determination of ckm phases through rigid polygons of flavor su(3) amplitudes
TRANSCRIPT
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PHYSICAL REVIEW D 1 AUGUST 1996VOLUME 54, NUMBER 3
0556
Determination of CKM phases through rigid polygons of flavor SU„3… amplitudes
Amol S. Dighe*Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637
~Received 15 September 1995!
Some new methods for the extraction of CKM phasesa andg using flavor SU~3! symmetry are suggested.Rigid polygons are constructed in the complex plane with sides equal to the decay amplitudes ofB mesons intotwo light ~charmless! pseudoscalar mesons. These rigid polygons incorporate all the possible amplitude trangles and, being overdetermined, also serve as consistency checks and in estimating the rates of some dmodes. The same techniques also lead to numerous useful amplitude triangles when octet-singlet mixingbeen taken into account and nearly physicalh,h8 are used.@S0556-2821~96!03115-3#
PACS number~s!: 12.15.Hh, 11.30.Er, 11.30.Hv, 13.25.Hw
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I. INTRODUCTION
In the standard model,CP violation is parametrized bythe Cabibbo-Kobayashi-Maskawa~CKM! @1# matrix. Theunitarity of the CKM matrix implies Vub* Vud1Vcb* Vcd
1Vtb* Vtd50, suggesting a unitarity triangle with its thresides as these threeV*V terms and the anglesa,b,g (a1b1g5p) which, in Wolfenstein’s parametri-zation @2#, take the form
a5ArgS 2Vtd
Vub*D , b52 Arg~Vtd!, g5Arg~Vub* !.
~1!
The current data from the measurements ofe8/e, B-B mix-ing, and uVubu/uVcbu @3# give the allowed ranges of thesthree phases~95% C.L.! as
21.00<sin2a<1.00, 0.21<sin2b<0.93,
0.12<sin2g<1.00. ~2!
The decays ofB mesons to light pseudoscalar meso(B→PP) give us access to the third row and third columnthe CKM matrix, where all the above phases lie. Expements will give us the magnitudes of the amplitudes of tdecay into various decay channels.~The data about the timedependence of the decays will be available, but we shalluse that here.! If theory expects the amplitudes of some thrdecays to form a triangle in the complex plane, constructthis triangle from the experimentally measured amplitudwill give us the relative phases between these amplitudfrom which information about the three phases above canobtained.
Assuming flavor SU~3! symmetry gives us numerous suctriangle relations and ways to determine these phases. Ssuch ways, with or without using any time-dependent infomation, have been suggested in@4–9# and the extent ofSU~3!-breaking effects has been estimated@10# to be about20%.
*Electronic address: [email protected]
54-2821/96/54~3!/2067~8!/$10.00
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The major contributions to the amplitudes of decays afrom tree- or penguin-type diagrams. The tree diagrams ivolve the processb→uW with the other quark in theBmeson acting as a spectator, whereas penguin diagramstaken to be dominated byt-quark exchange.~Corrections tothe t-quark dominance of theb→d andb→s QCD penguinamplitudes@11# have been neglected in this analysis.! Thephases contributed to various types of diagrams by the CKmatrix elements in the dominating term are as shownTable I.
Section II discusses the representation of decay amptudes in terms of an SU~3!-invariant basis. Section III givesthe ‘‘rigid polygon’’ relations between these amplitudesSections IV and V outline the strategies for extracting CKMphases from the relative phases of the amplitudes fuDSu51 andDS50, respectively. They also illustrate thesemethods with examples of particular decay modes and coment on their experimental feasibility. Section VI discussethe singlet-octeth mixing and the additional amplitudes in-troduced due to this. Section VII gives some of the corresponding useful amplitude triangles when approximaphysical particlesh,h8 are used instead ofh8 ,h1 . SectionVIII concludes.
