Data

VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000

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Possibility of Large Final State Interaction Phases in Light of B ! Kp and pp Data

Wei-Shu Hou1 and Kwei-Chou Yang2

1Department of Physics, National Taiwan University, Taipei, Taiwan 10764, Republic of China2Department of Physics, Chung Yuan Christian University, Chung-li, Taiwan 32023, Republic of China

(Received 20 December 1999; revised manuscript received 3 March 2000)

The newly observed B0! K

0p0 mode is quite sizable while p2p1 is rather small. Data also hint at

p2p0 * p2p1. Though consistent with zero, central values of CP violating asymmetries in K2p1,0

and K0p2 show an interesting pattern. Taking cue from these, we suggest that, besides g � arg�V �

ub�being large, the rescattering phase d in Kp and pp modes may be greater than 90±. If this is true,not only the above trends can be accounted for, but one would also find p0p0 � p2p1,0, and the CPasymmetry in B

0vs B0 ! p2p1 could be as large as 260%. These results can be tested in a couple

of years.

PACS numbers: 12.15.Ff, 11.30.Er, 13.25.Hw

The branching ratios �B� of all four B ! Kp modesas well as the p2p1 mode have recently been reportedby the CLEO Collaboration [1]. The measurements ofK2p1, K2p0, and K 0

p2 modes have been improved,while K

0p0 and p2p1 modes are newly observed. Di-

rect CP asymmetries (aCP’s) in five charmless hadronicmodes have also been obtained [2] for the first time, al-beit with large errors. The ratio K2p1� K 0

p2 � 1 sug-gests that the unitarity phase angle g (� argV �

ub) in theCabibbo-Kobayashi-Maskawa (CKM) matrix is of order90±. Surveys of observed and emerging charmless rare Bmodes suggest that g . 90± [3–6], which is in some con-flict with g � 60± 70± obtained [7] from the global CKMfit to data other than charmless hadronic B decays. Theratio K2p0�K2p1 � 0.65 confirms the expectation thatthe electroweak penguin (EWP) contribution to the K2p0

mode is constructive towards the leading strong penguin(P) contribution [3,5,8]. These results illustrate the infor-mation contained in charmless B decays that have beenemerging in the past three years.

However, the strength of the newly observed K0p0

mode, at �14.815.913.525.124.1� 3 1026, is hard to understand,

since EWP-P interference should be destructive. Onewould have expected that K

0p0�K2p0 � 1�3, or

K0p0 � 6 3 1026. The errors are still large, but if we

take the present central value seriously, since the K 0p0

mode is only weakly dependent on g, a natural possibilityis the presence of strong final state interaction (FSI)rescattering. What is more, we find that present aCP cen-tral values as well as the indication that p2p1 , p2p0

all offer some support for this possibility.The long awaited p2p1 mode is finally measured at

�4.711.821.5 6 0.6� 3 1026, which is rather small. The data

also hint at the p2p0 mode. Though not yet signifi-cant enough, preliminary CLEO findings give [1] p2p0 ��5.412.1

22.0 6 1.5� 3 1026, and the central values seem to in-dicate that p2p1 & p2p0. This could be brought aboutby large g and/or a large FSI phase. The two picturescan be distinguished by measuring p0p0 [4,8]. If small

0031-9007�00�84(21)�4806(4)$15.00

p2p1 is due to g . 90± while the FSI phase d # 30±

is small, then p0p0 & 1026 is expected. However, ifthe mechanism is due to a large FSI phase d . 90±, thenp0p0 can reach �5 3 1026, i.e., as large as p2p1.

It is known that aCP’s are very sensitive to FSI phases.Although the present accuracy of aCP’s in Kp modes islimited by statistics, the central values may already offer usa glimpse of the trend of the FSI phase. As we will show,we find a coherent picture where not only g is large, butlarge d is preferred as well, if the current central values aretaken at face value. Moreover, a large FSI phase d cannotonly be tested by finding p0p0 � p2p1, it can also betested in the near future by finding a rather large aCP inthe p2p1 mode. The size of FSI phases in B decaysis an issue of great theoretical dispute [9], which can besettled only by experiment. We shall consider elastic 2 !2 rescattering phases as the only long-distance FSI phases,returning to a more cautionary note towards the end ofour discussion. We do include short-distance rescatteringphases arising from quark-gluon diagrams. For simplicity,we shall also assume factorized amplitudes that feed theelastic rescattering.

