Flavor physics with Λ b baryons

Transcription

1 July 29, 2013

2 Introduction The LHC is a Λ b factory (Λ b = lightest bottom baryon, quark content udb, J P = ) < η < 5 f Λb /(f Bu + f Bd ) f Bs /(f Bu + f Bd ) p T (charm + µ) (GeV) [LHCb, arxiv: ]

3 Outline 1 The decay Λ b Λ l + l 2 The decay Λ b p l ν l 3 Lattice calculation with static b quarks 4 Lattice calculation with relativistic b quarks

4 The decay Λb Λ `+ ` (` = e, µ, τ ) First observed by CDF [arxiv: ] cdf.fnal.gov/events/cdfpictures.html Flavor physics with Λb baryons

5 Λ b Λ l + l Large contribution for all q 2 Large contribution near q 2 = 0 Large contribution near q 2 = m 2 J/ψ, m2 ψ, but small otherwise

6 b sl + l effective Hamiltonian Standard-Model contributions to effective operators:

7 b sl + l effective Hamiltonian with H eff = 4G F 2 V tb V ts O ( ) 9 = e 2 /(16π 2 ) sγ µ P L(R) b lγ µl, O ( ) 10 = e 2 /(16π 2 ) sγ µ P L(R) b lγ µγ 5l, i ( ) C i O i + C i O i O ( ) 7 = e/(16π 2 )m b sσ µν P R(L) b F (e.m.) µν, O 1 = c b γ µ P L b a s a γ µp L c b, O 2 = c a γ µ P L b a s b γ µp L c b [P R,L = (1 ± γ 5 )/2] Wilson coefficients in the Standard Model (at µ = m b ): (C i,eff include one-loop matrix elements of O 1,...,O 6 ) C 9,eff 4.2 +Y (q 2 ), C 10,eff 4.1, C 7,eff 0.30, C 9,eff 0, C 10,eff 0, C 7,eff 0 C , C 2 [Altmannshofer et al, arxiv: ] Some new-physics models predict sizable C i Rizzo, hep-ph/ ] [see, e.g., Cho and Misiak, hep-ph/ ;

8 Λ b Λ l + l decay amplitude Without long-distance contributions from O 1,2 etc., the decay amplitude is M = G F e 2 [ 4 2π V tbv 2 ts Λ sγ µ (C 9,eff P L + C 9,effP R )b Λ b ū l γ µv l + Λ sγ µ (C 10,eff P L + C 10,effP R )b Λ b ū l γ µγ 5v l 2m b q 2 Λ siσ µν q ν(c 7,eff P R + C 7,effP L )b Λ b ū l γ µv l ] Relative contributions from C i vs C i depend on polarization of Λ and Λ b. [Mannel and Recksiegel, hep-ph/970139; Chen and Geng, hep-ph/ ; Hiller and Kagan, hep-ph/ ] Hadronic matrix elements are written in terms of ten form factors, which are functions of q 2 : Λ s γ µ b Λ b = u Λ [ f V 1 γ µ f V 2 iσ µν q ν/m Λb + f V 3 q µ /m Λb ] uλb, Λ s γ µ γ 5 b Λ b = u Λ [ f A 1 γ µ f A 2 iσ µν q ν/m Λb + f A 3 q µ /m Λb ] γ5 u Λb, Λ s iσ µν q ν b Λ b = u Λ [ f TV 1 (γ µ q 2 q µ /q)/m Λb f TV 2 iσ µν q ν ] uλb, Λ s iσ µν q ν γ 5 b Λ b = u Λ [ f TA 1 (γ µ q 2 q µ /q)/m Λb f TA 2 iσ µν q ν ] γ5 u Λb These need to be calculated in lattice QCD.

9 Λ b Λ l + l : comparison with mesonic decays Λ b Λ l + l is a baryonic analogue of B K ( ) l + l. B Kl + l B K l + l Λ b Λ l + l Initial hadron has spin Final hadron has spin Final hadron stable in QCD First observation Belle, 2002 Belle, 2003 CDF, 2011 Polarization of the Λ/K influences the angular distribution of the four-body decays Λ b Λ( p + π ) l + l B K ( Kπ) l + l B K l + l form factors: forthcoming paper by Ron Horgan, Zhaofeng Liu,, Matthew Wingate B Kl + l form factors: talks by Andreas Kronfeld (Fri 5:30) and Chris Bouchard (Fri 5:50)