II. REPRESENTATION OF AMPLITUDES WITHIN SU „3…
Two main approaches have been taken@5–7,12# for find-ing the amplitude triangle~or quadrilateral! relations. Onemethod is to represent the amplitudes in terms of the basisT ~tree!, P ~penguin!, C ~color-suppressed tree!, E ~ex-change!, A ~annihilation!, and PA ~penguin annihilation!diagram contributions. An exhaustive list of all such amplitudes has been made in@6# and some quadrangle, triangle, orequivalence relations have been obtained. The contributioby E, A, and PA have been neglected since they are expected to be suppressed by a factor off B /mB55%. (E and
TABLE I. The weak phases for tree and penguin diagrams.
Diagram type uDSu50 uDSu51
Tree Arg(Vub* Vud)5g Arg(Vub* Vus)5gPenguin Arg(Vtb* Vtd)52b Arg(Vtb* Vts)5p
2067 © 1996 The American Physical Society
2068 54AMOL S. DIGHE
TABLE II. The coefficients of the six invariant amplitudes under SU~3!.
Group Decay mode Coefficients of Factor Linearno. C3 C6 C15 A3 A6 A15 combination
1 B1 K1 K0 1 21 21 0 1 3 1 p
B0 K0 K0 1 21 21 2 1 23 1 p
B0 p0 h8 1 21 21 0 1 25 2A3 p18 Bs K0 K0 1 21 21 2 1 23 1 p8
B1 K0 p1 1 21 21 0 1 3 1 p828 B1 K1 p0 1 21 7 0 1 3 2A2 t81c81p8
3 B1 p1 p0 0 0 8 0 0 0 2A2 t1c48 B1 K1 h8 1 21 29 0 1 3 A6 2t82c81p8
5 B1 p1 h8 22 2 26 0 22 26 A6 2t2c22p6 B0 p0 p0 1 1 25 2 21 1 A2 2c1p
Bs K0 p0 1 1 25 0 21 21 A2 2c1p
Bs K0 h8 1 1 25 0 21 21 A6 2c1p
68 B0 K0 p0 1 1 25 0 21 21 A2 2c81p8
B0 K0 h8 1 1 25 0 21 21 A6 2c81p8
78 Bs p0 h8 0 22 4 0 2 24 2A2 c8
8 B0 p1 p2 1 1 3 2 21 1 21 t1pBs K2 p1 1 1 3 0 21 21 21 t1p
88 B0 K1 p2 1 1 3 0 21 21 21 t81p8Bs K1 K2 1 1 3 2 21 1 21 t81p8
9 B0 h8 h8 1 23 3 6 3 23 3A2 c1p10 B0 K1 K2 0 0 0 22 0 22 1 0108 Bs p1 p2 0 0 0 2 0 2 21 0
Bs p0 p0 0 0 0 4 0 4 21 0118 Bs h8 h8 2 0 26 3 0 23 3/A2 2c812p8
t
A will also be helicity suppressed by a factormq /mb whereq5u,d,s.)
Another equivalent approach@7,12# is to represent theamplitudes in the basis of six SU~3!-invariant amplitudeswhose combinatorial coefficients will be the six invariaquantities formed by the combinations of the three-vecBi[(B1,B0,Bs), two pseudoscalar matricesM j
i ~one foreach pseudoscalar!, and H, the Hamiltonian for
b→q1q2q3 . H can be split using SU~3! into 3^3^ 353% 3%6%15 and thus its transformation properties canencoded intoHi(3), a vector,Hk
@ i j #(6), a traceless tensoantisymmetric in the upper two indices, andHk
( i j )(15), atraceless tensor symmetric in the upper two indices.
The tree amplitude, in terms of this basis, will be~modulothe CKM factors!
T5A3TBiH
i~3!Mkj M j
k1C3TBiM j
iMkj Hk~3!
1A6TBiHk
i j ~6!M jlM l
k1C6TBiM j
iHljk~6!Mk
l
1A15T BiHk
i j ~15!M jlM l
k1C15T BiM j
iHljk~15!Mk
l . ~3!
