Contribution of vector resonances to the decay
The fully differential angular distribution for the rare flavor-changing neutral current decay is studied. The emphasis is placed on accurate treatment of the contribution from the processes with intermediate vector resonances = , , , , , decaying into the pair. The dilepton invariant-mass dependence of the branching ratio, longitudinal polarization fraction of the meson, and forward-backward asymmetry is calculated and compared with data from Belle, CDF and LHCb. It is shown that inclusion of the resonance contribution may considerably modify the branching ratio, calculated in the SM without resonances, even in the invariant-mass region far from the so-called charmonia cuts applied in the experimental analyses. This conclusion crucially depends on values of the unknown phases of the and decay amplitudes with zero helicity.
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The investigation of the rare decay
induced by the flavor-changing neutral current (FCNC) transition is an important test of the standard model (SM) and its extensions (see Antonelli:2009 () for a review). The phenomenology of this decay mode has been discussed by many authors, e.g. see Refs. Melikhov:1998 (); Kruger:2000 (); Ali:2000 (); Kim:2000 (); Ali:2002 (); Kruger:2005 (); Lunghi:2007 (); Bobeth:2008 (); Bobeth:2010 (); Bobeth:2011 (); Egede:2008 (); Altmannshofer:2009 (); Korchin:2010 (); Egede:2010 (); Lunghi:2010 (); Bharucha:2010 (); Alok:2010 (); Alok:2011 (); Alok:2011a (); DescotesGenon:2011 (); Becirevic:2012 (); Altmannshofer:2011 (); Korchin:2011 (); Matias:2012 (); Das:2012 (); Das:2012a ().
This decay takes place in a very wide region of dimuon invariant mass squared, , namely . The light vector resonances , , (and their radial excitations) and the resonances , (and higher states) are also located in this region. Thus at the decay can go through the hadronic weak decay , followed by the dimuonic annihilation of vector meson . All resonances with make a contribution to this mechanism. Therefore both the nonresonant and resonant parts can contribute to the total amplitude of the decay .
Main attention in literature has been paid to description of the nonresonant amplitude of the decay in the region . In this region, using the QCD factorization (QCDf) Beneke:1999 (); Beneke:2000 (), one can perform a systematical calculation of non-factorizable corrections to “naive factorization approximation” (NFA) and spectator effects Beneke:2001a (); Beneke:2005 (). At larger dimuon masses, at about , the QCDf and the light-cone sum rules (LCSR) methods are not applicable. For the estimation of non-factorizable corrections, an operator product expansion in powers of can be used Grinstein:2004 (); Beylich:2011 (). In the region GeV GeV the non-factorizable effects due to soft-gluon emission have been included in Khodjamirian:2010 ().
Often the resonant contribution to amplitudes of rare decays of -meson is modeled in terms of the Breit-Wigner functions for the resonances Deshpande:1989 (); Lim:1989 (); Ali:1991 (); Ligeti:1996 (); Ligeti:1998 (). In these references the resonance corrections are added to the perturbative loops of charm quarks. Note an original approach of Ref. Kruger:1996 () for the inclusive process, in which dispersion relation exploiting experimentally measured cross section has been applied to account for the resonance terms (see also Beneke:2009 ()).
These approaches more often apply to the inclusive decays and use information on the and branching ratios for description of the resonant contribution. Sometimes, such approaches are extended to the exclusive decays Ali:2000 (); Ligeti:1996 (); Ligeti:1998 (), in which the branching ratios for exclusive decays and are used. In these studies, carried out in framework of the NFA, additional factors are introduced into the resonant terms to adjust the branching ratios for the decays, for instance,
where the right-hand side is taken from experiment.
Recall that, in general, the process is characterized not only by the branching ratio. The decay of a meson into a pair of vector mesons, , is described by three complex amplitudes Valencia:1989 (). In the transverse basis Dunietz:1991 (); Dighe:1996 (), these decay amplitudes correspond to linearly polarized states of vector mesons, which are polarized either longitudinally () or transversely to the direction of their motion, being polarized in parallel () or perpendicular () to each another. Overall, six real parameters describe three complex amplitudes , , and . They could be chosen to be, for example, the branching ratio, , , , and one overall phase . The phase convention is arbitrary for an isolated decay . Sometimes, this phase is chosen zero, . However, for certain decays, this phase can produce meaningful and observable effects, such as for with .
For example, in the decay , the phase of the amplitude has been measured with respect to the phase of the amplitude of the decay and is equal to Babar:2007 (); Babar:2008 (). For the other vector resonances , the corresponding relative phase has not been measured so far.
At present in decay modes to the light resonances, and , only the branching ratio and longitudinal polarization fraction of the meson are measured, while the decays to radial excitations, , , , , , have not been observed. At the same time, all amplitudes of the decay are known from experiment, while there is no information on decays to radial excitations of . For the and decays, the full angular analysis has been performed. As for the decays to the higher states, , , , , the experimental information is absent.
In the present paper for description of resonant contribution to the four-body decay the available information on the helicity amplitudes for is used. The fully differential angular distribution over the three angles and dimuon invariant mass is analyzed in the whole region . The amplitude of this decay consists of the nonresonant amplitude in the SM model and the resonant amplitude. For the first amplitude we use the NFA, in which hadronic matrix elements are parameterized in terms of form factors Ball:2005 (), and the Wilson coefficients are taken in the next-to-next-to-leading order (NNLO) approximation.