10 Λ b Λ l + l : comparison with mesonic decays B 0 K 0 ( K π + ) l + l Λ b Λ( p + π ) l + l K 0 K π + is a strong decay. Complicates theory. Strictly speaking, need B Kπ form factors (not B K ) [Lü and Wang, arxiv: ; Bećirević and Tayduganov, arxiv: ; Matias, arxiv: ; Blake, Egede, Shires, arxiv:121279; Döring, Meißner, Wang, arxiv: ]. Λ p π is a weak decay. Parityviolating. Protons emitted preferentially in direction of Λ spin vector. [Gutsche, Ivanov, Körner, Lyubovitskij, Santorelli, arxiv: ] Long lifetime of (electrically neutral) Λ complicates experiment (cτ Λ = 7.9 cm). At LHCb, 75% of the Λ p π decays happen outside the vertex locator [LHCb, arxiv: ]

11 The decay Λ b p l ν l

12 The CKM matrix element V ub B X u l ν (inclusive) PDG(BABAR, Belle, CLEO) 2012, Lange et al PDG(BABAR, Belle, CLEO) 2012, Gambino et al PDG(BABAR, Belle, CLEO) 2012, Andersen and Gardi 2006 B π l ν (exclusive) HFAG(BABAR, Belle, CLEO) 2011, Ball and Zwicky 2005 (LCSR) HFAG(BABAR, Belle, CLEO) 2011, Khodjamirian et al (LCSR) HFAG(BABAR, Belle, CLEO) 2011, HPQCD 2006 HFAG(BABAR, Belle) 2011, FNAL/MILC 2009 (BCL fit) B τ ν significance: Belle 4.0σ, BABAR: 3.8σ CKM unitarity global fits Belle 2013, FNAL/MILC 2011, HPQCD 2012 BABAR 2012, FNAL/MILC 2011, HPQCD 2012 CKMfitter (2013) UTfit (2013) V ub

13 The CKM matrix element V ub So far, all experimental data come from B factories (which are no longer running). Extraction of V ub from B πl ν l uses form factors from lattice QCD. New calculations in progress. Talks by Taichi Kawanai (Tue 2:20), Daping Du (Fri 4:30), Chris Bouchard (Fri 5:50) See also F. Bahr et al., arxiv: At LHCb, measurement of B πl ν l is difficult because of large pion background. Need more distinctive final state. Better: B s Kl ν l Talks by Andreas Kronfeld (Fri 5:30) and Chris Bouchard (Fri 5:50) Even better: Λ b pl ν l [Egede and Sutcliffe, 2013]

14 Λ b pl ν l Analysis of Λ b pµ ν µ at LHCb in progress. Thanks to Ulrik Egede and William Sutcliffe for discussions! To extract V ub from experimental data, need lattice QCD calculation of the form factors N + u γ µ b Λ b = u N [ f V 1 γ µ f V 2 iσ µν q ν/m Λb + f V 3 q µ /m Λb ] uλb, N + u γ µ γ 5 b Λ b = u N [ f A 1 γ µ f A 2 iσ µν q ν/m Λb + f A 3 q µ /m Λb ] γ5 u Λb (f V 3 and f A 3 do not contribute to decay rate in m l = 0 approximation)

15 V ub inclusive-exclusive tension: new physics? Possible new-physics explanation: right-handed current [Chen and Nam, arxiv: ; Crivellin, arxiv: ; Buras, Gemmler, Isidori, arxiv: ] H eff = G F 2 V ub (ūγ µb ūγ µγ 5b) ( lγ µ ν lγ µ γ 5ν) + G F V ub (ūγ µb + ūγ µγ 5b) ( lγ µ ν lγ µ γ 5ν) 2 Contributions to matrix elements: Process ūγ µb ūγ µγ 5b B π l ν l B l ν l B X u l ν l Λ b p l ν l

16 Lattice calculation with static b quarks William Detmold, C.-J. David Lin,, Matthew Wingate, arxiv: (Λ b Λ l + l ), arxiv: (Λ b p l ν l ) Credit: Lawrence Berkeley National Lab - Roy Kaltschmidt, photographer

17 Λ b decay form factors with static b quarks In the following, we discuss the case Λ b Λ. The case Λ b p is analogous. In the limit m b, the hadronic matrix elements with an arbitrary gamma matrix Γ in the current can be written as where Λ(p, s ) s ΓQ Λ Q (v, s) = ū Λ (p, s ) [F 1 + /v F 2] Γ u ΛQ (v, s) v = (four-velocity of Λ Q ) and the two form factors F 1, F 2 are functions of p v (the energy of the Λ in the Λ Q rest frame) [Mannel, Roberts, Ryzak, NPB 355, 38 (1991); Hussain, Körner, Kramer, Thompson, Z. Phys. C 51, 321 (1991)] This simplifies the lattice calculation and the data analysis Finite-m b corrections are of order O(Λ QCD /m b, p /m b )