Let us study first the Kp and pp modes withoutassuming a long-distance FSI phase. For the relevanteffective weak Hamiltonian, we refer to Refs. [8,10,11].We take q2 � m2

b�2 in penguin coefficients to gener-ate favorably large quark level absorptive parts [12].Adopting the factorization approach with Nc � 3 andassuming that FSI rescattering is negligible (i.e., set-ting the FSI phase d � 0), B and aCP for the Kp

modes are shown vs g in Figs. 1(a) and 1(b). We haverescaled the value of FBK

0 � 0.3 to fit the K 0p2 data,

and take FBp0 �FBK

0 � 0.9 for SU(3) breaking. We usems�mb� � 80 MeV since lower ms improves agreementwith data [4,6]. Nonfactorizable effects are usuallylumped into an effective Neff

c fi 3. In our case onlyK

0p0 and p0p0 have color suppressed tree contribution

and could be sensitive to Neffc if it is as small as Neff

C � 1.However, such a small value of Neff

c is not reasonable.

© 2000 The American Physical Society


FIG. 1. Branching ratio and CP asymmetry vs g for Kp andpp, respectively. In (a) and (b), solid, dashed, dot-dashed, anddotted lines denote B ! K2p1, K

0p2, K2p0, and K

0p0,

respectively, using ms � 80 MeV. For (c) and (d), solid,dashed, and dotted lines represent B ! p2p1, p2p0, andp0p0, respectively, with md � 2mu � 4 MeV. We have usedjVub�Vcbj � 0.087 and branching ratios are in units of 1025.

The current data give K2p1, K2p0, K0p2, K

0p0 �

1.88, 1.21, 1.82, 1.48 3 1025, respectively. The observa-tion K2p1�K 0

p2 * 1 gives a strong hint for g . 90±

if d is negligible. The K0p0 rate is weakly g depen-

dent since it receives only color suppressed tree �T � con-tribution. Thus, as seen from Fig. 1, the value of g haslittle impact on the K

0p0 rate. Without EWP, one ex-

pects both K2p0�K2p1 and K 0p0� K 0

p2 � �1�p

2�2,where 1�

p2 comes from the p0 isospin wave function.

The ratio K2p0�K2p1 � 0.65 confirms numerically theexpectation that the K2p0 mode is enhanced by the EWPcontribution. However, EWP-P interference is expectedto be destructive in the K 0

p0 amplitude, hence the ra-tio K

0p0� K

0p2 should decrease from 1�2 and read, for

ms � 80 MeV (ms large),

K0p0

K 0p2

�12

Ç1 2 r0

32a9

a4 1 a6R

Ç2� 0.36 �0.33� , (1)

where we have dropped the a2 term for dis-play purposes, r0 � fpFBK

0 �fKFBp0 � 0.9, and

R � 2m2K��mb 2 md� �ms 1 md�. Small ms can

enhance slightly the K0p0� K

0p2 ratio but does not help

much in understanding data.The aCP’s for K2p1,0 modes are dominated by

Im�V �usVub�a1 3 Re�V �

tsVtb� Im�a4 1 a6R� which peak at110% around g � 70±. But for K 0

p2,0 modes, whichdo not have a sizable T component, aCP’s are too smallto be measurable. Because of large errors in aCP’s so far,it may be premature to compare theoretical results withdata. We note, nevertheless, that the present aCP data [2]

give the central values for K2p1, K2p0, and K 0p2 as

�20.04, 20.27, and 10.17, respectively, which are notconsistent with the theoretical expectations of Fig. 1(b).

The T -P interference for p2p1 is anticorrelated [4,13]with the K2p1,0 case because the penguin KM factorsare Re�V �

tdVtb� � Al3�1 2 r� and Re�V �tsVtb� � 2Al2,

which are opposite in sign since 1 2 r . 0 by definition.Thus, K2p1,0 are enhanced for cosg , 0 while p2p1

is suppressed, as illustrated in Figs. 1(a) vs 1(c). Althoughexperimental error for the p2p0 mode is rather large,the central value [1] seems to suggest p2p1 & p2p0,hence g * 140± is favored if FSI can be neglected.The aCP for p2p1 is given in Fig. 1(d), which clearlyis opposite in sign to K2p1,0. It could be as largeas 220% at g � 120±. For this g value, one wouldexpect K2p1:K 0

