Angular Analysis of the Decay

Transcription

1 Angular Analysis of the Decay Λ b Λ[ Nπ]l + l Philipp Böer, Thorsten Feldmann, and Danny van Dyk Naturwissenschaftlich-Technische Fakultät - Theoretische Physik 1 Universität Siegen 4th Workshop on flavour symmetries and consequences in accelerators and cosmology, Brighton q et f FOR 1873 quark flavour physics and effective field theories D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

2 Effective Field Theory Flavor Changing Neutral Current (FCNC at parton level: Λ b ˆ= (u d b Λ ˆ= (u d s handle in effective theory (all fields heavier than b are integrated out matrix elements of operators O i represent physics below µ m b Wilson coefficients C i C i (M W, M Z, m t,... represent physics above µ m b Effective Hamiltonian H = 4G F 2 [V tb V ts i ] C i O i + O (V ub Vus + h.c. D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

3 Effective Field Theory Flavor Changing Neutral Current (FCNC at parton level: Λ b ˆ= (u d b Λ ˆ= (u d s handle in effective theory (all fields heavier than b are integrated out matrix elements of operators O i represent physics below µ m b Wilson coefficients C i C i (M W, M Z, m t,... represent physics above µ m b Decays B K l + l B s µ + µ B K l + l B K γ B X s l + l B X s γ Λ b Λ[ Nπ]l + l D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

4 Why Λ b Λ B K l + l is being measured with increasing precision. Why spend effort on Λ b Λ? pro arguments, sorted from weakest to strongest D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

5 Why Λ b Λ B K l + l is being measured with increasing precision. Why spend effort on Λ b Λ? pro arguments, sorted from weakest to strongest independent confirmation of results: same b sl + l operators, different hadronic matrix elements D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

6 Why Λ b Λ B K l + l is being measured with increasing precision. Why spend effort on Λ b Λ? pro arguments, sorted from weakest to strongest independent confirmation of results: same b sl + l operators, different hadronic matrix elements Γ Λ ev: small width approximation fully applicable (compare B K πl + l where non-res. P-wave contributions are unconstrained D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

7 Why Λ b Λ B K l + l is being measured with increasing precision. Why spend effort on Λ b Λ? pro arguments, sorted from weakest to strongest independent confirmation of results: same b sl + l operators, different hadronic matrix elements Γ Λ ev: small width approximation fully applicable (compare B K πl + l where non-res. P-wave contributions are unconstrained fewer hadronic matrix elements (2 in HQET limit, 1 in SCET limit D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

8 Why Λ b Λ B K l + l is being measured with increasing precision. Why spend effort on Λ b Λ? pro arguments, sorted from weakest to strongest independent confirmation of results: same b sl + l operators, different hadronic matrix elements Γ Λ ev: small width approximation fully applicable (compare B K πl + l where non-res. P-wave contributions are unconstrained fewer hadronic matrix elements (2 in HQET limit, 1 in SCET limit doubly weak decay: complementary constraints on b sl + l physics with respect to B K l + l D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

9 Kinematics and Decay Topology Λ b (p Λ(k [ N(k 1 π(k 2 ] l + (q 1 l (q 2 3 independent decay angles Momenta cos θ Λ k q cos θ l k q cos φ k q only for unpolarized Λ b q = q 1 + q 2 q = q 1 q 2 k = k 1 + k 2 k = k 1 k 2 D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

10 Λ b Λ Hadronic Matrix Elements In General using (overall 10 helicity form factors (FFs [compare Feldmann/Yip ] ε (λ Λ Γ Λ b fλ(q Γ 2 for lepton mass m l 0 this reduces to 8 independent FFs results for all FFs expected from the lattice in the long run Within HQET heavy quark spin symmmetry reduces matrix elements to 2 FFs at leading power known from Lattice QCD [Detmold/Lin/Meinel/Wingate ] Within SCET applicable when E Λ = O (m b matrix elements reduce further to 1 single FF estimates from SCET sum rules [Feldmann/Yip ] D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

11 Λ Nπ Hadronic Matrix Element Λ Nπ is a parity-violating weak decay branching fraction B[Λ Nπ] = (99.7 ± 0.1% [PDG, our naive average] equations of motions reduce independent matrix elements to 2 we choose to express them through decay width Γ Λ parity-violating coupling α within small width approximation, Γ Λ cancels α well known from experiment: α pπ = ± [PDG average] D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

