(Semi)leptonic B decays: Form factors and New Physics Searches Danny van Dyk Universität Zürich

Transcription

1 (Semi)leptonic B decays: Form factors and New Physics Searches Danny van Dyk Universität Zürich Page 1

2 Introduction and Motivation

3 QCD in Low Energy Processes One focus of theoretical physics at UZH is QCD in high-energy processes, such as in higgs or top events at the LHC. Conversely, my interests focus on QCD in low-energy processes, as needed in b hadron decays. For this talk I will focus on the two simplest classes of b hadron decays with a b ulν transition: purely leptonic decays, e.g.: B τ ν semileptonic decays, e.g.: B 0 π + µ ν Beyond these exclusive decay modes, it is also interesting to study the corresponding inclusive decay: B X uµν (Semi)leptonic B decays Page 3

4 Motivation: The CKM element V ub Why are these decays interesting? These decays can be used to determine the value of V ub, which is an important input to CKM fits. [CKMfitter Group: J. Charles et al. hep-ph/ ] η excluded area has CL > 0.95 sin 2β ε K α V ub γ α γ excluded at CL > 0.95 β α & m s m d m d 1.0 CKM f i t t e r EPS 15 γ ε K sol. w/ cos 2β < 0 (excl. at CL > 0.95) (Semi)leptonic B decays Page 4 ρ

5 Motivation: A Tension between V ub Determinations However, the determinations of V ub from the individual decay channels do not agree well: [HFAG 2014, ] V B πµν ub = (3.28 ± 0.29) 10 3 V B Xuµν ub = (4.41 ± 0.21) 10 3 This has a non-neglible impact on the CKM fit. η excluded area has CL > 0.95 sin 2β γ α m d m d & m s V ub ε K CKM f i t t e r EPS 15 sol. w/ cos 2β < 0 (excl. at CL > 0.95) α 0.1 γ β α ρ

6 Motivation: A Tension between V ub Determinations However, the determinations of V ub from the individual decay channels do not agree well: [HFAG 2014, ] V B πµν ub = (3.28 ± 0.29) 10 3 V B Xuµν ub = (4.41 ± 0.21) 10 3 This has a non-neglible impact on the CKM fit. η excluded area has CL > 0.95 sin 2β α γ γ α m d m d & m s V ub Λ b ρ V ub SL V ub τ ν ε K β CKM f i t t e r EPS 15 sol. w/ cos 2β < 0 (excl. at CL > 0.95) α

7 Effective Operator It is convenient to use an effective theory from which W has been removed. Schematically, this can be illustrated by replacing the W propagator (here in Feynman/ t Hoft gauge): ig µν q 2 M 2 W igµν M 2 W [ ( q O M 2 W )]. In the SM, the expansion gives rise to exactly one universal local effective operator of mass dimension six: O LL [u(x)γ µ (1 γ 5 )b(x)] [l(x)γ µ(1 γ 5 )ν(x)] which enters the effective Lagrangian via L eff G F 2 V ub O LL reminder: G F = g 2 4 2M 2 W (Semi)leptonic B decays Page 6

8 Hadronic Matrix Elements In order to compute the partial decay widths one needs knowledge of the process dependent hadronic matrix element of the operator. For the purely leptonic decay, one defines a B-meson decay constant f B via 0 u(x)γ µ γ 5 b(x) B(p) if B p µ For the semileptonic decay B πµ ν, the hadronic matrix element is not a constant anymore, but rather depends on the momentum transfer q p k. One parametrizes commonly π(k) u(x)γ µ b(x) B(p) f +(q 2 )(p + k) µ + terms q µ The function f + is called the vector form factor for this transitions. For massless leptons the unspecified terms are irrelevant, since q µ [lγ µ(1 γ 5 )ν] = (Semi)leptonic B decays Page 7

