Λ b → Λµ + µ − is unique for tworeasons: it is the only baryonic decay that has been studied; and the Λ baryon decaysweakly leading to new hadron-side observables. The angular distribution of Λ b → Λµ + µ − decays has been studied in Refs. [1, 2] for the case of unpolarised Λ b baryons. Theresulting angular distribution is described by 10 angular observables. The decay rateand lepton side angular distribution has also been studied in the SM and in severalextensions of the SM (NP models) in Refs. [3–12]. If the Λ b is produced polarised, amuch larger number of observables are measurable. These observables are explored inthis paper. The exploitation of production polarisation in radiative Λ b → Λ ( ∗ ) γ decayshas previously been studied in Refs. [13–16].In e + e − collisions, Λ b baryons can be produced with large longitudinal polarisations.The longitudinal polarisation of Λ b baryons and b -quarks produced via e + e − → Z ( → bb )decays has been studied by the LEP experiments in Refs. [17–19]. The production of Λ b baryons with longitudinal polarisation is forbidden in strong interactions, due toparity conservation. The Λ b can, however, be produced with transverse polarisation in pp collisions. In this paper, we focus on the transverse polarisation of the Λ b baryon.The transverse polarisation of Λ b baryons produced in pp collisions at √ s = 7 and8 TeV has been studied by the LHCb and CMS experiments in Refs. [20] and [21],respectively. The LHCb experiment measures P Λ b = 0 . ± . ± .
02 at √ s = 7 TeV.The CMS experiment measures P Λ b = 0 . ± . ± .
02 combining data from √ s = 7and 8 TeV. In both cases, the production polarisation is determined from the observedangular distribution of Λ b → J/ψ Λ decays. Whilst the measured transverse productionpolarisation is small, polarisations of O (10%) cannot be excluded. Polarised Λ b baryonscan also be obtained from decays of heavier b -baryons, for example in decays of theΣ ( ∗ ) b [22].The only existing measurements of the angular distribution of the Λ b → Λ(cid:96) + (cid:96) − decaycome from the LHCb experiment [23]. Due to the limited size of their dataset, LHCbonly studied a subset of the angular distribution that could be accessed from single angleprojections on the lepton- and hadron-side. With the much larger data sets that willbe available at the LHC experiments after run 2 of the LHC, the experiments will beable to probe the full angular distribution. However, the sheer number of observablesinvolved will most likely require an analysis of the moments of the angular distribution(see for example Ref. [24]) rather than the conventional approach of fitting for theangular observables. This approach is discussed in Sec. 6, where we provide the weightingfunctions needed to extract the observables.2 Angular distribution
The angular distribution of the Λ b → Λ(cid:96) + (cid:96) − decay has been previously studied in Refs. [1,2]. In this paper we extend those studies to include the case where the Λ b baryon isproduced with a transverse polarisation. We start by expanding the differential decayrate for the Λ b → Λ(cid:96) + (cid:96) − decay in terms of generalised helicity amplitudesd Γd q d (cid:126) Ω ∝ (cid:88) λ ,λ ,λ p ,λ (cid:96)(cid:96) ,λ (cid:48) (cid:96)(cid:96) ,J,J (cid:48) ,m,m (cid:48) ,λ Λ ,λ (cid:48) Λ , (cid:16) ( − J + J (cid:48) × ρ λ Λ − λ (cid:96)(cid:96) ,λ (cid:48) Λ − λ (cid:48) (cid:96)(cid:96) ( θ ) × H m,Jλ Λ ,λ (cid:96)(cid:96) ( q ) H † m (cid:48) ,J (cid:48) λ (cid:48) Λ ,λ (cid:48) (cid:96)(cid:96) ( q ) × h m,Jλ ,λ ( q ) h † m (cid:48) ,J (cid:48) λ ,λ ( q ) × D J ∗ λ (cid:96)(cid:96) ,λ − λ ( φ l , θ l , − φ l ) D J (cid:48) λ (cid:48) (cid:96)(cid:96) ,λ − λ ( φ l , θ l , − φ l ) × h Λλ p , h † Λλ p × D / ∗ λ Λ ,λ p ( φ b , θ b , − φ b ) D / λ (cid:48) Λ ,λ p ( φ b , θ b , − φ b ) (cid:17) , (1)which depends on five angles, (cid:126) Ω = ( θ l , φ l , θ b , φ b , θ ), and the dilepton invariant masssquared, q . The angular basis is illustrated in Fig. 1. The helicity basis is definedstarting from the normal vector between the direction of the Λ b baryon in the lab-frameand the beam-axis of the experiment (ˆ n = ˆ p Λ b × ˆ p beam ). This is an appropriate choicewhen considering transverse production polarisation of the Λ b baryon.Equation 1 involves three sets of helicity amplitudes: H m,Jλ Λ ,λ (cid:96)(cid:96) ( q ) describing the decayof the Λ b baryon into a Λ baryon with helicity λ Λ and a dilepton pair with helicity λ (cid:96)(cid:96) ; h m,Jλ ,λ describing the decay of the dilepton system to leptons with helicities λ and λ ;and h Λλ p , describing the decay Λ → pπ to a proton with helicity λ p . The index J refers tothe spin of the dilepton system, which can either be zero or one. When J = 0, λ (cid:96)(cid:96) = 0,and when J = 1, λ (cid:96)(cid:96) can take the values − , , +1. The helicity labels λ p , λ Λ , λ and λ can take the values ± /
2. Angular momentum conservation in the Λ b decay requires | λ Λ − λ (cid:96)(cid:96) | = 1 /
