# Spin analyzing power for polarized top decays with jets

###### Abstract

We perform perturbative QCD factorization of infrared radiations associated with an energetic quark from a polarized top quark decay, taking the semi-leptonic channel as an example. The resultant formula is expressed as a convolution of an infrared-finite heavy-quark kernel with a -quark jet function. Evaluating the heavy-quark kernel up to leading order in the coupling constant and adopting the jet function from QCD resummation, we predict the dependence of the spin analyzing power for a polarized top quark on the invariant mass of the -quark jet. It is observed that the spin analyzing power could be enhanced by a factor 2 compared to the inclusive case with the jet mass being integrated over. It is worthwhile to test experimentally the enhancement of the spin analyzing power due to the inclusion of jet dynamics.

###### pacs:

14.65.Ha, 13.88.+e, 13.38.-b, 12.38.Cy, 13.87.-a## I Introduction

A study of the top quark physics contributes to the understanding of the origin of the electro-weak symmetry breaking in the standard model and its extensions. Progress in this field from the Large Hadron Collider (LHC), a top quark factory, has been reviewed in Topreview1 (); Topreview2 (). Higher-order calculations of top production and decay processes were summarized in TopreviewNLO () (see also references therein). Especially, observables related to the spin of a top quark may reveal various new physics effects, which can be explored at LHC (for recent references, refer to New ()). Since a top quark decays almost into a quark and a boson before hadronization, one can extract the information on the spin of a ”bare top quark” from the angular distribution of its decay products. For this purpose, the spin analyzing power for a final state in a polarized top quark decay has been defined via

(1) |

where is the partial decay width, is the polarization vector of a polarized ensemble of top quarks at rest with , and is the angle of the particle momentum measured from the top quark spin. That is, relates the top quark spin to the angular distribution of the decay product .

QCD corrections to the polarized decay were investigated in TopdecayWNLO (), and those to the spin analyzing power were done for semi-leptonic decays in TopdecayNLOlep1 (); TopdecayNLOlep2 (), and for hadronic decays in TopdecayNLOhad (). The results have been summarized in Table 3 of Ref. Topreview2 (): charged leptons and down-type quarks exhibit the largest power (), a bottom quark has a second-largest power (), neutrinos and up-type quarks have a value of , and the less energetic non- jet shows less.energetic (). It was observed that next-to-leading-order (NLO) corrections to the spin analyzing power are not important, and the quark mass could be safely ignored in the above analysis of the polarized top quark decay TopdecayNLOhad (). Although the light-particle jets from the and quarks have been considered, the spin analyzing power associated with the -quark jet is not known yet. In particular, a jet mass can be measured, so it is interesting to examine how the spin analyzing power for a polarized top quark depends on the -quark jet mass. This is the motivation of the present work.

A fundamental framework for studying high-energy processes is the perturbative QCD (pQCD) factorization theorem Factorization1 (); Factorization2 (); Factorization3 (), in which cross sections and decay widths are factorized into convolutions of several subprocesses. Take top quark decays as an example. The subprocesses include jet functions which contain infrared radiations associated with energetic hadronic final states, soft functions which collect infrared gluons exchanged among the top quark and hadronic final states, and heavy-quark kernels from the difference between the QCD diagrams for the top quark decays and the effective diagrams for the jet and soft functions. In this paper we shall demonstrate the factorization of the -quark jet from the semi-leptonic top quark decay, that requires the eikonal approximation for particle propagators in infrared regions and on the Ward identity for organizing all diagrams with attachments of infrared gluons Li01 (). The soft functions will be neglected here due to the cancellation between virtual and real corrections, and to the fact that they have minor effects on jet mass distributions. The pQCD factorization formula for the semi-leptonic decay of a polarized top quark is then expressed as a convolution of the infrared-finite heavy-quark kernel with the -quark jet function.