The AiT are the terms that come from contracting the lig
quark part of theB vector directly with the HamiltonianThis would suggest that the light quark is an active partthe decay process and not just a spectator. Such amplit~corresponding toE, A, andPA) will be suppressed by afactor of f B /mB and hence can be neglected to a first aproximation. The tests for this approximation~which will
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come naturally from the method of grouping described be-low! are specified at the end of this section.
The values of the nonzero elements ofH(3), H(6), andH(15) are given in@7,12#. The convention used for the mem-bers of the pseudoscalar meson octet is
p15ud, p051
A2~dd2uu!, p252ud,
K15us, K05ds, K05ds,
K252us, h851
A6~2ss2uu2dd!. ~4!
An exhaustive list of all the coefficients inB→PP is givenin Table II.
The decay modes having the sameCi coefficients aregrouped together and given the same group number. Theunprimed~primed! group numbers correspond to the decaymodes withuDSu5 0 ~1!. A( j ) represents the amplitude forall the decay modes in that group. The advantage of usingthe amplitudes in terms of group numbers is that in case ofmultiple decay modes in a group, the ones easier to deteccan be used or a suitable average of all the modes within agroup can be taken to improve statistics. Comparisons ofbranching fractions of decay modes within a group also serveas tests of flavor SU~3! symmetry.
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54 2069DETERMINATION OF CKM PHASES THROUGH RIGID . . .
The factor column gives the factor by which the actudecay amplitude should be multiplied to get thegroup am-plitude: e.g.,
AT~48
n
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2070 54AMOL S. DIGHE
superposing the ‘‘particle’’ and ‘‘antiparticle’’ triangles inthe process of extraction of the CKM phases, we shallways superpose their penguin-only sides.
IV. CKM PHASES FROM zDSz51 TRIANGLES
In this case, the penguin-only modes for both particle aantiparticle decay have the phasedP1p, so that aligningthese amplitudes along the real axis is equivalent to rotathe amplitude triangles by an unknown but fixed phadP .
As a result of this rotation, the amplitude of a genedecay is now, from Eq.~8!,
AR~ i 8!52uVtb* VtsuuPu1uVub* VusuuTuei ~dT2dP!eig, ~12!
where the subscriptR stands for the amplitudes with thephase rotated such that the phase of the penguin-only amtude is 0 orp.
The amplitude for the antiparticle decay is
AR~ i 8!52uVtb* VtsuuPu1uVub* VusuuTuei ~dT2dP!e2 ig.~13!
Subtracting the two equations gets rid of the penguin conbution:
AR~ i 8!2AR~ i 8!52isinguVub* VusuuTuei ~dT2dP!
52isinguAT~ i 8!uei ~dT2dP!. ~14!
Now, if the tree contributions touAT( i 8)u anduAT( j )u can berelated for somej and furthermore if the penguin contribution to A( j ) can be neglected~as is the case whenever thtree contribution is not color suppressed@8#!, then the mea-surement of the decay rate gives us directly the valueuAT( i )u, henceuAT( i 8)u, and from Eq.~14! we obtain thevalue of sing anddT2dP .
A. D„18-28-48…
From Table II,
A~48!52A~18!2A~28!. ~15!
Constructing theCP-conjugate triangle withA(48), A(28),and A(18) and superposingA(18) and A(18) ~this is pos-sible sinceuA(18)u5uA(18)u; this will introduce a discretetwofold ambiguity, leading to a discrete twofold ambiguiin sing), the line joining the remaining vertices of these twtriangles is
AR~28!2AR~28!52isinguAT~28!uei ~dT2dP!. ~16!
Also,
uAT~28!u5uAK1p0T u5
uVusuuVudu
f Kf p
uAp1p0T u5
uVusuuVudu
f Kf p
uAT~3!u.
~17!