2. The factor ( − J + J (cid:48) originates from the structure of the Minkowskimetric tensor, see Ref. [25] for details. The remaining index, m = V, A , denotes thedecay of the dilepton system by either a vector or an axial-vector current. The term ρ λ Λ − λ (cid:96)(cid:96) ,λ (cid:48) Λ − λ (cid:48) (cid:96)(cid:96) is the polarisation density matrix for the transverse polarisation of the Λ b . The matrix is a two-by-two matrix (with Tr( ρ ) = 1) given by ρ +1 / , +1 / ( θ ) = (1 + P Λ b ) cos θ ,ρ +1 / , − / ( θ ) = P Λ b sin θ ,ρ − / , − / ( θ ) = (1 − P Λ b ) cos θ ,ρ − / , +1 / ( θ ) = P Λ b sin θ . (2)Finally, the D jm,m (cid:48) ( φ, θ, − φ ) are Wigner- D functions. An explicit form of the Wigner- D functions is given in Appendix A. 3 p { lab } ⇤ b ˆ z ⇤ = ˆ p { ⇤ b } ⇤ ˆ n ˆ y ⇤ = ˆ n ⇥ ˆ p ⇤ b ⇤ ˆ y ⇤ ˆ z ⇤ ⇤ ˆ z ` ¯ ` ˆ y ` ¯ ` ✓ ˆ z ` ¯ ` = ˆ p { ⇤ b } ` ¯ ` ˆ y ` ¯ ` = ˆ n ⇥ ˆ p { ⇤ b } ` ¯ ` p⇡ ˆ z ⇤ ˆ x ⇤ ˆ y ⇤ b ⇤ b rest-frame⇤ rest-frame p⇡ ˆ z ⇤ ✓ b ˆ z { ⇤ } ⇤ = ˆ p { ⇤ } ` ¯ ` ˆ x ⇤ ⇤ b rest-frame ˆ x ` ¯ ` ` ¯ ` ˆ x ⇤ , ˆ x ` ¯ ` ˆ y ⇤ b ˆ y ` ¯ ` ` ¯ ` ⦿ ˆ p { ⇤ b } ⇤ ⇤ b rest-frame Figure 1: The Λ b → Λ(cid:96) + (cid:96) − decay is described by five angles: the angle, θ , betweenthe direction of the Λ baryon and the normal vector ˆ n in the Λ b rest-frame; and twosets of helicity angles, describing the decays of the Λ baryon ( θ b , φ b ) and the dileptonsystem ( θ l , φ l ). For transverse production polarisation ˆ n is chosen to be ˆ p Λ b × ˆ p beam .The helicity angles are then defined with respect to this normal vector through thecoordinate systems (ˆ x Λ , ˆ y Λ , ˆ z Λ ) and (ˆ x (cid:96) ¯ (cid:96) , ˆ y (cid:96) ¯ (cid:96) , ˆ z (cid:96) ¯ (cid:96) ). The ˆ z axis points in the direction ofthe Λ /dilepton system in the Λ b rest-frame. The angle between the two decay planes inthe Λ b rest frame is χ = φ l + φ b . The angles θ l , θ b and χ are sufficient to parameterisethe angular distribution of the decay in the case of zero production polarisation4 .1 Lepton system amplitudes There are two sets of amplitudes for the dilepon system, with either a vector or anaxial-vector current, h V,Jλ ,λ = ¯ (cid:96) ( λ ) γ µ (cid:96) ( λ ) ε ∗ µ ( λ − λ ) h A,Jλ ,λ = ¯ (cid:96) ( λ ) γ µ γ (cid:96) ( λ ) ε ∗ µ ( λ − λ ) , (3)where γ µ is a Dirac γ -matrix and ε µ is a polarisation vector. These amplitudes evaluateto [1] h V, / , +1 / = 0 , h A, / , +1 / = 2 m l = (cid:113) q (1 − β l ) ,h V, / , − / = 0 , h A, / , − / = 0 , (4) h V, / , +1 / = 2 m l = (cid:113) q (1 − β l ) , h A, / , +1 / = 0 ,h V, / , − / = − (cid:112) q , h A, / , − / = (cid:112) q β l , where m l is the lepton mass and β l is the lepton velocity in the dilepton rest frame( | (cid:126)p l | /E l ), i.e. β l = (cid:115) − m l q . (5)The amplitudes with J = 0 vanish in the case that the lepton mass is zero (when β l = 1).Under the Parity transformation h V,J − λ , − λ = h V,Jλ ,λ h A,J − λ , − λ = − h A,Jλ ,λ . (6) On the hadron side, the Λ decay amplitudes can be expressed in terms of the well known Λ asymmetry parameter [26] α Λ = | h b , | − | h b − , | | h b , | + | h b − , | = 0 . ± . . (7)The hadron side amplitudes are normalised such that | h b , | + | h b − , | = 1 . (8)5 .3 Helicity and transversity amplitudes After replacing the lepton and hadron-side amplitudes with the expressions given inSecs. 2.2 and 2.1, the angular distribution can be expanded in terms of 10 helicityamplitudes, H m, / , +1 , H m, − / , − , H m, / , , H m, − / , , H A, / , and H A, − / , ,P Λ b , α Λ and a set of kinematic factors that come from the lepton-side amplitudes. Forthe remainder of this paper it is convenient to absorb a common factor of (cid:112) q from thelepton-side amplitudes into these helicity amplitudes, i.e. (cid:112) q H m,Jλ Λ ,λ (cid:96)(cid:96) = H (cid:48) m,Jλ Λ ,λ (cid:96)(cid:96) . (9)By absorbing this factor, the only kinematic dependence outside of H (cid:48) m,Jλ Λ ,λ (cid:96)(cid:96) ( q ) comesfrom factors of β l .The helicity amplitudes can be replaced by a corresponding set of transversity am-plitudes for the decay that separate the vector and axial-vector contributions on thehadron-side: the amplitudes A R , L (cid:107) and A R , L (cid:107) depend only on the vector contribution to H (cid:48) λ Λ ,λ (cid:96)(cid:96) ( i.e. on (cid:104) Λ | ¯ sγ µ b | Λ b (cid:105) ); and the amplitudes A R , L ⊥ and A R , L ⊥ depend only on theaxial-vector contribution to H (cid:48) λ Λ ,λ (cid:96)(cid:96) ( i.e. on (cid:104) Λ | ¯ sγ µ γ b | Λ b (cid:105) ). To do this, we start byre-writing the original helicity amplitudes as H (cid:48) { R , L } ,Jλ Λ ,λ (cid:96)(cid:96) = 1 √ (cid:16) H (cid:48) V,Jλ Λ ,λ (cid:96)(cid:96) ± H (cid:48) A,Jλ Λ ,λ (cid:96)(cid:96) (cid:17) , (10)where the indices L and R refer to left- and right-handed chiralities of the dileptonsystem, respectively. This is followed by the replacements A { R , L }⊥ = 1 √ (cid:16) H (cid:48) { R , L } , / , +1 − H (cid:48) { R , L } , − / , − (cid:17) ,A { R , L }(cid:107) = 1 √ (cid:16) H (cid:48) { R , L } , / , +1 + H (cid:48) { R , L } , − / , − (cid:17) ,A { R , L }⊥ = 1 √ (cid:16) H (cid:48) { R , L } , / , − H (cid:48) { R , L } , − / , (cid:17) ,A { R , L }(cid:107) = 1 √ (cid:16) H (cid:48) { R , L } , / , + H (cid:48) { R , L } , − / , (cid:17) ,A ⊥ t = 1 √ (cid:16) H (cid:48) A, / , − H (cid:48) A, − / , (cid:17) ,A (cid:107) t = 1 √ (cid:16) H (cid:48) A, / , + H (cid:48) A, − / , (cid:17) . (11)Here, the subscript t refers to the time-like polarisation vector of the dilepton system.6 Observables
