Radiative and rare decays Shoji Hashimoto (KEK) @ BaBar-Lattice workshop, SLAC, Sep 16, 2006

Radiative and rare decays

Shoji Hashimoto (KEK)

@ BaBar-Lattice workshop, SLAC, Sep 16, 2006

Sep. 16, 2006 Shoji Hashimoto (KEK)

Decay modes of interest

• Radiative: BK*, • only q2=0 is relevant• SU(3) breaking ratio /K* is also interesting

for |Vtd/Vts|.

• Rare: BK(*)ll, K(*) • may contain interesting new physics effects• K(*)ll suffer from long distance contribution


Form factors

.)().)((

))(.()()(

)()()1(),(

,)(.*

)(

.*)()(

.*)(2)(),(

,)(2

)(),(

22

2*2

3

*22*22

*215

2

222

2

2*2

1

22

05

*2

kpmm

qqpqT

kppmmqT

kpqiTpBbqqkV

qq

mmkp

mm

qqiA

qq

qqAmmi

qq

qqAmipBbqkV

kpmm

qVpBbqkV

VB

VB

VB

VB

VB

V

VB

Main operators for BK*llalso appear in Bl

Main operator for BK*

Heavy quark scaling:V~M1/2, A1~M-1/2, A2~M1/2

Heavy quark symmetry: V(q2)=2T1(q2), A1(q2)=2iT2(q2)

from O9, O10

O7


Previous lattice calculations

• Bernard, Hsieh, Soni, PRL72 (1994) 1402.• UKQCD (Burford et al.), NPB447, 425 (1995).• APE (Abada et al.), PLB365, 275 (1996).

• BK*, quenched

• UKQCD (Flynn et al.), NPB461, 327 (1996).• Bl, also K*, quenched• most extensive study, so far.

• UKQCD (Del Debbio et al.), PLB416, 392 (1998).• Reanalysis of the above data; testing parametrizations.

• SPQcdR (Abada et al.), Lattice 2002, hep-lat/0209116.• finer lattice than UKQCD; still quenched.


Some results

• All quenched.• Lattice cutoff ~ typically

2.7 GeV.

• Use (O(a)-improved) Wilson fermion for heavy quark; need heavy quark extrapolation.

UKQCD (1996) confirmed the HQS relations

Heavy quark symmetry: V(q2)=2T1(q2), A1(q2)=2iT2(q2)


Some more results

• Form factor shape• constant, pole, or

dipole?

• 1/M extrapolation• Linear or quadratic?

again, from UKQCD (1996)


Not much progress since then

1. More complicated than BP. May want to do B first.2. Statistical noise is much larger. Even worse for lighter

light quarks and heavier heavy quarks. extrapolations are harder.

3. No solid theoretical guide for the chiral extrapolation of vector particles.

4. Decay width of K* is significant (50 MeV): not sure how to treat on the lattice.

5. Only q2=0 is relevant for BK*. Must introduce model dependence when extrapolate.

WHY?WHY?


Then, what we have been doing…

1996-2006:• Dynamical fermion simulations become a

standard. No more quenching.• Chiral extrapolation is more delicate than we

naively thought. We don’t care much about it in the quenched theory (a wrong theory, anyway), but full theory is a different story.

• To perform reliable chiral extrapolation, one must come closer to the chiral limit.


An example

Pion decay constant

• Early Wilson fermion simulations are limited in a heavy pion mass region.

• Staggered fermion groups went ahead to reduce sea quark mass, but a complicated ChPT analysis is necessary.

• Exact chiral symmetry simulation is now available. Can get to small enough sea quark masses.

2 2

2 21 ln

2 (4 )fNf m m

f f

JLQCD (2002)

JLQCD (2006)


Step by step

• Instead of going to very complicated quantities like BK*ll, one should (I want to) make sure that the chiral extrapolation goes right using easier quantities.1. Pion decay constant, PCAC relation: the simplest quantities.2. Pion EM form factor, kaon form factors: simplest non-trivial

form factors. Detailed comparison with ChPT is possible.3. Heavy-light decay constants: need the B*B coupling to control

the chiral extrapolation.4. Heavy-to-heavy form factors: more symmetry than heavy-to-

light; easier to get precise results.5. Heavy-to-light form factors, but BP.6. Heavy-to-light form factors, then BV.


Possibilities in the future

Besides the steady progress in the standard direction, I am interested in…

• K*ll at low recoil. Long distance effect?

• Higher recoil region by the moving frame technique.


K*ll at low recoil

• Lattice QCD is most suitable for high q2 region.

• But, long distance contrib arises when q2 is around the charmonium resonances.

b

s

c

c

l

l

O1, O2

Morozumi, Lim, Sanda (1989)

LEET or SCET

resonances

Lattice calculation (?)


