Determination of gamma from charmless B^+/- -> M^0 M^+/- decays using U-spin

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by  Amarjit Soni

Determination of γ from charmless B± M0M± decays using U-spin
→
Amarjit Soni and Denis A. Suprun
High Energy Theory Group, Brookhaven National Laboratory, Upton, NY 11973
(Dated: February 2, 2008)
U-spinmultipletapproachisappliedtothefullsetofcharmlesshadronicB± M0M±decaysfor
thepurposeofpreciseextractionoftheunitarityangleγ. Eachofthefourdata→sets,P0P±,P0V±,
V0P± andV0V±,withP pseudoscalarandV vector,canbeusedtoyieldaprecisevalueofγ.
≡ ≡
6 ThecrucialadvantageofthismethodoverthecommonSU(3)symmetrybasedquark-diagrammatic
0 approach is that no assumptions regarding relative sizes of topological decay amplitudes need to
0 be made. As a result, this method avoids an uncontrollable theoretical uncertainty that is related
2 to the neglect of some topological diagrams (e.g., exchange and annihilation graphs) in the SU(3)
n approach. Application of the U-spin approach to the current data yields: γ = 54+−1121 ◦. We find
a thatimprovedmeasurementsofφπ± andK∗0K± branchingratioswouldleadto(cid:0)apprec(cid:1)iablybetter
J
extraction of γ. In this method, which is completely data driven, in a few years we should be able

8 toobtain a model independentdetermination of γ with an accuracy of O(few degrees).
2

4 Precise determinations of the angles of the unitarity from SU(3) fits [5]. The effects of the η η mix-
′
v −
triangle (UT) remains an important but difficult goal in ing on the theoretical error in γ were also found
2
Particle Physics. Though methods for direct determina- to be small (< 1 ) [2]. Since in the U-spin ap-
1 ◦
0 tions of all the angles are now known, we are still quite proach graphi∼cal topologies are not used, estimate
1 farawayfromhavinglargeenoughsampleofB’sthatare of U-spin breaking effects on γ extraction may be
1 needed [1]. The main challenge in extracting the angles amenabletocalculationalframeworkssuchasQCD
5 from the data is of course that weak decays take place factorization, pQCD, soft collinear effective theory
0 in the presence of strong interactions (i.e. QCD) which (SCET), or QCD sum rules [6].
/
h in this energy regime has important, non-perturbative
The fact that the U-spin approach does not make
p effects. Fortunately, QCD respects flavor symmetries.
•
- Use of these symmetries presents an important avenue use of graphical topologies, of course means that
p electroweak penguins are automatically fully con-
to extract results, though often at the expense of some
e
tained within this approach. This is in sharp con-
h accuracy. In the context of the angle γ of the UT, in
trast to the case of α extraction with the use of
: factSU(3)flavorsymmetryhasalreadybeenputto very
v isospin. Therein electroweak penguins complicate
good use [2, 3]. In this paper we propose to use U-spin
i
X for determining γ from charmless B decays. the extraction of the CKM phase α and present
±
a serious limitation. In this respect the U-spin
r TherearesubstantialdifferencesbetweenU-spinmulti-
a approach has an advantage over even the isospin
pletapproachandotherphenomenologicalmethods,such
method.
as SU(3) based approach, of understanding the current
B decay data. Needless to say, the standard B DK methods
• →
of direct γ extractionare theoretically the cleanest
U-spin multiplet method has the significant advan- (error of O(.1%) [1]) and should ultimately pro-
•
tage that, unlike SU(3) fits to charmless B decays, vide the most accurate determination of γ. But
quarkdiagrammatictopological approachis notin- this accuracy will only be attained after very large
voked at all. Thus, we do not need to make any datasamplesbecomeavailable,perhapsmanyyears
assumptionsabouttherelativesizesofvariouscon- down the road. The U-spin approach that we are
tributingtopologicaldiagramsandsonoamplitude using here, on the other hand, can provide a fairly
need be neglected [4]. accurate value of γ (error of O(few percent)) with
modest increase of luminosities.
U-spin is a flavor symmetry similar to isospin.
•
Since it is a subgroup of SU(3) flavor symmetry, Furthermore, since the U-spin approach automat-
•
we expect it to be at least as accurate as SU(3) ically includes all penguin contributions, whereas
and possibly better. the B DK method do not, a comparison of the
→
valuesofγ fromthetwomethodsprovidesacrucial
It is important to emphasize that noticeable flavor test for new physics.
