CALT-68-2159 hep-ph/9802XXX February 1998 WEAK PHASE γ FROM COLOR-ALLOWED B → DK RATES Michael Gronau1 California Institute of Technology Pasadena, CA 91125 ABSTRACT
arXiv:hep-ph/9802315v1 12 Feb 1998
The ratios of partial rates for charged B decays to the recently observed D 0 K mode and to the two DCP K ?nal states (CP = ±) are shown to constrain the ? weak phase γ ≡ Arg(Vub ). The smaller color-suppressed rate, providing further information about the phase, can be determined from these rates alone. Present estimates suggest that, while the ?rst constraints can already be obtained in a high luminosity e+ e? B factory, measuring the color-suppressed rate would require dedicated hadronic B production experiments.
PACS codes: 12.15.Hh, 12.15.Ji, 13.25.Hw, 14.40.Nd
Permanent Address: Physics Department, Technion – Israel Institute of Technology, 32000 Haifa, Israel.
The CLEO Collaboration has recently observed the decay B ? → D 0 K ? and its charge conjugate . This is the ?rst observation of a decay mode described by the quark ? subprocess b → c?s involving the Cabibbo-Kobayashi-Maskawa (CKM) factor VcbVus . u The reported branching ratio, 0.055 ± 0.014 ± 0.005, measured relative to B ? → D 0 π ? , is in agreement with the Standard Model expectation. The observed decay plays an crucial role in a method proposed some time ago [2, 3] to determine the CP violating ? ? weak phase γ, the relative phase between Vcb Vus and Vub Vcs . The purpose of this Letter is to reexamine this method in view of its importance, and to suggest some variants to overcome its di?culties. A complementary variant was proposed in Ref. . ? The other processes involved in the method are B ? → DCP K ? , B ? → D 0 K ? and their charge conjugates. Partial D decay rates into CP-eigenstates (such as K + K ? ) are about an-order-of-magnitude smaller than into states of speci?c ?avor (such as K ? π + ). Thus, by combining a few CP modes, the decays B ? → DCP K ? should be observed ? in near future high statistics experiments. The third process, B ? → D 0 K ? , mediated ? by b → u?s and involving the CKM factor Vub Vcs , is harder to measure. It is usually c assumed to have a “color-suppressed” branching ratio, about two orders-of-magnitude smaller than that of B ? → D 0 K ? . Let us recall the arguments on which this estimate is based. The e?ective Hamiltonians for b → c?s and b → u?s transitions are u c GF ? He? (b → c?s) = √ VcbVus [c1 (?)(?u)(?b) + c2 (?)(?u)(?b)] , u s c c s 2 (1)
GF ? He? (b → u?s) = √ Vub Vcs [c1 (?)(?c)(?b) + c2 (?)(?c)(?b)] , c s u u s (2) 2 respectively, where c1 (mb ) = 1.13, c2 (mb ) = ?0.29 . (?b) = cγ ? (1 ? γ5 )c etc. are c ? left-handed color-singlet quark currents. The ratio of the corresponding CKM factors ? ? is |Vub Vcs /Vcb Vus | = 0.4 ± 0.1 . The hadronic matrix elements of the four-fermion operators, depending on the scale ?, are very di?cult to calculate. The conventional description of strangeness-conserving decays such as B 0 → D ? π + assumes that “colorallowed” operator matrix elements factorize . Nonperturbative e?ects, arising from soft gluon exchange , require the use of a free parameter to describe decay amplitudes. This parameter, ?tted by data, determines the ratio of color-suppressed and color-allowed amplitudes, a2 /a1 ≈ 0.26 . This value depends on unmeasured form factors of B mesons into light mesons for which a model must be assumed . These form ? factors dominate color-suppressed amplitudes of processes such as B 0 → D 0 π 0 . Using ? ?avor SU(3) , this value of a2 /a1 can also be used to study B → DK decays. An ap? plication to relations between B → DK (given by a B → D form factor) and B → DK (given by a B → K form factor), in which ?nal states carry opposite charm, is more questionable. Nevertheless, one often assumes that r≡
? ? |VubVcs | a2 |A(B ? → D 0 K ? )| ≈ 0.1 . ≈ ? |A(B ? → D 0 K ? )| |VcbVus | a1
