# Constraining weak annihilation using semileptonic decays

###### Abstract

The recently measured semileptonic decay rate can be used to constrain weak annihilation (WA) effects in semileptonic and decays. We revisit the theoretical predictions for inclusive semileptonic decays using a variety of quark mass schemes. The most reliable results are obtained if the fits to decay distributions are used to eliminate the charm quark mass dependence, without using any specific charm mass scheme. Our fit to the available data shows that WA is smaller than commonly assumed. There is no indication that the WA octet contribution (which is better constrained than the singlet contribution) dominates. The results constrain an important source of uncertainty in the extraction of from inclusive semileptonic decays.

An intriguing hint of a possible conflict in the -factory data, which may be a sign of physics beyond the standard model, is the roughly difference between the value of obtained from a global fit to the CKM parameters using extracted from the asymmetry in and related modes, and measured directly from inclusive semileptonic decays, calculable using an operator product expansion (OPE) Chay:1990da (); [][[Erratum; ibid.B297(1993)477].]Bigi:1992su; Bigi:1993fe (); Manohar:1993qn (). At order in the OPE, four-quark operators, the so-called weak annihilation (WA) terms, give a significant contribution in a phase space region which affects all inclusive measurements to some extent. Hence, a reliable estimate of the WA contribution is necessary to determine whether there is, in fact, a conflict with standard model predictions.

The WA contribution to the total decay rate Bigi:1993bh (); Voloshin:2001xi (), and to the charged lepton (or neutrino) energy spectrum Leibovich:2002ys () are calculable in terms of the matrix elements of local four-quark operators. However, there is so far no first-principles derivation of the WA contribution to the double or triple differential spectra. For this reason, the extraction of the WA contribution from the differential spectra is model dependent, and any model-independent bound on the magnitude of the WA matrix elements is important. It was pointed out by Voloshin Voloshin:2001xi () that the same matrix elements that enter decay can be constrained by the semileptonic rate difference between and mesons, since the and matrix elements are related by heavy quark symmetry. In this paper we revist the theoretical calculations of semileptonic decays and extract bounds on the WA contribution to and decays.

At order , there are dimension-6 four-quark operators in the OPE for the semileptonic decay rate, the WA operators,

(1) |

where and is the heavy quark effective theory charm quark field. The matrix elements of these operators are defined by

(2) |

where labels the flavor of the light quark in the meson. Compared to the dimension-3, 5, and other dimension-6 operators, the matrix elements of these operators are enhanced by , and contribute to the semileptonic decay widths of the three mesons as Voloshin:2001xi ()

(3) |

relative to the semileptonic width at lowest order in the OPE and at tree-level in ,

(4) |

If one assumes factorization and the vacuum saturation approximation, then the WA contribution vanishes, since or depending on whether or not is the same as . Deviations from the factorization ansatz are usually estimated at the 10% level Voloshin:2001xi ().

The above analysis also holds for decays, with the replacement of meson quantities by the corresponding meson ones. The matrix elements in the and sectors are related by heavy quark symmetry. The same matrix element estimate for decays (i.e., ), along with MeV [][[seealso\url{http://www.latticeaverages.org}].]Laiho:2009eu, implies that the four-quark operators contribute % to the total rate, making it difficult to accurately determine the WA contribution from decays. However, the WA contribution to decay is formally enhanced relative to decay, and is comparable to the leading order decay rate, , due to the enhancement. Thus studying WA effects in decay is a good way to constrain the matrix elements of the four-quark operators. Note that to determine the WA contribution to decays at the 1% level only requires the four-quark matrix elements to % accuracy. Even if this contribution to decays is comparable to the leading order rate, this does not necessarily mean that the expansion breaks down, since the WA contribution is the only enhanced contribution at .

It has long been known that the difference in the semileptonic branching ratios Amsler:2008zzb (); Asner:2009pu ()

(5) |

is mainly due to the lifetime difference, and that the semileptonic widths are equal to within . Recently CLEO-c measured Asner:2009pu () the semileptonic branching ratio

(6) |

The expressions for semileptonic decays are well known in the literature. Schematically,

where is defined in Eq. (3), and

(7) | |||||

where and are the matrix elements of dimension-5 and 6 two-quark operators, are the matrix elements of time-ordered products, and . These may all be determined from fits to various decay spectra Bauer:2004ve (); Bauer:2002sh (); Buchmuller:2005zv (). The complete expression including corrections is complicated. In our analysis, we include the radiative corrections to order vanRitbergen:1999gs () in the terms, and power corrections to order Gremm:1996df (); Blok:1994cd (). For the leading term in the OPE, we include the effect of a nonzero strange quark mass, since it affects the semileptonic width by %, and set elsewhere. For nonzero , the contribution to has a divergence as . In the OPE with , this divergence is effectively absorbed into the matrix elements of WA operators. Including the WA contribution converts the spectrum into a plus distribution Ligeti:2007sn (), which integrates to zero and gives the contribution to the total semileptonic rate in Eq. (7).