9 B τ ν In B τ ν only one observable, the partial decay width Γ, emerges: Γ(B τ ν) = G2 F V ub 2 fbm 2 B Mτ 2 8π ( ) 2 1 M2 τ. MB 2 decay most frequent for τ final state due to helicity suppression value f B crucial for V ub determination available from lattice QCD or QCD sum rules (this talk) (Semi)leptonic B decays Page 8

10 B 0 π + µ ν In the semileptonic decay we have the partial differential decay width dγ(b 0 π + µ ν) dq 2 = G2 F V ub 2 192π 3 λ(m2 B, M 2 π, q 2 ) 3 2 [f+(q 2 )] 2. all experiments bin rate in q 2 rate reduces at large q 2 value and shape of f +(q 2 ) crucial lattice QCD works only for large q 2 Light-Cone Sum Rules (this talk) only at small q 2 ) -2 (GeV 2 B/ q HFAG PDG 2014 Belle untagged Belle hadronic tag BaBar untagged (6 bins) BaBar untagged (12 bins) Theory prediction used in fit 2 LCSR (low q ): Bharucha Theory prediction not used in fit 2 LQCD (high q ): FNAL/MILC Fitted BCL param. (3+1 par.) [HFAG 2014, ] q 2 (GeV ) (Semi)leptonic B decays Page 9

11 Determining Hadronic Matrix Elements from QCD Sum Rules

12 Setup The current j 5 (x) m b u(x)iγ 5 b(x) matches the quantum numbers of the B meson. It it suitable to interpolate the B in a hadronic matrix element. In QCD 2-point sum rules, one considers a specific 2-point function: F(p 2 d 4 x ) = e ip x 0 T {j (2π) 4 5 (x), j 5 (0)} 0 F(p 2 ) has poles and branchcuts at Rep 2 > 0, corresponding to on-shell states that are compatible with the interpolating current. Suggests a dispersive representation of F. [D.Rosenthal, Dissertation 2015] (Semi)leptonic B decays Page 11

13 Setup The current j 5 (x) m b u(x)iγ 5 b(x) matches the quantum numbers of the B meson. It it suitable to interpolate the B in a hadronic matrix element. In QCD 2-point sum rules, one considers a specific 2-point function: F(p 2 d 4 x ) = e ip x 0 T {j (2π) 4 5 (x), j 5 (0)} 0 F(p 2 ) = i 2π = 1 π C 0 0 F(s)ds s p 2 Im s [F(s)] ds s p 2 ρ(s)ds s p 2 [D.Rosenthal, Dissertation 2015] (Semi)leptonic B decays Page 11

14 Dispersive Representations Use two independent representations for F: F OPE (p 2 ), expressed in terms of perturbative coefficients and vacuum condensates valid if p 2 m 2 b, p2 m b Λ QCD F had, expressed in terms of hadronic matrix elements (e.g. f B ) F(p 2 ) = d 4 x (2π) 4 e ip x 0 T {j 5 (x), j 5 (0)} 0 F OPE (p 2 ) = m 2 b ρ OPE (s)ds s p 2 F had (p 2 ) = f BM 2 B 2 MB 2 + ρ(s)ds p2 (M B +2M π) 2 s p (Semi)leptonic B decays Page 12

15 OPE Side ρ OPE (s) = k C k (s) 0 O k 0 expansion in universal vacuum condensates 0 O k 0 of increasing mass dimension, beginning with unity quark-quark condensate 0 qq 0, with q = u, d gluon condensate αs π 0 GµνGµν 0 take values of condesates from low(er)-energy QCD processes perturbative coefficients C k known up to NNLO biggest sources of uncertainty: α s(µ), m b Calculation has saturated : Results quite reliable, and improvements are work intensive and small (Semi)leptonic B decays Page 13

16 Hadronic Side F had (p 2 ) = f BM 2 B 2 MB 2 + ρ(s)ds p2 (M B +2M π) 2 s p 2 f B f B F t B F t π +... B π parametrizes correlator in terms of unknown constants and functions e.g. F t : one of the B ππ form factors (Semi)leptonic B decays Page 14

