aa r X i v : . [ h e p - ph ] J u l Flavor Physics and CP Violation Conference, Bled, 2007 Direct CP Violation in B Decays
M. Gronau
Physics Department, Technion - Israel Insitute of Technology, 32000 Haifa, Israel
We discuss several aspects of direct CP asymmetries in B decays, which are very useful in spite of hadronic un-certainties in asymmetry calculations. 1) Asymmetries in decays to D ( ∗ ) K ( ∗ ) , π + π − , ρ + ρ − , providing precisiontests for the CKM phase γ . 2) Null tests in B + → J/ψK + , π + π , where a nonzero asymmetry provides evidencefor New Physics. 3) Isospin and broken flavor SU(3) relations among CP asymmetries in B → Kπ, ππ predicting A CP ( B → K π ) and A CP ( B → π π ). 4) The significance of A CP ( B → K + π − ) = A CP ( B + → K + π ). 5)A potentially stringent constraint on γ from A CP ( B + → K + π ) and R c ≡ B + → K + π ) / Γ( B + → K π + ).6) The role of direct CP asymmetries in b → s ¯ qq decays for studying the origin of potential New Physics.
1. Importance of direct CP violation
It took 35 years from the discovery of tiny CP viola-tion in K - ¯ K mixing by Christenson, Cronin, Fitchand Turlay [1] to an observation of direct CP viola-tion in K → ππ by the KTeV [2] and NA48 [3] collab-orations. While this observation was very importantby itself, ruling out the Superweak hypothesis for CPviolation [4], hadronic uncertainties involved in calcu-lating this effect prohibited a precise quantitative testof the CKM framework [5].Tremendous effort has been devoted by the CLEOcollaboration at Cornell, by BaBar at SLAC, Belle atKEK, and by the CDF and D0 collabotaions at Fer-milab, to measure direct CP violation in hundreds ofcharged and neutral B decay modes. A small sample ofthe measured asymmetries is plotted in Fig. 1 [6]. Theasymmetry in B → K + π − , involving the smallest ex-perimental error, provides unambiguous evidence fordirect CP violation. +1.0 0.0 - CP Asymmetry in Charmless B Decays
HFAG March 2007
PDG2006BABARBelleCLEONew Avg.CDF K ∗ + K + K − K ∗ + π + π − K + K S K S K + K − K + K + π − π K + π + π − K ∗ ( ) π + K + K K K K + π K π + K + π − K ∗ + π − K ∗ + π K ∗ π + Figure 1: A sample of direct CP asymmetries [6].
Calculations of direct CP asymmetries involve un-certainties from weak hadronic matrix elements andstrong final state phases. To illustrate these uncer-tainties, consider for instance the decay B → K + π − which has a dominant penguin amplitude and a CKM-suppressed tree amplitude, as shown in Fig. 2. The bd W suudT bd gu, c, tW suudP Figure 2: Penguin and tree amplitudes in B → K + π − . amplitudes for this decay process and its charge-conjugate are given in terms of suitably defined mag-nitudes | P | , | T | , a CP-conserving strong phase δ , anda CP-violating weak phase γ ≡ arg( − V ∗ ub V ud /V ∗ cb V cd ), A ( B → K + π − ) = | P | e iδ + | T | e iγ ,A ( B → K − π + ) = | P | e iδ + | T | e − iγ . (1)A calculation of the CP asymmetry in terms of γ , A CP ( K + π − ) ≡ Γ( B → K − π + ) − Γ( B → K + π − )Γ( B → K − π + ) + Γ( B → K + π − )= − | T /P | sin δ sin γ | T /P | + 2 | T /P | cos δ cos γ , (2)requires computing | T /P | and δ . This is extremelydifficult, as these quantities involve non-perturbativelong-distance effects. In QCD calculations based ona heavy quark expansion [7, 8, 9] one faces uncer-tainties in these quantities from chirally enhanced1 /m b -suppressed terms including annihilation contri-butions from penguin operators, α s -suppressed termsand “charming penguin” terms [10]. Some of thesecontributions can be traced back to incalculable softrescattering amplitudes from (¯ sc )(¯ cd ) intermediate fpcp07 143 Flavor Physics and CP Violation