18 Static b quarks on the lattice Set v = (1, 0, 0, 0). Action: S = a 3 x Q(x) [ Q(x) U 0 (x aˆ0)q(x aˆ0) ] Propagator: [Eichten and Hill, PLB 240, 193 (1990)] Exponential improvement of signal-to-noise ratio achieved by smearing U 0 [Della Morte et al., hep-lat/ ] We use one level of HYP smearing, (α 1, α 2, α 3 ) = (,, ) [Della Morte, Shindler, Sommer, hep-lat/ ]

19 Actions for light quarks and gluons RBC/UKQCD ensembles with a 0.11 fm (coarse), a 8 fm (fine), L 2.7 fm [Aoki et al., arxiv:101892] Thank you for providing them! u, d, s quarks: domain-wall action [Kaplan, hep-lat/ ; Furman and Shamir, hep-lat/ ] gluons: Iwasaki action [Iwasaki and Yoshie, PLB 143, 449 (1984)] seven data sets, four partially quenched Set L 3 T am (sea) u,d am (sea) s a (fm) am (val) u,d am (val) s m (vv) π C (17) (4) C (17) (4) C (17) (5) C (17) (5) F (12) (3) F (12) (4) F (17) (7) (MeV)

20 Baryon interpolating fields, static-light current baryon interpolating fields: Λ Qα = ɛ abc (Cγ 5) βγ d a β ũ b γ Q c α, Λ α = ɛ abc (Cγ 5) βγ ũ a β d b γ s c α (tilde = Gaussian smearing) O(a) improved current, matched to continuum HQET in MS scheme: [ ] ( J Γ = U(m b, a 1 ) Z 1 + c(ma) Γ m s a ) (pa) Q Γ s + c }{{} 1 (w0 MF ) 2 Γ a Q Γ γ s RG evolution Z, c (ma) Γ, c (pa) Γ calculated using tadpole-improved one-loop lattice perturbation theory by Tomomi Ishikawa et al. [arxiv: ] See talk on B 0 - B 0 mixing by Tomomi Ishikawa (Tue, 5:20)

21 Three-point functions Backward three-point function: C (3,bw) αδ (Γ, p, t, t t ) = e ip (y x) Λ Qα (x 0 + t, y) J Γ (x 0 + t, y) Λ δ (x 0, x) y Forward three-point function: C (3,fw) δα (Γ, p, t, t ) = e ip (x y) Λ δ (x 0, x) J Γ (x 0 t +t, y) Λ Qα (x 0 t, y) y No sequential DW propagators needed Inexpensive to do many source-sink separations t

22 Ratios Compute the ratio R(Γ, p, t, t ) = [ ] 4 Tr C (3,fw) (Γ, p, t, t ) C (3,bw) (Γ, p, t, t t ) [ ] [ ] Tr C (2,Λ) (p, t) Tr C (2,Λ b) (t) For Γ = (any product of γ µ s), this gives E Λ + m Λ [F 1 + F 2] 2 + (excited-state contribs), if [Γ, γ 0 ] = 0, E Λ R(Γ, p, t, t ) = E Λ m Λ [F 1 F 2] 2 + (excited-state contribs), if {Γ, γ 0 } = 0. E Λ Define F + = F 1 + F 2 F = F 1 F 2

23 Ratios To improve statistics, average over multiple gamma matrices and define R +(p, t, t ) = 1 [ R(1, p, t, t ) + R(γ 2 γ 3, p, t, t ) 4 ] +R( γ 3 γ 1, p, t, t ) + R(γ 1 γ 2, p, t, t ), R (p, t, t ) = 1 [ R(γ 1, p, t, t ) + R(γ 2, p, t, t ) 4 ] +R(γ 3, p, t, t ) + R( iγ 5, p, t, t ) For given p 2, then average over the direction of p ; denote the direction-averaged quantities by R ±( p 2, t, t )