Expanding out the sum in Eq. 1, gives 34 different angular termsd Γd q d (cid:126) Ω = 332 π (cid:16) (cid:88) i =0 K i ( q ) f i ( (cid:126) Ω) (cid:17) d Γd q d (cid:126) Ω = 332 π (cid:16) (cid:0) K sin θ l + K cos θ l + K cos θ l (cid:1) + (cid:0) K sin θ l + K cos θ l + K cos θ l (cid:1) cos θ b +( K sin θ l cos θ l + K sin θ l ) sin θ b cos ( φ b + φ l ) +( K sin θ l cos θ l + K sin θ l ) sin θ b sin ( φ b + φ l ) + (cid:0) K sin θ l + K cos θ l + K cos θ l (cid:1) cos θ + (cid:0) K sin θ l + K cos θ l + K cos θ l (cid:1) cos θ b cos θ +( K sin θ l cos θ l + K sin θ l ) sin θ b cos ( φ b + φ l ) cos θ +( K sin θ l cos θ l + K sin θ l ) sin θ b sin ( φ b + φ l ) cos θ +( K cos θ l sin θ l + K sin θ l ) sin φ l sin θ +( K cos θ l sin θ l + K sin θ l ) cos φ l sin θ +( K cos θ l sin θ l + K sin θ l ) sin φ l cos θ b sin θ +( K cos θ l sin θ l + K sin θ l ) cos φ l cos θ b sin θ + (cid:0) K cos θ l + K sin θ l (cid:1) sin θ b sin φ b sin θ + (cid:0) K cos θ l + K sin θ l (cid:1) sin θ b cos φ b sin θ + (cid:0) K sin θ l (cid:1) sin θ b cos (2 φ l + φ b ) sin θ + (cid:0) K sin θ l (cid:1) sin θ b sin (2 φ l + φ b ) sin θ (cid:17) . (12)Integrating this expression over (cid:126) Ω yields the differential decay rate as a function of q ,dΓd q = 2 K + K . (13)This can be used to define a set of normalised angular observables M i = K i K + K . (14)7 Angular terms
The first ten angular terms are K = (cid:16) | A L (cid:107) | + | A L ⊥ | + | A R (cid:107) | + | A R ⊥ | (cid:17) + (1 + β l ) (cid:16) | A L (cid:107) | + | A L ⊥ | + | A R (cid:107) | + | A R ⊥ | (cid:17) + (1 − β l )Re (cid:16) A R (cid:107) A ∗ L (cid:107) + A R ⊥ A ∗ L ⊥ + A R (cid:107) A ∗ L (cid:107) + A R ⊥ A ∗ L ⊥ (cid:17) + (1 − β l ) (cid:16) | A (cid:107) t | + | A ⊥ t | (cid:17) ,K = (1 + β l ) (cid:16) | A R (cid:107) | + | A R ⊥ | + | A R (cid:107) | + | A L ⊥ | (cid:17) + (1 − β l ) (cid:16) | A R (cid:107) | + | A R ⊥ | + | A L (cid:107) | + | A L ⊥ | (cid:17) + (1 − β l )Re (cid:16) A R (cid:107) A ∗ L (cid:107) + A R ⊥ A ∗ L ⊥ + A R (cid:107) A ∗ L (cid:107) + A R ⊥ A ∗ L ⊥ (cid:17) + (1 − β l ) (cid:16) | A (cid:107) t | + | A ⊥ t | (cid:17) ,K = − β l Re (cid:16) A R ⊥ A ∗ R (cid:107) − A L ⊥ A ∗ L (cid:107) (cid:17) K = α Λ Re (cid:16) A R ⊥ A ∗ R (cid:107) + A L ⊥ A ∗ L (cid:107) (cid:17) + α Λ (1 + β l )Re (cid:16) A R ⊥ A ∗ R (cid:107) + A L ⊥ A ∗ L (cid:107) (cid:17) + α Λ (1 − β l )Re (cid:16) A R ⊥ A ∗ L (cid:107) + A R (cid:107) A ∗ L ⊥ + A R ⊥ A ∗ L (cid:107) + A R (cid:107) A ∗ L ⊥ (cid:17) + α Λ (1 − β l )Re (cid:16) A ⊥ t A ∗(cid:107) t (cid:17) ,K = α Λ (1 + β l )Re (cid:16) A R ⊥ A ∗ R (cid:107) + A L ⊥ A ∗ L (cid:107) (cid:17) + α Λ (1 − β l )Re (cid:16) A R (cid:107) A ∗ R ⊥ + A L (cid:107) A ∗ L ⊥ (cid:17) + α Λ (1 − β l )Re (cid:16) A R ⊥ A ∗ L (cid:107) + A R (cid:107) A ∗ L ⊥ + A R ⊥ A ∗ L (cid:107) + A R (cid:107) A ∗ L ⊥ (cid:17) + α Λ (1 − β l )Re (cid:16) A ⊥ t A ∗(cid:107) t (cid:17) ,K = − α Λ β l (cid:16) | A R (cid:107) | + | A R ⊥ | − | A L (cid:107) | − | A L ⊥ | (cid:17) ,K = √ α Λ β l Re (cid:16) A R ⊥ A ∗ R (cid:107) − A R (cid:107) A ∗ R ⊥ + A L ⊥ A ∗ L (cid:107) − A L (cid:107) A ∗ L ⊥ (cid:17) ,K = √ α Λ β l Re (cid:16) A R ⊥ A ∗ R ⊥ − A R (cid:107) A ∗ R (cid:107) − A L ⊥ A ∗ L ⊥ + A L (cid:107) A ∗ L (cid:107) (cid:17) ,K = √ α Λ β l Im (cid:16) A R ⊥ A ∗ R ⊥ − A R (cid:107) A ∗ R (cid:107) + A L ⊥ A ∗ L ⊥ − A L (cid:107) A ∗ L (cid:107) (cid:17) ,K = √ α Λ β l Im (cid:16) A R ⊥ A ∗ R (cid:107) − A R (cid:107) A ∗ R ⊥ − A L ⊥ A ∗ L (cid:107) + A L (cid:107) A ∗ L ⊥ (cid:17) . (15)These terms are accessible even if the Λ b baryon is unpolarised and have been previouslystudied in Refs. [2, 27]. There is a straightforward relationship between our observables8nd those of Ref. [2], with K ss = K , K cc = K , K c = K , K ss = K , K cc = K , K c = K , K sc = K , K s = K , K sc = K and K s = K .The remaining 24 terms are only non-vanishing if P Λ b is non-zero. Terms K through K have a similar dependence to K through K . These are K = − P Λ b Re (cid:16) A R (cid:107) A ∗ R ⊥ + A L (cid:107) A ∗ L ⊥ (cid:17) + P Λ b (1 + β l )Re (cid:16) A R (cid:107) A ∗ R ⊥ + A L (cid:107) A ∗ L ⊥ (cid:17) − P Λ b (1 − β l )Re (cid:16) A R (cid:107) A ∗ L ⊥ + A R ⊥ A ∗ L (cid:107) − A R (cid:107) A ∗ L ⊥ − A R ⊥ A ∗ L (cid:107) (cid:17) + P Λ b (1 − β l )Re (cid:16) A (cid:107) t A ∗⊥ t (cid:17) ,K = − P Λ b (1 + β l )Re (cid:16) A R (cid:107) A ∗ R ⊥ + A L (cid:107) A ∗ L ⊥ (cid:17) + P Λ b (1 − β l )Re (cid:16) A R (cid:107) A ∗ R ⊥ + A L (cid:107) A ∗ L ⊥ (cid:17) − P Λ b (1 − β l )Re (cid:16) A R (cid:107) A ∗ L ⊥ + A R ⊥ A ∗ L (cid:107) − A R (cid:107) A ∗ L ⊥ − A R ⊥ A ∗ L (cid:107) (cid:17) + P Λ b (1 − β l )Re (cid:16) A (cid:107) t A ∗⊥ t (cid:17) ,K = P Λ b β l (cid:16) | A R (cid:107) | + | A R ⊥ | − | A L (cid:107) | − | A L ⊥ | (cid:17) ,K = − α Λ P Λ b (cid:16) | A R (cid:107) | + | A R ⊥ | + | A L (cid:107) | + | A L ⊥ | (cid:17) + α Λ P Λ b (1 + β l ) (cid:16) | A R (cid:107) | + | A R ⊥ | + | A L (cid:107) | + | A L ⊥ | (cid:17) + α Λ P Λ b (1 − β l ) (cid:16) | A (cid:107) t | + | A ⊥ t | (cid:17) − α Λ P Λ b (1 − β l )Re (cid:16) A R (cid:107) A ∗ L (cid:107) + A R ⊥ A ∗ L ⊥ − A R (cid:107) A ∗ L (cid:107) − A R ⊥ A ∗ L ⊥ (cid:17) ,K = − α Λ P Λ b (1 + β l ) (cid:16) | A R (cid:107) | + | A R ⊥ | + | A L (cid:107) | + | A L ⊥ | (cid:17) + α Λ P Λ b (1 − β l ) (cid:16) | A R (cid:107) | + | A R ⊥ | + | A L (cid:107) | + | A L ⊥ | (cid:17) − α Λ P Λ b (1 − β l )Re (cid:16) A R (cid:107) A ∗ L (cid:107) + A R ⊥ A ∗ L ⊥ − A R (cid:107) A ∗ L (cid:107) − A R ⊥ A ∗ L ⊥ (cid:17) + α Λ P Λ b (1 − β l ) (cid:16) | A (cid:107) t | + | A ⊥ t | (cid:17) ,K = α Λ P Λ b β l Re (cid:16) A R ⊥ A ∗ R (cid:107) − A L ⊥ A ∗ L (cid:107) (cid:17) . (16)The observables K and K are trivially related to K and K through K = − P Λ b K and K = − P Λ b K and can therefore be used as an experimental consistency check orto determine P Λ b . The observables K , K , K and K have a similar structure to K , K , K and K but, unlike in those observables, the amplitudes with λ (cid:96)(cid:96) = 0 enterwith a different relative sign to those with λ (cid:96)(cid:96) = ± K through K also involve new combinations of amplitudes that9re not accessible if the Λ b baryon is unpolarised. They are K = − √ α Λ P Λ b β l Re (cid:16) A R (cid:107) A ∗ R (cid:107) − A R ⊥ A ∗ R ⊥ + A L (cid:107) A ∗ L (cid:107) − A L ⊥ A ∗ L ⊥ (cid:17) ,K = − √ α Λ P Λ b β l Re (cid:16) A R (cid:107) A ∗ R ⊥ − A R ⊥ A ∗ R (cid:107) − A L (cid:107) A ∗ L ⊥ + A L ⊥ A ∗ L (cid:107) (cid:17) ,K = − √ α Λ P Λ b β l Im (cid:16) A R (cid:107) A ∗ R ⊥ − A R ⊥ A ∗ R (cid:107) + A L (cid:107) A ∗ L ⊥ − A L ⊥ A ∗ L (cid:107) (cid:17) ,K = − √ α Λ P Λ b β l Im (cid:16) A R (cid:107) A ∗ R (cid:107) − A R ⊥ A ∗ R ⊥ − A L (cid:107) A ∗ L (cid:107) + A L ⊥ A ∗ L ⊥ (cid:17) ,K = √ P Λ b β l Im (cid:16) A R (cid:107) A ∗ R (cid:107) + A R ⊥ A ∗ R ⊥ + A L (cid:107) A ∗ L (cid:107) + A L ⊥ A ∗ L ⊥ (cid:17) ,K = − √ P Λ b β l Im (cid:16) A R (cid:107) A ∗ R ⊥ + A R ⊥ A ∗ R (cid:107) − A L (cid:107) A ∗ L ⊥ − A L ⊥ A ∗ L (cid:107) (cid:17) ,K = − √ P Λ b β l Re (cid:16) A R (cid:107) A ∗ R ⊥ + A R ⊥ A ∗ R (cid:107) + A L (cid:107) A ∗ L ⊥ + A L ⊥ A ∗ L (cid:107) (cid:17) ,K = √ P Λ b β l Re (cid:16) A R (cid:107) A ∗ R (cid:107) + A R ⊥ A ∗ R ⊥ − A L (cid:107) A ∗ L (cid:107) − A L ⊥ A ∗ L ⊥ (cid:17) ,K = √ α Λ P Λ b β l Im (cid:16) A R (cid:107) A ∗ R ⊥ + A R ⊥ A ∗ R (cid:107) + A L (cid:107) A ∗ L ⊥ + A L ⊥ A ∗ L (cid:107) (cid:17) ,K = − √ α Λ P Λ b β l Im (cid:16) A R (cid:107) A ∗ R (cid:107) + A R ⊥ A ∗ R ⊥ − A L (cid:107) A ∗ L (cid:107) − A L ⊥ A ∗ L ⊥ (cid:17) ,K = − √ α Λ P Λ b β l Re (cid:16) A R (cid:107) A ∗ R (cid:107) + A R ⊥ A ∗ R ⊥ + A L (cid:107) A ∗ L (cid:107) + A L ⊥ A ∗ L ⊥ (cid:17) ,K = √ α Λ P Λ b β l Re (cid:16) A R (cid:107) A ∗ R ⊥ + A R ⊥ A ∗ R (cid:107) − A L (cid:107) A ∗ L ⊥ − A L ⊥ A ∗ L (cid:107) (cid:17) ,K = α Λ P Λ b (1 − β l )Im (cid:16) A R ⊥ A ∗ R (cid:107) + A L ⊥ A ∗ L (cid:107) + A R ⊥ A ∗ L (cid:107) − A R (cid:107) A ∗ L ⊥ (cid:17) + α Λ P Λ b (1 − β l )Im (cid:16) A ⊥ t A ∗(cid:107) t (cid:17) ,K = α Λ P Λ b (1 + β l )Im (cid:16) A R ⊥ A ∗ R (cid:107) + A L ⊥ A ∗ L (cid:107) (cid:17) + α Λ P Λ b (1 − β l )Im (cid:16) A R ⊥ A ∗ L (cid:107) − A R (cid:107) A ∗ L ⊥ (cid:17) + α Λ P Λ b (1 − β l )Im (cid:16) A ⊥ t A ∗(cid:107) t (cid:17) ,K = α Λ P Λ b (1 − β l ) (cid:16) | A R ⊥ | − | A R (cid:107) | + | A L ⊥ | − | A L (cid:107) | (cid:17) + α Λ P Λ b (1 − β l )Re (cid:16) A R ⊥ A ∗ L ⊥ − A R (cid:107) A ∗ L (cid:107) (cid:17) + α Λ P Λ b (1 − β l ) (cid:16) | A ⊥ t | − | A (cid:107) t | (cid:17) ,K = α Λ P Λ b (1 + β l ) (cid:16) | A R ⊥ | + | A L ⊥ | − | A R (cid:107) | − | A L (cid:107) | (cid:17) + α Λ P Λ b (1 − β l )Re (cid:16) A R ⊥ A ∗ L ⊥ − A R (cid:107) A ∗ L (cid:107) (cid:17) + α Λ P Λ b (1 − β l ) (cid:16) | A ⊥ t | − | A (cid:107) t | (cid:17) ,K = α Λ P Λ b β l (cid:16) | A R ⊥ | − | A R (cid:107) | + | A L ⊥ | − | A L (cid:107) | (cid:17) ,K = α Λ P Λ b β l Im (cid:16) A R ⊥ A ∗ R (cid:107) + A L ⊥ A ∗ L (cid:107) (cid:17) . (17)10he angular terms K and K are zero in the massless lepton limit. Λ b → J/ψ Λ
The angular distribution of the Λ b → J/ψ Λ decay is a limiting case of Eq. 1, with a purevector current in the dilepton system. In this limit, the expression collapses to the onegiven in Refs. [28, 29] with β l ∼
1. The amplitudes a ± and b ± in Ref. [28, 29] are relatedto the ones in this paper by a − = H (cid:48) V, − / , , a + = H (cid:48) V, / , ,b − = H (cid:48) V, / , +1 , b + = H (cid:48) V, − / , − . (18) The values of the normalised angular observables can be determined experimentally froman analysis of the moments of the angular distribution, M i = 332 π (cid:90) (cid:88) j =0 M j f j ( (cid:126) Ω) g i ( (cid:126) Ω)d (cid:126)
Ω (19)if the weighting functions g i ( (cid:126) Ω) are chosen such that they satisfy (cid:90) f j ( (cid:126) Ω) g i ( (cid:126) Ω)d (cid:126)