17 Relating both Sides Quark-Hadron Duality (QHD) [M. Shifman hep-ph/ ] [QHD] was first formulated at the dawn of the QCD era by Poggio, Quinn and Weinberg [1], who suggested that certain inclusive hadronic cross sections at high energies, being appropriately averaged over an energy range, had to (approximately) coincide with the cross sections one could calculate in the quark-gluon perturbation theory. (emphasis mine) (Semi)leptonic B decays Page 15

18 Relating both Sides Find some artifical threshold s 0 > mb, 2 from which on the integral over the OPE spectral density ρ OPE coincides with the hadronic continuum: ds ρ(s)! = (M B +2M π) 2 s p 2 s 0 ds ρope (s) s p 2 Consequently: fbm 2 B 2 s0 MB 2 = ρ OPE (s) p2 m b 2 s p 2 Also, apply Borel transform from p 2 to M 2 to increase numeric stability of the sum rule. Obtain F(p 2 ) F(1/M 2 ) by 1 [ X p X 2 e M 2, p 2] n 0 if n 0 Final 2-point sum rule (2ptSR): f 2 B = 1 M 2 B s0 m 2 b M B 2 s ds e M 2 ρ OPE (s) (Semi)leptonic B decays Page 15

19 How to determine s 0? Sum rule: f 2 B = 1 M 2 B s0 m 2 b M B 2 s ds e M 2 ρ OPE (s) result for f 2 B depends artificially on Borel parameter M 2! differentiate both sides w.r.t. 1/M 2 and rewrite M 2 B = s0 m b 2 s0 m 2 b ds s ρ OPE (s) exp s M 2 ds ρ OPE (s) exp s M (Semi)leptonic B decays Page 16

20 Bayesian Approach [Imsong,Khodjamirian,Mannel,DvD ] use informative priors for M 2, as well as the QCD and OPE parameters construct a (purely theoretical) likelihood, which constrains the first moment of the sum rule to within 1% of M 2 B 0.25 the posterior shows the priors mostly unchanged, and a 0.20 peaking structure for s computer posterior-predictive distribution of f B using the full posterior PDF B->pi::sp_0^B@DKMMO (Semi)leptonic B decays Page 17

21 Light-Cone Sum Rules (LCSRs) When switching from 2ptSRs to LCSRs, we change the correlation function: F(q 2, (p + q) 2 d 4 x ) = e iq x π(p) T {J µ(x), j (2π) 4 5 (0)} 0 pµ coeff. only F had (q 2, (p + q) 2 ) = f Bf +(q 2 )MB 2 + (p + q)2 M 2 B ρ(q 2, s)ds s (p + q) 2 Changes: replace one j 5 with weak current J µ final state is on-shell π now dispersive integrals now in (p + q) 2 F OPE (q 2, (p + q) 2 ) = OPE in x 2 0 now m 2 b ρ OPE (q 2, s)ds s (p + q) 2 vaccum condensates replaced by Light-Cone Distribution Amplitudes, similar to PDFs of π (Semi)leptonic B decays Page 18

22 Form Factor Results [Imsong,Khodjamirian,Mannel,DvD ] LCSR e.g. f +(q 2 ), as well as 1st and 2nd derivatives w.r.t. q 2 q 2 = 0 and q 2 = 10 GeV 2 includes 6-by-6 correlation matrix f +(0) = ± 0.020, f +(10 GeV 2 ) = ± 0.032, with a correlation coefficient of correlation smaller between f + and derivatives Fit to BCL Ansatz f +(q 2 ) = f+(0) 1 q2 M [ B b 1 (z(q 2 ) z(0)) + b 2 O (z 2)] 3 parameters: b + 1 normalization f +(0) two shape params b 1, b f Bπ + (0) [10 1 ] b f + Bπ (0) [10 1 ] parametrization as in [Bourrely,Caprini,Lellouch ] (Semi)leptonic B decays Page 19