p2:K2p0:K 0p0 � 1:0.86:0.63:0.33,

which, however, deviates considerably from the presentobservation of �1:0.97:0.64:0.79, mainly in K 0

p0.Since large g is favored but cannot explain data com-

pletely, could a large FSI phase d alone work? In Fig. 2,we show the branching ratios of Kp and pp vs d withg � 64± [7]. Large d * 100± could help explain [4,8]p2p1 & p2p0, but it fails badly in the Kp modessince K2p1�K 0

p2 , 0.8, while K 0p0 is only half of

K2p1 and actually gains little from the K2p1 mode viaFSI rescattering. We therefore conclude that large d alonecannot account for data and large g is still favored, butlarge d in this context should be further explored.

Before studying the case of having g and d both large,let us make explicit our treatment of FSI phases in pp

and Kp final states. Following the notation of [14], wedecompose the B ! pp amplitudes as

A�B 0! p2p1� � A0eid0 1 A2eid2 ,

p2A�B 0

! p0p0� � A0eid0 2 2A2eid2,p

2A�B2 ! p2p0� � 3A2eid2,(2)

where A0,2 correspond to final state isospin 0 and 2, andd0,2 are FSI phases. For Kp modes, we decompose theamplitudes into A3�2�1�2� for DI � 1 transitions to I �3�2�1�2� final states, and B1�2 for DI � 0 transitions toI � 1�2 final states

FIG. 2. Branching ratios for Kp and pp vs d for g � 64±

with the same notation as in Fig. 1.

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A�B 0! K2p1� � A3�2eid3�2 2 �A1�2 2 B1�2�eid1�2 ,

p2A�B 0

! K 0p0� � 2A3�2eid3�2 1 �A1�2 2 B1�2�eid1�2 ,

A�B2 !K0p2� � 2A3�2eid3�2 1 �A1�2 1 B1�2�eid1�2 ,

p2A�B2 !K2p0� � 2A3�2eid3�2 1 �A1�2 1 B1�2�eid1�2,

(3)

and d3�2,1�2 are FSI phases. Short distance quark-antiquark rescattering effects have been included, whichlead to small and calculable perturbative phases for A3�2,A1�2, and B1�2. Because in Kp modes the strength ofEWP is comparable to T , it is known [15] that SU(3) rela-tions of Ref. [14] do not hold. We refrain from discussingSU(3) but restrict ourselves to SU(2) (isospin), where di inEqs. (2) and (3) are nominally elastic FSI phases but theymodel long-distance rescattering. The phase differencesare observable, which we denote as dKp � d3�2 2 d1�2and dpp � d2 2 d0. Unlike the aforementioned caseof EWP �T in the amplitudes Ai and Bi , electroweakeffects in FSI rescattering are negligible compared tostrong FSI.

We plot in Fig. 3 B and aCP vs d for Kp andpp , respectively, for several large g values. FromFig. 3(a), we see that K2p1 ! K 0

p0 FSI rescatteringis magnified by large g, while K2p0 ! K

0p2 rescat-

tering enhances K 0p2. This is because for cosg ,

0, the T contribution (hence A3�2) changes sign,leading to a marked change in the rescattering pat-tern. Taking g � 110± and dKp � 90±, we obtainK2p1:K 0

p2:K2p0:K 0p0 � 1:1.1:0.61:0.47, and the

relative size of K 0p0 has been enhanced by �30%. Such

FIG. 3. Branching ratio and CP asymmetry for Kp andpp vs d. For curves from (a) up (down) to down (up) forK2p1, K

0p2,0 (K2p0) at d � 180±; (b) up (down) to down

(up) for K2p1,0 (K0p2,0) at d � 90±; (c) down to up at

d � 180±; and (d) down to up for p2p1 (p0p0) at d � 160±

(20±), are for g � 130±, 110±, and 90±, respectively.