•
symmetry breaking effects in decay amplitudes do
not necessarily lead to large uncertainties in γ ex- We will show that there are four separate sets of two-
traction. For instance, SU(3) breaking effects of bodydecaysofchargedB’seachofwhichcangiveavalue
about 20% that are related to the ratio of decay of γ. Existing data already allows determination of γ
constants f and f , only lead to a small (2 , or withanaccuracyinthesameballparkasothermethods
K π ◦ 0
3%) theoretical uncertainty in determination of γ beingused. Weidentifymodes(φπ± andK∗ K±)whose
2
improvedexperimentalmeasurementsshouldappreciably InisospinanalysisofB decays[10]theeffectiveHamil-
improve the accuracy on γ with this method. In the era tonian transforms as either ∆I = 1 or ∆I = 3. While
2 2
of the current B-factories, with the planned luminosities electroweak penguins, violate isospin due to the charge
ofafewab 1,themethodshouldallowustodetermineγ difference between u and d quarks, they do not violate
−
withanaccuracyofafewdegrees. Furthermore,asbetter U-spin. There are only three topological diagrams that
experimentalinformation,attheseluminosities,becomes maycontributetochargedBdecays: tree,penguin(QCD
availableforalltherelevantdatasets,thismethodshould andelectroweak),andannihilation. The effective Hamil-
give an understanding of its inherent systematic error. tonianofanyofthesedecaytypestransformsasaU-spin
Let us very briefly recapitulate some elementary as- doublet, ∆U = 1. This makesU-spin aparticularlycon-
2
pects of U-spin [3, 4, 7, 8]. Recall that the U-spin sub- venient approach that allows the complete description
groupofSU(3)issimilartotheI-spin(isospin)subgroup of chargedB decays without making additional assump-
exceptthatthequarkdoubletswithU =1/2,U3 = 1/2 tions on the size of individual topological diagrams and
±
are without neglecting any of them, including annihilation.
WhiletheSU(3)basedapproach[2,3]doesnotinherently
1 1 d 1 1 s¯
(cid:20) ||122−221ii(cid:21)=(cid:20) ||sii (cid:21), (cid:20) ||122−212ii(cid:21)=(cid:20) −||d¯ii(cid:21). (1) rneihquiliarteioingncoornintgribauntniiohnisla,tiinonp,reaxccthicaengoeneanhdaspetongduointhaant-
to limit the number of parameters and keep SU(3) fits
B+ isaU-spinsinglet,whilechargedcharmlessmesons
stable. This advantage of the U-spin approach makes it
π+(ρ+), K+(K +) belong to U-spin doublets. Neutral
∗ particularly appealing; in the long run, it should signif-
mesons may get contributions from a U-spin triplet and
icantly reduce theoretical uncertainties associated with
two U-spin singlets. Strange neutral mesons K0(K 0)
∗ this method.
andtheirantiparticlesarepureU-spintriplets. Themul-
tiplet decompositionsofother neutralmesonscanbe de-
Since the initial B+ meson is a U-spin singlet and
termined to be
the effective Hamiltonian always transforms like a U-
π0 = −21|1 0i+ √23|0 0i8 , sdpoiunbldeotsu.blTeth,etyhecafinnbael Mfor0mMed+ isntattherseemduisffterbeentUw-sapyisn.