The resonant amplitude is expressed in terms of the invariant amplitudes for the decays . The information on the latter is taken from experiment if available, or from theoretical estimations. As mentioned above, the phase for an isolated decay is arbitrary. This phase may produce observable effects in the decay via the interference with the nonresonant amplitude. We investigate influence of the phases for each resonance on the differential branching ratio, longitudinal polarization fraction of and forward-backward asymmetry .
We also study two aspects of the resonant amplitude. The first one is related to the fact that the vector mesons are off their mass shells, therefore an off-mass-shell extension of the on-mass-shell amplitudes is proposed. The second one is the choice of the vector-meson-dominance (VMD) model which describes the transition vertex . We use two versions of the VMD model (called further VMD1 and VMD2) which result in rather different vertices. In particular, in the VMD2 model, the vertex is suppressed compared to the VMD1 vertex in the region .
Results of the present calculations are compared with the recent data from Belle, CDF and LHCb experiments. Usually in these experiments the resonance contributions are removed by putting cuts on the invariant dimuon mass near the resonance mass . This assumption is used in the analyses of all ongoing and planned experiments.
The paper is organized as follows. In Section 2.1 the fully differential angular distribution is discussed. Nonresonant and resonant amplitudes in the transverse basis are specified in Section 2.2. Results for the dependence of observables on the invariant mass squared are presented in Section 3. Conclusions are drawn in Section 4. In A calculation of the amplitudes for the off-mass-shell vector meson is described.
2 Angular distributions and amplitudes for the decay
2.1 Differential decay rate
The decay , with on the mass shell 111This means the narrow-width approximation for the propagator: ., is completely described by four independent kinematic variables: the dimuon invariant mass squared, , and the three angles , , . In the helicity frame (Fig. 1), the angle is defined as the angle between the directions of motion of in the rest frame and the in the rest frame. The azimuthal angle is defined as the angle between the decay planes of and in the rest frame. The differential decay rate in these coordinates is given by
where the angular terms are defined as
and the amplitude terms as
where , is the mass of the muon, is the mass of the meson, , and
Here , the dependent on products of the six transversity amplitudes , and , where and refer to the chirality of the leptonic current, as well as the seventh transversity amplitude . The latter amplitude is related to the time-like component of the virtual gauge boson, which does not contribute to the decay rate in the case of massless leptons and can be neglected if the lepton mass is small in comparison to the invariant-mass of the leptonic pair. Further, , , where is the mass of the meson, and
The longitudinal, parallel, and perpendicular partial widths are given, respectively, by
The familiar muon-pair invariant-mass spectrum for decay can be recovered after integration over all angles as
The fraction of meson polarization is
Integrating Eq. (1) over the variables and , we obtain
Integration of Eq. (1) over and yields
where is the muon forward-backward asymmetry,
and the normalized forward-backward asymmetry
is given as
Finally, the one-dimensional angular distribution in the angle between the lepton and meson planes takes the form
where the asymmetry is sensitive to new physics from right-handed currents beyond the standard model, and the amplitude is sensitive to complex phases in the hadronic matrix elements. Sometimes is called transverse asymmetry Kruger:2005 ().
2.2 Resonant and nonresonant transverse amplitudes
The effects of the long-distance contribution from the decays , where , , , , , mesons, followed by in the decay are included through the VMD approach, as shown in Fig. 2.
There is no unique way of introducing the transition, and one can use various versions of VMD models which yield different transition vertices. In one of VMD models (see Feynman (), chapter 6)
where is the metric tensor, is the photon (meson) four-momentum and is the effective electric charge of the quarks in the meson :
The decay constant of the neutral vector meson can be extracted from electromagnetic decay width, using
We will call this version VMD1. The vertex in Eq. (14) follows from the transition Lagrangian
Another model (called hereafter VMD2) originates from
The term does not contribute being contracted with the leptonic current , and vertex (19) is suppressed compared to (14) at small , i.e. in the region far from the vector-meson mass shell. Of course on the mass shell, , the VMD2 and VMD1 are equivalent.
These two versions of VMD model have been discussed earlier in Refs. Klingl:1996 (); O'Connell:1997 (). Note also that VMD2 vertex follows from the Resonance Chiral Theory Ecker:1989 () and has been applied Eidelman:2010 () when studying the reaction .
Parameters of vector resonances are collected in Table 1.
The nonresonant amplitudes are calculated in the NFA, with the short-distance NNLO Wilson coefficients, and nonperturbative transition form factors.
Then the total amplitudes including nonresonant and resonant parts take the form
is the usual Breit-Wigner function for the meson resonance shape with the energy-dependent width , , , is the mass (width) of a meson and the form factors enter as
In the above formulas the definition , are used, and  is the running bottom (strange) quark mass in the scheme at the scale .
The SM Wilson coefficients have been obtained in Altmannshofer:2009 () at the scale GeV to NNLO accuracy and equal
, where is quark-loop function given in Ref. Beneke:2001a (). Note that in the framework of the SM .
Further, , , , , , , are the transition form factors. In the numerical estimations, we use the form factors from the LCSR calculation Ball:2005 ().
The energy-dependent widths of light vector resonances , and are chosen as in Ref. Korchin:2010 (). The up-dated branching ratios for resonances decays to different channels are taken from PDG:2010 (). For the resonances , , we take the constant widths.
In order to calculate the resonant contribution to the amplitude of the decay, one has to know the amplitudes of the decays ,