Ω = (cid:18) π (cid:19) δ ij . (20)In this case, the moments can be extracted from data using Monte Carlo integration.The statistical uncertainty and correlation between the moments can be determinedfrom the single sample covariance or by bootstrapping the measurement (see for exampleRef. [30]).The weighting functions for M – M are g ( (cid:126) Ω) = (3 − θ l ) , g ( (cid:126) Ω) = 3 cos θ l cos θ b , (21) g ( (cid:126) Ω) = (5 cos θ l − , g ( (cid:126) Ω) = cos θ l sin θ l sin θ b cos( φ l + φ b ) ,g ( (cid:126) Ω) = cos θ l , g ( (cid:126) Ω) = sin θ l sin θ b cos( φ l + φ b ) ,g ( (cid:126) Ω) = (3 − θ l ) cos θ b , g ( (cid:126) Ω) = cos θ l sin θ l sin θ b sin( φ l + φ b ) ,g ( (cid:126) Ω) = (5 cos θ l −
1) cos θ b , g ( (cid:126) Ω) = sin θ l sin θ b sin( φ l + φ b ) . These weighting functions have been previously derived in Ref. [24]. The weightingfunctions for the polarisation-dependent terms can be derived in a similar manner, they11re g ( (cid:126) Ω) = (3 − θ l ) cos θ , g ( (cid:126) Ω) = cos θ l sin θ l sin θ cos φ l ,g ( (cid:126) Ω) = (5 cos θ l −
1) cos θ , g ( (cid:126) Ω) = sin θ sin θ l cos φ l , (22) g ( (cid:126) Ω) =3 cos θ l cos θ , g ( (cid:126) Ω) = cos θ l sin θ l cos θ b sin θ sin φ l ,g ( (cid:126) Ω) = (3 − θ l ) cos θ b cos θ , g ( (cid:126) Ω) = sin θ sin θ l cos θ b sin φ l ,g ( (cid:126) Ω) = (5 cos θ l −
1) cos θ b cos θ , g ( (cid:126) Ω) = cos θ l sin θ l cos θ b sin θ cos φ l ,g ( (cid:126) Ω) =9 cos θ cos θ l cos θ b , g ( (cid:126) Ω) = sin θ sin θ l cos θ b cos φ l ,g ( (cid:126) Ω) = cos θ l sin θ l sin θ b cos θ cos( φ l + φ b ) , g ( (cid:126) Ω) = (5 cos θ l −
1) sin θ b sin θ sin φ b ,g ( (cid:126) Ω) = sin θ l sin θ b cos θ cos( φ l + φ b ) , g ( (cid:126) Ω) = (3 − θ l ) sin θ b sin θ sin φ b ,g ( (cid:126) Ω) = cos θ l sin θ l sin θ b cos θ sin( φ l + φ b ) , g ( (cid:126) Ω) = (5 cos θ l −
1) sin θ b sin θ cos φ b ,g ( (cid:126) Ω) = sin θ l sin θ b cos θ sin( φ l + φ b ) , g ( (cid:126) Ω) = (3 − θ l ) sin θ b sin θ cos φ b ,g ( (cid:126) Ω) = cos θ l sin θ l sin θ sin φ l , g ( (cid:126) Ω) = sin θ b sin θ cos(2 φ l + φ b ) ,g ( (cid:126) Ω) = sin θ sin θ l sin φ l , g ( (cid:126) Ω) = sin θ b sin θ sin(2 φ l + φ b ) . The weighting functions are not unique and a more compact set can be formed by ex-ploiting the fact that the integral of sin θ b over dcos θ b is π/ e.g. the weighting functionsfor M and M can be written in a shorter form as g ( (cid:126) Ω) = π sin θ cos ( φ b + 2 φ l ) ,g ( (cid:126) Ω) = π sin θ sin ( φ b + 2 φ l ) . (23)More compact expressions can also be found for many of the other observables. Note,the different sets of weighting functions can lead to different experimental precision onthe normalised moments. In general, the longer form of the weighting functions providesthe best precision. In order to describe the SM contribution to the decay amplitudes, an effective field theoryapproach is used. The Hamiltonian for the decay is factorised into local four-fermionoperators and Wilson coefficients (see for example Ref. [31]). The Wilson coefficientsdescribe the short-distance contributions from the heavy SM particles.Numerical values for the SM predictions, in the case that P Λ b = 1, are provided inAppendix B in two q ranges: at large hadronic recoil, in the range 1 < q < /c ,and at low hadronic recoil, in the range 15 < q <
20 GeV /c . To evaluate SM pre-dictions for the different angular observables we use the EOS flavour tool [32]. At lowhadronic recoil, the SM calculations employ an operator product expansion of the four-quark contributions to the matrix element in powers of Λ QCD / (cid:112) q [33]. At large recoil, EOS uses some of the known α s corrections to charm loop processes. However, potentially12arge contributions from hard spectator scattering [34] and soft gluon emission [35] areneglected. The form-factors for the Λ b → Λ transition are taken from a recent LatticeQCD calculation in Ref. [27]. These form-factors enable the observables to be computedwith high-precision. The form-factors at large hadronic recoil have also been calculatedin the framework of light-cone-sum-rules, see for example Refs. [36] and [37]. The SMWilson coefficients are computed in EOS to NNLO in QCD. The Λ b lifetime and CKMmatrix elements are taken from the latest experimental values [26]. The quark massesare taken in the MS scheme.Tables 2 and 3 in Appendix B also provide 68% confidence level intervals for theSM predictions. To evaluate these intervals: the form-factors from Ref. [27] have beenvaried within their full covariance matrix; the Λ b lifetime, the Λ asymmetry parameterand CKM matrix elements are varied within their experimental precision [26, 38]; thescale dependence of Wilson coefficients C i ( µ ) is explored by varying the scale, µ , in therange m b / < µ < m b ; and in keeping with Ref. [39] a 3% correction to the amplitudesfrom hadronic matrix elements is considered (see also Ref. [40]). At low hadronic recoil the observables are precisely predicted in the SM. The uncertain-ties on the predictions are worse at large recoil, where a large extrapolation in q of theform-factors is needed. Figures 2–9 in Appendix C demonstrate how the observablesdepend on NP contributions to the Wilson coefficients. In the large-recoil region thereis sensitivity to C NP9 from both the polarised and unpolarised observables. Interestingly,the observables M and M can also distinguish between two of the possibilities thatare favoured by global fits to