η = q23|1 0i+ √32|0 0i8− 31|0 0i1 , While the charmless charged meson M+ can only be-
η′ = 2√13|1 0i+ 16|0 0i8+ 2√32|0 0i1 , (2) long to a doublet, the neutral meson M0 can be a linear
ρ0 = −21|1 0i+ √23|0 0i8 , caonmyb∆inSat=io0n,oBf+threeMd0iMffe+rendtecmayulatmippleltitsu.deAscaanrbeesuelxt-,
φω == 1−121|10 +0i−1 √063|00 80+i8+1q0230|010i,1 , pcorrersesespdoinndtetromtsheo→fUt-hsrpeienatmripplliettu,dUes-:spAind1,sAind0g,leBt0d.0T0he8y,
√2| i √6| i √3| i | i
and SU(3) singlet 0 0 1 contributions into the decay
| i
where two U-spin singlets are defined as amplitude. Each of these three amplitudes consists of
a “u-like” and a “c-like” part (Eq. 6). Similarly, any
0 0 8 1 ss¯+dd¯ 2uu¯ , ∆S = 1 decay amplitude can be written in terms of
|0 0i1 ≡ √16|uu¯+dd¯−+ss¯ i. (3) |thre|e other amplitudes: As1, As0, B0s. The assumption of
| i ≡ √3| i U-spin symmetry implies that the difference between Ad
1
One may decompose the ∆S = 1 and ∆S = 0 effec- and As1 comes only through the difference in the CKM
tive Hamiltonians into mem|bers|of the same two U-spin matrix elements. Thus, the complete amplitudes for U-
doublets multiplying given CKM factors. For practical spin final states are given by
purposes, using CKM unitarity, it is convenient to write
the effective Hamiltonian so that it involves only the u
and c quarks:
∆S =0: ¯b d¯ = V V Ou+V V Oc , (4)
|∆S|=1: HHe¯ebff→→ff s¯ = Vuu∗∗bbVuusdOsud +Vcc∗∗bbVccsdOscd . (5) ∆S =0: BAd00d,1 == VVuu∗∗bbVVuuddABu00u,1++VVc∗bc∗VbVcdcBdA0cc0,1, , (6)
∆S =1: As = V V Au +V V Ac ,
The assumptionof U-spinsymmetry implies thatU-spin | | B0s,1 = Vu∗bVusB0u,1+V c∗Vb cBs c0,.1
doubletoperatorsOu andOu areidentical,aswellasthe 0 u∗b us 0 c∗b cs 0
d s
Oc and Oc operators. The subscripts d and s may be
d s
omitted. Hadronic matrix elements of these two opera-
tors, Ou and Oc, will be denoted Au and Ac and will be
referredto as “u-like” and“c-like”amplitudes [9], where
the latter includes electroweak penguin contributions. ThenwefindthatphysicaldecayamplitudesforV0P+
Note that these amplitudes multiply different CKM fac- and V0V+ modes may be decomposed into U-spin am-
tors in ∆S =1 and ∆S =0 processes. plitudes [11],
| |
3
so for the CP-asymmetries. In any case, with 16 data
points and 12 fit parameters one can perform a fit and
A(K∗0K+), A(K∗0K∗+) = 2√2Ad1 , extract the preferred values for all parameters.
A(ρ0π+), A(ρ0ρ+) = 3−Ad Ad ,
0 1 In the other 3 subsets some modes have not yet been
−
A(ωπ+), A(ωρ+) = Ad0 Ad1+√2B0d , observedbut upper limits on their branching ratioswere
− −
A(φπ+), A(φρ+) = √2Ad0+√2Ad1+B0d ,(7) reported. Needless to say, direct CP asymmetries for
A(K 0π+), A(K 0ρ+) = 2√2As , these modes have not been determined yet. For some of
∗ ∗ 1
A(ρ0K+), A(ρ0K +) = 3−As+As , these modes a central value and a large uncertainty are
∗ 0 1
A(ωK+), A(ωK +) = As+As+√2Bs , known. Fortheothers,whereonlyanupperlimitat90%
∗ 0 1 0
−
A(φK+), A(φK +) = √2As √2As+Bs , confidencelevelisreported,onecantakecentralvalueas
∗ 0 1 0
− equalto0andapproximatelyestimatetheuncertaintyby
whereA1, A0 andB0 correspondtofinalstateswithvec- dividing the upper limit value by 2. For example, from
tor mesons V0 in the U-spin triplet, in the octet U-spin (B+ ωρ+)<16 we crudely estimate that (ωρ+)=
B → B
singlet and in the SU(3) singlet, respectively. Naturally, 0.0 8.0 [13]. The data from upper limits helps in two
the formulae for related V0P+ and V0V+ decay modes way±s. First of all, it provides additional data points,
arethesame,asseenintheaboverelations. However,the makingaU-spinfitfeasible. Second,itallowsustoverify
actual values for each of the U-spin amplitudes are con- thatthe resultingfitisconsistentwiththe currentupper
stant only within each of the two subsets. They accept limits.