It has been known that the overlap of collinear and soft dynamics produces large double logarithms in a jet function, which should be summed up to all orders. References on the resummation of various double logarithms can be found in Collins.review (); Factorization.old1 (); Factorization.old2 (); Factorization.old3 (); Factorization.review (). Recently, the QCD resummation technique for light-particle jets have been developed Lijetl (); Lijet (), by means of which jet substructures, such as jet mass distributions and energy profiles, can be calculated. Substituting the quark jet function derived in Lijet () into the pQCD factorization formula for the polarized top quark decay, we show that the spin analyzing power increases with the jet mass quickly, and is possibly enhanced by a factor 2 in a wide range of the jet mass. This result is attributed to the observation that the shape of the jet function puts more weight on the contribution from the kinematic region with higher -quark jet momentum, where the spin analyzing power is larger. It is worthwhile to test experimentally the enhancement of the spin analyzing power due to the inclusion of jet dynamics.

The pQCD factorization of the -quark jet function from the semi-leptonic decay of a polarized top quark is performed up to NLO by employing the eikonal approximation in Sec. II. The procedures that rely on the application of the Ward identity for the all-order factorization are outlined. The doubly differential width for the angular distribution of the -quark jet with a specified invariant mass is presented in Sec. III. We propose a parametrization for the quark jet function derived in Lijet () in order to simplify numerical analysis. The choices of physical parameters and our predictions are summarized. Section IV is devoted to a conclusion.

## Ii Formalism

We consider the semi-leptonic top quark decay

(2) |

where , , , and are the momenta of the top quark, the quark, the charged lepton, and the neutrino, respectively, as indicated in Fig. 1 for the leading-order (LO) amplitude. In this section we shall identify the leading infrared contributions, and perform their factorization into the -quark jet function.

### ii.1 Factorization at LO

The averaged decay amplitude squared at LO in both the electro-weak coupling and the strong coupling, depicted in Fig. 2, is written as

(3) |

with the leptonic and hadronic tensors

(4) |

In the above expression is the gauge coupling of the weak interaction, is the Cabibbo-Kobayashi-Maskawa matrix element, , and are the momentum, the mass, and the decay width of the boson, respectively, arises from the summation over the polarizations of the boson, is the projection matrix, is the spin projector for the top quark, and are the masses of the top quark and the quark, respectively, and the lepton masses have been dropped.

The three-body phase-space integral for the top quark decay is given by

(5) | |||||

with being the energy of the particle , . The angle is the polar angle of the quark with respect to the top quark spin as indicated in Fig. 3, the azimuthal angle has been integrated out, and denotes the angle between the - plane and the - plane.

We insert the identity

(6) |

into the differential decay width, where () is the jet invariant mass (energy), and the jet (-quark) direction is defined by (). The three -functions in Eq. (6), together with the -quark phase space, are absorbed into the LO -quark jet function , which will be specified later. We further factorize the fermion flow in Eq. (4) by applying the Fierz transformation

(7) |

with being the identity matrix and . The vectors

(8) |

are introduced, which lie on the light cone and satisfy . The third term in Eq. (7), especially the component , gives the leading-power contribution. The matrix goes into the trace for the -quark jet function, and the matrix goes into the trace for the heavy-quark kernel. We also employ the identity

(9) |

to factorize the color flow, where goes into the -quark jet function, and goes into the heavy-quark kernel. The fermion and color traces between the -quark jet function and the heavy-quark kernel are then separated.

Figure 2 becomes Fig. 4, where the cross vertex stands for the insertion of the Gamma matrices from the Fierz transformation. The differential decay width is factorized at LO into

(11) | |||||

with the LO heavy-quark kernel

(12) | |||||

and the function

(13) |

In the above expression the dimensionless variables are defined as

(14) |

and the LO jet function is equal to the -function, . It is easy to see that Eq. (12) is consistent with the fixed-order calculations in TopdecayNLOlep1 (); TopdecayLO ().

