Indications for Large Rescattering in Rare B Decays
a r X i v :h e p -p h /0312201v 1 16 D e c 2003
EPJ manuscript No.
(will be inserted by the editor)
Indications for Large Rescattering in Rare B Decays
George W.S.Hou
Department of Physics,National Taiwan University,Taipei,Taiwan 10764
Received:date /Revised version:date
Abstract.The sign of A CP (K ?π+)<0,the evidence for ˉB
0→π0π0,and the possibly sizable A ππand S ππin ˉB
0→π+π?all suggest that ?nal state rescattering may be needed in ˉB →P P decay,which is echoed by large color suppressed ˉB
0→D 0h 0modes.An SU(3)formalism of 8?8→8?8rescattering in P P ?nal states leads to interesting predictions,in particular allowing for small ˉB 0→K ?K +.PACS.11.30.Hv Flavor symmetries –13.25.Hw Decays of bottom mesons
1Motivation
Around 1999,the emergence of large Kπ/ππratio in B
decay lead to the suggestion [1]that maybe γ≡φ3≡
arg V ?ub >～90?
,in contrast to the CKM ?t (to other data)of ～60?.The pattern of Kπ,ππdata can then be under-stood within factorization.As the ?nal results from CLEO came out,it was further speculated [2]that rate and (di-rect)A CP pattern could hint at rescattering in ?nal state (FSI).If one makes a ?nal state isospin decomposition,FSI phases are in principle present.As A CP s depend crit-ically on absorptive parts,the CP invariant FSI phases could easily shift direct CP patterns.
The discovery of color suppressed ˉB
0→D 0h 0de-cays [3]above factorization predictions suggests that FSI may have to be taken seriously.More recently [4],the 3σ
e?ect of A CP (K ?π+)<0,the evidence for ˉB
0→π0π0,and the possibly sizable A ππand S ππin ˉB
0→π+π?etc.,all could be hinting at presence of sizable rescattering in ˉB
→P P ?nal states,where P stands for an octet pseu-doscalar.We thereby revisit the FSI speculation.
Treating the color suppressed ˉB
0→D 0h 0modes as an exercise,we developed [5]an SU(3)based 3?8→3?8rescattering in DP ?nal states.Here we report the results [6]on extending the formalism to P P ?nal states.
2Ansatz:Multimode Fit with FSI Phases
Since factorization seems to account for the rates of lead-ing decays,we adopt the simple and physical picture of (naively)factorized amplitudes A f l followed by FSI,i.e.
i ;out |H W |B =
l
S 1/2
il A f l .
(1)
We use naive factorization not just for sake of simplicity,but because more sophisticated treatment in,say,QCD
factorization introduces hadronic parameters,and one may incur double counting.Note that l is summed over quasi-elastic channels in Eq.(1).We assume that the large can-cellations between numerous in elastic channels generate only the “perturbative”FSI phase accounted for by the penguin absorptive part.
We treat ˉB
→P P ?nal states only,since V P modes are not yet settled (both experiment and theory).Also,we are yet unable to treat η′hence take η～=η8.Thus,we consider 8?8→8?8rescattering.Since only the 1,one of the 8s,and the 27are symmetric,the S 1/2matrix in Eq.(1)takes up the form
S 1/2=e iδ27|27 27|+e iδ8|8 8|+e iδ1|1 1|,
(2)
hence there are just two physical phase di?erences,which we take as δ≡δ27?δ8and σ≡δ27?δ1.These FSI phases redistribute A f l according to Eq.(1).Alternatively,they can be viewed as a simple two parameter model extension beyond the usual B →P P amplitudes.
It is important to point out that the σphase appears only in the π?π+,π0π0,K ?K +,K 0
2George W.S.Hou:Indications for Large Rescattering in Rare B Decays
Table1.World average inputs and?tted outputs;data in brackets are not used in?t,whileη8K(π)?entries are forηK(π)?. Horizontal lines separate rescattering subsets.Fit1or2stand forφ3free or?xed at60?.Settingδ=σ=0but keeping other parameters?xed give the results in parentheses;the?tted parameters andχ2min.are given in Table2.