different values in V0P+ and V0V+ subsets. In the case of V0P+ decays, for instance, 6 out of 8
Thus,eightV0P+ decaysaredescribedby12parame- modes have been observed and provide 12 data points.
ters: six U-spin amplitudes |Au0,1|, |Ac0,1|, |B0u| and |B0c|, The remaining two decays, K∗0K+ and φπ+, have not
five relative strong phases between them and the weak
yet been observed. At present only the upper limits
Vph0aVse+γm. oTdhees,satomoe. statementisseparatelyvalidforeight f0o.0r+t1h.3e+se0.6tw(o<m5.o3d)e[s14a]reankdnow(nB:+B(Bφ+π+→) <K∗00.K41+[)15=].
tudInestfhoerPsa0mPe+waanydoPn0eVc+andedceacyommpodoseesipnhtyosUic-aslpianmapmli-- Fro−m0.0t−h0e.s0e measurements weBcan c→rudely estimate that
plitudes. Then we find [12]: 0B.(0B+0→.2. KT∗o0Km+ak)e=su0re.0t0h+−a10t..4030thaenfidtBis(Bco+ns→isteφnπt+w)it=h
A(K0K+), A(K0K +) = 2√2Ad , the±upper limits on the K∗0K+ and φπ+ branching ra-
∗ 1
A(π0π+), A(π0ρ+) = 3−Ad Ad , tios we add these two data points to the fit. Thus, the
A(ηπ+), A(ηρ+) = 2√2Ad+02−√2A1d Bd , 12-parameter V0P+ U-spin fit features 14 data points,
√3 0 √3 1− 0 making γ extraction possible.
A(η π+), A(η ρ+) = 1 Ad+ 1 Ad+2√2Bd ,
′ ′ √3 0 √3 1 0 (8) Similarly, in the V0V+ sector 5 modes have been de-
A(K0π+), A(K0ρ+) = 2√2As1 , tected and the first measurement of their CP asymme-
A(π0K+), A(π0K∗+) = 3−As0+As1 , tries has been attempted (though, again, with rather
A(ηK+), A(ηK +) = 2√2As 2√2As Bs , large errors) for a total of 10 data points. The other
∗ √3 0− √3 1− 0
A(η K+), A(η K +) = 1 As 1 As+2√2Bs . 3 modes have not yet been observed but the upper lim-
′ ′ ∗ √3 0− √3 1 0 its were reported, allowing estimates of their branching
Just as the two subsets of M0M+ that were considered ratios. The total number of V0V+ data points rises to
before,P0P+ andP0V+ arealsoseparatelydescribedby 13.
similar 12 parameters. The least is known about P0V+ decays. Not even an
Charmless hadronic decays of the B+ meson to the upperlimitisknownforK¯0K∗+. Oftheremaining7de-
two-meson final states that contain vector V or pseu- cays modes only 4 have been detected, providing 8 data
doscalarP mesonscomprisefoursubsets: P0P+,V0V+, points. For the other three an estimate of the branch-
V0P+, and P0V+. Each of the subsets comprises eight ing ratio can be made using current upper limits. Thus,
decays, with all possible combinations of two charged there are only 11 data points and a reasonable 12 pa-
mesons (e.g., π+ and K+ in the pseudoscalaroctet) and rameter U-spin fit cannot be performed. To avoid this
four neutral ones (K 0, ρ, ω, and φ in the vector octet). problem, one can make a joint U-spin fit to two M0M+
∗
Thus there are altogether 16 relevant decays of B of decaysubsets,e.g.afittobothV0P+ andP0V+ decays.
±
each of the four types. Each of the subsets, again, is de- With γ being the only common parameter for both pa-
scribedby12parameters,namely,6U-spinamplitudes,5 rameter sets, there are 11 completely free P0V+ U-spin
relativestrongphasesbetweenthem,andtheweakphase parameters(amplitudesandstrongphases)thatdescribe
γ which is the only common parameter among four pa- 11P0V+ datapoints. Thereisjustenoughdatatomake
rameter sets. the joint fit work.