Conference, Bled, 2007 states. A clear distinction between calculable shortdistance contributions and incalculable soft contribu-tions is particularly challenging for the strong phase δ . While observing direct CP violation in B decays isimportant by itself, it then seems that these asymme-tries (like the one in K → ππ ) cannot provide accuratetests for the mechanism of CP violation, originatingin the Standard Model in the phase γ of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The purpose ofthis talk is to show that, in fact, direct asymmetriesmeasured in certain B decay modes do provide preci-sion tests for the CKM framework, in spite of theoret-ical difficulties in calculating these asymmetries. γ in B → D ( ∗ ) K ( ∗ ) We recall well-known examples of direct CP asym-metries in a whole class of processes B → D ( ∗ ) K ( ∗ ) ,where D and ¯ D decay to a variety of common fi-nal states. These decays provide a clean determina-tion of the weak phase γ [11]. The trick here lies inrecognizing the measurements which yield this funda-mental CP-violating quantity through interference oftree-level ¯ b → ¯ cu ¯ s and ¯ b → ¯ uc ¯ s amplitudes. A broadand up-to-date review of CP violation in the B mesonsystem, including numerous references for studies of B → D ( ∗ ) K ( ∗ ) , can be found in Ref. [12]. γ in B → π + π − , ρ + ρ − In B → π + π − and B → ρ + ρ − , mixing-inducedasymmetries ( S ) and direct asymmetries ( C ≡ − A CP )are both needed to fix γ in a rather precise methodbased on isospin symmetry [13]. Measurementsof these asymmetries and conservative assumptionsabout flavor SU(3) breaking yield the currently mostprecise value γ = (72 ± ◦ [12, 14, 15], in agreementwith γ = (66 ± ◦ obtained from ∆ m d / ∆ m s [16]. Interesting applications of direct CP asymmetriesare null tests of the CKM framework in decays whereasymmetries are expected to be very small. Two pro-cesses dominated by a single CKM phase are B + → J/ψK + [17, 18] and B + → π + π , where the Stan-dard Model predicts vanishingly small asymmetries,much smaller than one percent. This includes elec-troweak penguin contributions in B + → π + π [19]. Anonzero asymmetry observed in one of these modes ata percent level (as small as it can be with future an-ticipated precision [6]) would be a clean signature forNew Physics. While in general calculating strong phases is verydifficult (though phases are known in e.g. B + → K + K − K + mediated by c ¯ c resonant states [20]), asym-metry measurements provide information for study-ing the dynamics of hadronic decays. Thus, certaincharmless B decays have been predicted to lead tolarger asymmetries than others, because they involvetwo interfering amplitudes with different weak phases,whose ratio is expected to be dynamically enhanced.This may follow from an enhancement of the smaller ofthe two amplitudes, or a suppression of the larger am-plitude. Such effects were noted in an approach usingflavor SU(3) symmetry [21, 22], and may probablybe realized in QCD calculations [7, 8]. We mentionfour examples for this enhancement effect using thelanguage of flavor SU(3) amplitudes: (1) B + → π + η ,where an amplitude P smaller than T involves a fac-tor 2, (2) B + → K + η , where a potentially domi-nant P amplitude involves destructive interference be-tween a few quark-level penguin amplitudes [23], (3) B + → K + ρ , where subdominant T V and C P ampli-tudes add up constructively, (4) B → ρ ± π ∓ , whichinvolves constructive interference of P V T V and P P T P terms. Table I CP asymmetries involving large | ∆ S | = 1 tree-to-penguin ratios or large ∆ S = 0 penguin-to-tree ratios [6]. π + η K + η K + ρ ρ ± π ∓ − . ± . − . ± .