24 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

25 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

26 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

27 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

28 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

29 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

30 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

31 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

32 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

33 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

34 Ratios: example results Λ Q Λ p 2 = 4 (2π/L) 2 F43 data set t/a = R+ R t (fm)

35 Extrapolation to infinite source-sink separation Evaluate at mid-point t = t/2 and compute R ±( p 2 EΛ, t) = R ±( p E Λ ± m 2, t, t/2) Λ t F±(E Λ) Λ Q Λ p 2 = 4 (2π/L) 2 C Λ Q Λ p 2 = 4 (2π/L) 2 F t (fm) R R + Fit leading excited-state contamination using R i,n ± (t) = F i,n ± + A i,n ± exp[ δ i,n t], 2.0 t (fm) where i labels data set and n labels momentum. At each momentum, fit done simultaneously for all 7 data sets, with constraints on lattice-spacing and quark-mass dependence of δ i,n R R +

36 Λ Q Λ form factors 2.0 C14 F 2.0 C24 F 2.0 C54 F F + F + F E Λ m Λ (GeV) E Λ m Λ (GeV) E Λ m Λ (GeV) 2.0 F23 F 2.0 F43 F 2.0 F63 F F + F + F E Λ m Λ (GeV) E Λ m Λ (GeV) E Λ m Λ (GeV) Dipole fit model: Y ± F ± = ( X ± + E Λ m Λ ) [1 + d±(ae Λ) 2 ], 2 X ± = X ± + c l,± [ (mπ) 2 (m phys π ) 2] 2.0 C53 F + c s,± [ (mηs ) 2 (m phys η s ) 2] E Λ m Λ (GeV) F +

37 Λ Q p form factors 2.0 C14 F 2.0 C14 F 2.0 C54 F F + F + F E N m N (GeV) E N m N (GeV) E N m N (GeV) 2.0 F23 F 2.0 F43 F 2.0 F63 F F + F + F E N m N (GeV) E N m N (GeV) E N m N (GeV) Dipole fit model: Y ± F ± = ( X ± + E N m N ) [1 + d±(ae N) 2 ], 2 X ± = X ± + c l,± [ (mπ) 2 (m phys π ) 2] Note: here, E N computed from m N using relativistic dispersion relation

38 Λ Q Λ form factors at m π = m phys π, m ηs = m phys η s, a = Λ Q Λ E Λ m Λ (GeV) F F + Error bands shown here include the following estimates of systematic uncertainties: renormalization: 6% finite volume: 3% chiral extrapolation: 3% continuum extrapolation: 2%

39 Λ Q p form factors at m π = m phys π, m ηs = m phys η s, a = Λ Q p E N m N (GeV) F F + Error bands shown here include the following estimates of systematic uncertainties: renormalization: 6% finite volume: 3% chiral extrapolation: 3% continuum extrapolation: 2%

40 Λ b Λµ + µ differential branching fraction db/dq 2 (10 7 GeV 2 ) Λ b Λ µ + µ CDF 2012 (9.6 fb 1 ) LHCb 2013 ( fb 1 ) form factors extrapolated q 2 (GeV 2 ) [CDF data: Public Note 10726, LHCb data: arxiv: ] Inner error band from uncertainty in static form factors p 2 + Λ 2 QCD Outer error band includes uncertainty from static approximation m b

41 Λ b p µ ν µ differential decay rate 4 dγ/dq 2 Vub 2 (ps 1 GeV 2 ) Λ b p µ ν µ q 2 (GeV 2 ) Only the kinematic range where we have lattice data is shown here. Integrated rate: q 2 max 1 dγ(λ b p µ ν µ) dq 2 = 15.3 ± 2.4 ± 3.4 ps 1 V ub 2 14 GeV 2 dq 2 V ub extraction with theory uncertainty 15% (dominated by static approximation), which is already smaller than the exclusive-inclusive discrepancy

42 Lattice calculation with relativistic b quarks Work in progress. All results are very preliminary. Credit: Oak Ridge National Laboratory Flavor physics with Λb baryons

43 Relativistic form factors To eliminate the 1/m b systematic uncertainty, replace static heavy-quark action by relativistic heavy-quark action and perform direct calculation of the relativistic form factors X q γ µ b Λ b = u X [ f V 1 γ µ f V 2 iσ µν q ν/m Λb + f V 3 q µ /m Λb ] uλb, X q γ µ γ 5 b Λ b = u X [ f A 1 γ µ f A 2 iσ µν q ν/m Λb + f A 3 q µ /m Λb ] γ5 u Λb, X q iσ µν q ν b Λ b = u X [ f TV 1 (γ µ q 2 q µ /q)/m Λb f TV 2 iσ µν q ν ] uλb, X q iσ µν q ν γ 5 b Λ b = u X [ f TA 1 (γ µ q 2 q µ /q)/m Λb f TA 2 iσ µν q ν ] γ5 u Λb