### ii.2 Factorization at NLO

We then consider radiative corrections to the above LO decay width, which involves two dramatically different scales, the top-quark mass and the -quark mass . Since the -quark mass will be neglected eventually, and the QCD scale will not be differentiated here. There are two leading infrared regions for a loop momentum , in which the radiative gluon is off-shell by . The collinear region corresponds to collimated to , namely, . Another is the soft region defined in the heavy-quark effective theory, where the gluon momentum scales like with . Collinear gluons can be factorized from the top quark decay amplitude into the -quark jet function with the jet cone of radius . Soft gluons, except those emitted into the jet cone, will be ignored as explained in the Introduction. Hence, we focus on the factorization of the -quark jet function below.

For the factorization at NLO, we first consider a virtual gluon emitted by the top quark and attaching to the quark. In the collinear region, where the loop momentum carries a large component of along the quark momentum, we have the hierarchy . It implies that the collinear gluon can be factorized into the NLO virtual -quark jet function by means of the eikonal approximation, under which the virtual top quark propagator is simplified into . It means that the collinear gluon has been detached from the top quark line, and collected by a Wilson line in the direction of . The separation of the fermion flow and the color flow between the jet function and the heavy-quark kernel also follows Eqs. (7) and (9), respectively. Associating the three -functions in Eq. (6) with the NLO jet function, the differential decay width with the above virtual correction is written, at leading power of , as the convolution

(15) | |||||

The infrared-finite NLO virtual heavy-quark kernel is defined as the difference between the left-hand side and the first term on the right-hand side via the above expression.

We then consider the QCD correction, where a real gluon is emitted by the top quark and attaches to the quark. The hierarchy () for a real gluon automatically holds in the collinear (soft) region, implying the factorization of the infrared gluon into the -quark jet function through the eikonalization of the top-quark propagator. In this case we insert the identity

(16) |

to define the NLO real -quark jet function , where the dependence is introduced by requiring the real gluon to be emitted within the jet cone of radius , and denotes the direction of the total momentum of the quark and the real gluon. The separation of the fermion flow and the color flow is also achieved by applying Eqs. (7) and (9). Combining the three -functions in Eq. (16) and the phase spaces for the quark and the gluon, together the normalization factor , we construct .

The difference between the original diagram and the effective diagram with the eikonal approximation gives the NLO real heavy-quark kernel . For this difference, it is understood that the infrared contribution in the top quark decay specifically associated with the quark has been subtracted. The top quark decay is then calculated as its final states are composed of a jet, a real gluon, and a lepton pair at this order. Consequently, we insert the identity in Eq. (6), and associate the three -functions in Eq. (6) with the LO -quark jet function, forming . Therefore, the differential decay width is expressed as the convolution

where the dependence of arises from the subtraction of from the QCD diagram. We should include the NLO diagrams into the complete heavy-quark kernel with the radiative gluons being exchanged among the top quarks. This category contains the virtual diagrams, such as the self-energy correction to the top quark, and the real diagrams, where gluons are exchanged between the top quarks on both sides of the final-state cut. These diagrams are also characterized by the scale , i.e., dominated by the contribution from not collimated to .

### ii.3 Factorization at All Orders

To extend the jet factorization to all orders, we follow the procedures outlined in Li01 (): we first employ the collinear replacement for the metric tensor of a gluon propagator Li01 ()

(18) |

where the vertex () is located on the -quark line (all other lines in the QCD diagrams for the differential decay width). It is easy to see that Eq. (18) picks up the leading collinear contribution. Applying the Ward identity to the contractions of Li01 (), it can be shown that all the contractions are summed into the factorized form containing the NLO jet functions derived in the above subsections. The factor in Eq. (18) corresponds to the Feynman rules for the top-quark propagator under the eikonal approximation, namely, for the Wilson lines, which are demanded by the gauge invariance of the jet function as a matrix element of a nonlocal operator. Hence, a gluon collimated to the quark is factorized from the diagrams, leading to . At last, we establish the all-order factorization by induction Li01 ()

(19) | |||||

The above steps are basically the same as in Li01 (), so the detail will not be repeated in this work.