The factor off K / f p comes from taking into account the firsorder SU~3! breaking under the assumption of factorizatioThe dominating contribution to uAK1p0
T u(uAp1p0T u)
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is from the tree diagram, in whichK1(p1) is formedpurely through a weak current@W1→K1(p1)#. Withfactorization, this implies a multiplicative term^K1usgmg5uu0&(^p1udgmg5uu0&) in the amplitude, whichis proportional tof K( f p). This is precisely where the firstorder SU~3! breaking appears@10#.
B1→p1p0 proceeds only via theI52 channel, and sothe QCD penguin amplitude does not contribute here.~TheQCD penguin contribution is a pureDI51/2 operator.! Sincethe electroweak penguin contribution can be neglected iDS50 channels@8#, uAT(3)u'uA(3)u, and the measurementof uA(3)u along with Eq.~16! gives the value of sing anddT2dP up to a twofold ambiguity.
If factorization holds here to a fair extent, then
uAR~28!2AR~28!u<2uAT~28!u'uVusuuVudu
2 f Kf p
uA~3!u.
~18!
This provides a weak partial test~necessary, but not suffi-cient! for factorization.
The same triangle has been suggested by Deshpande aHe in @7#. It is restated here for the sake of completeness anto illustrate the method. The decay modes involved herhave at least one charged particle in the final states@in group(18), we can chooseB1→K0p1# and the branching frac-tions are expected to be;1025; so this triangle will be ex-perimentally easy to construct. The difficulty of separatingh8 haunts it, though~as it does all the modes in this sectionthat involveh8).
B. D„28-78-88…
We already know the phase ofAR(28) from the construc-tion of D(18-28-48). Now we can use
A~28!5A~78!1A~88! ~19!
and construct this triangle on top of the earlier one. Constructing theCP-conjugate triangle and orienting it using theinformation about the phase ofAR(28) from the constructionin Sec. IV A, we get the phases ofAR(88) andAR(88). Now
AR~88!2AR~88!52isinguAT~88!uei ~dT2dP! ~20!
and, assuming factorization,
uAT~88!u5uAB0→K1p2T u5
uVusuuVudu
f Kf p
uAB0→p1p2T u
5uVusuuVudu
f Kf p
uAT~8!u. ~21!
If we can neglect the QCD penguin contribution inA(8),then uAT(8)u'uA(8)u and we get the value of sing anddT2dP up to a discrete twofold ambiguity. The argumentabout the factorf K / f p in Sec. IV A and a similar weak par-tial test for factorization,
uAR~88!2AR~88!u<uVusuuVudu
2 f Kf p
uA~8!u, ~22!
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to
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54 2071DETERMINATION OF CKM PHASES THROUGH RIGID . . .
is valid in this case also.Alternatively, knowingg andAR(28),AR(28) from Sec.
IV A, the measurement ofuA(78)u and uA(78)u will enableus to determineAR(88) and AR(88) simply by geometry.
This triangle has been suggested in@8# as a part of thequadrilateralFGOD ~Fig. 1! with a slightly different ap-proach. Both the decay products in the mode (78) are neutralparticles. So we might come across the problem of low aceptance rate here. The branching fraction for this modealso expected to be very small@13#.
V. CKM PHASES FROM DS50 TRIANGLES
When the polygon is oriented such that the penguin-onamplitude is along the real axis, the amplitude@Eq. ~7!# of ageneric decay becomes
AR~ j !5uVtb* VtduuP~ j !u1uVub* VuduuT~ j !uei ~dT2dP!ei ~p2a!,~23!
since Arg(Vub* Vud /Vtb* Vtd)5p2a.The corresponding antiparticle amplitude oriented in
similar manner will be
AR~ j !5uVtb* VtduuP~ j !u1uVub* VuduuT~ j !uei ~dT2dP!ei ~2p1a!.~24!
When the tree contribution inb→uud is not color sup-pressed, the penguin contribution is expected to b;l('0.2) times the tree contribution@8# and the uP( j )uterm can be neglected. The angle between these two amtudes will then be 2a.