44 Actions Reuse the existing domain-wall propagators for light and strange quarks For the b quarks, adopt the three-parameter RHQ action used by RBC/UKQCD [Aoki et al., arxiv: ] S = a 4 x Q (m 0 + γ ζγ a a 2 ζ 2 + a 4 c P iσ µν F µν ) Q See the following talks for more details: Oliver Witzel on f B, f Bs (Tue 2:00) Taichi Kawanai on B πl ν l (Tue, 2:20) Ben Samways on g B Bπ (Tue, 4:40) in addition to Λ b p and Λ b Λ, also compute Λ b Λ c [Λ b Λ c first studied by Bowler et al. (UKQCD), hep-lat/ ; Gottlieb and Tamhankar, hep-lat/ ]

45 Mostly nonperturbative renormalization Heavy-light current written as follows: [El-Khadra, Kronfeld, Mackenzie, Ryan, Simone, hep-ph/ ] where = nonperturbative J Γ = Z QQ V Z qq V ρ Γ [ ] q Γ Q + O(a) Determination of Z QQ V : see talk by Taichi Kawanai (Tue, 2:20). Thanks for providing preliminary results! = one loop Computed using PhySyHCAl framework ( by Christoph Lehner [arxiv: ] See talk by Christoph Lehner (Tue, 2:40). Thanks for providing preliminary results and for discussions!

46 Relativistic form factors from ratios Similar method as used in static calculation (current insertion at mid-point t = t/2, then extrapolate to t = ). Have constructed combinations of ratios that directly give the 8 form factors: R V 1 (t) R V 2 (t) R A 1 (t) R A 2 (t) t t t t R TV 1 (t) t R TV 2 (t) t R TA 1 (t) t R TA 2 (t) t f V 1, f V 2, f A 1, f A 2, f TV 1, f TV 2, f TA 1, f TA 2 (f V 3 and f A 3 do not contribute to decay rates in massless lepton approximation)

47 Example results: R A 1 (t) p 2 = 1 (2π/L) 2 C54 p 2 = 1 (2π/L) 2 F R A Λ b Λ c Λ b p Λ b Λ Λ b Λ c Λ b p Λ b Λ t (fm) t (fm) Caveats: for Λ b Λ c, Z cc V O(a) improvement not yet included computed only at tree level, ρ factors missing

48 Example results: R A 2 (t) p 2 = 1 (2π/L) 2 C54 Λ b p Λ b Λ Λ b Λ c p 2 = 1 (2π/L) 2 F43 Λ b p Λ b Λ Λ b Λ c R A t (fm) t (fm) Caveats: for Λ b Λ c, Z cc V O(a) improvement not yet included computed only at tree level, ρ factors missing

49 Example results: R TV 1 (t) p 2 = 1 (2π/L) 2 C p 2 = 1 (2π/L) 2 F43 R TV Λ b p Λ b Λ Λ b Λ c Λ b p Λ b Λ Λ b Λ c t (fm) t (fm) Caveats: for Λ b Λ c, Z cc V O(a) improvement not yet included ρ factors for tensor currents missing computed only at tree level, ρ factors missing

50 Example results: R TV 2 (t) p 2 = 1 (2π/L) 2 C p 2 = 1 (2π/L) 2 F43 R TV Λ b Λ c Λ b p Λ b Λ Λ b Λ c Λ b p Λ b Λ t (fm) t (fm) for Λ b Λ c, Z cc V O(a) improvement not yet included ρ factors for tensor currents missing computed only at tree level, ρ factors missing

51 Conclusions / Outlook Lattice calculation of heavy light baryon form factors is less expensive than lattice calculation of light light baryon form factors (sequential propagators for heavy quark only) Substantial excited-state contamination successfully removed through wide range of source-sink separations and t extrapolation Plan to update predictions of decay rates using relativistic form factors, also predict angular distributions Plan to calculate relativistic form factors on new RBC/UKQCD Möbius DW ensemble at physical pion mass using all-mode-averaging [Blum, Izubuchi, Shintani, arxiv: ] Λ b Λµ + µ : theory uncertainty will be dominated by long-distance contributions (B K ( ) µ + µ has the same problem). LHCb measurement is statistics limited; higher precision results and angular analysis forthcoming [Watson, QFTHEP 2013] Λ b pµ ν and Λ b Λ cµ ν: theory predictions (in low-recoil region) will be very precise. Awaiting experimental data from LHCb [Egede and Sutcliffe, 2013]