The terms in are all independent of the flavor of the spectator quark in the heavy meson, and so give equal contributions to for all three (or ) mesons. The leading term in which depends on the flavor of the spectator quark, and thus produces a difference in the semileptonic partial widths, is . depends on two independent matrix elements in the flavor limit, since the operators in Eq. (Constraining weak annihilation using semileptonic decays) have an singlet and octet part, each of which yield one invariant with the two fields. We can write the decay rates of the three mesons in terms of these two parameters as

(8) | |||||

Here we normalized the weak annihilation contribution to the observed semileptonic decay rate,

(9) |

where and and are dimensionless numbers proportional to the singlet and octet matrix elements,

(10) |

The size of is then (approximately) the fraction of the meson semileptonic width due to WA.

The difference is suppressed by . Neglecting this correction, it is straightforward to extract from the measured difference of the semileptonic widths of the and the , as proposed in Refs. Voloshin:2001xi (); Voloshin:2002je (). This difference combined with any of the individual semileptonic widths also allows to be extracted, but this requires a reliable computation of Becirevic:2008us (). Since the charm mass is not particularly large compared with nonperturbative QCD scales, this computation suffers from both large perturbative and corrections, limiting the precision with which WA can be studied in charm decays.

The leading perturbative corrections to the semileptonic decay widths are given by a perturbation series multiplying the free-quark decay width given in Eq. (4). This perturbation series depends on the choice of scheme for , and could be determined from the decay data using the method of Ref. Hoang:2005zw (). However, it is well-known that the perturbation series for this leading term in the OPE is badly behaved when the rate is expressed in terms of the charm quark pole or masses. Including the known results up to order vanRitbergen:1999gs () and using gives the series

(11) |

and

(12) |

respectively. Here counts the order in the perturbation series, and the BLM subscript refers to the terms in the perturbation series. [In Eqs. (11) – (15) we set for simplicity; this has no effect on our discussion.] These series are poorly behaved, and do not appear to converge.

The bad behavior of these perturbation series is understood theoretically from decays, and arises from a poor choice for the heavy quark mass. A better behaved series is obtained by using a threshold mass scheme, such as the Hoang:1998ng (); Hoang:1998hm (); Hoang:1999ye (), kinetic Czarnecki:1997sz () or PS Beneke:1998rk () mass schemes. As observed already in Hoang:1998ng (), the perturbation series relating to the mass is reasonably well-behaved, and extracting using the method of Ref. Hoang:2005zw ()

(13) |

The series is less well-behaved in the PS or kinetic schemes (defining both with a 1 GeV factorization scale). For the PS scheme we find

(14) |

while for the kinetic scheme, as previously noted Kamenik:2009ze (), the series is considerably worse

(15) |

In addition to the uncertainties in the above series, there will be additional uncertainties in extracting a charm quark threshold mass from other physical quantities, such as moments of decay spectra Bauer:2004ve (); Bauer:2002sh (). Since the charm quark mass is an intermediate quantity which is not required for our analysis, we can minimize this source of theoretical uncertainty by bypassing any choice of charm mass scheme, and instead directly relate the semileptonic decay widths to the values of and extracted from a global fit to decay spectra. From Eq. (11) and the relation between the quark pole and masses Hoang:1998hm ()

(16) |

we find the reasonably well-behaved perturbation series

(17) |

using GeV, GeV, , and, as in the previous expressions, we have continued to set to zero. We will therefore use this method to determine the semileptonic widths theoretically.

The masses and and HQET parameters , and , as well as their correlated uncertainties, are obtained using a fit to the decay spectra Bauer:2004ve (); Bauer:2002sh (). The values for decay are related to those for decay by renormalization group evolution between and . is not renormalized due to reparametrization invariance Luke:1992cs (), while , with . Radiative corrections to the terms are computed in Refs. Manohar:1997qy (); Bauer:1997gs (); Manohar:2010sf (). Since they are small and were not included in the decay fits, we neglect them here.

