Boosting B meson on the lattice

Transcription

1 Boosting B meson on the lattice Shoji Hashimoto INT Workshop Effective Field Theory, QCD, and Heavy Hadrons, U. of Washington, Seattle; April 28, 2005.

2 Beyond the small recoil processes A big limitation of the lattice calculation = spatial momentum of initial and final hadrons must be small (< 1 GeV/c) to avoid large discretization errors. B πlν form factor restricted in the large q 2 region. B K * γ, ργ are beyond the reach. No way to treat the light-cone distribution amplitude / shape function.

3 Example: semi-leptonic decays Iijima CKM2005: Dingfelder CKM2005 Only the highest bin is usable; others introduce model dependence.

4 Another example: radiative decays CKM2005 Interesting physics: V td /V ts from penguin diagram, strong constraint on new physics Theoretically clean? Is the current estimate robust enough? No way to improve theoretical uncertainty?

5 Need boost! Moving NRQCD? A possible solution is to discretize after boosting. Moving NRQCD (Foley, Davies, Dougall, Lepage at Lattice 2004) was proposed recently as a method to calculate B πlν near the maximum recoil. GOOD NEWS! GOOD NEWS! The idea is not new. Why does it revived? How much recoil needed? Any limitations? Can we boost the B meson to the light-cone?

6 Light-cone distribution amplitude B ππ on the lattice? ππ final state is not gold-plated : more than one particle in the final state. Emitted π is energetic: large discretization error is expected for particles with momentum larger than 1 GeV/c. QCD-based calculation? Possible with the QCD factorization / SCET Only the leading term in 1/m b ; higher orders have to be modeled. Factorized nonperturbative part is left for us to calculate!: light-cone distribution amplitude (LCDA).

7 Factorization Slide stolen from CKM2005

8 This talk What can we do for the large recoil processes? 1. Introduction 2. Moving HQET/NRQCD Properties and limitations 3. Light-cone distribution amplitude Light-like separation possible!? Disclaimer: no numerical results, yet.

9 Moving HQET/NRQCD Properties and limitations Statistical noise Physical Reach

10 Moving HQET Mandula, Ogilvie (1992) Generalization of the usual lattice HQET Write the b quark momentum as p b =m b u+k and discretize the residual momentum k. Notation HQET: The original field is recovered by The lattice version: Evolution equation

11 The first application Isgur-Wise function Mandula, Ogilvie, heplat/ v =0, 0.25, 0.5, 0.75 very noisy ξ(v v ) t=3 t=4 ξ(v v ) v v

12 Where does the disease come from? Dispersion relation Evolution The heavy quark propagator increases in time for k ~ v; unphysical mode π/a increases most. Physical mode k ~ 0 is selected by the Fourier transform of the heavylight meson operator: k Q + k q = 0 Exponentially large cancellation is needed for large t. Statistical error grows exponentially with a slope E Qq (E QQ +E qq )/2 Lepage (1991)

13 A possible solution Introducing 1/m Q terms = Moving NRQCD Dispersion relation The large negative energy mode is lifted by the kinetic term. SH and Matsufuru (1996) Still limited in a small v range

14 Larger velocity? Explored in the recent work (Davies, Dougall, Foley and Lattice 2004) Works fine up to v ~ 0.8. magic smearing? lat04 Heavy-light binding energy: lat04 Then, is there any limitation in v?

15 Heavy-light system Intuitively, light degrees of freedom should also move fast in the boosted frame: momentum (Λ QCD /m B )p B =Λ QCD γv Lorentz contraction means a wider distribution in the momentum space. How is it realized in the lattice calculation? A toy model: E tot = E v (k Q ) + E q (k q ) k Q + k q = 0 boost ±Λ QCD Λ QCD γ(v±1) v = 0.8 v = 0.5 v = 0

16 How large velocity needed? To avoid large discretization effect, Λ QCD is necessary. Namely, v < 0.7~0.8. The 1/m b correction would help to slightly reduce v. Need finer lattice to boost further. B πlν at maximum recoil (q 2 =0) 1+ 1 GeV 2.64 GeV 0 1 v 1.0 GeV GeV GeV B K * γ p K* v 2.57 GeV GeV GeV GeV 0.94

17 Moving HQET/NRQCD Can boost the B meson: The largest recoil point is within reach if we allow ~ 1 GeV/c momentum for the final state. For both semi-leptonic and radiative. Limitations come from: Statistical noise Physics (accompanying light degrees of freedom) Other options: Moving Fermilab Moving light (Boyle, Mackenzie) when you want to boost pion. Boost both heavy and light? The physics limitaion remains (= broad momentum distribution).

18 Light-cone distribution amplitude Light-like separation possible? Other possibilities?

19 Light-cone distribution amplitude Separation z is light-like: z ± =(z 0 ±z 3 )/ 2 [z,0] denotes a path-ordered Wilson line. Appears in the hard spectator amplitude, convoluted with the hard scattering kernel. At the leading order, often appears in the form Poorly known: λ B = 350 ± 150 MeV (BBNS) λ B ~ 600 MeV (LCSR: Ball-Kou, 2003) λ B = 460 ± 110 MeV (QCDSR: Braun et al, 2004)

20 Light-cone separation? No light-cone on the Euclidean lattice. Lorentz boost changes the scale of k + (k + u + k + ), but z - is still on the light-cone. A previous proposal: (Aglietti et al, 1998, Abada et al, 2001) with a scalar propagator S(x;0) ~ 1/q 2. Then, π (1-u)p π up π q+up π for large Q 2 =-q 2. After all, I have got stuck is obtained. Extract Φ π from here.

21 What to calculate? Any amplitude instead of the LCDA itself. (Not necessarily be a physical one.) The simplest process: B lνγ (*) At the leading order, it is given as proportional to the inverse moment 1/λ B.

22 Boost! Need a large momentum insertion from the photon. Then, the internal light quark becomes energetic. Boost back to the rest frame of the photon. Or, the soft photon limit (red shift). Not possible: it requires v=1 for the massless photon. Consider a virtual (spacelike) photon instead. Q 2 must be large enough to keep v < 0.8.

23 4-momentum injection How do you treat the photon in the external state? EM current is inserted everywhere when we calculate the matrix element. Working on an Euclidean lattice. Only space-like momentum q µ can be injected. Expected functional form: The usual pole form. F V,A can be extracted through the exponential fall-off in the Euclidean time.

24 The problem is more general Usually, the ground state hadron is extracted from the Euclidean t dependence ~ exp(-mt). Three-momentum can be specified, but the time component is not. Need a method for the case that there is no hadron in the final state but momentum must be injected. B lνγ related to LCDA when photon is energetic. Background to B lν when photon is soft; the same problem for D decay, though less pronounced. π γγ (*) Related to pion LCDA (Lepage-Brodsky!) when photon is enegetic. Input to the light-by-light amplitude in muon g-2, when photon is soft.

25 Recipe 1. Calculate the 3-point function with a space-like four-momentum (q 2 =-Q 2 ) inserted. Boost the b quark so that the photon becomes soft. 2. Fit the data with the expected functional form to obtain the form factor. 3. Extrapolate to the Q 2 0 limit. 4. Compare with the perturbative calculation. Then, 1/λ B is obtained at the leading order; higher orders can also be included if the data are precise enough.

26 Summary Boosting may open new applications of lattice QCD: semi-leptonic with large recoil, radiative decays. Even more exciting if we can calculate LCDA. Probably possible through physical processes like B lνγ or π γγ. A method to treat the four-momentum insertion, yet it s space-like. Real works are yet to be done