K?π+18.2±0.819.4+1.0
?1.2
(19.7)18.5±0.6(18.1)?9±4?6+2?3(9)?4±1(7)
K0η8[<4.6(90%CL)]3.4+0.8
?0.6(4.1)3.9+1.0
?0.8
(4.6)—24+9?4(0)15+3?2(0)
K0π?20.6±1.319.6+2.2
?1.4(18.5)21.6±0.6(20.9)1±68+2?1(0)5±0(0)
K?π012.8±1.111.6+0.5
?1.0(12.1)11.0±0.3(10.9)1±12?19+4?7(7)?14+1?2(6)
K?η8[3.2±0.7]3.6+0.8
?0.7(4.2)4.6+1.0
?0.8
(5.4)[?32±20]33+15
?9
(?9)19+6?4(?5)
π?η8[3.9±0.8]1.2+0.1
?0.3(1.4)1.5+0.0
?0.1
(1.8)[?51±19]75+25
?18
(?32)42+9?6(?19)
K?K0<2.2(90%CL)1.7+0.3
?0.2
(1.5)1.3±0.1(1.0)—?84+9?14(?4)?79±3(?3)
K0K0<1.6(90%CL)1.5+0.3
?0.6(1.5)1.1±0.1(1.0)—?63+141
?24
(?4)?86+6?1(?3)
π0η8—0.4+0.2
?0(0.3)0.2±0.0(0.2)—?3±0(?4)?3±0(?4)
η8η8—0.2±0.0(0.1)0.2±0.1(0.1)—?10+96
?75(?6)?91+14
?3
(?6)
Fit1Fit2No FSI FSI:χ2min./d.o.f.is50/7(65/8)for Fit1(2),as seen in the last column of Table2.
From Table2we see that,whether one keepsφ3?xed or free,the?tted FSI phasesδandσare rather sizable, while disallowing them gives much poorerχ2.Let’s see what drives these phases.
Theδdependence of A CP s for K?π+and K?π0are plotted in Fig.1.The former now has some signi?cance, but opposite in sign w.r.t.QCD factorization predictions. We see that FSI can bring about a sign change,which disfavors sinδ<0.Together with“restraint”from K?π0 mode,δ～60?is more or less settled between the two. Note that A CP(K?π0)
Theπ?π+andπ0π0rates are sensitive to bothδand σ.They are plotted in Fig.2(a)withδ?xed to?t values of
o
δ
?
?
A
C
P
(
K
?
π
+
)
o
δ
?
?
A
C
P
(
K
?
π
)
Fig.1.A CP(K?π+,0)vs.δ.Solid(dashed)line is for Fit1 (Fit2),and shaded bands are1σexperimental ranges.
George W.S.Hou:Indications for Large Rescattering in Rare B Decays 3
o
σB r (π+π?,π0π0)
0o
90o 180o 270o 360o
σ
01234
B r (K +K ?)
Fig.2.Rates (×106)for (a)π+π?,π0π0,(b)K ?K +vs.σ.Solid (dashed)line is for Fit 1(Fit 2)with δ?xed at 67?(63?).Horizontal bands are 1σexperimental ranges or upper limit.
Table 2.Both ?ts clearly favor large σ,to account for the smallness of π?π+rate by rescattering into π0π0,which both [8]BaBar and Belle now have evidence for!
The rate of K ?K +<6×10?7is very suppressed,which could challenge FSI.From Eq.(2),if δ1,δ8,δ27are all randomly sizable,K ?K +～π?π+>10?6would be
expected.However,as seen in Fig.2(b),for |δ?σ|<～50?
,the K ?K +rate can be comfortably below the present limit,but δ,σcan be separately large .The reason is due to the smallness of 27in the I =0ππ→ππamplitude.K 0