12 Operator Basis Semileptonic/Radiative Operators SM and chirality-flipped operators O 9(9 = α e 4π [ sγµ P L(R b][ lγ µ l] O 7(7 = em b 4π [ sσµν P R(L b]f µν O 10(10 = α e 4π [ sγµ P L(R b][ lγ µ γ 5 l] Hadronic Operators contribute via intermediate off-shell photon O 1 = [ cγ µ P L T A b][ sγ µ P L T A c] O 2 = [ cγ µ P L b][ sγ µ P L c] D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

13 Hadronic Contributions all exclusive b sl + l processes face problem of hadronic contributions hadronic operators give rise to e.g. b s cc, hadronizes to Λ b ΛJ/ψ( l + l systematically include effects via hadronic two-point function T (q 2 C 7 O 7 T (q 2 different approaches to obtain T (q 2, depending on kinematics B K l + l : methods and domain of validity small q 2 mb 2 : QCD Factorization (QCDF [Beneke/Feldmann/Seidel hep-ph/ and hep-ph/ ] large q 2 mb 2 : Operator Product Expansion (OPE [Grinstein/Pirjol hep-ph/ ] Λ b Λl + l : the low recoil OPE can be directly applied at q 2 m 2 b D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

14 Angular Distribution of Λ b Λ[ Nπ]l + l we define the angular distribution as 8π 3 d 4 Γ dq 2 d cos θ l d cos θ Λ dφ K (q2, cos θ l, cos θ Λ, φ when considering only SM and chirality-flipped operators K = 1 (K 1ss sin 2 θ l + K 1cc cos 2 θ l + K 1c cos θ l ( + cos θ Λ K 2ss sin 2 θ l + K 2cc cos 2 θ l + K 2c cos θ l ( + sin θ Λ sin φ K 3sc sin θ l cos θ l + K 3s sin θ l ( + sin θ Λ cos φ K 4sc sin θ l cos θ l + K 4s sin θ l K n K n(q 2 D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

15 Angular Observables matrix elements parametrized through 8 transversity amplitudes A λ χ M A R 1, A R 1, A R 0, A R 0, and (R L λ dilepton chirality χ transversity state, similar as in B K l + l M third component of dilepton angular momentum D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

16 Angular Observables matrix elements parametrized through 8 transversity amplitudes A λ χ M A R 1, A R 1, A R 0, A R 0, and (R L λ dilepton chirality χ transversity state, similar as in B K l + l M third component of dilepton angular momentum express angular observables through transversity amplitudes, e.g. K 1cc = 1 [ A R A R (R L ] K 2c = α [ A R A R 1 2 (R L ]. full list in the backups D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

17 Simple Observables start with integrated decay width Γ = 2K 1ss + K 1cc define further observables X as weighted (ω X integrals X = 1 d 3 Γ Γ d cos θ l d cos θ Λ dφ ω X (cos θ l, cos θ Λ, φd cos θ l d cos θ Λ dφ A leptonic forward-backward asymmetry K 1c A l FB = 3 2 2K 1ss + K 1cc with ω A l FB = sgn cos θ l D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

18 Simple Observables start with integrated decay width Γ = 2K 1ss + K 1cc define further observables X as weighted (ω X integrals X = 1 d 3 Γ Γ d cos θ l d cos θ Λ dφ ω X (cos θ l, cos θ Λ, φd cos θ l d cos θ Λ dφ A leptonic forward-backward asymmetry K 1c A l FB = 3 2 2K 1ss + K 1cc with ω A l FB = sgn cos θ l B fraction of longitudinal dilepton pairs F 0 = 2K 1ss K 1cc 2K 1ss + K 1cc with ω F0 = 2 5 cos 2 θ l D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

19 Simple Observables start with integrated decay width Γ = 2K 1ss + K 1cc define further observables X as weighted (ω X integrals X = 1 d 3 Γ Γ d cos θ l d cos θ Λ dφ ω X (cos θ l, cos θ Λ, φd cos θ l d cos θ Λ dφ C hadronic forward-backward asymmetry A Λ FB = 1 2K 2ss + K 2cc 2 2K 1ss + K 1cc with ω A l FB = sgn cos θ Λ D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

