# Top Jets in the Peak Region: Factorization Analysis with NLL Resummation

## Abstract

We consider top-quarks produced at large energy in collisions, and address the question of what top-mass can be measured from reconstruction. The production process is characterized by well separated scales: the center-of-mass energy, , the top mass, , the top decay width, , and also ; scales which can be disentangled with effective theory methods. In particular we show how the mass measurement depends on the way in which soft radiation is treated, and that this can shift the mass peak by an amount of order . We sum large logs for and demonstrate that the renormalization group ties together the jet and soft interactions below the scale . Necessary conditions for the invariant mass spectrum to be protected from large logs are formulated. Results for the cross-section are presented at next-to-leading order with next-to-leading-log (NLL) resummation, for invariant masses in the peak region and the tail region. Using our results we also predict the thrust distribution for massive quark jets at NLL order for large thrust. We demonstrate that soft radiation can be precisely controlled using data on massless jet production, and that in principle, a short distance mass parameter can be measured using jets with precision better than .

^{5}

###### Contents

- I Introduction
- II Formalism
- III Renormalization and Anomalous Dimensions
- IV SCET Results
- V HQET Results
- VI Short-Distance Top Jet Mass and the Top Quark Pole
- VII Soft Function Models with Perturbative Corrections
- VIII Numerical Analysis up to Next-to-Leading Log Order
- IX Conclusion
- A Summary of Feynman diagrams in SCET
- B Summary of Feynman diagrams in bHQET
- C Plus function and imaginary part identities
- D General RGE with plus and delta functions
- E Analytic results for in the peak & tail cross-section
- F Cross-section in the ultra-tail region
- G Analytic results for the function for thrust

## I Introduction

The top quark is the heaviest known fermion of the Standard Model and couples strongly to the Higgs sector. The most recent CDF and DØ measurements obtained a top mass, unk (2007), with uncertainty. For the standard model a precise top mass determination is important for precision electroweak observables which test the theory at the quantum level, and which constrain extensions of the theory such as supersymmetry. In Ref. Fleming et al. (2008a) we derived a factorization theorem for the invariant mass distribution of high energy top jets for , which allows in principle a determination of with uncertainty better than . Such accuracy is possible because the factorization theorem separates the perturbative and non-perturbative contributions in terms of field theory Wilson coefficients and matrix elements. A virtue of our approach is that the non-perturbative matrix elements are universal and in some cases are straightforward to extract from other processes. In addition the factorization theorem provides a unique prescription for determining the Wilson coefficients and perturbative matrix elements at any order in the expansion. This level of control allows us to make stable predictions for the invariant mass distribution in terms of a short-distance top quark mass, which is not limited in precision by .

Determining the top mass with jet reconstruction methods in general faces issues
such as i) defining an observable that is sensitive to the top mass, ii) soft
gluon interactions and color reconnection, iii) uncertainties from higher order
perturbative corrections, iv) the large top quark width , and other finite lifetime effects, v) final state radiation,
vi) initial state radiation, vii) treatment of beam remnants, viii) underlying
events, and ix) parton distributions. In Ref. Fleming
et al. (2008a) we
addressed the definition of a suitable top quark mass and issues i) through
v) in the framework of electron-positron collisions at high energies ,
where is the center of mass energy and is the top mass.^{6}

Our analysis of top jets uses effective theory techniques to exploit the hierarchy of scales , and separate dynamical fluctuations. This hierarchy provides a systematic power counting in and , and gives a clear interpretation to elements in the factorization theorem. In Ref Fleming et al. (2008a) we focused on developing the formalism and describing the main conceptual points in the factorization theorem for the invariant mass distribution in the peak region. The same formalism also yields a factorization theorem for the invariant mass distribution in the tail region above the peak. Here we use models for the soft function that are consistent for both the peak and tail regions, and carry out detailed calculations of perturbative quantities in the factorization theorem. We verify that the matching conditions which define the Wilson coefficients at the scales and are infrared safe, compute one-loop perturbative corrections to the matrix elements, and carry out the next-to-leading-log renormalization group summation of large logs. For the peak region these are logs between the scales , , , and , while away from the peak they are between , , and the variables and described below.

As an observable sensitive to the top mass, we considered in Ref. Fleming et al. (2008a) the double differential invariant mass distribution in the peak region around the top resonance:

(1) |

where

(2) |

Here and represent a prescription to associate final state hadronic four momenta to top and antitop invariant masses respectively. For simplicity we call the top and antitop jets, and the invariant mass of the top and antitop jets respectively. The distribution in Eq. (1) has a width which can be larger than the top quark width . The restriction defines the peak region, which is the region most sensitive to the top quark mass . Here the dynamics is characterized by energy deposits contained predominantly in two back-to-back regions of the detector with opening angles of order associated with the energetic jets or leptons coming from the top and antitop decays, plus collinear radiation. The region between the top decay jets is populated by soft particles, whose momentum is assigned to one of or . The tail region is defined by invariant masses starting just past the peak where the cross-section begins to fall off rapidly, namely where and either or . Farther out, when , we have an ultra-tail region where the cross-section is very small. We do not consider the region where . The observable in Eq. (1) in the peak and tail regions is the main focus of our analysis. We also briefly consider the cross-section in the ultra-tail region.