The errors from the fits include the experimental uncertainties, as well as additional theoretical uncertainties due to neglected higher order corrections, as given in Ref. Bauer:2004ve (); Bauer:2002sh (). We treat the and decay calculations as independent. Thus the decay fit results will be held fixed (at order ) while we vary the order of the decay results between tree-level and . While this may be formally inconsistent, numerically, the and corrections are significantly larger for than for decay.

Using the value of obtained as discussed above, and fitting to the experimentally measured rates in Eqs. (Constraining weak annihilation using semileptonic decays) and (6) gives the WA annihilation parameters and . Figure 1 shows the 90% CL contours at tree level, order , and order . The best fit parameters at order are

(18) |

where the error is from the order fit. The series of corrections to are flavor independent, and lead to a shift in depending on the order in , but do not affect , which can be determined from . cancels in this difference, so is not affected by the convergence of the expansion. The shift in between and is 0.06, which is smaller than other uncertainties. The expansion in is also not as rapidly convergent as in the meson system, so there are significant uncertainties which mainly affect . We find that the and terms in Eq. (Constraining weak annihilation using semileptonic decays) contribute roughly and to the semileptonic widths. These corrections are much larger than the corresponding ones for the hadron masses, because of the larger coefficients of , , and . One could estimate the uncertainty corresponding to these large corrections by including an additional error of in , which is half the term. In the limit, the meson sector of QCD has a symmetry Veneziano:1979ec () and this implies that (see, e.g., Ref. Manohar:1998xv ()), which is shown as the black line in Fig. 1.

Neglecting Cabibbo-suppressed terms, the correspondence between our notation and that of Ref. Voloshin:2001xi () is

(19) |

and the same equation with and the non-singlet replaced by the singlet ones. Taking Laiho:2009eu () gives , which is somewhat smaller than (although consistent with) the simple estimate in Voloshin:2001xi (); Voloshin:2002je ().

The linear combinations and that contribute to the decay rates in Eq. (8) are

(20) |

where only the fit uncertainty is quoted, as discussed above. The 90% confidence level contours in these variables are shown in Fig. 1. While there are significant uncertainties in the fit result for the WA contribution in Eqs. (18) and (20), it still has important implications for and decays and the determination of .

It has often been assumed that the WA term where the light quark in the operator matches that in the heavy meson is much larger than when the light quarks differ, i.e., . Indeed, the central values of our results suggest that the WA contribution to decay is larger than that to decay. The WA matrix element in which the light quark field of the operator is contracted with the spectator quark in the heavy meson is helicity suppressed by , where is the lepton mass, and gives a contribution of relative order to the decay width. Other diagrams, in which the spectator quark is not annihilated by the four-quark operator, are of relative order . In a quark model, they would contain additional suppression factors from gluon exchange to connect the spectator light quark with the rest of the diagram, but nothing as small as .

The meson lifetimes also depend on the WA matrix elements through both the semileptonic and non-leptonic decay rates. The non-leptonic rates depend on two additional color octet operators, and the behavior of the perturbation series is even worse than for the semileptonic case. Neglecting the color octet matrix elements and violation (as before), one would predict Voloshin:2001xi ()

(21) |

where , and we have used and for the numerical values. The branching ratios Eqs. (Constraining weak annihilation using semileptonic decays) and (6) and the lifetimes yield . This shows that there must be some other large contribution to the nonleptonic decay rates, e.g., large color octet matrix elements, corrections, or higher order terms, so the total widths do not provide a useful bound on .

It is often stated that the difference between the and semileptonic rates can be used to constrain the impact of WA on the extraction of from decays. However, , while individually and , which determine , depend on both and . We find no evidence that , so the width difference will not strongly constrain the WA contribution to . While the uncertainties in our analysis are substantial, it gives strong indication that the WA contribution to the rate is less than the estimate Voloshin:2001xi () often used. Our conclusions are unchanged if breaking or higher order corrections are included, since these will only shift the estimate of the WA contribution by % of its value. The discrepancy in mentioned in the introduction cannot be explained away using WA.

Our results imply that the WA contribution to decays, which is a factor smaller than the corresponding contribution to decays, is around 1%. If we use heavy quark symmetry for the bag parameters instead of the matrix elements, scaling with gives , still smaller than past estimates. A recent CDF measurement cdfnote () of the meson and baryon lifetimes also indicates that spectator effects in the hadron decays may be smaller than previously thought.

###### Acknowledgements.

We thank Frank Tackmann for helpful conversations, and C. S. Park and S. Stone for correspondence about Ref. Asner:2009pu (). The work of ZL was supported in part by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under contract DE-AC02-05CH11231. ML was supported in part by the Natural Sciences and Engineering Research Council of Canada.## References

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