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enhancement occurs only when dKp * 60±. Note thatwith larger g, say 130±, K2p1 � K

0p2 is still possible

if dKp � 90±. For even larger dKp, in principle, onecan have K

0p0 . K2p0 but then K

0p2 . K2p1 as

well. The preferred combination of g and dKp can bebetter determined as data improve. For aCP , as shown inFig. 3(b), one has K2p1:K 0

p2:K2p0:K 0p0 � 20.04

�20.04�, 0.13 (0.17), 20.16 �20.27�, 0.2, respectively,for g � 110± and dKp � 90±, where the numbers inparentheses are the experimental central values. Althoughthese numbers should not be taken too seriously at present,we note that if current experimental aCP central valuescontinue to hold, they can be accounted for by having g

and dKp both large. Without final state rescattering, theaCP’s for K2p1,0 are positive and very close to eachother, while the aCP’s for K

0p2,0 would be practically

zero. As illustrated here, with large final state rescattering,the aCP’s for K2p1,0 change sign, while K

0p2,0 modes

gain aCP’s that are opposite to K2p1,0 modes. Thesetrends can be tested in the near future.

For the pp modes, as shown in Fig. 3(c), dpp * 90±

could give p2p1 & p2p0, which is hinted by presentdata. For larger g values, one has less need for a largedpp phase. Because FSI p2p1 ! p0p0 rescatter-ing feeds the p0p0 mode, p0p0 � p2p1 & p2p0

becomes possible. Observing large p0p0 would bea good indication [4,8] for large FSI. The aCP’s offeran even better test. We plot in Fig. 3(d) the aCP’sfor pp modes. For dpp � 90±, the aCP in p2p1

can reach �260%, which is 2–3 times larger than thecase without the dpp phase. We stress once again that,from Fig. 2, although a large dpp * 90± with smallg � 64± may explain the small observed p2p1 rate andp2p1 & p2p0, the ratio K2p1� K

0p2 � 3�4 is not

very sensitive to dKp and would be too low compared towhat is observed.

Before we conclude, we comment on some uncertaintiesin the present study. First, in the factorization approach,ms always appears together with a6. We have used theset of effective Wilson coefficients of [10]. To fit, forexample, the K2p0 mode to data, a larger ja6j would beaccompanied by a larger ms and vice versa. The set ofai’s adopted does not change our conclusions. Second,the p2p0 rate is insensitive to g and independent of d

and is proportional to jFBp0 Vub j

2. Although our p2p0

result is below the present experimental central value, thelatter is not yet firm and we have just concerned ourselveswith the p2p0�p2p1 ratio. Third, factorization in thepp modes has been shown [16] to follow from pQCD inthe heavy quark limit. One could extract the effective NC

from this study [10]. However, there are no significantchanges in Kp and pp modes without considering thelong-distance FSI phase d, which is the position taken inRef. [16]. Note that the strong phases generated by hardgluon rescattering off the spectator quark calculated in


[16] is destructive with the hard quark-antiquark rescat-terings in the penguin loop, resulting in aCP�p2p1� �24% 3 sing, which is much smaller than shown inFig. 1(d). Thus, p0p0 � p2p1,0 and aCP�p2p1� aslarge as 260% would definitely indicate the existenceof large (elastic) FSI phase d arising from long-distanceeffects, something that is argued [16] to be (1�mB) powersuppressed. That would be rather interesting, since Regge[9] and other [17] arguments give phase differences oforder 10±–20±. As B factories at SLAC and KEK havealready turned on, together with the recent commissioningof the CLEO III detector at Cornell, one should have atleast 10 times the present data in two years. It is thus verylikely that the FSI effects discussed here will be tested inthe near future. Finally, we stress that our illustration withelastic strong FSI phase difference d is only phenomeno-logical. While d � 90± is in principle possible [9], FSI inB decays are expected to be highly inelastic [17]. A largestrong phase could well be accompanied by deviations inthe magnitude of amplitudes from factorized ones that wehave used.

In conclusion, large g is favored by data if one con-siders Kp together with pp data. Although a large FSIphase d * 90± with the current g � 64± can account forthe smallness of p2p1, it fails to explain the observedK2p1� K 0

p2 � 1. The strength of the observed K 0p0

mode, the hint that p2p0 * p2p1, together with thecurrent experimental central values for aCP in the Kp

modes, all seem to suggest that on top of g * 100±, thelong-distance phase d could be as large as 90± for bothKp and pp modes. If true, it would not only uphold theabove indication and hints, one would find an enhancedp0p0 rate comparable to p2p1, and aCP in the lattermode as large as 260%. These results should be testablein the next two years.

This work is supported in part by the National ScienceCouncil of Republic of China under Grants No. NSC-89-

2112-M-002-036 and No. NSC-89-2112-M-033-010. Wethank M. Suzuki for useful communications.

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