23 V ub Results [Imsong,Khodjamirian,Mannel,DvD ] data: Belle+BaBar, 6 bins q 2 12 GeV data vs inclusive: barely 99% prob. f + Bπ (0) [10 1 ] V ub [10 3 ] 68%, 95%, 99% prob. contours for 2010 data 68%, 95%, 99% prob. contours for 2013 data central value and 68% CL interval for GGOU [HFAG 2014, ] 2013 data: Belle+BaBar, 6 bins q 2 12 GeV data increases tension 1D marginals: V ub 2010 = ( ) 10 3 V ub 2013 = ( ) (Semi)leptonic B decays Page 20

24 Global Analysis of Data on b u Transitions

25 Search for NP in Semileptonic b u Transitions Effective Field Theory Modify the effective b ulν vertex by adding a different chirality in the hadronic current: L eff b u = G FVub eff [ ] C V,LL O V,LL + C V,RL O V,RL 2 where O V,RL = [u(x)γ µ (1 + γ 5 )b(x)] [l(x)γ µ(1 γ 5 )ν(x)] Consequently: Vub B τν 2 Vub eff 2 CV,L C V,R 2 Vub B πlν 2 Vub eff 2 CV,L + C V,R 2 Vub incl. 2 Vub eff ( 2 C V,L 2 + C V,R 2) B τ ν τ BaBar 2009 SL tag BaBar 2012 had. tag Belle 2012 had. tag Belle 2015 SL tag B 0 π + µ ν µ BaBar 2010 untagged BaBar 2012 untagged Belle 2010 untagged Belle 2013 full recon. B X ulν BaBar+Belle HFAG avg. (GGOU) (Semi)leptonic B decays Page 22

26 Strategy [Feldmann, Müller, DvD ] fit C V,L and C V,R from data choose global phase as phase of V eff ub C V,L C V,L is real-valued in the fit introduce nuisance parameters for exclusive decays, using informative priors B τ ν: B decay constant f B from 2ptSRs [Gelhausen,Khodjamirian,Pivovarov,Rosenthal ] B 0 π + µ ν : form factor f +(q 2 ) from LCSRs [Imsong,Khodjamirian,Mannel,DvD ] use several fit scenarios and perform statistical comparison 1 no right-handed currents: C V,R = 0, fit C V,L equivalent to determination of V ub = V eff ub C V,L 2 real-valued right-handed currents 3 complex-valued right-handed currents (Semi)leptonic B decays Page 23

27 Search for NP in Semileptonic b u Transitions Result: Scenario 1 [Feldmann, Müller, DvD ] find C V,L = 1.02 ± 0.05 at 68% probability corresponds to V ub = (4.07 ± 0.20) 10 3 at 68% prob. χ 2 = for 28 degrees of freedom excellent fit with p value of 91% however: form factor pull does not enter χ 2 3 form factor parameters pull of 3σ (Semi)leptonic B decays Page 24

28 Search for NP in Semileptonic b u Transitions Result: Scenario 2 (w/ real-valued C V,R ) [Feldmann, Müller, DvD ] CV,R C V,L contours at 68% and 95% prob. blue stripes, negative slope: B πµν blue stripes, positive slope: B τν green rings: B X ulν orange areas: combination gray area: add. hypo. measurement of A FB in B s K + ( K π)lν (SM ±10%) (Semi)leptonic B decays Page 25

29 Search for NP in Semileptonic b u Transitions Result: Scenario 2 (w/ real-valued C V,R ) [Feldmann, Müller, DvD ] CV,R C V,L contours at 68% and 95% prob. blue stripes, negative slope: B πµν blue stripes, positive slope: B τν green rings: B X ulν orange areas: combination gray area: add. hypo. measurement of A FB in B s K + ( K π)lν (SM ±10%) (Semi)leptonic B decays Page 25

