General Analysis of B -> K^(*) l^+ l^- Decays at Low Recoil
We analyze the angular distributions of and decays in the region of low hadronic recoil in a model-independent way by taking into account the complete set of dimension-six operators . We obtain several novel low-recoil observables with high sensitivity to non-standard-model Dirac structures, including CP-asymmetries which do not require flavor tagging. The transversity observables are found to be insensitive to hadronic matrix elements and their uncertainties even when considering the complete set of operators. In the most general scenario we show that the low recoil operator product expansion can be probed at the few-percent level using the angular observable . Higher sensitivities are possible assuming no tensor contributions, specifically by testing the low-recoil relation . We explicitly demonstrate the gain in reach of the low-recoil observables in accessing the ratio compared to the forward-backward asymmetry, and probing CP-violating right-handed currents . We give updated Standard Model predictions for key observables in decays.
Flavor Changing Neutral Current (FCNC) decays of beauty hadrons have a high sensitivity to New Physics (NP) since the corresponding Standard Model (SM) contributions are loop and flavor suppressed. In addition, the large value of the -quark mass facilitates the control of power corrections.
The large number of complementary observables and the excellent accessibility at contemporary high energy experiments, in particular for muons, highlights the exclusive FCNC decays . In the kinematic region of low hadronic recoil, where the emitted is soft in the -rest frame, a local Operator Product Expansion (OPE) can be performed Grinstein:2004vb (); Beylich:2011aq (). Together with the improved Isgur-Wise relations Grinstein:2002cz (); Grinstein:2004vb (); Bobeth:2010wg (), this results in a simple structure of the transversity decay amplitudes at leading order in Bobeth:2010wg ()
factorizing into universal short-distance coefficients and form factors . This feature allows to extract short-distance couplings without long-distance pollution, and vice versa, as well as to test the performance of the OPE Bobeth:2010wg (); Bobeth:2011gi ().
The opposite kinematical region of large recoil has been subjected to the question of optimized observables as well, e.g., Kruger:2005ep (); Bobeth:2008ij (); Egede:2008uy (); Altmannshofer:2008dz (); Lunghi:2010tr (); Becirevic:2011bp (); Matias:2012xw (); Das:2012kz (). Several proposals exploit specifically that QCD factorization (QCDF) Beneke:2001at (); Beneke:2004dp () at leading order maintains universal short-distance coefficients for and , while Eq. (1) is broken for at lowest order, and for all at order .
The additional benefit of the low recoil region is the strong parametric suppression of the subleading corrections to the decay amplitudes at the order of a few percent Grinstein:2004vb (); Bobeth:2010wg (). Together with an angular analysis Kruger:1999xa () this enables a rich flavor physics program, complementing the large recoil region. One application is to extract form factor ratios from data, as has recently been demonstrated in Hambrock:2012dg (); Beaujean:2012uj ().
The key questions addressed in this work are:
To which extent is Eq. (1) and its benefits preserved in the presence of operators beyond the SM ones?
What are the optimal low recoil observables model-independently?
What is their sensitivity to NP?
What is the sensitivity to potential corrections to the OPE?
To answer the above questions we perform a most general, model-independent analysis of the decays and . In terms of semileptonic dimension-six operators this concerns the chirality-flipped partners of the SM ones, (pseudo-)scalar and tensor operators. We compute various decay distributions and asymmetries.
The plan of the paper is as follows: The effective theory including the operator basis is given in Section II. We present low recoil observables and relations from different operator sets in Section III and Section IV for and , respectively. In Section V we study the sensitivity of the low recoil observables to even small NP effects. The sensitivity to OPE corrections is worked out in Section VI as well as a brief discussion of S-wave backgrounds. We conclude in Section VII.
In several appendices we give formulae and subsidiary information. In Appendix A we discuss the full angular decay distribution in decays. In Appendix B we present the angular observables in terms of the transversity amplitudes for the complete set of semileptonic operators. In Appendix C we detail the transversity amplitudes that parametrize the tensor contribution to the matrix element. An update of the SM predictions for the key observables in and decays is given in Appendix D.
Ii The Effective Hamiltonian
Rare semileptonic decays are described by an effective Hamiltonian
Here, denotes Fermi’s constant, the fine structure constant and unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix has been used. The subleading contribution proportional to has been neglected.
The renormalization scale , which appears in the short-distance couplings and the matrix elements of the operators , is of the order of the -quark mass. In the following we suppress the dependence of the Wilson coefficients on the scale .