20 Simple Observables start with integrated decay width Γ = 2K 1ss + K 1cc define further observables X as weighted (ω X integrals X = 1 d 3 Γ Γ d cos θ l d cos θ Λ dφ ω X (cos θ l, cos θ Λ, φd cos θ l d cos θ Λ dφ C hadronic forward-backward asymmetry A Λ FB = 1 2K 2ss + K 2cc 2 2K 1ss + K 1cc with ω A l FB = sgn cos θ Λ D combined forward-backward asymmetry A lλ K 2c FB = 3 4 2K 1ss + K 1cc with ω A l FB = sgn cos θ Λ sgn cos θ l D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

21 At Low Recoil: Amplitudes when q 2 = O ( m 2 b ingredients Low Recoil OPE expected valid only for q 2 -integrated quantities [Grinstein/Pirjol hep-ph/ ], [Beylich/Buchalla/Feldmann ] to leading power in the 1/m b expansion there are only two independent form factors Λ Γ Λ b ū Λ [ ξ1 Γ + ξ 2 /vγ ] u Λb with v: Λ b velocity D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

22 At Low Recoil: Amplitudes when q 2 = O ( m 2 b ingredients Low Recoil OPE expected valid only for q 2 -integrated quantities [Grinstein/Pirjol hep-ph/ ], [Beylich/Buchalla/Feldmann ] to leading power in the 1/m b expansion there are only two independent form factors all amplitudes factorize schematically Λ Γ Λ b ū Λ [ ξ1 Γ + ξ 2 /vγ ] u Λb with v: Λ b velocity see backups for full expr A R(L C R(L ± f + corrections for only SM-like operators coefficients unify C R(L ( ( Λ corrections suppressed: O α s and O C7 m b ± CR(L Λ m b C 9 D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

23 At Low Recoil: Observables when q 2 = O ( m 2 b observables are bilinears of C R(L ± some are the same as in B K l + l [Bobeth/Hiller/DvD and ] ρ ± 1 = 1 ( C R ± 2 + C± L 2 ρ 2 = 1 ( C R + C R C C L L new complementary combinations (also in B K πl + l, see talk by Gudrun Hiller ρ ± 3 = 1 ( C R ± 2 C± L 2 ρ 4 = 1 ( C R + C R +C C L L D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

24 At Low Recoil: Observables when q 2 = O ( m 2 b observables are bilinears of C R(L ± some are the same as in B K l + l [Bobeth/Hiller/DvD and ] ρ ± 1 = 1 ( C R ± 2 + C± L 2 ρ 2 = 1 ( C R + C R C C L L new complementary combinations (also in B K πl + l, see talk by Gudrun Hiller ρ ± 3 = 1 ( C R ± 2 C± L 2 ρ 4 = 1 ( C R + C R +C C L L which observables probe these new combinations? hadronic forward-backward asymmetry A Λ FB is sensitive to Re ( ρ ± 3 combined forward-backward asymmetry A lλ FB is sensitive to Re (ρ 4 D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

25 Sketch of Hypothetical Constraints in NP Model assume C 9(9 free floating, and C 10(10 = CSM 10(10 ( 4, 0 4 existing constraints ρ ± 1 blue banded constraints 2 C9 ' C9 black square: SM point D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

26 Sketch of Hypothetical Constraints in NP Model assume C 9(9 free floating, and C 10(10 = CSM 10(10 ( 4, existing constraints ρ ± 1 ρ 2 blue banded constraints insensitive (not shown C9 ' C9 black square: SM point D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

27 Sketch of Hypothetical Constraints in NP Model assume C 9(9 free floating, and C 10(10 = CSM 10(10 ( 4, 0 C9 ' existing constraints ρ ± 1 ρ 2 blue banded constraints insensitive (not shown new constraints ρ 3 purple banded constraints C9 black square: SM point D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

28 Sketch of Hypothetical Constraints in NP Model assume C 9(9 free floating, and C 10(10 = CSM 10(10 ( 4, 0 C9 ' existing constraints ρ ± 1 ρ 2 blue banded constraints insensitive (not shown new constraints ρ 3 ρ 4 purple banded constraints gold hyperbolic constraint C9 black square: SM point D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

