Abstract
By using the recent experimental measurements of and branching ratios, we find that factorization is unable to reproduce the observed s even taking into account the uncertainties of the input parameters. Charming and GIM penguins allow to reconcile the theoretical predictions with the data. Because of these large effects, we conclude, however, that it is not possible, with the present theoretical and experimental accuracy, to determine the CP violation angle from these decays. Contrary to factorization, we predict large asymmetries for several of the particle–antiparticle s, in particular , and . This opens new perspectives for the study of CP violation in systems.
RM3TH/013
ROMA1310/01
CHARMING PENGUINS STRIKE BACK
M. Ciuchini, E. Franco, G. Martinelli, M. Pierini, L. Silvestrini
Dip. di Fisica, Univ. di Roma Tre and INFN, Sezione di Roma Tre,
Via della Vasca Navale 84, I00146 Roma, Italy
Dip. di Fisica, Univ. “La Sapienza” and INFN,
Sezione di Roma, P.le A. Moro, I00185 Rome, Italy.
1 Introduction
The theoretical understanding of nonleptonic two body decays is a fundamental step for testing flavour physics and CP violation in the Standard Model and for detecting signals of new physics [1]–[5]. The increasing accuracy of the experimental measurements at the factories [6, 7] calls for a significant improvement of the theoretical predictions. In this respect, important progress has been recently achieved by systematic studies of factorization made by two independent groups [8, 9]. These studies, while confirming the physical idea [10] that factorization holds for hadrons containing heavy quarks, , give the explicit formulae necessary to compute quantitatively the relevant amplitudes at the leading order of the expansion. They also examine some of the contributions entering at higher order in . The question which naturally arises is whether in practice the powersuppressed corrections, for which quantitative estimates are missing to date, may be phenomenologically important for decays. This problem was previously addressed in refs. [11, 12, 13]. In particular, the main conclusion of refs. [11] was that nonperturbative penguin contractions of the leading operators of the effective weak Hamiltonian, and , although formally of , may be important in cases where the factorized amplitudes are either colour or Cabibbo suppressed. The most dramatic effect of these nonfactorizable penguin contractions manifested itself in the very large enhancement of the branching ratios, as was also emerging from the first measurements by the CLEO Collaboration [14]. In this case, the effect was triggered by Cabibboenhanced penguin contractions of the operators and , usually referred to as charming penguins. Since the original publications, about three years ago, several other decay channels have been measured [15, 16, 17] and the precision of the measurements is constantly improving. With respect to previous analyses, it is now possible to attempt a more quantitative study of charming penguin effects and of the corrections expected to the factorized predictions. We now present the main conclusions of our new analysis.
Factorization with and from other determinations
Using the available experimental information on and on the CP angle provided by the unitarity triangle analysis (UTA) [18], the branching ratios predicted with the factorized amplitudes, including the corrections computed according to ref. [9], fail to reproduce the experimental branching ratios that are systematically larger than the theoretical predictions. In addition, , which depends on the semileptonic form factor , is about a factor of 2 larger than its experimental value ^{1}^{1}1 Unless explicitly stated the s always refer to the average of particles and antiparticles, e.g. .. We note that the value of within factorization is essentially fixed by the measured rate. Thus, contrary to the statement of ref. [9], the predicted value of is independent of the theoretical assumptions on the value of . This holds essentially true also for the s since the value of the “semileptonic” form factor at zero momentum transfer is correlated to by the approximate symmetry.
Factorization fitting
Even if one ignores the value of from UTA, which is only justified if there are contributions to mixing due to physics beyond the Standard Model, there are serious difficulties in reproducing the experimental results. In particular, and are much smaller than their experimental values. Moreover, the value of extracted from a fit to the data, , is in total disagreement with that from the UTA, [18]. In addition, in order to enhance the rates, the preferred values of and are incompatible with the latest theoretical estimates, , [19] and , [20], whereas must have a rather low value, instead of that extracted from inclusive [21] and exclusive [22] semileptonic decays, . We conclude that, even relaxing the constraint on , it is very difficult to reconcile the predictions from factorization with the experimental and theoretical findings. For this reason any attempt to extract, within factorization, the value of from ratios of s, for which the discrepancies with the experiments can be accidentally hidden, is not very useful. We think that a preliminary step is to understand the missing dynamical effects.