A. D„1-5-3…
We have
A~3!52A~5!22A~1! ~25!
and superposing theCP-conjugate triangle such thatAR(1)and AR(1) overlap, we get~up to a discrete twofold ambi-guity! the relative phases ofAR(5) and AR(5), which is2a.
uA(1)u is expected to be very small and one might haveworry about low statistics here. But we know thaA(1)5AP(1) andA(18)5AP(18), and from Eq.~9! we get
A~1!5Vtd
VtsA~18!. ~26!
If the value ofuVtdu/uVtsu is known through other means, wecan use the data fromA(18) to determine the magnitude ofA(1) to be used in the construction of this triangle. Equations (3) and (5) have one charged particle in their finastate and aT contribution~in theDS50 mode! which wouldindicate a sizable branching fraction and higher acceptanfor both of them.
B. D„1-9-6…
The triangle
A~9!52A~1!2A~6!, ~27!
c-is
ly
a
e
pli-
tot
-l
ce
with its CP-conjugate triangle, gives the phases ofAR(6),
AR(6), AR(9), andAR(9).This triangle relation is not directly useful for finding any
of the CKM phases since the tree contribution toA(6) andA(9) is color suppressed and hence of the same ordermagnitude as the penguin contribution. The penguin contrbution here, therefore, cannot be neglected. But the informtion about the phases will be used in the construction of thnext triangle.
C. D„9-5-8… or D„6-3-8…
The relation
A~9!1A~8!52A~5! ~28!
or
A~8!2A~6!5A~3!, ~29!
with the information about the phase ofAR(6) or AR(9)from Sec. V B, gives the phase ofAR(8) and theCP-conjugate triangle gives the phase ofAR(8). Thephasedifference between these two is 2a, similar to Sec. V A.
n~6-3-8! is the same as thep-p isospin triangle in@14#.With ~3!, ~5!, and ~8! having T contributions~and, conse-quently, a charged particle in the final state!, the branchingfractions and the acceptance for these modes is expectedbe on the higher side.
n(18-68-78), a part of the quadrilateral suggested in@8#,andn(18-68-118) are possible, but are not very useful sincethe tree contributions toA(78), A(68), andA(118) are colorsuppressed and the electroweak penguin contributions aexpected to be significant.
VI. PHYSICAL h AND h8
The SU~3! eigenstatesh8 andh1 are different from thephysical particlesh andh8. Taking into account the mixingangle of'20° @15#, these physical states are very close to@6,16#
h51
A3~ss2uu2dd!, ~30!
h851
A6~2ss1uu1dd!. ~31!
Since we have contributions from the singlet componenhere, the trace of the pseudoscalar matrixM is no longerzero. There will, therefore, be additional terms in the amplitude, whose coefficients would have been zero had we bedealing with only the pseudoscalar octet mesons.
The additional terms in the amplitude will be
E3BiM jiH j~3!Mk
k1D3BiHi~3!M j
jMkk1D6BiHk
i j ~6!M jkMl
l
1D15BiHki j ~15!M j
kMll . ~32!
There are no terms with amplitudesE6 or E15 since theircoefficients would involveHj
i j (6) andHji j (15), respectively,
all of which are zero.
2072 54AMOL S. DIGHE
TABLE III. Coefficients with physicalh andh8 for uDSu51.
Decay mode Coefficients of Factor LinearC3 C6 C15 E3 combination
B1 K1 h 0 1 7 1 2A3 t81c81s8B1 K1 h8 3 1 1 4 A6 t81c813p814s8Bs p0 h 0 2 24 0 A6 2c8
Bs p0 h8 0 2 24 0 A3 2c8
B0 K0 h 0 21 3 1 2A3 c81s8B0 K0 h8 3 21 23 4 A6 c813p814s8Bs h h 1 0 24 21 3/A2 2c81p82s8Bs h h8 22 0 5 21 23/A2 1
2c822p82s8Bs h8 h8 2 0 22 4 3/A2 c812p814s8
al
cthe
The termsDi will be suppressed byf B /mB since theseterms will correspond to some annihilation diagrams~for thesame reason as for theAi ’s!. So the only significant additional term we have here is theE3 term.