b → s(cid:96) + (cid:96) − processes: where C NP9 (cid:39) − C NP10 = 0and where C NP9 = − C NP10 (cid:39) − C NP9 isreduced.In Ref. [2], the authors point out that the observables at low hadronic recoil placeconstraints on six combinations of Wilson coefficients ρ ± = | C V ± C (cid:48) V | + | C ± C (cid:48) | ρ = Re (cid:0) C V C ∗ − C (cid:48) V C (cid:48)∗ (cid:1) − i Im (cid:0) C V C (cid:48)∗ V + C C (cid:48)∗ (cid:1) ρ ± = 2Re (cid:0) ( C V ± C (cid:48) V )( C ± C (cid:48) ) ∗ (cid:1) ρ = | C V | − | C (cid:48) V | + | C | − | C (cid:48) | − i Im (cid:0) C V C ∗ − C (cid:48) V C (cid:48)∗ (cid:1) , (24)where C V contains contributions from C and C . The primed coefficients correspondto right-handed currents whose contribution is vanishingly small in the SM. The short-distance dependence of K – K on ρ ± , ρ ± , ρ and ρ is provided for completeness inAppendix D.If the Λ b is unpolarised, the decay rate is insensitive to the short-distance contributionIm( ρ ) but provides sensitivity to ρ ± , Re( ρ ), ρ ± , Re( ρ ) and Im( ρ ). The polarisedobservables also depend on these short-distance contributions but have different form-factor dependencies. This permits a new set of checks of the OPE and the form-factors.The short-distance combination Im( ρ ) can also be determined from M , M , M and13 . Furthermore, in K – K the short-distance contributions ρ +1 and ρ − always appeartogether as a sum. Using the polarised observables , ρ +1 and ρ − can be separated, e.g. by using K + 2 α Λ P Λ b K = 16 s − | f V ⊥ | ρ +1 ,K − α Λ P Λ b K = 16 s + | f A ⊥ | ρ − , (25)where f V ⊥ and f A ⊥ are helicity form-factors (see for example Ref. [44]). A similar trickcan be used to separate ρ +3 and ρ − using K and K . It is also possible to form newshort-distance relationships, in which the form-factors cancel by taking ratios of the K i , K K = 2 Re( ρ )Im( ρ ) , K K = − Im( ρ )Im( ρ ) , K K = − Re( ρ )Im( ρ ) P Λ b . (26)The short-distance combinations ρ and ρ can then be determined up-to their overallnormalisation, independent of the hadronic form-factors, using Eq. 26 and the relation-ship K K = − α Λ Re( ρ )Re( ρ ) (27)from Ref. [2]. Similarly, one can form short-distance relationships that depend only on ρ ± and ρ ± P Λ b K + α Λ K K − K = − ρ − ρ − , P Λ b K − α Λ K K + K = ρ +3 ρ +1 . (28)Alternatively, it is possible to form ratios that depend only on the form-factors and noton the short-distance physics. For example, K K = 12 (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f V ⊥ − ( m Λ b − m Λ ) (cid:112) q f A f A ⊥ (cid:33) ,K K = 12 (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f V ⊥ + ( m Λ b − m Λ ) (cid:112) q f A f A ⊥ (cid:33) P Λ b (29)allow the ratios f V /f V ⊥ and f A /f A ⊥ to be determined independent of the ρ i . At very large hadronic recoil ( q (cid:28) /c ), the angular distribution of the Λ b → Λµ + µ − decay is sensitive primarily to the Wilson coefficients C and C (cid:48) due to a pole-likeenhancement of the amplitudes. The observable K is proportional to Re( C C (cid:48) ) andcan therefore provide a null test of the size of C (cid:48) (in the same way as the S observablein the B → K ∗ µ + µ − decay). In this case, however, the observable is suppressed bythe size of P Λ b . 14 Expected experimental precision
Table 1 indicates the typical precision on the angular moments that could be achievedat the LHCb experiment. The experimental precision has been estimated using pseudo-experiments corresponding approximately to the expected signal yield in the current andin a future LHCb dataset. Experimental backgrounds and non-uniform angular accep-tance have been neglected in this estimate. However, these are expected to have only asmall impact on the experiments sensitivity. The sensitivity that can be achieved withthe large datasets that will be available at an upgraded LHCb experiment is interestingevent for modest values of P Λ b .Table 1: Expected experimental precision on the angular moments of the Λ b → Λµ + µ − decay at the LHCb experiment. The four columns correspond to: the observed yield of300 Λ b → Λµ + µ − candidates with 15 < q <
20 GeV /c in the LHC run 1 dataset [23];an expected yield of ∼ ∼ − of integrated luminosity with an upgraded LHCbexperiment; and an expected yield of ∼
50 000 candidates in 300 fb − with the proposedLHCb phase II upgrade.Obs. Run 1 Run 2 Upgrade Phase II Obs. Run 1 Run 2 Upgrade Phase II M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M In this paper we have derived an expression for the angular distribution of the Λ b → Λµ + µ − in the case of non-zero production polarisation. This extends the number ofobservables in the decay from 10 to 34. These observables can be determined from15oments of the Λ b → Λµ + µ − angular distribution. Explicit expressions have beenprovided for the observables in terms of the angular moments to enable an experimentto determine the new observables from their dataset. A phenomenological analysis hasalso been performed to illustrate how these observables might vary in extensions of theStandard Model. The analysis shows that there is interesting new sensitivity that canbe gained if the Λ b baryon is produced polarised.