OPE technique

Not possible to insert time-like momentum to the charm loop.

Grinstein and Pirjol, PRD70, 114005 (2004).

• Proposal to expand the charm loop by operator products: /mb, mc/mb.

• Then, it becomes the usual semileptonic topology; easier to calculate on the lattice.

Impact of the higher order terms on the AFB analysis, for example?

b

s

c

c

l

l

b

s

l

l


Moving frame?

• 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.• Bl K(*)ll form factor restricted in the large q2

region.

• BK*, are beyond the reach.

• A possible solution is to discretize after boosting.


Moving NRQCD

• Moving HQET: Mandula-Ogilvie (1992)

• Moving NRQCD: SH, Matusfuru (1996), Sloan (1998), Foley, Davies, Dougall, Lepage (2004~)

• Work ongoing by Wong, Davies, Lepage (at Lattice 2006)

• How much recoil needed?• Any limitations? Can we

boost the B meson to the light-cone?

Isgur-Wise function (SH, Matsufuru, 1996)


Moving HQET

Mandula, Ogilvie (1992)• Generalization of the

usual lattice HQET• Write the b quark

momentum as

pb=mb u+kand discretize the residual momentum k.

• Notation

• HQET:

The original field is recovered by

• The lattice version:

• Extension to NRQCD is straightforward.


How much boost possible?

• Intuitively, light degrees of freedom should also move fast in the boosted frame: momentum (QCD/mB)pB=QCDv

• Lorentz contraction means a wider distribution in the momentum space.

• How is it realized in the lattice calculation?

A toy example:

Etot = Ev(kQ) + Eq(kq)

kQ + kq = 0

boost

v = 0v = 0.5

v = 0.8QCD

QCD(v1)


How large velocity needed?

• To avoid large discretization effect,

is necessary. Namely, v < 0.7~0.8.

• Need finer lattice to boost further.

• Bl at maximum recoil (q2=0)

• BK*

GeV11

1QCD

v

vp v2.64 GeV 01.0 GeV 0.750.5 GeV 0.930.0 GeV 0.999

pK* v2.57 GeV 01.0 GeV 0.670.5 GeV 0.850.0 GeV 0.94

Seems feasible. On-going work by Wong et al for (non-)perturbative matching of the parameters.

Questions,

and partial answers


SU(3) breaking

1. Can lattice QCD calculate the ratio of (and also the individual) form factors T1(BK*)/T1(B) and if so, what precision can be achieved. Related to that, is there a new program to calculate the ratio of (and also the individual) longitudinal to transverse decay constants for K* and ?

• Yes, we can, but currently with model uncertainty in the q2 extrapolation. Precision???

• Decay constants of K* and , to input in the LCSR calc, are much easier to calculate. • A related work: Becirevic,

Lubicz, Mescia, Tarantino, JHEP05 (2003) 007.

I don’t know about new programs.


Annihilation diagram

2. Can lattice QCD reliably estimate the u quark contribution to the B decay (annihilation diagram and u quark loop).

• I don’t have any good idea. Need double insertion of operators.

• The best one can do is to measure Bl.

from Beyer, Melikov, Niktin, Stech, PRD64, 094006 (2001).


Inclusive

3. In the light of Super-B factories, is the inclusive ratio of bd/ bs easier to understand theoretically?

• Probably. I’m not sure how much uncertainty cancels in the ratio, but some of them should.

• Don’t give up the exclusive modes.


Asymmetry

4. Since we are entering precision measurement of asymmetris (CP, isospin) in the BK* decay, can lattice QCD say anything about these quantities?

• Basically, all we can provide is the form factors. Since the single form factor dominates the BK* rate, the form factor is irrelevant in the CP asymmetry.


Asymmetry (cont)

• Grinstein, Grossman, Ligeti, Pirjol, PRD71, 011504R (2005), argued that the contribution of wrong helicity photon could be ~10%. Namely, suppressed only by QCD/mb rather than ms/mb.

comes from charm loop.

1.0~3 7

2

b

QCD

mC

C


Another ratio

5. How about the ratio of form factors for BsK*/Bs? Could this be calculated in principle better than B/BK*?

• Yes. I think so. Lattice calculation would be easier, because the chiral extrapolation for the spectator quark is not necessary.


Vts/Vub

6. Is it possible to have a theoretically precise estimator of Vts/Vub by measuring ratio of branching fractions BKll/Bl at high q2?

• Yes, if we can control the long distance contribution using the OPE method.


7. Are SU(3) breaking in the form factors more or less managable than for the vector meson case?

• Yes. A lot easier than the decaying particles.

Documents

Radiative and rare decays Shoji Hashimoto (KEK) @ BaBar-Lattice workshop, SLAC, Sep 16, 2006