The coefficients ofCi and E3 for decays involving thephysical statesh andh8 are given in Tables III and IV.
All the amplitudes can be explicitly written in terms ofour complex numberst, c, p, ands as in thelinear combi-nation column of Tables II and III. (s is the additional am-plitude due to the contribution of the singlet. It is the samethe contribution from the two-gluon diagramP1 @17# and thecorresponding electroweak penguin,cdPEW.) Thus, theknowledge of the magnitudes~obtained from the measurements! and the relative phases~obtained from the construction of triangles! of t, c, p, and s ~or any four of theirindependent linear combinations! will give us the amplitudesof all the decay modes. We therefore need only three inpendent connected triangles~each triangle shares a side wiwith at least one of the others! to get the amplitudes anphases of all the other amplitudes with the sameuDSu.
Some examples of useful triangle relations are givenSec. VII. All the triangles have discrete twofold ambiguitiassociated with them.
Methods for estimating the first-order SU~3!-breaking ef-fects are indicated in@10#. One way to guage the effects oSU~3! breaking on the amplitude triangles is to check if ttriangle relations remain valid even when the first-ord
-
f
as
--
de-thd
ines
fheer
SU~3!-breaking terms are introduced@17#. Phase space ef-fects have to be taken into account especially when the finstate particles contain one or more heavyh8.
VII. AMPLITUDE TRIANGLES WITH h AND h8
A. zDSz51
The triangles
~1! A~B1→K0p1!1A6A~Bs→p0h!
5A2A~B0→K0p0!, ~33!
~2! A~B1→K0p1!1A3A~B0→K0h!53
A2A~Bs→hh!,
~34!
~3! 2A3A~B1→K1h!13
A2A~Bs→hh!
52A~B0→K1p2! ~35!
are three connected amplitude triangles, sufficient to predithe amplitudes and phases of all the remaining decays of ttype uDSu51.
TABLE IV. Coefficients with physicalh andh8 for DS50.
Decay mode Coefficients of Factor LinearC3 C6 C15 E3 combination
B1 p1 h 2 21 5 1 2A3 t1c12p1sB1 p1 h8 2 2 2 4 A6 t1c12p14sB0 p0 h 22 1 3 21 A6 22p2sB0 p0 h8 21 21 3 22 2A3 2p22sBs K0 h 0 21 3 1 2A3 c1s
Bs K0 h8 3 21 23 4 A6 c13p14s
B0 h h 1 22 2 1 3/A2 c1p1sB0 h h8 2 21 1 5 23A2 2c12p15sB0 h8 h8 1 1 21 4 3A2 c1p14s
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54 2073DETERMINATION OF CKM PHASES THROUGH RIGID . . .
The triangle in Eq.~35!, when constructed on the top othe triangle in Eq. ~34!, gives the phase ofAR(B
0
→K1p2). The CP-conjugateAR(B0→K2p1) and the
method of Sec. IV B giveg anddT2dP .All the amplitudes above have penguin contribution
which are substantial in thisuDSu51 mode, and so thebranching ratios will be high, but the presence of many netral particles in the final state might pose acceptance prlems.
Some more triangles, e.g.,
~4! A~B1→K0p1!13
A2A~Bs→h8h8!
5A6A~B0→K0h8!, ~36!
~5! 2A~Bs→hh!1A~Bs→h8h8!52A~Bs→hh8!~37!
can be constructed which will serve to validate our assumtions collectively, if not individually. The remaining ampli-tudes can be generated from the information gained throuEqs.~33!–~37!.
B. DS50
The triangle
~1! 3A~B1→K1K0!12A6A~B0→p0h!
52A3A~B0→p0h8! ~38!
has a penguin-only side and hence will be useful in definithe orientations of all the other amplitudes.
~2! 2A2A~B1→p1p0!1A3A~B1→p1h!