Factorization and charming penguins
The inclusion of charming penguin effects, which will be explained in detail in sec. 2, considerably improves the situation for the channels, with values of and well compatible with other determinations. In contrast to the case, charming penguins are not Cabibbo enhanced in decays and are thus expected a priori to play a minor role. For this reason they should be consistently neglected, together with all other corrections. This would leave the problem of a too large predicted unsolved. A natural question is then whether the inclusion of effects in can improve the agreement of the predictions with the experimental data. In particular, besides the charming penguins, penguin contractions of and (GIM penguins in the notation of ref. [11]), which are Cabibbo suppressed in , might play an important rôle. We show that, for numerical values of the charming and GIM penguin amplitudes of the expected size, –, we can easily reproduce the experimental data for both and decays while respecting the constraints from the UTA. The sensitivity of to effects casts serious doubts on the possibility of extracting from the coefficient of the term obtained from CP asymmetry measurements. On the other hand, we find that the value of the rate asymmetry,
(1) 
could be unexpectedly large and call our experimental colleagues for separate measurements of the and s. In particular, we find , . We also find –. In the latter case, as discussed in the following, the results are subject to other effects on which we do not have control. For this reason we do not quote an error. We simply signal that there is room for a large asymmetry also in decays.
2 Results
In this section we describe and discuss more in detail the different cases which have been considered in our analysis.
The physical amplitudes for and are more conveniently written in terms of RG invariant parameters built using the Wick contractions of the effective Hamiltonian [23]. In the heavy quark limit, following the approach of ref. [9], it is possible to compute these RG invariant parameters using factorization. The formalism has been developed so that it is possible to include also the perturbative corrections to order , i.e. at the nexttoleading order in perturbation theory. We present results obtained with this formalism with the addition of the nonperturbative corrections to factorization described below in this section. An alternative framework is provided by the approach of ref. [8]. This method differs in the treatment of the terms; unlike the method of ref. [9], the calculations are only valid at the leading logarithmic order and it is not clear how the independence of the final result from the renormalization scale of the operators of the effective Hamiltonian is recovered. Moreover the Sudakov suppression of the endpoint region, advocated in [8], is still rather controversial from both the theoretical and phenomenological point of view. For these reasons we prefer to postpone the analysis with the approach of ref. [8] until the theoretical situation will become clearer.
In the leading amplitudes, we have taken into account the SU(3) breaking terms by using the appropriate decay constants, and , and form factors, and . Strictly speaking, the form factors should be evaluated at the invariant mass of the emitted meson (, or ). The difference is however of higher order in and not Cabibbo or colour enhanced and can safely be neglected (it is also numerically immaterial) [24]. As for corrections, we have assumed instead symmetry and neglected Zweigsuppressed contributions. In this approximation, by symmetry one can show that all the Cabibboenhanced corrections to decays can be reabsorbed in a single parameter . Several corrections are contained in : this parameter includes not only the charming penguin contributions, but also annihilation and penguin contractions of penguin operators. It does not include leading emission amplitudes of penguin operators (–) which have been explicitly evaluated using factorization. Had we included these terms, this contribution would exactly correspond to the parameter of ref. [23]. The parameter () encodes automatically not only the effect of the annihilation diagrams considered in [25], but all the other contributions of with the same quantum numbers of the charming penguins. In this respect it is the most general parameterization of all the perturbative and nonperturbative contributions of the operators and ( and ), including the worrying highertwist infrared divergent contribution to annihilation discussed in ref. [26]. The parameter is of and has the same quantum numbers and physical effects as the original charming penguins proposed in [11], although it has a more general meaning. In some of the previous analyses, see for example [27], penguin contractions of the operator , computed by using perturbation theory and factorization, are enhanced by taking a low effective scale for . This procedure produces a physical effect similar to that coming from the nonperturbative charming penguins that we are using here, since they have the same quantum numbers.
If one also includes decays we have several other parameters, for example and , in the formalism of ref. [23]. A closer look to shows that this term is due either to Zweig suppressed annihilation diagrams (called CPA and DPA in ref. [11]) or to annihilation diagrams which are colour suppressed with respect to those entering . For this reason we have put to zero. will be discussed later on.