In the SM processes are mainly governed by the operators which will be referred to as the SM operator basis. Beyond the SM chirality-flipped ones , collectively denoted here by SM’, may appear. The SM and SM’ operators are written as Bobeth:2007dw (); Altmannshofer:2008dz (); Kruger:2005ep ()
Furthermore, we allow for scalar and pseudo-scalar operators, referred to as S and P,
which includes the chirality-flipped ones, as well as tensor operators, referred to as T and T5,
Current-current and QCD penguin operators , as well as the chromo-magnetic dipole operator have to be included for a consistent description of decays, for definition see Chetyrkin:1996vx (). The matrix elements of contribute to processes via quark-loop effects. The latter are taken into account by means of the effective Wilson coefficients . The effective Wilson coefficients are renormalization group invariant up to higher orders in the strong coupling constant . In the case of exclusive decays the corrections in the large- and low-recoil region from QCDF Beneke:2001at (); Beneke:2004dp (); Bobeth:2007dw () or SCET Ali:2006ew (); Lee:2006gs () and the low-recoil OPE Grinstein:2004vb (); Bobeth:2010wg (); Bobeth:2011gi (), respectively, should be included in the . We evaluate at which takes into account most of the NLO QED corrections Bobeth:2003at (); Huber:2005ig ().
Iii at low recoil
We study decays in the low recoil region for a generalized operator basis and detail the relevant observables and their relations. In Section III.1 we give the results using SM operators only. In Section III.2, III.3, III.4 we include either SM’, S and P or T and T5 operators, respectively. Interference effects are worked out in Section III.5.
The main results of this section are summarized in Table 1, where the low recoil relations between the observables and the amount of their violations is given. Our results are based on the angular distribution presented in Appendix A, and the angular observables in Appendix B.
iii.1 SM operators
The amplitude of the exclusive decays can be treated at low recoil using an OPE and further matching onto HQET Grinstein:2004vb (). After application of the improved Isgur-Wise relations Grinstein:2004vb (), one finds for the transversity amplitudes Bobeth:2010wg (); Bobeth:2011gi (), see also Eq. (1),
The short-distance coefficients read
where denotes the matrix elements of the 4-quark operators, see Bobeth:2011gi () for details. Here, the matching correction arises from the lowest order improved Isgur-Wise relations. Its -dependence compensates the one of the dipole form factors .
The term in Eq. (8) involves uncertainties from corrections at order . However, since generically (in the SM , and ) the coefficient can be regarded as strongly short-distance dominated whereas yields only a numerically subleading contribution to observables. It follows that the subleading power corrections enter the amplitude at the few percent level.
The factorization into short-distance coefficients and form factors, Eq. (6), allows to identify suitable combinations of the observables appearing in the angular distribution of , see Appendix A for details. The angular observables depend on two short-distance parameters only,
as well as the long-distance free ratios
Here we point out a further nontrivial observable, which does depend neither on form factors nor on short-distance physics:
and which equals one. Note that this observable can be obtained via . However, by using the definition Eq. (16) directly different appear. This offers additional advantages in the experimental extraction from the angular distributions.
In addition, long-distance free CP asymmetries can be formed, which are related to the CP asymmetry of the decay rate, of the forward-backward asymmetry, and of , respectively Bobeth:2011gi ().
Furthermore, several short-distance free ratios of form factors (9) can be obtained
To sum up, using SM-type operators only – which may or may not receive contributions from beyond the SM – the low recoil OPE predicts at leading order in
iii.2 Chirality-flipped operators
Taking into account the chirality flipped operators the universal structure of the transversity amplitudes (6) is broken in part. One obtains in the (SM+SM’) model
Here is defined analogously to , i.e., denotes the matrix element of the chirality-flipped 4-quark operators.
The angular observables in (SM+SM’) read
where still holds and and have been generalized to
Switching off the chirality flipped operators one recovers (and analogously for ), such that .
They remain long-distance free. Furthermore, the low recoil predictions obtained in the SM basis
In Fig. 1(a) we show and . While both equal one at low recoil in SM+SM’, at large recoil both observables exhibit a nontrivial -dependence and depend on short- and long-distance contributions. However, to lowest order form factors drop out in cf. Matias:2012xw () and . We show the residual uncertainty from the form factors and subleading corrections by the shaded bands. The SM is represented by the thin (blue) band, whereas the lighter shaded (gold) one corresponds to a scenario with , , . For numerical input, see Appendix D.
