aa r X i v : . [ h e p - ph ] S e p S-waves and the extraction of b s Sheldon Stone ∗ Department of Physics, Syracuse UniversitySyracuse, N. Y., U. S. A., 13244E-mail: [email protected]
The CP Violating asymmetry in B s mixing ( b s ) is one of the most promising measurements wherephysics beyond the Standard Model could be revealed. As such, analyses need to be subjected togreat scrutiny. The mode B s → J / yf has been used, and the mode B s → ff proposed for futuremeasurements. These modes both have two vector particles in the final state and thus angularanalyses must be used to disentangle the contributions from CP + and CP − configurations. Theangular distributions, however, could be distorted by the presence of S-waves masquerading aslow mass K + K − pairs, that could result in erroneous values of b s . The S-waves could well be theresult of a final state formed from an s -quark s -quark pair in a 0 + spin-parity state, such as the f ( ) meson. Data driven and theoretical estimates of the B s decay rate into the CP + final state J / y f ( ) are given, when f → p + p − . The S-wave contribution in J / yf should be taken intoaccount when determining b s by including a K + K − S-wave amplitude in the fit. This may changethe central value of current results and will also increase the statistical uncertainty. Importantly,the J / y f ( ) mode has been suggested as an alternative channel for measuring b s . Flavor Physics and CP Violation - FPCP 2010May 25-29, 2010Turin, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ -waves
Sheldon Stone
1. Introduction
Measurements of Charge-Parity ( CP ) violation in the B s meson system are sensitive to thepresence of heavy, as yet undiscovered, particles. The CP violating angle − b s , the so called“phase of B s − B s mixing" is a particularly important place to look for physics beyond the StandardModel, since the expected asymmetry is very small, 2 b s = . ± .
002 [1], thus allowing theeffects of any new physics to be more easily observed. Both CDF [2] and D0 [3] have investigated − b s using B s → J / yf decays. Central values have been found far from the expected StandardModel values, but the errors are large and the results are not statistically significant.Since the final state consists of two spin-1 particles, it is not a CP -eigenstate, yet it is wellknown that CP violation can be measured using angular analyses [4]. Except for one very recentanalysis [5], previous determinations have ignored the possibility of an S-wave K + K − system inthe region of the f . Not accounting for the S-wave can bias the result, and the resulting quotederror is smaller than if the S-wave components are allowed in the fit.
2. Evidence for S-waves
In fact, there is a great deal of evidence for S-waves in many decays where vectors are dom-inant. Consider the reaction D + s → K + K − p + . This mode has long been known to have large fp + and K ∗ K components [6]. CLEO has looked explicitly at the low mass K + K − region in the K + K − p + Dalitz plot [7]. Fig. 1 shows the K + K − mass projection in the region near 1 GeV. Thesignal is extracted by fitting the D s yield in each bin of K + K − , so no background remains in theplot. There is an additional component of signal beneath the f . To estimate the size of this com-ponent, the CLEO data are fit to a Breit-Wigner to describe the f , convoluted with a Gaussianfor detector resolution, and in addition a linear component that we take as an S-wave based onDalitz plot studies. The fraction of S-wave depends on the mass interval considered. For ± f mass there is a 6.3% S-wave contribution, which rises to 8.9% for a ±
15 MeVinterval. (Note that these fractions depend on the experimental resolution.)If the S-wave here is the 0 + f (980) state, then we should see an f signal peak in the D + s → p + p + p − final state, since the f (980) decays into p + p − as well as K + K − . A BaBar Dalitz plotanalysis shows a large f signal [8].S-wave effects have also been observed in semileptonic charm decays. FOCUS observed aninterfering S-wave amplitude in the D + → K + p − m + n channel with a rate fraction of (2.7 ± K ∗ in the K − p + invariant mass region between 0.8–1.0 GeV [9]. BaBarmeasured an S-wave fraction in D + s → K + K − e + n decays of ( . + . − . ) % for K + K − invariantmasses between 1.01–1.03 GeV [10]. The K + K − invariant mass spectrum when weighted by the K + decay angle shows a clear distortion due to the interference (Fig. 2).The analogous channel to J / yf in B decay J / y K ∗ is well known to have an S-wave K p component in the K ∗ mass region. This interference, in fact, has been used by BaBar to measurecos ( b ) and thus remove an ambiguity in the value of b from the sin ( b ) measurement. TheS-wave component in the region of the K ∗ is measured as ( . ± . ) % of the P-wave for 0 . < m ( K p ) < . ( b ) , where b is the CP violating angle measured in B → J / y K S decays, for example. Visual2 -waves Sheldon Stone
Figure 1:
Dalitz plot projections for K + K − invariant mass in D + s → K + K − p + from the CLEO Collaboration[7]. The signal is extracted individually in each mass bin, thus there is no background. The data are fitwith a Breit-Wigner signal function for the f convoluted with a Gaussian for detector resolution and linearrepresentation of an S-wave component (dashed line). The solid curve shows the sum. (Only the data isascribed to CLEO, the fits have been added.) Figure 2: K + K − invariant mass distribution weighted by the measured value of the cosine of the decayangle from BaBar [10] in the channel D + s → K + K − e + n . -waves Sheldon Stone θ K K + π – Figure 3:
The cosine of the decay angle of the kaon from the K ∗ decay in B → J / y K ∗ decay. Thehistogram is Monte Carlo without S-P wave interference, and the points the BaBar data [11]. evidence of the S-wave can be seen in Fig. 3 where there is an obvious asymmetry of the decayangle distribution of the kaon in the K ∗ rest-frame.Perhaps it may be hoped that the S-wave K + K − under the f in J / yf is smaller due to therelatively narrow width ( G ) of the f (4.3 MeV) compared to the K ∗ (51 MeV), but even so, thequestion is how much does the presence of the S-wave amplitude affect the extraction of b s ?Similar considerations apply to the measurement of CP violation in the process B s → ff . Here theproblem is exacerbated by the presence of two f ’s in the final state. The decay diagrams for bothof these processes are shown in Fig. 4. In both cases the ss forms a f . Other manifestations of ss quarks are the h , h ′ and f (980) mesons. The first two are pseudoscalars, while the last is a scalar. b W - c } (cid:13) y s } c J s s f } (cid:13) ss f W-t,c,u bs ss } f (a) (b) g Figure 4:
Decay diagrams (a) B s → J / yf , and (b) B s → ff . The formalism for the time dependent B s and B s decay rates as a function of the decay angulardistributions is given in Ref. [12]. Addition of the S-wave amplitudes was done by Xie et al. [13].The number of terms to consider increases from 6 to 10. Another approach has been given by Azfar et al. [14]. Adding the S-wave terms in the fit can only increase the experimental error. The sizeof the effect depends on many factors including the magnitude and phase of the S-wave amplitude, b s , values of the strong phases, detector acceptances, biases, etc..While S-waves are a nuisance in analyzing the J / yf final state, they can also be used ben-eficially. When the f materializes as a p + p − there cannot be an iso-vector r state as ss pair isisoscalar. Near the f mass the f ( ) can materialize as a p + p − pair in the 0 + state and this J / y f state is useful for b s measurements [15]. The final state is a CP + eigenstate, thus no angu-4 -waves Sheldon Stone lar analysis is necessary! Note, that the modes J / yh and J / yh ′ can also be used, but they involvephotons in the decay and thus have lower efficiency at hadron colliders.Predictions of the ratio R f / f ≡ G ( B s → J / y f , f → p + p − ) G ( B s → J / yf , f → K + K − ) (2.1)have been given based on D s decay data, and purely from theory. Stone and Zhang using D + s → f p + decays where f was detected in both K + K − and p + p − modes predicted R f / f ≈
20% [15].CLEO made an estimate of R f / f = ( ± ) % based on the ratio of the branching fractions for D + s → f e + n to D + s → f e + n at q = D s and the final statehadron is largest, which best approximates B s → J / y decays [16].Theoretical predictions for R f / f are difficult, however there are a few heroic attempts. Colan-gelo, De Fazio and Wang use Light Cone Sum Rules to make two predictions [17]. For the first theyuse their calculations of B ( B s → J / y f ) which are ( . ± . ) × − at leading order (LO) and ( . ± . ) × − at non-leading order (NLO), combined with the measured B ( B s → J / yf ) =( . ± . ) × − to predict R f / f =24% (LO) and R f / f =41% (NLO). Secondly, they use the form-factor calculation of Ball and Zwicky [18] to predict R Lf / f =13% (LO) and R Lf / f =22% (NLO),where R L refers to only longitudinal f production, so since transverse f production is about 46%this lowers their R f / f predictions for the second case by almost a factor of two [6]. The experi-mental predictions above for R f / f based on D s decays are also enhanced by normalizing to f finalstates that are mostly longitudinal. Thus they should be lowered.In a later paper using QCD factorization Colangelo, De Fazio and Wang [19] predict B ( B s → J / y f ) = ( . ± . ) × − using CDSS form-factors [20], and ( . ± . ) × − using Ball andZwicky [18] . These predictions are somewhat smaller than those given above, but still have R f / f as 36% or 20%. Within the framework of QCD factorization O. Leitner et al. [22] give a range ofpredictions for R f / f that are in the 30-50% range. Thus predictions for R f / f have a rather widerange from 7-50%.The only reported experimental search for J / y f ; f → p + p − was done by BELLE using23.6 fb − of data taken on the ¡ ( S ) resonance, about 1/5 of their total accumulated data sample.They find R f / f < .
5% at 90% c.l. [21]. Their data however show a hint of signal with a centralvalue about half of the upper limit. It will be quite interesting to see which experiment finds thesignal first. CDF has recently put the S-wave amplitudes in their fits for b s in the J / yf channel.They find that the fitted fraction of K + K − S-wave in the signal region is < .
7% at 95% c.l. [5].They do not however, report any result for a direct search using the f → p + p − channel.In conclusion, S-waves are ubiquitous, they appear whenever looked for and must be takeninto account in B s → J / yf measurements of amplitudes, phases, and CP violation. Kudos to CDFfor including S-waves in their most recent fits. In addition it appears to be wise to add S-waveamplitudes in the analysis of B → K ∗ m + m − and surely in B s → ff . Furthermore, especially sinceangular analysis is not required, B s → J / y f may be a useful mode to add to the statistical precisionon the measurement of − b s and will provide a useful systematic check [23]. Acknowledgments
I thank the U. S. National Science Foundation for support and L. Zhang for his collaboration5 -waves
Sheldon Stone on this work.
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