10 Acknowledgements
We would like to thank Danny Van Dyk, Georgios Chatzikonstantinidis, Tim Gershonand Nigel Watson for their useful feedback on the manuscript. We would also like tothank Danny Van Dyk for his help in implementing the observables in the
EOS flavourtool. T. B. acknowledges support from the Royal Society (United Kingdom). M. K.acknowledges support from the Science & Technology Facilities Council (United King-dom). 16 ppendicesA Wigner D -functions The Wigner D -functions are D Jm (cid:48) ,m ( α, β, γ ) = e − im (cid:48) α d Jm (cid:48) ,m ( β ) e − imγ (30)where the α , β and γ correspond to the Euler rotation angles needed to rotate betweenthe reference frame of the mother particle and the helicity frame of its daughters. Therelevant small d -functions are d / / , / ( β ) = cos( β/ ,d / / , − / ( β ) = − sin( β/ ,d , ( β ) = cos ( β/ ,d , − ( β ) = sin ( β/ ,d , ( β ) = cos( β/
2) sin( β/ ,d , ( β ) = cos( β ) , (31)with d Jm (cid:48) ,m ( β ) = d J − m, − m (cid:48) ( β ) = ( − m − m (cid:48) d Jm,m (cid:48) ( β ) . (32) B Numerical results
Standard Model predictions for the angular observables with P Λ b = 1 are provided inTables 2 and 3. Predictions are provided in two q ranges: at large hadronic recoil, in therange 1 < q < /c , and at low hadronic recoil, in the range 15 < q <
20 GeV /c .The SM predictions are evaluated using the EOS flavour-tool. For any other choice of P Λ b , predictions for M – M can be achieved by multiplying the values in Tables 2 and3 by the new value of P Λ b . 17able 2: Predictions from EOS for the angular observables of the Λ b → Λµ + µ − decaywith P Λ b = 1 in the range 1 < q < /c . The SM calculation is describedin the text. The observables M and M vanish due to the small size of the muonmass. Observables that depend on the imaginary part of the product of two transversityamplitudes also tend to be vanishingly small, due to the small strong phase differencebetween pairs of amplitudes in the SM.Obs. Value 68% interval Obs. Value 68% interval M .
459 [0 . , . M .
000 [ − . , . M .
081 [0 . , . M − .
025 [ − . , − . M − .
005 [ − . , − . M − .
003 [ − . , . M − .
280 [ − . , − . M .
002 [0 . , . M − .
045 [ − . , − . M .
002 [0 . , . M − .
366 [ − . , − . M − .
147 [ − . , − . M .
071 [0 . , . M .
132 [0 . , . M .
001 [ − . , . M − .
001 [ − . , − . M .
243 [0 . , . M .
004 [0 . , . M − .
052 [ − . , − . M .
089 [0 . , . M .
003 [0 . , . M − .
089 [ − . , − . M .
004 [ − . , . M .
000 [0 . , . M .
029 [0 . , . M .
000 [0 . , . M − .
001 [ − . , − . M .
000 [0 . , . M − .
003 [ − . , . M .
075 [0 . , . M .
002 [0 . , . M .
007 [0 . , . M − .
005 [ − . , − . M .
000 [ − . , . EOS for the angular observables of the Λ b → Λµ + µ − decaywith P Λ b = 1 in the range 15 < q <
20 GeV /c . The SM calculation is describedin the text. The observables M and M vanish due to the small size of the muonmass. Observables that depend on the imaginary part of the product of two transversityamplitudes also tend to be vanishingly small, due to the small strong phase differencebetween pairs of amplitudes in the SM.Obs. Value 68% interval Obs. Value 68% interval M .
351 [0 . , . M .
187 [0 . , . M .
298 [0 . , . M − .
022 [ − . , − . M − .
236 [ − . , − . M − .
100 [ − . , − . M − .
195 [ − . , − . M .
000 [0 . , . M − .
154 [ − . , − . M − .
001 [ − . , − . M − .
064 [ − . , − . M − .
299 [ − . , − . M .
240 [0 . , . M .
337 [0 . , . M − .
292 [ − . , − . M − .
001 [ − . , − . M .
034 [0 . , . M .
001 [0 . , . M − .
191 [ − . , − . M .
221 [0 . , . M .
151 [0 . , . M − .
187 [ − . , − . M .
102 [0 . , . M .
000 [0 . , . M .
021 [0 . , . M − .
001 [ − . , − . M .
000 [0 . , . M .
000 [0 . , . M − .
001 [ − . , − . M − .
046 [ − . , − . M .
000 [0 . , . M − .
053 [ − . , − . M − .
002 [ − . , − . M .