11 0 . +0 . − . − . ± . Table 1 quotes measured CP asymmetries for thesefour final states [6]. All four asymmetries are nonzeroat a level of about 3 σ with central values between 13and 31 percent. Somewhat higher precision is requiredin these asymmetry measurements for claiming unam-biguous evidence of direct CP violation in these modes.
2. CP asymmetries in B → Kπ Γ and A CP The decays B , + → Kπ , which occur in four distinctfinal states, can be expressed in terms of three isospinamplitudes [24]. The initial states are I ( B ) = 1 / I ( Kπ ) = 1 / , / I = 0 ,
1. Denoting∆ I = 0 and ∆ I = 1 amplitudes by B and A or A ′ , thephysical amplitudes for the two pairs of B + and B decays are related to each other by isospin reflection,implying [25] A ( K π + ) = B + A, − A ( K + π − ) = B − A, (3) −√ A ( K + π ) = B + A ′ , √ A ( K π ) = B − A ′ . fpcp07 143 lavor Physics and CP Violation Conference, Bled, 2007 A ( K π + ) − A ( K + π − )+ √ A ( K + π ) −√ A ( K π ) = 0 , (4)which has important physical implications in terms oftwo approximate sum rules for decay rates Γ [26] andCP rate differences ∆ [27]:Γ( K + π − ) + Γ( K π + ) ≈ K + π ) + Γ( K π )] , ∆( K + π − ) + ∆( K π + ) ≈ K + π ) + ∆( K π )] , (5)where ∆( Kπ ) ≡ Γ( ¯ B → ¯ K ¯ π ) − Γ( B → Kπ ) . (6)We now present the shortest proofs for these sumrules, using the dominance of a ∆ I = 0 penguin am-plitude P (part of the isospin amplitude B ) in all B → Kπ decays. Evidence for penguin-dominance isprovided by the four measured B → Kπ decay rateswhich are equal within 2 σ [6], R ≡ Γ( B → K + π − )Γ( B + → K π + ) = 0 . ± . ,R c ≡ B + → K + π )Γ( B + → K π + ) = 1 . ± . ,R n ≡ Γ( B → K + π − )2Γ( B → K π ) = 0 . ± . . (7)The non-penguin amplitudes are calculated to bemuch smaller than P , | non- P | / | P | ∼ . P (or B ) cancel trivially, while termswhich are linear in P cancel because of (4). Thus,the only terms which may violate the two sum rulesare quadratic in non-penguin amplitudes, and can beshown to amount to a few percent of each side of thetwo sum rules (5).Indeed, the sum rule for decay rates Γ holds withinexperimental errors which are currently about 5% ofeach side [6, 28]. The sum rule for CP rate asymme-tries ∆, expected to hold within a similar precision,leads to a prediction for the asymmetry in B → K π in terms of the other three asymmetries which havebeen measured with higher precision (see Table II), A CP ( B → K π ) = − . ± . . (8)This prediction, which is expected to hold within afew percent, can be improved by reducing errors in A CP ( K π + ) , A CP ( K + π ). While the value (8) is con-sistent with experiment, higher accuracy in asymme-try measurements is required for testing this predic-tion following from the rather precise ∆ relation (5). A CP ( K + π ) = A CP ( K + π − ) a puzzle? The measurement of a nonzero CP asymmetry in B → K + π − provides the first evidence for an inter-ference between a dominant penguin amplitude P and Table II CP asymmetries in B → Kπ [6]. B → K + π − B + → K + π B + → K π + B → K π − . ± .
012 0 . ± .