It is di?cult to associate a theoretical uncertainty with this estimate, which is based largely on empirical observations in the ?S = 0 sector, rather than on ?rm theoretical 2
grounds. We will usually assume that the ratio of amplitudes r cannot be greater or smaller than 0.1 by a factor larger than two. We note, however, that larger values cannot be excluded. As we will see, the precision of determining the weak phase γ improves as r increases. One of the questions addressed in the present Letter is how to determine this quantity experimentaly. ? An essential di?culty in measuring the rate of B ? → D 0 K ? was pointed out by ? 0 from B ? → D 0 K ? is identi?ed through its ? Atwood, Dunietz and Soni . When a D hadronic decay mode (such as K + π ? ), the decay amplitude interferes with a comparable doubly Cabibbo suppressed decay amplitude of a D 0 from B ? → D 0 K ? . (Here Eq. (3) ? is assumed). This forbids a direct measurement of Γ(B ? → D 0 K ? ). Using two di?erent + ? ?nal states to identify a neutral D meson, (e.g. K π and K + π ? π 0 ), may allow a determination of r and γ from the branching ratios of these processes and their chargeconjugates . The products of corresponding B and D decay branching ratios are expected to be about two orders of magnitude smaller than B(B ? → D 0 K ? )B(D 0 → K ? π + ), at a level of 10?7 . The number of events, expected in future e+ e? colliders, is likely to be too small to allow a precise determination of r and γ . Such precision can potentially be achieved in dedicated hadronic B production experiments , which are expected to yield an order of a few thousand events of this kind . Let us study the information about γ obtained from measuring only the more abundant processes B ? → D 0 K ? , B ? → DCP K ? and their charge-conjugates. We will de? rive a simple sum rule from which the suppressed rate of B ? → D 0 K ? can, in principle, be determined from the above less suppressed rates, without involving an interference with B ? → D 0 K ? . New constraints on the weak phase γ will be shown to be obtained by measuring only the two ratios of partial decay rates into CP-even-and-odd and into ?avor states, combining particles and antiparticles. We will look into the prospects of carrying out these studies in a future very high luminosity e+ e? B factory. De?ning decay amplitudes by their magnitudes, strong and weak phases, ? ? ? A(B + → D 0 K + ) = Aei? , A(B + → D 0 K + ) = Aei? eiγ , (4)
we ?nd (disregarding common phase space factors) ? ? Γ(B + → D 0 K + ) = Γ(B ? → D 0 K ? ) = A2 , ? Γ(B + → D 0 K + ) = Γ(B ? → D 0 K ? ) = A2 , (5) (6)
1 ? ? Γ(B ± → D1 K ± ) = [A2 + A2 + 2AA cos(δ ± γ)] , (7) 2 1 ? ? (8) Γ(B ± → D2 K ± ) = [A2 + A2 ? 2AA cos(δ ± γ)] , 2 ? where δ ≡ ? ? ?. D1,2 are the two neutral D meson CP-eigenstates, D1,2 = (D 0 ± √ ? 0 )/ 2. D One obtains the following sum rule ? Γ(B ? → D1 K ? ) + Γ(B ? → D2 K ? ) = Γ(B ? → D 0 K ? ) + Γ(B ? → D 0 K ? ) . 3 (9)
A similar sum rule is obeyed by the charge-conjugated processes. In principle these sum ? rules allow a determination of Γ(B ? → D 0 K ? ) = Γ(B + → D 0 K + ) from measurements of the other larger rates. Using Eq. (3) we note, however, that the second rate on the right-hand-side is expected to be about two orders-of-magnitude smaller than the ? ?rst rate. Therefore, a useful determination of Γ(B ? → D 0 K ? ) requires very precise ? 0 ? ? ? measurements of Γ(B → D K ) and of Γ(B → DCP K ). The ratio of amplitudes r can be obtained from the charge-averaged ratio for decays into D meson CP and ?avor states S≡ Γ(B + → D1 K + ) + Γ(B ? → D1 K ? ) + Γ(B + → D2 K + ) + Γ(B ? → D2 K ? ) , ? Γ(B + → D 0 K + ) + Γ(B ? → D 0 K ? ) S = 1 + r2 . (10)
