LAT07 .....A. Soni

Challenges of non-leptonic weak decays on the lattice

Amarjit Soni

HET,BNL

(soni@bnl.gov,adlersoni@gmail.com)

LAT07 .....A. Soni

Outline• Introduction & Motivation

• Recapitulate: Crutches are unavoidable

• Key idea -> common path to several crutches

• Lattice considerations

• HLChPT & its implications

• Outlook

LAT07 .....A. Soni

Introduction & Motivation• Via the UT, Lattice Weak Matrix Elements

effort played a pivotal role in attaining a milestone in our understanding of CPV phenomena

• For the future,though, exceedingly limited menu is/should be a very serious concern

LAT07 .....A. SoniYoungjoon Kwon

LAT07 .....A. SoniYoungjoon Kwon

LAT07 .....A. Soni

Past performance & future prospects

• Some gloating over the past is justified.

• Note, though, that program all initiated in ’80’s

• Refinements therein are certainly useful

• But expt/phenom needs are NOT frozen in

time….moves often speedily (esp.expt)

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Focus on B->DM:

LAT07 .....A. Soni

LAT07 .....A. Soni

ILLUSTRATES power of B decays

• For 2 or more common modes of D0 , \bar D0 both weak and strong phases can be extracted 2 modes mean 4 observables versus

4 unknowns [rBM ,wk phase γ,δstrong (mode1),

δstrong (mode2)]. Thus FOR THE FIRST TIMEa data driven method is able to tame DIRECT CPV. Availablity of dozens of FS allows for numerous similar useful applications

LAT07 .....A. Soni

M= K, K*, pi, rho…. For charged and neutral B are of interest

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Special note: Br(B->\barD0 M),is very difficult to measure

experimentally due to technical(exptal)

complications

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Crutches are unavoidable

I. Infinite volume….lessons learned via:• D->K pi…Bernard, Simone,A.S.’90;• Abada et al, ’90• Maiani + Testa,’90• Ciuchini,Franco,Martinelli,Silvestrini,’96• II. Finite volume (K)….Lellouch, Luscher’,’00;• Lin,Martinelli,Sachrajda,Testa,’01• Christ, Kim, Yamazaki ,’06• Kim, Sachrajda, Sharpe,’06

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Useful crutches

• QCDF

• pQCD

• SCET

• HLChpt

• ….

Key idea common to all these crutches is that

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Application of HLChPT to B>DM(Christopher Aubin + A.S)

• For HL mesons, (following Wise;Donoghue+Burdman) marry HQET & ChPT….Now U(2m) spin-flavor symmetry on

the HQ side and standard SU(n)LXSU(n)R for m heavy and n light flavors.• For definiteness m=2; n=3• For now, just to provide a concrete,though crude, 0th

order step use LO HLChPT….NLO clearly highly desirable (i.e need a good student!)… Meantime…

LAT07 .....A. Soni

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Weak NL Decays of Q(path analogous now to BDSPW’85)

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For B->D only the scalar piece is relevant & in LO leads tothe very simple (expected) relation:

Similar relations in LO for the B-> \bar D case.An important diference is that for π -> K

B0 -> BS

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B->D (M) decays -> New “B”-parameters {to avoid confusion call B1-params (for decays)}

B,BSD0

For M=K , use BS ; also i=1,2Vac Sat leads to (e.g. for M=K):

1(or 1/3) MBs f Bs MD f D [contrast with (8/3)M2 f2 for the ΔB =2]

Qi from Heff ΔB=1, b->cu l OR b->u bar-c l

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Some Lattice Considerations• Relevant important decays involve no

quark loops, no mixing with LDO -> less worrysome even with mixed action approaches

However, partial cancellation{a la G+L’74; A+M’74} between contributions of Q1 & Q 2 puts more demand on accuracy.

• B1’s are simultaneously (along with the usual B’s) accessible & represent basically a new application

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Hand-waving estimate• Sqrt(rBM) ~ [Vub /(Vcb Vus) ~0.4]/{Neff}

• THUS what is relevant for (L)QCD

is a precise determination of {Neff} in

the above.

. Neff = [c2 + (c1/N)]/ [(c2/N) + c1 ]

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How well are rBM currently known?

• Sqrt(rBK ) (VS) ~ 0.16 +- ~ 0.08 {CRUDE GUESS} ( Large N + partial cancellation)

• Direct experimental determination (so far)has gotten the 1-σ error on rBK to be ~130% , i.e rBK =0.012+-0.015 (See hep-ex/0607065 &/or HFAG’07)

• Since of central interest is ratio which scales as {1/N} (in amplitude), concern reg. HLChpt may be addrssed by varying the momentum of M and/or lattice masses

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Possible Impact• Lattice determination of Neff can help in improving constraint on γ (using input

value of Vub ) Of course,eventually,experimentalist will simultaneously

extract both γ & rBK , as is

guaranteed by the method, then with input

from the lattice of Neff , exptal knowledge of rBK allows a new way to determine Vub /(Vcb

Vus)

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Summary & OUTLOOK

.Non-leptonic weak decays continue to be challenging…HUGE POTENTIAL• Specific example: B->D^0 (\bar D0) K very useful for UT

phase γ and also for |Vub| possibly providing a new (NL) avenue to |Vub|

• HLChPT provides a starting point; LOChPT gives all relevant decay ΔB=1 amplitudes in terms of

charcteristic two fig. 8 (B1) parameters. Simulateously accessible along with usual mixing ( ΔB=2)

B-parameters• Numerous continuum ideas and tons of available data

should provide useful bridges.