We give now the explicit expression of the amplitude as an illustrative example. In terms of the parameters defined in [23], this amplitude reads
(2)  
Using the approach of [9], we have
(3) 
where denotes the factorized matrix element, and the parameters are defined in [9]. The tilded parameters represent corrections; in channels the only Cabibboenhanced correction is given by . This term has no arguments since we take it in the symmetry limit.
We use input parameters (like , , the form factors) with errors, and extract output quantities (like the s, the asymmetries, but also , or the form factors when they are not used as inputs) with their uncertainties. Let us explain how we used the input errors and extracted the output uncertainties. We proceed with the usual likelihood method, by generating the input quantities weighted by their probability density function (p.d.f.). In the case of theoretical quantities this is assumed to be flat, whereas the experimental quantities are extracted with Gaussian distributions. Probability density functions, averages and standard deviations are then obtained by weighting the output quantities by the likelihood factor
(4) 
where are the standard deviations of the experimental s, , given in table 1. In cases where the experimental input has a systematic error dominated by theoretical uncertainties, we should extract the latter with a flat distribution [18]. We have instead combined the errors in quadrature and extracted all the experimental quantities with gaussian distributions. Within the present accuracy, and taking into account the unknown nonperturbative parameters, this procedure is fully justified. We have also verified that by extracting the theoretical errors with a gaussian distribution, we obtain very similar results. For more details on the likelihood procedure, the reader is referred to [18], where all aspects are discussed at length.
Results with factorization
We start by considering the case in which we use factorization and take the CKM parameters and from other experimental determinations. We discuss first since in this case, due to isospin symmetry, we do not have the complications due to penguin contractions. Thus, at fixed , the prediction for only depends on (trivial dependences as from will be omitted in this discussion). By using the theoretical estimate and uncertainty of from [19], and taking into account the uncertainties on , we predict in this case in very good agreement with the experimental average given in table 1. A complementary exercise is to use as input and the experimental value of in order to extract the value of . In this case we find , in very good agreement with lattice and QCD sum rules estimates. This exercise shows that we do not need to rely on theoretical calculations for the form factors. Indeed also for we only need which cannot differ too much from one. Moreover it is likely that a large part of the uncertainties of the theoretical predictions cancel in this ratio.
Here and in all the other cases where and are taken from other experimental determinations, we use as equivalent input parameters the values of and given in table 1 from the UTA analysis of ref. [18]. These values correspond to
(5) 
By using either from theory or from the fit to and assuming factorization, we then predict as a function of only. Besides, in order to analyze all decays, we only need to which the previous considerations apply. Alternatively we may take only from the experiments and fit the value of . In the first case, the results are given in table 2 labeled as “ UTA” and show a generalized disagreement between predictions and experimental data. In the second case, the value of is fitted and the results are labeled as “ free”. In this case the disagreement is reduced for and , and also for , but it remains sizable for and . The pattern :::=2:2:1:1, which is suggested by the data, and is well reproduced when the contribution of the charming penguins is large, as discussed in the following, is lost in this case. Moreover the fitted value of is in striking disagreement with the results of the UTA. Although one may question on the quoted uncertainty of the UTA result, it is clearly impossible to reconcile the two numbers. Thus either there is new physics or corrections are important. We now discuss the latter possibility.
UTA  free  UTA  free  

Factorization with Charming and GIM penguins
We now discuss the effects of charming penguins, parameterized by . is a complex amplitude that we fit on the s. In order to have a reference scale for its size, we introduce a suitable “Bag” parameter, , by writing
(6) 
where is the Fermi constant. We use for both and channels since, as mentioned before, for charming penguins we work in the limit. is a ClebshGordan parameter depending on the final () channel. In the case where and are taken from the UTA, by fitting the channels and only, we find
(7) 
Note that the size of the charming penguin effects is of the expected magnitude. As for the phase , it is very instructive to consider its distribution, which is displayed in fig. 1: the preferred value of has a sign ambiguity since we are fitting the average of the and s (or of the and s). The ambiguity can be resolved by measuring separately particle and antiparticle s.
By using the distribution on the left of fig. 1, we compute the mean value of with the result and leave the sign undetermined. This is a reasonable procedure, given the approximate symmetry of the distribution and the large uncertainty. In view of the discussion of the particleantiparticle asymmetry which we present at the end of this paper, we note here that the value of could be rather large. In table 3 we give the corresponding predicted values and uncertainties for the relevant branching ratios (label “Charming”). We observe a remarkable improvement for the channels and a large shift in the value of ^{2}^{2}2 This effect was already noticed in [11]., in spite of the fact that in the latter case penguin effects are not Cabibbo enhanced (the amplitude is however colour suppressed). The predicted value for remains however much larger than the experimental one.