026 0 . ± . − . ± . a small tree amplitude T with a nonzero relative strongphase δ between the two amplitudes. [See Eqs. (1) and(2)]. Such an interference occurs also in B + → K + π ,in which a spectator d -quark in B → K + π − is re-placed by a u -quark. No asymmetry has been observedin B + → K + π . An assumption that other contribu-tions to this asymmetry are negligible has raised somequestions about the validity of the CKM framework.In fact, a color-suppressed tree amplitude C , also oc-curring in B + → K + π [21] (see Fig. 3), resolves this“puzzle” if this amplitude is comparable in magnitudeto T . A too naive assumption, | C | ≪ | T | , has been Figure 3: Color-suppressed tree amplitude in B + → K + π made in Ref. [21] followed by numerous other works.More recent studies, including a global SU(3) fitfor all charmless B decays to two pseudoscalars, haveshown that | C | ∼ | T | [22, 29, 30]. For consistency be-tween the two CP asymmetries in B → K + π − and B + → K + π , the strong phase difference between C and T must be negative and cannot be very small [31].This, and the somewhat large value of C , seem tostand in contrast to QCD calculations using a factor-ization theorem [7, 9]. While this may be considereda difficulty for QCD calculations, by no means shouldit be considered evidence for New Physics as arguedsometimes. A CP ( K + π ) and R c The asymmetry A CP ( K + π ) and the ratio of rates R c defined in (7) involve the decay amplitude for B + → K + π , which seems to confront QCD calcula-tion with a difficulty. The smallness of the measuredasymmetry and of the measured value of R c − γ which we discuss now.Including color-favored and color-suppressed elec-troweak penguin contributions, P EW and P cEW , onehas A ( K + π ) = P + T + C + P EW + P cEW , fpcp07 143 Flavor Physics and CP Violation Conference, Bled, 2007 A ( K π + ) = P. (9)A small 1 /m b -suppressed annihilation amplitude A from a current-current operator has been neglectedin the two processes [7, 9, 21, 32]. One introducestwo calculable parameters for ratios of amplitudes, r c ≡ | T + C | / | P | and δ EW ≡ | P EW + P cEW | / | T + C | .The parameter r c is given by [33] r c = √ V us V ud f K f π s (¯ B + → π + π )(¯ B + → K π + ) = 0 . ± . , (10)where the error includes an uncertainty from SU(3)breaking. The parameter δ EW is defined by [34] P EW + P cEW = − δ EW e − iγ ( T + C ) ,δ EW = − c + c c + c | V ∗ tb V ts || V ∗ ub V us | = 0 . ± . . (11)Here the error is dominated by the current uncertaintyin | V ub | / | V cb | , including also a smaller error from SU(3)breaking estimated using QCD factorization.The asymmetry A CP ( K + π ) and the ratio of rates R c are given, to first order in r c , by A CP ( K + π ) = − r c sin γ sin δ c + O ( r c ) , (12) R c − − r c (cos γ − δ EW ) cos δ c + O ( r c ) , where δ c is the strong phase difference between T + C and P . Eliminating δ c it is now straight forward toprove the following sum rule [31] (cid:18) A CP ( K + π )sin γ (cid:19) + (cid:18) R c − γ − δ EW (cid:19) = (2 r c ) + O ( r c ) . (13)This sum rule implies that at least one of the twoterms whose squares occur on the left-hand-side mustbe sizable, of the order of 2 r c = 0 .
4. The firstterm, | A CP ( B + → K + π ) | / sin γ , is already smallerthan ≃ .
1, using the current 2 σ bounds on γ and | A CP ( B + → K + π ) | . Thus, the second term mustprovide a dominant contribution. For R c ≃
1, thisimplies γ ≃ arccos δ EW ≃ (53 . ± . ◦ . This range isexpanded by including errors in R c and A CP ( B + → K + π ). For instance, bounds 0 . < R c < . γ < ◦ . Cur-rent values of A CP ( K + π ) and R c lead to an up-per limit γ ≤ ◦ at 90% confidence level [31]. Thisbound is consistent with the value of γ obtained from B → π + π − and B → ρ + ρ − , as mentioned above, butis not yet competitive with its precision.