To verify that the jet function in Eq. (19) absorbs the correct logarithms, Eq. (12) for virtual corrections and Eq. (22) for real corrections in the “unresolved” region in TopdecayNLOhad () are expanded in the small limit, and the large logarithms , with the energy fraction of the quark , are identified. The coefficients of these large logarithms are then compared to those of , with , in the expansion of the Sudakov exponent up to NLO (see Appendix A of Lijet ()), where the variable arises from the Mellin transformation in the jet mass space. It is seen that both the double and single logarithms have the same coefficients in the two references, except an additional single logarithm proportional to in TopdecayNLOhad (). The dimensionless parameter was introduced by slicing the phase space of real corrections into the “resolved” and “unresolved” regions, and the contribution from the former was obtained from a numerical integration. We believe that the additional logarithm proportional to should be canceled, once the “resolved” contribution is also included in the comparison, since the total result is independent of . An equivalent postulation is that can be made vanish by choosing an appropriate value for the arbitrary .

As stated in the Introduction, the NLO corrections to the spin analyzing power, being integrated over all final-state phase space, are not important TopdecayNLOhad (). With the large logarithms having been organized into the jet function, the remaining NLO corrections to the heavy-quark kernel are expected to be negligible. Therefore, we will adopt the LO heavy-quark kernel for numerical analysis below. The differential decay width is then given by

(20) |

where the dependence has been integrated over, and the constant represents

(21) |

## Iii Results

We re-express Eq. (20) in terms of the polar angle of the -quark jet, and the angle between the momenta of the charged lepton and of the -quark jet,

(22) |

with

(23) |

and . Further integrating over the energy fractions and , Eq. (22) gives

(24) |

where , , and are written as

(25) |

with , , and

(26) |

The dependencies of and in Eq. (25) are implicit.

The spin analyzing power for a final state in a polarized top quark decay can be defined via the average of

(27) |

with which the usual expression in Eq. (1) leads to . Following the same reasoning, we can study the spin analyzing power for the -quark jet with a specified invariant mass,

(28) |

That is, the function is related to the decay width for producing a -quark jet of mass , and the function is related to angular distribution of the -quark jet. In this paper, we will evaluate , , and , and discuss the effect on the spin analyzing power from the inclusion of the jet function.

It has been observed that the quark mass is negligible in the analysis of the polarized top quark decay according to TopdecayNLOhad (), implying the approximation of the -quark jet function by the light-quark jet function. Moreover, it was noticed that the differential cross sections for jet production scale with the ratio Lijet (). To simplify the numerical calculation, we are allowed to parameterize the light-quark jet function,

(29) |

where fixes the dimension of the jet function. The normalization constant , determined by , does not affect the evaluation of the spin analyzing power as indicated by Eq. (28). Fitting Eq. (29) to Figs. 4 and 5 in Lijet (), which have involved non-perturbative contribution from the region with large Mellin moments, we obtain , and , and . It should be stressed that the parameterized jet function must have some deviation from the exact one, and the soft function has been neglected here. Hence, results presented below can be regarded as being derived from a theoretically well motivated phenomenological study.

(a) | (b) |

We adopt the following inputs PDG.data ()

(30) |

and vary the jet radius to examine its influence by choosing , , and . The jet mass dependencies of and with three different values of are displayed in Figs. 5(a) and 5(b), respectively. The peaks arise from the convolutions of the jet function with the kernels and that contain the Breit-Wigner structure in the -boson propagator. Increasing the jet radius, the peaks of and shift toward the large mass region, and both distributions become broader, because more infrared radiations are included. The peak positions (), almost identical for and , are found to be 2.1 (2.5 GeV), 6.5 (4.4 GeV), and 1.3 (6.3 GeV) with , 0.7, and 1.0, respectively. They are of the same order of magnitude as the mass ratio between the quark and the top quark, for .