If one fits the channels, and simultaneously, one finds a better agreement for but a rather small value for (column “Charming with ” of table 3). This happens at the price of reducing the fitted value of the form factor, , which is pushed down by . In fact the latter has an experimental error much smaller than , and therefore governs the fit. However, we do not think that this is the correct procedure: theoretically, is on much more solid grounds than , since it is not affected by penguins or annihilations, and thus is much more suitable to constrain .
Charming  Charming  Charming  Charming  Charming  Charming  

with  + GIM  with  + GIM  
In order to reduce the predicted without affecting , one may include other effects of the same order of the charming penguins, as for example the GIM penguins introduced in ref. [11]. In this case we fit all the s given in table 1. With GIM and charming penguins included, we find
(8) 
where the notation is selfexplaining. We have given the absolute value of since, as in the previous case, the sign ambiguity persists when we include GIM penguins. The distribution is also shown in fig. 1. The results for the s can be found in table 3 with the label “Charming+GIM”. They show that the extra GIM parameter improves the agreement for the measured s. We do not claim, however, to be able to predict : our results instead show that accurate predictions for decays can only be obtained by controlling quantitatively the corrections, which is presently beyond the theoretical reach. Estimates for charming penguin effects can also be obtained by using some phenomenological model, as for example done in ref. [12]. We observe that the sensitivity of the s to the value of is lost, with the present experimental accuracy, once penguin effects are introduced. Indeed when one tries to fit () and simultaneously, one finds that the value of is essentially undetermined. From the above discussion it clearly emerges that one of the important step for the improvement of this kind of analyses is a more precise measurement of .
Particle–Antiparticle asymmetries for the Branching Ratios
The large absolute values of , and the sizable effects that penguins have on the s, stimulated us to consider whether we could find observable particleantiparticle asymmetries as the one defined in eq. (1). We find large effects in , and , as shown in fig. 2. As discussed before, for our predictions suffer from very large uncertainties due to contributions which cannot be fixed theoretically. For this reason, the values of the asymmetry reported in table 4 are only an indication that a large asymmetry could be observed also in this channel. The sign ambiguity of is reflected in the asymmetry . This ambiguity can be solved only by an experimental measurement or, but this is extremely remote, by a theoretical calculation of the relevant amplitudes. For each channel, we give the absolute value of the asymmetry in table 4. Note that within factorization all asymmetries would be unobservably small, since the strong phase is a perturbative effect of [9]. The possibility of observing large asymmetries in these decays opens new perspectives. These points will be the subject of a future study.
Charming  Charming + GIM  Charming  Charming + GIM  

Conclusion
We have analyzed the predictions of factorization for and decays. We note that the normalization of all the other s is essentially fixed by the value of and symmetry. Even taking into account the uncertainties of the input parameters, we find that factorization is unable to reproduce the observed s. The introduction of charming and GIM penguins [11] allows to reconcile the theoretical predictions with the data. It also shows however that it is not possible, with the present theoretical and experimental accuracy, to determine the CP violation angle . Contrary to factorization, we predict large asymmetries for several of the particle–antiparticle s, in particular , and . This opens new perspectives for the study of CP violation in systems.
Acknowledgments
We thank G. Buchalla and C. Sachrajda for useful discussions on our work. M.C. thanks the TH division at CERN where part of this work has been done. L.S. thanks Hsiangnan Li for very informative discussions and J. Matias for pointing out a misprint.
References
 [1] M. Ciuchini, E. Franco, G. Martinelli, A. Masiero and L. Silvestrini, Phys. Rev. Lett. 79 (1997) 978, hepph/9704274.
 [2] R. Barbieri and A. Strumia, Nucl. Phys. B 508 (1997) 3, hepph/9704402.
 [3] R. Fleischer and T. Mannel, hepph/9706261.
 [4] A. F. Falk, A. L. Kagan, Y. Nir and A. A. Petrov, Phys. Rev. D 57 (1998) 4290, hepph/9712225.