3. SU(3) relations ∆( Kπ ) = − ∆( ππ ) One may prove two useful relations between CP ratedifferences within the CKM framework [35, 36]:∆( K + π − ) = − ∆( π + π − ) , (14)∆( K π ) = − ∆( π π ) . (15) A slightly over-simplified proof of these relations goesas follows. [A precise proof, including electroweak pen-guin terms and justifying an assumption about negligi-ble E + P A terms can be found in Ref. [12]).] Writing A ( K + π − ) = P + T, (16)where P and T contain strong and weak phases, onehas in the flavor SU(3) limit A ( π + π − ) = − λP + λ − T, (17)where λ ≡ V us /V ud = − V cd /V cs . Similarly, √ A ( K π ) = P − C, √ A ( π π ) = − λP − λ − C. (18)The CP rate differences in the two pairs of processesare given by interference terms between P and T andbetween P and C , which are equal in magnitude andhave opposite signs in B → Kπ and B → ππ . Thisproves (14) and (15). Table III Direct CP asymmetries in B → ππ [6]. B → π + π − B → π π . ± .
07 0 . +0 . − . Using branching ratios from [6] and asymmetries inTables II and III, Eq. (14) reads(¯ K + π − ) A CP ( K + π − ) = − (¯ π + π − ) A CP ( π + π − ) , ( − . ± . − = ( − . ± . − . (19)Both the signs and the magnitudes agree well, pro-viding evidence for the success of flavor SU(3). Wenote that using SU(3) breaking factors f K /f π for both T and P , as assumed in [35], would imply a fac-tor ( f K /f π ) on the right-hand-side of (19) leadingto a worse agreement. Some reduction of errors in A CP ( K + π − ) and A CP ( π + π − ) is required in order todetermine well the pattern of SU(3) breaking in P T .The relation (15) and the value of A CP ( K π ) in (8)obtained using isospin symmetry imply a predictionfor A CP ( π π ), A CP ( π π ) = − A CP ( K π ) (¯ K π )(¯ π π ) = 1 . ± . . (20)The error is dominated by errors in A CP ( K π + ) and A CP ( K + π ). An SU(3) breaking factor f K /f π in C would lower this prediction by a factor f π /f K . A largepositive CP asymmetry in B → π π implies compa-rable sides in the ¯ B → ππ isospin amplitude triangle,but a squashed B → ππ isospin triangle. This has asimplifying effect on the isospin analysis in B → ππ ,where a discrete ambiguity disappears in the limit ofa flat B → ππ triangle [13]. fpcp07 143 lavor Physics and CP Violation Conference, Bled, 2007
4. CP violation in b → s ¯ qq A class of b → s ¯ qq penguin-dominated B decays toCP-eigenstates has attracted recently considerable at-tention. This includes final states XK S and XK L ,where X = φ, π , η ′ , ω, f , ρ , K + K − , K S K S , π π ,for which measured mixing-induced asymmetries ± S (for CP eignstates with eigenvalues η CP = ∓
1) and direct asymmetries C ≡ − A CP are quoted in TableIV [6]. Table IV CP asymmetries in b → s ¯ qq for η CP = ∓ X φ π η ′ ω ± S . ± .
18 0 . ± .
21 0 . ± .
07 0 . ± . C . ± .
13 0 . ± . − . ± . − . ± . X ρ f (980) K + K − K S K S ± S . ± .
57 0 . ± .
17 0 . +0 . − . . ± . C . ± . − . ± .
13 0 . ± . − . ± . Whereas a value S = − η CP sin 2 β = 0 . ± .
025 [6]is expected approximately [37, 38], the actual averageof all corresponding measured entries in Table IV issin β eff ≡ h− η CP S i = 0 . ± .
05. A question of-ten raised is “ is this . σ discrepancy caused by NewPhysics? ”. In a similar manner, one may calculatethe average of all direct CP asymmetries obtaining avalue h A CP i = 0 . ± .
04, which is small and consis-tent with zero.
Does this mean that there is very littleplace for New Physics?