Since in (SM+SM’) two additional long-distance free ratios
can be constructed. They obey
We point out a further nontrivial observable, which depends neither on form factors nor on short-distance physics:
where in (SM+SM’)
For an analogous comment as on applies, see text below Eq. (16).
The transverse asymmetries are driven by the real and imaginary part of , written as
where vanishes for including vanishing chirality-flipped four-quark operators. Real-valued SM+SM’ Wilson coefficients can still induce somewhat suppressed, finite values of through the absorptive contributions in the matrix elements of the four-quark operators and , by . Note that in the SM in the low recoil region Bobeth:2011gi (). In any case, are null tests of the SM. A SM background to right-handed currents arises at higher order in the OPE including and counting -terms as such and enters with additional parametric suppression by or Grinstein:2004vb (); Beylich:2011aq ().
Combinations result in further useful observables which do not depend on form factors either:
Both equal in (SM+SM’).
Since are naive T-odd these angular observables give optimal access to CP violation in the presence of small strong phases Bobeth:2008ij (). Since both are also CP-odd, can be measured from -meson samples without tagging and give rise to a further long-distance-free CP asymmetry defined as
for and , respectively. Here, the barred quantities are obtained by conjugating the weak phases. In terms of the short-distance coefficients reads
The generalization of Bobeth:2011gi () is given by
Due to the presence of and , the generalization of the CP asymmetries and leads to a doubling
In this case the CP asymmetry of the decay rate can not be related to any of the and is not long-distance free. However, from (25) it is straightforward to read off strategies to relate the , to the . In particular can be extracted from ratios involving , whereas requires . In analogy to Eq. (2.37) of Ref. Bobeth:2011gi (), the set has to be restricted to for and to for .
In (SM+SM’) short-distance free ratios of angular observables exist for as given in Eq. (17), and additionally
Due to , however, no short-distance free ratios can be formed which involve . Hence, the observables and are no longer short-distance free Bobeth:2010wg () either
and the method used in Hambrock:2012dg () to extract form factor ratios would yield . With current data the correction factor is within at 2 .
Furthermore, we obtain the relation in (SM+SM’)
which can be checked experimentally.
iii.3 Scalar and pseudo-scalar operators
The (S+P) operators modify the angular observables only. The respective NP contributions are driven by , where denotes the axial-vector form factor and . We find that only receives generically unsuppressed contributions,
Helicity-suppressed () contributions from interference terms SMS arise in . For the explicit expressions see Appendix B.
We find that in the presence of (S+P) operators the low recoil relations
hold, and remain long-distance free. Since vanish in the considered scenario , and is ill-defined as in the SM-like scenario.
The helicity-suppressed contributions to break the relation at through a finite . In this case ceases to be free of form factors, and rather depends on . Moreover, the relation is broken at if there is additionally CP violation beyond the SM. With the exception of using , the ratio can be extracted by means of the methods proposed in Eqs. (17) and (41).
The (pseudo-)scalar contributions to break the relation , valid only in the (SM+SM’) basis for , see also Appendix B. At the same time, contributions to the longitudinal polarization of mesons are induced, see Eq. (97). These contributions prohibit that and , the lepton forward-backward asymmetry, can be extracted simultaneously from a fit to Eq. (99), the angular distribution in . Note that and can be extracted from Eq. (96), the distribution in , in a model-independent way. Discrepancies between the extracted values of from Eqs. (96) and (99) would indicate BSM physics. (We assume here that S-wave contributions from have been removed from the data, see Section VI.2.) Note also that interference terms (SM+SM’)S contribute to via due to (95).
iii.4 Tensor operators
The tensor operators (T+T5) give rise to additional tensor transversity amplitudes . Here the labels and denote the transversity state of the polarization vectors which comprise the rank-two polarization tensor that was used in the computation. We obtain for pairs , , the total angular momenta , respectively. For the definition of the transversity amplitudes and their general results, see Appendix C and Eqs. (118)-(120), respectively.
At low recoil, after application of the improved Isgur-Wise relations, we obtain
and the upper and lower sign refers to and ,
In the presence of tensor operators T and T5 in addition to (SM + SM’) the angular observables receive i) contributions which do not interfere with other operators in , and ii) helicity-suppressed interference contributions in , and iii) no contributions in from the additional six transversity amplitudes . We find