All8B+ P0P+ decayshaveactuallybeenobserved TableIshowstheresultsoftheU-spinfitstofoursub-
and their b→ranching ratios and CP asymmetries have sets of M0M+ decays and their combinations. The top
been measured, though, with the present statistics in part of the table shows three fits to individual subsets
most cases the errors are rather large. This is especially (V0P+, P0P+, V0V+) and one joint fit. As was men-
4
TABLE I: Results of the U-spin fits to various subsets of charmless B± M0M± decays. The bottom panel shows γ as
determinedfrom directmeasurementsinB D(∗)K(∗) decays,from indire→ct constraintson theapexoftheunitaritytriangle,
→
and from SU(3) fitsto charmless PP decays.
Fit Subset Modes χ2/dof γ
1. V0P+ K∗0K+ ρ0π+ωπ+φπ+ K∗0π+ρ0K+ωK+φK+ 3.97/2 30+17 ◦
−18
2. P0P+ K0K+ π0π+ηπ+η′π+ K0π+π0K+ηK+η′K+ 3.01/4 (cid:0)68+59(cid:1)◦
−14
3. V0V+ K∗0K∗+ ρ0ρ+ωρ+φρ+ K∗0ρ+ρ0K∗+ωK∗+φK∗+ 0.05/1 (cid:0)40+136(cid:1)◦
−35
4A. P0V+ K0K∗+ π0ρ+ηρ+η′ρ+ K0ρ+π0K∗+ηK∗+η′K∗+ insufficient(cid:0)data (cid:1)
◦
4B. (V0P+ P0V+) 4.03/2 30+17
−18
◦
5. (V0P+SP0P+) 10.02/7 (cid:0)54+12(cid:1)
−11
◦
6. (V0P+SP0P+ V0V+) 10.47/9 (cid:0)54+12(cid:1)
−11
Direct measuremSents, BaBSar [18] (67 (cid:0)28 13(cid:1) 11)◦
± ± ± ◦
Direct measurements, Belle [19] 68+14 13 11
Indirect constraints, CKMFitter [20] (cid:0) −5157.±3+7.3±◦ (cid:1)
−12.9
Indirect constraints, UTFit [21] ((cid:0)57.9 7.4(cid:1))◦
SU(3) fitsto VP decays [5] 66.2+3.±8 0.1 ◦
−3.9±
SU(3) fitsto PP decays [5] (cid:0) (59 9 2)◦(cid:1)
± ±
tioned before, the only way to explore P0V+ data is to but slightly smaller uncertainties for the weak phase:
make a joint fit, for example, the (V0P+ P0V+) one. γ =(54+11) .
10 ◦
Amongthe firstfourfits, theV0P+ oneSstandsout. It Weal−soexploredthejoint(V0P+ P0P+)fitinsome
is the only fit that features a deep minimum at its pre- detail with the purpose of estimatinSg the expected im-
ferredvalueofγ,withbothupperandloweruncertainties provement of γ extraction as higher statistics on B de-
stayingunder20 . TheP0P+ fithas,ontheotherhand, cays get accumulated. We tried to identify some specific
◦
ashallowminimumwithverylargeupperuncertainty. Of modes where smaller uncertainties on branching ratios
the other twoU-spinfits inthe toppartofthe table, the would help reduce the error on γ.
V0V+ one produces a very shallow minimum, leaving γ Wefoundthatsettingastricterupperlimitontheφπ+
practically undetermined. As was mentioned before, the branching ratio is of particular importance. The current
current data on P0V+ is insufficient for a U-spin fit. In- upper limit (B+ φπ+) < 0.41 [15] is based on 89
stead, P0V+ subset was combined with the other VP million BB¯ Bpairs. →Both B factories will each accumu-
decays (V0P+), making the joint fit possible. However, late in excess of 500 million BB¯ pairs by the summer
the effect of the P0V+ data appears to be insignificant; of 2006. The available statistics on (φπ+) decays will
B
thejointfitproducespracticallyidenticalresultstothose increase by about a factor of 10, leading to uncertainties
of the V0P+ fit. One can draw the conclusion that the that are about 3 times smaller than the current ones.
quality of the current data for P0V+ and V0V+ is un- With the new data point of (φπ+) = 0.00 0.07 the
likely to significantly affect joint fits. joint (V0P+ P0P+) U-spinBfit features a ra±ther deep
This is confirmed in the lower part of the table. The minimum witSh uncertainties on γ at the level of 8◦. The
best U-spin fit is achieved when V0P+ and P0P+ data improvementsinγ extractionduetostricterupperlimits
are combined (30 data points) and fitted with 23 pa- on K∗0K+ are somewhat smaller.