29 Optimized Observables Goals construct observables that predominantly test short-distance physics... that probe ratios of form factors Short-distance sensitive observables in SM + SM basis K 2cc = α Re (ρ 4 K 1c Re (ρ 2 Form-factor ratios in SM basis K 2cc K 1c = αρ 1 2 Re (ρ 2 K 1cc = 2α Re (ρ 2 K 2c ρ 1 in SM + SM only two ratios can be probed 2K 2ss K 2cc 1 + f V 0 f A 0 f V f A 2K 4sc f 0 V f 0 A K 2cc f V f A in SM basis also f V 0 /f A 0, f V /f A can be probed D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

30 Numeric Results preliminary results low recoil region 14.18GeV 2 q GeV 2 scale µ = 4.2GeV uncertainties: form factors, CKM, quark masses B = (5.2 ± F 0 = 0.46 ± 0.2 A l FB = A Λ FB = A lλ FB = ± agrees with B measurement [LHCb , our naive average] B = (6.2 ± D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

31 Conclusion rare decay Λ b Λ( Nπl + l mediated by b sl + l self-analyzing through weak Λ Nπ decay offers complementary new constraints on b sl + l physics decay induces angular distribution with 10 (or more angular observables three forward-backward asymmetries, two of which probe new constraints: A Λ FB and AlΛ FB predictions for observables at low recoil using HQET form factor relations Outlook at low q 2 : spectator interaction basically unknown wait for lattice results on all 10 form factors for Λ b Λ D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

32 Backup Slides D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

33 Angular Observables K 1ss = 1 [ ] A R A R A R A R (R L K 1cc = 1 [ ] A R A R (R L ( K 1c = Re A R 1 A R 1 (R L K 2ss = α ( 2 Re A R 1 A R 1 + 2A R 0 A R 0 + (R L ( K 2cc = α Re A R 1 A R 1 + (R L K 2c = α [ ] A R A R 1 2 (R L K 3sc = α ( Im A R 2 1 A R 0 A R 1 A R 0 + (R L K 3s = α ( Im A R 2 1 A R 0 A R 1 A R 0 + (R L K 4sc = α ( Re A R 2 1 A R 0 A R 1 A R 0 + (R L K 4s = α ( Re A R 2 1 A R 0 A R 1 A R 0 (R L D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

34 At Low Recoil: Amplitudes when q 2 = O ( m 2 b A L(R 1 A L(R 0 = 2NC L(R + f V s = + 2NC L(R + f V 0 m Λb + m Λ s q 2 A L(R 1 A L(R 0 = +2NC L(R f A s+ = 2NC L(R f A 0 m Λb m Λ s+ q 2 with s ± = (m Λb ± m Λ 2 q 2 ( C R(L + = (C 9 + C 9 + 2κm bm Λb C R(L = κ as in [Grinstein/Pirjol hep-ph/ ] q 2 (C 7 + C 7 ± (C 10 + C 10 ( (C 9 C 9 + 2κm bm Λb q 2 (C 7 C 7 ± (C 10 C 10 D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

35 At Low Recoil: ρs when q 2 = O ( m 2 b ρ ± 1 = C 79 ± C C 10 ± C 10 2 ρ 2 = Re (C 79 C 10 C 7 9 C 10 i Im (C 79 C C 10 C 10 ρ ± 3 = 2 Re ((C 79 ± C 7 9 (C 10 ± C 10 ( ρ 4 = C 79 2 C C 10 2 C 10 2 i Im (C 79 C10 C 7 9 C 10. D. van Dyk (Siegen Λ b Λ( Nπl + l / 17

36 Form Factor Uncertainties reproduce ξ 1,2 data points from the lattice [Detmold/Liu/Meinel/Wingate ] two points: q 2 = 13.5GeV 2 and q 2 = 20.5GeV 2 correlation across q 2 taken into account correlation across ξ 1,2 not taken into account at subleading power, 6 further FFs emerge: χ 1,...,6 order of magnitude m Λ /m Λb use uncorrelated gaussian priors D. van Dyk (Siegen Λ b Λ( Nπl + l / 17