The CP asymmetries of decays into D1 K and D2 K, normalized by the rate into the D meson ?avor state, Ai ≡ Γ(B + → Di K + ) ? Γ(B ? → Di K ? ) , ? Γ(B + → D 0 K + ) + Γ(B ? → D 0 K ? ) A ≡ A2 ? A1 = 2r sin δ sin γ . It is convenient to de?ne two charge-averaged ratios for the two CP-eigenstates Ri ≡ for which we ?nd The factor of 2 in the numerator of R1,2 is used to normalize these ratios to values of approximately one. Rewriting R1,2 = sin2 γ + (r ± cos δ cos γ)2 + sin2 δ cos2 γ , one obtains the two inequalities sin2 γ ≤ R1,2 , i = 1, 2 . (16) (15) R1,2 = 1 + r 2 ± 2r cos δ cos γ . (14) 2[Γ(B + → Di K + ) + Γ(B ? → Di K ? )] , ? Γ(B + → D 0 K + ) + Γ(B ? → D 0 K ? ) i = 1, 2 , (13) i = 1, 2 , (11)
are equal in magnitude and have opposite signs. They yield a combined asymmetry (12)
The quantities S, A and Ri hold information from which r, δ and γ can be determined up to discrete ambiguities. r is given by S, and γ is obtained from Ri and A: Ri = 1 + r 2 ± 4r 2 cos2 γ ? A2 cot2 γ . (17)
Plots of Ri as function of γ for a few values of r and A, and the precision in r, Ri and A required to measure γ to a given level, are given in Ref. . The accuracy of this method of determining γ depends on the actual value of r. The larger this ratio, the more precisely can γ be determined. 4
If r is as small as estimated in Eq. (3), then a useful determination of this quantity from S requires that the rates in the numerator and denominator of Eq. (10) are measured to better than 1% which is unattainable in near future experiments. This demonstrates the di?culty of looking for the color-suppressed process. Consider, for instance, a sample of 300 million B + B ? pairs thought to be produced in an upgraded version of CESR . Using B(B ? → D 0 K ? ) = 3 × 10?4  and B(D 0 → K ? π + ) = 0.04 , one expects a total of about 7000 identi?ed D 0 K ? and ? D 0 K + events. (Use of other D decay modes compensates for suppression due to detection e?ciencies). This would yield a 1.2% measurement of the sum of rates in the denominator of Eq. (10). To estimate the precision of the numerator, in which the D meson decays to CP-eigenstates, we use  B(D 0 → π + π ? ) + B(D 0 → K + K ? ) = 6 × 10?3 for two positive CP states. This implies a combined sample of about 1000 identi?ed D1 K + and D1 K ? events. (Detection e?ciencies may decrease this number somewhat). Let us assume a similar number of D2 K + and D2 K ? events, identi?ed by D decay ?nal states such as KS π 0 , KS ρ0 and KS φ. (The combined decay branching ratio into these states and others are actually larger than into positive CP-eigenstates , however detection e?ciencies are smaller due to the larger number of ?nal particles). This determines the numerator to within 2.2%, so that the combined statistical error on S is 2.5%. Systematic uncertainties are likely to increase this error, although some of them cancel in the ratio of rates. Assuming that the total error in S is 5%, a 90% c.l. upper limit r < 0.25 could be obtained from this measurement. Another way for learning r is by measuring the rate of the rare process B ? → (K + π ? )D K ? combined with its charge-conjugate. 300 million B + B ? pairs lead to a few tens of events, for which a large error in the combined branching ratio is expected. The amplitude of this process consists of two interfering contributions carrying an unknown ? relative phase. The two terms describe the color suppressed process B ? → D 0 K ? ? 0 ? followed by Cabibbo-allowed D decay, and B → D K followed by a doubly Cabibbo suppressed D decay. The magnitude of the second amplitude is expected to be known to a few percent at the time of the experiment. Comparison of this amplitude with the measured one could be used to constrain r. In the likely case that the two amplitudes are equal within experimental errors, destructive interference would be assumed to obtain an ? upper limit on r, r < 2 B(D 0 → K + π ? )/B(D 0 → K + π ? ) = 0.18. Here current central values were used and all experimental errors were neglected. This upper limit, increased somewhat by experimental errors in branching ratios, is about the limit obtained from S. Assuming that r is too small to be measured from S (i.e. r ≤ 0.2), one may still obtain useful constraints on the weak phase γ from the asymmetries Ai and the two ratios Ri . The information obtained from these pairs of quantities is complementary to each other. While the asymmetries become larger for large values of sin δ sin γ, the deviation of Ri from 1 + r 2 ≈ 1 increases with cos δ cos γ. One thousand identi?ed Di K ± events allow a 3σ asymmetry measurement at a level of 10% or larger. For r ? 