rameters (two sets of six U-spin amplitudes and five Finally,wescaleddownuncertaintiesinalldatapoints
strong phases, plus the weak phase γ). The joint tothelevelscorrespondingto1billionBB¯ pairs. TheU-
(V0P+ P0P+)fitprefersavalueofγthatisinbetween spinfit becomesmuchdeeper atits minimum andγ gets
the valuSes favored by V0P+ and P0P+ fits, namely, extracted with a 6◦ uncertainty. Theoretical uncertain-
γ = (54+1121)◦. The addition of the V0V+ subset does tiesassociatedwiththismethodareexpectedtobesmall
not chan−ge this result, as expected. soithasthe potentialtoputratherstringentconstraints
The above results are based on the latest world aver- on the weak phase γ [17].
age values for branching ratios and CP asymmetries in Summarizing, with current statistics, the best U-spin
charged charmless B decays [16]. When the individual fits allow the extraction of γ with a reasonable accu-
values from BaBar and Belle are very different, we em- racy and the preferred value is γ = (54+12) which is
11 ◦
ployedthePDGscalingfactorStoboostuncertaintieson quiteconsistentwiththe currentindirectd−eterminations
theweightedaverages. Thismodificationonlyslightlyaf- that expect γ to lie between 42 and 73 [20, 21]. Note
◦ ◦
fects the final result. The joint U-spin (V0P+ P0P+) that the intrinsic theoreticaluncertainty associatedwith
fit to the unscaled data prefers the same centSral value possible U-spin breaking effects is expected to be rather
5
small and that U-spin symmetry is the only assumption method is that the extraction of γ is completely model
that is made in this approach[22]. Clearly, as data with independentandentirelydatadriven. Notealsothatun-
higher statistics becomes available, the statistical uncer- liketheuseofisospinforα,electroweakpenguinsarenot
tainties on γ will become even smaller. At the moment aprobleminourapproach. Penguincontributionsareen-
thedifferencebetweenthefourvaluesofγ extractedfrom tering in an important way in this U-spin approach for
the four subsets is not very meaningful due to large un- getting γ. That means that this method is sensitive to
certainties (Table I). When all branching ratios and CP newphysicsintheloops. Incontrast,recallthatthestan-
asymmetries in chargedB decays are experimentally de- dard B DK methods [1] involve only tree B decays.
→
terminedwithhighaccuracy,U-spinapproachshoulden- Comparisonofγ fromthesetwomethodsisthereforeim-
able extractionof γ quite precisely from each of the four portant for uncovering new physics.
subsets of data. The resulting spread in γ values should
be small and could perhaps be used to indicate the sys- We thank J. G. Smith for helpful discussions. This
tematic errors inherent in the method due to residual researchwassupportedinpartbyDOEcontractNos.DE-
U-spin breaking effects. The crucial advantage of the FG02-04ER41291(BNL).
[1] For recent reviews see: M.-H. Schune, plenary talk andthe1/(√3)factorintothedefinitionofBd,s (Ref.[3]
0
at ICHEP 2005, Lisbon; A. Soni, hep-ph/0509180; usedadifferentconvention,withthe1/(2√3),√3/2and
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[2] C.-W. Chiang et al.,Phys.Rev. D 70, 034020 (2004). Bd,s, respectively). 1 0
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[12] Again, weabsorb the1/(2√3)factor intothedefinitions
[4] This advantage of U-spin has also been emphasized in
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Phys. Rev. D 66, 097501 (2002); A. Soni and J. Zupan, [21] M. Bona et al. [UTfit Group], JHEP 0507, 028
hep-ph/0510325. (2005). Updated results may be found on the web site
[9] SinceboththeuandthecpieceintheH alsoinvolve http://utfit.roma1.infn.it/.
eff
piecesofthet-quark,useoftreeandpenguinterminology [22] SU(3) breaking effects are argued to be O(few degrees),
for these, which is sometimes done, may be misleading. see M. Gronau and J. Zupan, Phys. Rev. D 70, 074031
[10] M. Gronau and D. London, Phys. Rev. Lett. 65, 3381 (2004); see also [5]. However, effect of ignoring some
(1990). graphs is not included in these studies.
[11] Weabsorbthe1/(2√3)factorintothedefinitionsofAd,s
0,1