000 [0 . , . Variation of observables with NP contributions
Figures 2–9 show the variation of M – M under two possible modifications of the SMWilson coefficients: a scenario where there is a NP contribution to Re( C ) or Re( C );and a scenario where there is a NP contribution to Re( C ) or Re( C (cid:48) ). - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M Figure 2: Variation of the observables M – M of the Λ b → Λµ + µ − decay from theirSM central values in the large-recoil region (1 < q < /c ) with a NP contributionto Re( C ) or Re( C ). The SM point is at (0 , .2 - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M Figure 3: Variation of the polarisation dependent angular observables of the Λ b → Λµ + µ − decay from their SM central values in the large-recoil region (1 < q < /c )with a NP contribution to Re( C ) or Re( C ). The SM point is at (0 , P Λ b = 1 is used. 21 .2 - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M Figure 4: Variation of the observables M – M of the Λ b → Λµ + µ − decay from their SMcentral values in the low-recoil region (15 < q <
20 GeV /c ) with a NP contributionto Re( C ) or Re( C ). The SM point is at (0 , .2 - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M - - - - - ) NP9 C Re( - ) N P C R e ( M Figure 5: Variation of the polarisation dependent angular observables of the Λ b → Λµ + µ − decay from their SM central values in the low-recoil region (15 < q <
20 GeV /c ) with a NP contribution to Re( C ) or Re( C ). The SM point is at (0 , P Λ b = 1 is used.23 .2 - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M Figure 6: Variation of the observables M – M of the Λ b → Λµ + µ − decay from theirSM central values in the large-recoil region (1 < q < /c ) with a NP contributionto Re( C ) or Re( C (cid:48) ). The SM point is at (0 , .2 - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M Figure 7: Variation of the polarisation dependent angular observables of the Λ b → Λµ + µ − decay from their SM central values in the large-recoil region (1 < q < /c )with a NP contribution to Re( C ) or Re( C (cid:48) ). The SM point is at (0 , P Λ b = 1 is used. 25 .2 - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M Figure 8: Variation of the observables M – M of the Λ b → Λµ + µ − decay from their SMcentral values in the low-recoil region (15 < q <
20 GeV /c ) with a NP contributionto Re( C ) or Re( C (cid:48) ). The SM point is at (0 , .2 - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M - - - - - ) NP9 C Re( - ) N P ' C R e ( M Figure 9: Variation of the polarisation dependent angular observables of the Λ b → Λµ + µ − decay from their SM central values in the low-recoil region (15 < q <
20 GeV /c ) with a NP contribution to Re( C ) or Re( C (cid:48) ). The SM point is at (0 , P Λ b = 1 is used.27 Short-distance dependence at low hadronic recoil
In the limit of low hadronic recoil, and neglecting lepton mass dependent effects, the K i functions can be written in terms of the short-distance dependent ρ -functions of Ref. [2]as K = 4 s + (cid:18) | f A ⊥ | + ( m Λ b − m Λ ) q | f A | (cid:19) ρ − + 4 s − (cid:18) | f V ⊥ | + ( m Λ b + m Λ ) q | f V | (cid:19) ρ +1 ,K = 8 s + | f A ⊥ | ρ − + 8 s − | f V ⊥ | ρ +1 ,K = 32 √ s + s − f A ⊥ f V ⊥ Re( ρ ) ,K = − α Λ √ s + s − (cid:32) f A ⊥ f V ⊥ + ( m Λ b − m Λ ) q f V f A (cid:33) Re( ρ ) ,K = − α Λ √ s + s − f A ⊥ f V ⊥ Re( ρ ) ,K = − α Λ s + | f A ⊥ | ρ − − α Λ s − | f V ⊥ | ρ +3 ,K = − α Λ √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ − ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Re( ρ ) K = 8 s + α Λ ( m Λ b − m Λ ) (cid:112) q f A f A ⊥ ρ − − s − α Λ ( m Λ b + m Λ ) (cid:112) q f V f V ⊥ ρ +3 ,K = 16 α Λ √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ + ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Im( ρ ) , (33)28nd K = − P Λ b √ s + s − (cid:32) f A f V ( m Λ b − m Λ ) q − f A ⊥ f V ⊥ (cid:33) Re( ρ ) ,K = 32 P Λ b √ s + s − f A ⊥ f V ⊥ Re( ρ ) ,K = 8 P Λ b s + | f V ⊥ | ρ − + 8 P Λ b s − | f A ⊥ | ρ +3 ,K = − α Λ P Λ b s − (cid:18) | f V ⊥ | − | f V | ( m Λ b + m Λ ) q (cid:19) ρ +1 − α Λ P Λ b s + (cid:18) | f A ⊥ | − | f A | ( m Λ b − m Λ ) q (cid:19) ρ − ,K = − α Λ P Λ b s − | f V ⊥ | ρ +1 − α Λ P Λ b s + | f A ⊥ | ρ − ,K = − α Λ P Λ b √ s + s − f A ⊥ f V ⊥ Re( ρ ) ,K = − α Λ P Λ b s − ( m Λ b + m Λ ) (cid:112) q f V f V ⊥ ρ +1 + 8 α Λ P Λ b s + ( m Λ b − m Λ ) (cid:112) q f A f A ⊥ ρ − ,K = − α Λ P Λ b √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ − ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Re( ρ ) ,K = 16 α Λ P Λ b √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ + ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Im( ρ ) ,K = 16 P Λ b √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ − ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Im( ρ ) ,K = − P Λ b √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ + ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Re( ρ ) ,K = − P Λ b s − ( m Λ b + m Λ ) (cid:112) q f V f V ⊥ ρ +3 − P Λ b s + ( m Λ b − m Λ ) (cid:112) q f A f A ⊥ ρ − ,K = − α Λ P Λ b √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ − ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Im( ρ ) ,K = 8 α Λ P Λ b s − ( m Λ b + m Λ ) (cid:112) q f V f V ⊥ ρ +1 + 8 α Λ P Λ b s + ( m Λ b − m Λ ) (cid:112) q f A f A ⊥ ρ − ,K = 16 α Λ P Λ b √ s + s − (cid:32) ( m Λ b + m Λ ) (cid:112) q f V f A ⊥ + ( m Λ b − m Λ ) (cid:112) q f A f V ⊥ (cid:33) Re( ρ ) ,K = − α Λ P Λ b √ s + s − ( m Λ b − m Λ ) q f A f V Im( ρ ) ,K = 4 α Λ P Λ b s − ( m Λ b + m Λ ) q | f V | ρ +1 − α Λ P Λ b s + ( m Λ b − m Λ ) q | f A | ρ − ,K = 4 α Λ P Λ b s − | f V ⊥ | ρ +1 − α Λ P Λ b s + | f A ⊥ | ρ − ,K = − α Λ P Λ b √ s + s − f A ⊥ f V ⊥ Im( ρ ) . (34)29he remaining K i ’s vanish in the low-recoil and zero lepton mass limits. In Eqs. 33and 34: f V , f A , f V ⊥ and f A ⊥ are the vector and axial-vector helicity form-factors for the Λ b → Λ transition; m Λ b and m Λ are the masses of the Λ b and Λ baryon, respectively; and s ± = ( m Λ b ± m Λ ) − q . The four contributing tensor form-factors have been removedby exploiting Isgur-Wise relationships [44] to relate the tensor form-factors to the vectorand axial-vector form-factors. 30 eferences [1] T. Gutsche et al. , Rare baryon decays Λ b → Λ l + l − ( l = e, µ, τ ) and Λ b → Λ γ :differential and total rates, lepton- and hadron-side forward-backward asymmetries ,Phys. Rev. D87 (2013) 074031, arXiv:1301.3737 .[2] P. B¨oer, T. Feldmann, and D. van Dyk,
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