0.1, such asymmetries are expected if δ is sizable, namely δ > 30? . It is needless to emphasize the importance of nonzero CP asymmetry measurements, however one should foresee a possibility of small ?nal state phases. Upper limits on the corresponding ?nal state
? phase-di?erence in B → Dπ decays are already at a level of 20? . Assuming that ? the ?nal state phase-di?erence between B → DK and B → DK is not larger, the only information about γ would be derived from Ri . A particularly interesting case is Ri < 1, holding for either i = 1 or i = 2. Using Eq. (16), this implies new bounds on γ. The condition Ri < 1 (i = 1 or 2), equivalent to | cos δ cos γ| > r/2, holds for values of δ and γ which are not too close to 90 degrees. Taking cos δ ≈ 1 and r ? 0.1, this condition is ful?lled by all the currently allowed vallues of γ, 30? ≤ γ ≤ 150? , excluding a narrow band around γ = 90? . That is, for all values outside this narrow band one of the two ratios of rates R1 or R2 must be smaller than one, R1 < 1 for γ > 90? and R2 < 1 for γ < 90? . Using r = 0.1, δ = 0 in Eq. (14), we ?nd for γ = 150 (30), 140 (40), 130 (50), 120 (60), 110 (70), 100 (80) degrees the following values of R1 (R2 ): 0.84, 0.86, 0.88, 0.91, 0.94, 0.98, respectively. Measuring these values for R1 or R2 would exclude by Eq. (16) the following ranges of γ: 66? ?114? , 68? ?112? , 70? ?110? , 73? ?107?, 76? ?104? , 81? ?99? , respectively. For another choice of parameters, r = 0.2, δ = 20? , the measurements of Ri corresponding to the above values of γ would be 0.71, 0.75, 0.80, 0.85, 0.91, 0.97, respectively. These values exclude somewhat larger ranges of γ than in the case r = 0.1. Assuming the above number of events, the statistical error of measuring R1 and R2 is 3.4%. Particularly interesting is the ratio of rates R1 . The systematic errors in this ratio, in which the numerator and denominator involve similar three charged pion and kaon ?nal states, are expected to cancel. A few percent accuracy in Ri is su?cient for excluding a sizable range of values of γ for the above two choices of parameters. The excluded range grows with increasing values of r, for which smaller values of Ri are obtained. In conclusion, we have shown that the ratios of rates Ri , for charged B decays to the two DCP K ?nal states and to the already observed D 0 K mode, lead to new constraints on the weak phase γ. The smaller color-suppressed rate, which would lend further information about this phase, can be determined from a sum rule involving these rates. The estimate of Eq. (3) suggests that this may be beyond the capability of future e+ e? B factories, and would have to await dedicated hadronic B production experiments. This method is complementary to the one suggested in Ref. . The two methods seem to be comparable in their statistical power for determining γ from the color-suppressed rate of ? B ? → D 0 K ? , which requires in both cases very high statistics hadronicly produced B experiments. With less statistics, already available in high luminosity e+ e? experiments, the present method can be used to set new bounds on γ through measurements of the more abundant processes B ± → DCP K ± . I wish to thank J. Rosner, A. So?er, S. Stone, M. Wise and F. W¨rthwein for u useful discussions. I am grateful to the Caltech High Energy Theory Group for its kind hospitality. This work was supported by the United States Department of Energy under Grant No. DE-FG03-92-ER40701.
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