These two questions, considering averages over sev-eral processes, should be considered with care becausein the Standard Model both ∆ S ≡ S + η CP sin 2 β and C ≡ − A CP are process-dependent. The small-ness of the asymmetries A CP relative to ∆ S may berelated to small strong phases δ , because A CP and ∆ S are proportional to sin δ and cos δ , respectively [17].Calculations of these asymmetries involve hadronicuncertainties at a level of several percent, of order λ [39, 40, 41, 42]. It has been pointed out some timeago [43] that if New Physics contributions are at thislow level, it becomes difficult to separate them fromhadronic uncertainties within the CKM framework.The importance of direct CP asymmetries measuredin this class of processes may be demonstrated throughtwo features of C and ∆ S , by which one can distin-guish New Physics effects from hadronic uncertaintiesin the Standard Model and learn about the origin ofNew Physics effects: • Within the Standard Model C and ∆ S can beshown to lie on a circle [17], (cid:18) ∆ S cos 2 β (cid:19) + C = (2 ξ sin γ ) , (21)whose radius depends on a process-dependent ra-tio of tree and penguin amplitudes ξ ∼ O ( λ ). The locus on the circle is fixed by a strong phase δ . In most cases one expects | δ | < π/ S > • Once the measured values of C and ∆ S dis-agree with calculations of ξ and the strongphase δ beyond hadronic uncertainties, we willhave solid evidence for New Physics. At thispoint one would seek signatures characterizingclasses of models rather than studying effectsof specific models of which quite a few exist[38, 44, 45, 46, 47, 48]. A useful way for clas-sifying extensions of the Standard Model is bythe isospin behavior, ∆ I = 0 or ∆ I = 1, of thenew effective operators.Recently it has been shown [25] that the isospinstructure of potential New Physics operators con-tributing to b → s ¯ qq can be determined by studying C and ∆ S in B → XK together with two other kindsof asymmetries: direct asymmetries A CP in isospin-reflected decays B + → XK + , and isospin-dependentCP-conserving asymmetries defined by A I ≡ Γ( XK + ) − Γ( XK )Γ( XK + ) + Γ( XK ) . (22)A study of the currently measured four kinds of asym-metries has shown that potential New Physics contri-butions to these processes must be small. Some reduc-tion of errors in the measured asymmetries is requiredfor identifying a signature for New Physics and for auseful implementation of this method. We refer thereader to Ref. [25] for details of this analysis.
5. Conclusion
The importance of direct CP violation is demon-strated by using direct asymmetries in B → D ( ∗ ) K ( ∗ ) , π + π − , ρ + ρ − for a determination of the weakphase γ . Asymmetries in B + → J/ψK + , π + π pro-vide unambiguous signatures for New Physics. In spiteof the difficuly of calculating strong phases, measuredasymmetries provide useful information about the dy-namics of hadronic decays.The different asymmetries measured in B + → K + π and B → K + π − cannot be easily explainedwithin QCD calculations, but should not be con-sidered evidence for New Physics. An isospin sumrule among the four B → Kπ asymmetries predicts A CP ( K π ) = − . ± . A CP ( K + π ) andsmall R c − γ . Theflavor SU(3) prediction A CP ( K + π − ) /A CP ( π + π − ) = − (¯ π + π − ) / (¯ K + π − ) works well. The ratio of thetwo asymmetries fixes the pattern of SU(3) breaking,which is useful for a precise determination of γ . Fla-vor SU(3) predicts a large positive direct asymmetryin B → π π , which has an implication on the B → ππ fpcp07 143 Flavor Physics and CP Violation Conference, Bled, 2007 isospin analysis. Direct CP asymmetries in b → s ¯ qq play a central role in studying New Physics operators,in particular for learning their isospin structure. Acknowledgments
I would like to thank the Local Organizing Commit-tee for a very interesting and smoothly running confer-ence in a beautiful setting. I am grateful to JonathanRosner for a long and fruitful collaboration and to sev-eral other short-term collaborators. This work wassupported in part by the Israel Science Foundationunder Grant No. 1052/04 and by the German-IsraeliFoundation under Grant No. I-781-55.14/2003.
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