The dependence of the spin analyzing power on the jet mass is shown in Fig. 6, with its behavior at small being highlighted in Fig. 6(a). It indicates that all curves agree with obtained in the literature at . The spin analyzing power then increases rapidly with the jet mass, stabilizes around (depending on the jet radius) in a wide mass region , and slowly decreases at large . At last, the spin analyzing power vanishes quickly at as expected, which corresponds to the kinematic boundary. Another important feature is that is insensitive to the choice of the jet radius in the middle mass region. Although the jet production is dominated by the mass region around , it is still possible to determine the spin of a polarized top quark by measuring the angular distribution of the -quark jet with a bit higher mass. In particular, the broader mass distribution at a larger jet radius warrants this possibility against lower event rates. We compare in Eq. (28) based on the jet factorization with the usual spin analyzing power based on the parton model. Replacing the jet function by its LO expression , namely, regarding the quark as a parton, we get the spin analyzing power . This value is close to the fixed-order (LO or NLO) results in TopdecayNLOlep1 (); TopdecayNLOlep2 () for semi-leptonic decays and in TopdecayNLOhad () for hadronic three-body decays. Integrating and over the jet mass, our formalism reduces to the usual spin analyzing power, giving , , and for , 0.7, and 1.0, respectively. The above checks confirm the consistency of our formalism.

(a) | (b) |

The possible enhancement of the spin analyzing power by considering the jet function is explained as follows. We first investigate the jet-mass dependence of the ratio in Fig. 7(a), and observe that for a given decreases with from an initial value at . Plotting the dependence of and , it is seen that the curve exhibits a peak at for a wide range of , namely, is a typical jet energy fraction in a top quark decay. It accounts for the usual spin analyzing power derived in the literature, because the initial value is about as . The initial value increases with , and reaches around as . To enhance the spin analyzing power, the large region must be selected. We then display the behavior of the -quark jet function with in Fig. 7(b), where it exhibits weaker suppression at slightly higher than zero, say, , as increases. That is, Fig. 7(b) shows the known broadening of a jet in the mass distribution with the jet energy.

Since with gets more weight as convoluted with the jet function, the spin analyzing power grows with . To confirm that the jet energy dependence plays an important role, we test a model jet function, which is energy independent, . Inserting this simple model into Eq. (25), we obtain the spin analyzing power that always decreases with . It is worthwhile to test experimentally the enhancement of the spin analyzing power through the inclusion of jet dynamics.

(a) | (b) |

## Iv Conclusion

In this paper we have performed pQCD factorization of the -quark jet function from the semi-leptonic decay of a polarized top quark by means of the eikonal approximation and the Ward identity. The resultant formula is expressed as the convolution of the infrared-finite heavy-quark kernel with the jet function, which is dominated by infrared radiations. Adopting the LO heavy-quark kernel and parameterizing the jet function from QCD resummation, we have predicted the dependence of the spin analyzing power associated with the -quark jet on its invariant mass. It has been verified that our formalism reproduces the fixed-order results by integrating over the jet mass. Our investigation indicated that the spin analyzing power could be possibly enhanced by a factor 2 as measuring the angular distribution of the -quark jet with a specified invariant mass. The mechanism responsible for the enhancement has been elaborated, which is attributed to the broadening of a jet in the mass distribution with the jet energy. This broadening puts more weight on the contribution from the region with higher jet energy, that intends to give a larger spin analyzing power. The above observation is rather insensitive to the choice of the jet radius, and worth of experimental confrontation in view of deeper understanding of the effect from including jet dynamics into the polarized top quark decay.

As future works, NLO corrections to the heavy-quark kernel need to be taken into account in order to improve the precision of our predictions. At this level, the dependence of the top quark mass definition on renormalization schemes (for instance, the mass msbarmass1 (); msbarmass2 () or the pole mass), becomes an essential issue. It is possible to extend the current formalism to hadronic decays of a polarized top quark, whose analysis is more complicated. Another important direction is to develop the formalism for a boosted top quark at LHC Boost1 (); Boost2 (), that mainly decays into a single jet. Substructures for a polarized top quark jet, which improve the extraction of the information on the top quark polarization, can be calculated.

## Acknowledgment

We thank Z.G. Si for useful discussions. This work was supported in part by the National Science Council of R.O.C. under Grant No. NSC-101-2112-M-001-006-MY3, and by the National Center for Theoretical Sciences of R.O.C..

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