5A6A~B0→p0h! ~39!
constructed on top of the above triangle gives the phaseAR(B
1→p1p0). The CP-conjugate triangle gives thephase ofAR(B
2→p2p0) and the phase difference betweethese is 2a. The same procedure can be used with the infomation obtained from the phases ofAR(B
1→p1h) andAR(B
2→p2h), the phase difference between which2a.
Both B1→p1p0 andB1→p1h have aT contributionand, hence, are expected to have sizable branching fractand acceptances.
These triangles, along with thep2p isospin triangle
~3! 2A2A~B1→p1p0!1A2A~B0→p0p0!
52A~B0→p1p2!, ~40!
will enable us to predict the amplitudes and phases of allother decay modes withDS50. The isospin triangle alsoenables one to get the phases ofAR(B
0→p1p2) andAR(B
0→p1p2), the phase difference between whicshould be 2a.
f
s
u-ob-
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gh
ng
of
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is
ions
the
h
~4! A~B1→K1K0!2A3A~Bs→K0h!53
A2A~B0→hh!
~41!
gives the phase ofAR(Bs→K0h), which isdT2dP1g sincethe only contribution here is from theC diagram. @ThePEW contribution is expected to be;l('0.2) times theCcontribution here@8#.# TheCP-conjugate triangle gives thephase ofAR(Bs→K0h) and dT2dP2g, and thusg is ob-tained along withdT2dP . ~This is one instance wheredT2dP is obtained in theDS50 mode. But this one will beplagued by low statistics and more neutral particles in thfinal state.!
Some additional triangles such as
~5! 2A~B1→K1K0!13A2A~B0→h8h8!
5A6A~Bs→K0h8!, ~42!
~6! A~B0→hh!12A~B0→h8h8!522A~B0→hh8!~43!
can be constructed for consistency checks. All the remaininamplitudes may be constructed using the information gainefrom the above triangles and thelinear combinationcolumnof Table IV.
VIII. CONCLUSIONS
Using only the time-independent information about therates ofB mesons decaying into light pseudoscalars, we cadetermine the angles of the CKM unitarity triangle undeflavor SU~3! symmetry. Here we neglect the annihilation-type diagrams which are expected to be suppressedf B /mB . The amplitudes are represented in terms of aSU~3!-invariant basis. Rigid amplitude polygons are constructed which are overdetermined and hence can servether for multiple ways of determininga and g, as consis-tency checks, as tests for the approximations made, orestimate the amplitudes for decays hard to detect experimetally. The tests for the assumptions of flavor SU~3! symme-try, factorization, and annihilation diagram suppression aralso built in.
The expected branching fractions of most of the decamodes are of the order of (1026–1025) @16# and withinreach of current and upcoming experiments. The methodgrouping helps in improving statistics by using the information from more than one decay mode or by allowing one tomeasure, say, a mode with charged decay products insteadneutral ones. The knowledge of the ratios of magnitudes oCKM elements can be used to estimate the decay ratessome modes with lower branching fractions.
The physical particlesh and h8 are different from theSU~3! singleth1 or octeth8 . Taking into account this mix-ing, the same methods have been applied to the appromately physicalh andh8, which will be the actual particlesto be detected. The decay modes withh or h8 as one of thedecay products form a sizable portion of charmlessB decays
rand-hisn-
2074 54AMOL S. DIGHE
and hence taking into account the deviation of the physistates from the octet or singlet states is important. All tdecay amplitudes to the approximately physical particlesexpressed explicitly in terms of four SU~3!-invariant quanti-ties and amplitude triangle relations are found which arerectly useful to obtain the CKM phases, validate our assumtions, and provide self-consistency tests.
calheare
di-p-
ACKNOWLEDGMENTS
I would like to thank Aaron Grant and Mihir Worah fohelpful discussions. I am particularly indebted to JonathRosner for getting me interested in this topic, critically reaing the manuscript, and giving helpful suggestions. Twork was supported in part by the U.S. Department of Eergy under Contract No. DE FG02 90ER40560.
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