 [5] S. Baek and P. Ko, Phys. Rev. Lett. 83 (1999) 488, hepph/9812229.
 [6] The Babar Physics Book, edited by P.F. Harrison and H.R. Quinn, SLAC report SLACR504, October 1998.
 [7] F. Takasaki, in Proc. of the 19th Intl. Symp. on Photon and Lepton Interactions at High Energy LP99 ed. J.A. Jaros and M.E. Peskin, Int. J. Mod. Phys. A 15S1 (2000) 12, hepex/9912004.
 [8] Hn Li and H.L. Yu, Phys. Rev. Lett. 74 (1995) 4388 and Phys. Rev. D 53 (1996) 2480; C.H. Chang and Hn Li, Phys. Rev. D 55 (1997) 5577; T.W. Yeh and Hn Li, Phys. Rev. D 56 (1997) 1615; H.Y. Cheng, Hn Li and K.C. Yang, Phys. Rev. D 60 (1999) 094005; for a recent review see Hn Li, hepph/0103305.
 [9] M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914; Nucl. Phys. B 591 (2000) 313; hepph/0007256.

[10]
J.D. Bjorken, Nucl. Phys. B (Proc. Suppl.) 11 (1989)
325;
M. J. Dugan and B. Grinstein, Phys. Lett. B 255 (1991) 583;
H. D. Politzer and M. B. Wise, Phys. Lett. B 257 (1991) 399.  [11] M. Ciuchini, E. Franco, G. Martinelli and L. Silvestrini, Nucl. Phys. B 501 (1997) 271, hepph/9703353. M. Ciuchini, R. Contino, E. Franco, G. Martinelli and L. Silvestrini, Nucl. Phys. B 512 (1998) 3, hepph/9708222.
 [12] P. Colangelo, G. Nardulli, N. Paver and Riazuddin, Z. Phys. C 45 (1990) 575. C. Isola, M. Ladisa, G. Nardulli, T. N. Pham and P. Santorelli, hepph/0101118.
 [13] A. J. Buras and R. Fleischer, Phys. Lett. B 341 (1995) 379, hepph/9409244.
 [14] R. Godang et al. [CLEO Collaboration], Phys. Rev. Lett. 80 (1998) 3456, hepex/9711010.
 [15] D. CroninHennessy et al. [CLEO Collaboration], hepex/0001010. D. M. Asner et al. [CLEO Collaboration], hepex/0103040.
 [16] G. Cavoto, BaBar Collaboration, talk given at the XXXVI Rencontres de Moriond QCD.
 [17] B. Casey, Belle Collaboration, talk given at the XXXVI Rencontres de Moriond QCD.
 [18] M. Ciuchini et al., hepph/0012308.
 [19] A. Abada, D. Becirevic, P. Boucaud, J. P. Leroy, V. Lubicz and F. Mescia, heplat/0011065. See also L. Del Debbio, J. M. Flynn, L. Lellouch and J. Nieves [UKQCD Collaboration], Phys. Lett. B 416 (1998) 392, heplat/9708008.
 [20] A. Khodjamirian, R. Ruckl, S. Weinzierl, C. W. Winhart and O. Yakovlev, Phys. Rev. D 62 (2000) 114002, hepph/0001297. For an earlier determination of the form factors, see also P. Ball, JHEP9809 (1998) 005, hepph/9802394.
 [21] LEP Working group on , http://battagl.home.cern.ch/battagl/vub/vub.html.
 [22] B.H. Behrens et al., CLEO Collaboration, Phys. Rev. D61 (2000) 052001.
 [23] A. J. Buras and L. Silvestrini, Nucl. Phys. B 569 (2000) 3, hepph/9812392.
 [24] W. N. Cottingham, H. Mehrban and I. B. Whittingham, hepph/0102012.
 [25] Y. Keum, H. Li and A. I. Sanda, hepph/0004004; Y. Y. Keum, H. Li and A. I. Sanda, Phys. Rev. D 63 (2001) 054008, hepph/0004173.
 [26] M. Beneke, hepph/0009328.

[27]
A. Ali and C. Greub,
Phys. Rev. D 57 (1998) 2996,
hepph/9707251;
A. Ali, G. Kramer and C. Lu, Phys. Rev. D 58 (1998) 094009, hepph/9804363; Phys. Rev. D 59 (1999) 014005, hepph/9805403.