30 Search for NP in Semileptonic b u Transitions Result: Scenario 2 (w/ real-valued C V,R ) [Feldmann, Müller, DvD ] best fit point: C V,L = 1.025, Re (C V,R ) = χ 2 increases to for 27 degrees of freedom still very good fit with p value of 81% form factor pull decreases to 2σ C V,R compensates need to adjust form factor solutions Wilson coefficients: ) C V,L = 1.02 ± 0.05 and Re (C V,R 0.10 at 68% probability ) Re (C V,R = 1.02 ± 0.05 and C V,L 0.10 at 68% probability looses against scenario 1 in posterior odds with 1 : (Semi)leptonic B decays Page 25

31 Search for NP in Semileptonic b u Transitions Result: Scenario 3 (w/ complex-valued C V,R ) [Feldmann, Müller, DvD ] Re CV,R 0.0 Im CV,R C V,L Re C V,R (Semi)leptonic B decays Page 26

32 Search for NP in Semileptonic b u Transitions Result: Scenario 3 (w/ complex-valued C V,R ) [Feldmann, Müller, DvD ] Re CV,R 0.0 Im CV,R C V,L Re C V,R (Semi)leptonic B decays Page 26

33 Search for NP in Semileptonic b u Transitions Result: Scenario 3 (w/ complex-valued C V,R ) [Feldmann, Müller, DvD ] best fit point: C V,L = 1.025, C V,R = i0.000 χ 2 remains at for now 26 degrees of freedom still very good fit with p value of 77% form factor pull stays as 2σ C V,R compensates need to adjust form factor marginal solutions of the Wilson coefficients are not disjoint looses against scenario 2 in posterior odds with 1 : 3.62 (against scenario 1 with 1 : 100) (Semi)leptonic B decays Page 26

34 Comments on V ub from Λ b plν

35 Physik Institut 3 10 L V ub inclusive B πlν Λ b pµν (LHCb) combined ε R [LHCb ] The LHCb measurement of V ub in Λ b plν is the first measurement of V ub in a baryon decay. The different spin structure between meson and baryon decays gives rise to complementary information on the Wilson coefficients C V,L(R). But: The V ub extraction relies heavily on the normalization to the decay Λ b Λ cµν! In fact, LHCb is only sensitive to the ratio V ub /V cb! Accurate and precise predictions of the ratio of (binned) hadronic matrix elements are required, and currently only available from lattice QCD. However: We can test the lattice results by applying continuum QCD methods (Semi)leptonic B decays Page 28

36 Testing the Λ b Λ c Form Factors Inclusive calculation of 2pt correlation function [T. Mannel, DvD ] { } T Γ (ε) d 4 x e i(v x)ε Λ b T b v(x)γc v(x), c v(0)γb v(0) Λ b v: velocity of the b and c quark in the Λ b rest frame ε: excitation energy of the intermediate state Contour integral of T (ε) encompasses all, i.e. the elastic and all inelastic, contributions to the imaginary part of the forward matrix element bounds the sum of exclusive form factors at the zero recoil point (when Λ c rests in Λ b rest frame) (Semi)leptonic B decays Page 29

37 Zero-Recoil Sum Rule CDF 1.0 Difference between inclusive bounds and lattice values at zero recoil for both vector and axialvector currents: CDF F inel G inel For both currents we find that the lattice results for the Λ b Λ c transitions exceed the inclusive bounds at 55% probability (Semi)leptonic B decays Page 30

38 Conclusion

39 Conclusion there is a tension between exclusive and inclusive determinations of V ub V ub from exclusive decays requires knowledge of hadronic decay constants and form factor QCD sum rules can calculate these quantities parametric correlations have been studied for the first time the tension could be due to New Physics global analysis suggests that right-handed currents alone cannot explain the tension LHCb s measurement of Λ b pµ ν important for V ub /V cb determination; complementary probe of NP however: zero-recoil sum rules suggest that lattice form factors are too large (Semi)leptonic B decays Page 32