The result for the double differential cross-section in the peak region to all orders in is given by Fleming et al. (2008a)

(3) |

where, as indicated, power corrections are suppressed by , , , or . Here is the short-distance top quark mass we wish to measure, and for convenience we have defined

(4) |

where are of natural size in the peak region. In Eq. (I) the normalization factor is the total Born-level cross-section, the and are perturbative coefficients describing hard effects at the scales and , are perturbative jet functions that describe the evolution and decay of the the top and antitop close to the mass shell, and is a nonperturbative soft function describing the soft radiation between the jets. To sum large logs and will be evolved to distinct renormalization scales , as we discuss in section II.3 below. For the tail region Eq. (I) becomes

(5) |

so the only changes are that the soft-function becomes calculable, and we have an additional nonperturbative correction from the power expansion of the soft-function which we will include in our analysis. The result in Eq. (I) was derived by matching QCD onto the Soft Collinear Effective Theory(SCET) Bauer et al. (2001b, a, 2002b, 2002a); Bauer and Stewart (2001) which in turn was matched onto Heavy Quark Effective Theory(HQET) Manohar and Wise (2000); Isgur and Wise (1989, 1990); Grinstein (1990); Eichten and Hill (1990); Georgi (1990) generalized for unstable particles Fadin and Khoze (1987); Beneke et al. (2004a, b); Hoang and Reisser (2005) as illustrated in Fig. 1. The decoupling of perturbative and nonperturbative effects into the jet functions and the soft function was achieved through a factorization theorem in SCET and HQET, aspects of which are similar to factorization for massless event shapes Korchemsky and Sterman (1995, 1999); Bauer et al. (2003, 2004). The result in Eq. (I) is an event shape distribution for massive particles, and can be used to determine common event shapes such as thrust or jet-mass distributions. Note that a subset of our results can also be used to match results with the event shape cross-sections for massless jets, namely by using our SCET ultratail cross-section and taking the limit .

In general the functions and depend on exactly how and , or equivalently and , are defined. The factorization theorem in Eq. (I) holds in the form shown when all the soft radiation is assigned to either or , and the probability of radiation being assigned to or increases to unity when we approach the top or antitop direction Fleming et al. (2008a). Finally, the definition should be inclusive in the hard jets and leptons from the top decay. One possibility for defining in Eq. (I) is a hemisphere mass definition, where contain everything to the left or right of the plane perpendicular to the thrust axis. In this case our is identical to the soft function of Refs. Korchemsky and Sterman (1999); Korchemsky and Tafat (2000); Bauer et al. (2004) that appears in the factorization theorem for massless event shapes in the dijet region. For studies of soft-functions in massless event shapes see Refs. Korchemsky and Sterman (1995); Korchemsky (1998); Korchemsky and Sterman (1999); Korchemsky and Tafat (2000); Berger et al. (2003a); Bauer et al. (2003, 2004); Lee and Sterman (2007, 2006); Berger and Sterman (2004); Berger et al. (2003b); Belitsky et al. (2001). The are inclusive in the jets from the top decay and collinear radiation and can be defined by forward matrix elements Fleming et al. (2008a). Other definitions to associate all soft radiation to the top and antitop jets can be used which modifies the required function, but for the class of masses defined above leaves unchanged.

The use of a short-distance mass definition in the jet function and a short-distance gap parameter in the soft function Hoang and Stewart (2008) are crucial for obtaining predictions that remain stable when higher order perturbative corrections are included. In Ref. Fleming et al. (2008a) we showed that suitable mass schemes for reconstruction measurements can only differ from the pole mass by an amount , and we proposed a jet-mass scheme which satisfies this criteria. We will refine the criteria for this jet-mass scheme here. In Eq. (I) the jet-mass, , only appears in the calculable Wilson coefficients and jet functions . The greatest sensitivity to is in . Through these jet functions, influences the spectrum of the mass distribution in the peak region. The spectral distribution and location of the peak are also affected by nonperturbative effects in the soft function . In Ref. Hoang and Stewart (2008) a gap parameter scheme based on moments of the partonic soft function was devised to avoid perturbative ambiguities in the definition of the partonic endpoint where the variables in Eq. (I) approach zero. Methods for using Eq. (I) to extract are discussed in detail in Ref. Fleming et al. (2008a).

In this paper we determine the functions , , at one-loop order in , and carry out the summation of large logs between the scales in Eq. (I). The derivation of results for the top jet-mass scheme are discussed in detail. We also show that there are constraints on the allowed soft functions, and implement a consistent method to include perturbative corrections in . In our numerical analysis we extend the work in Ref. Fleming et al. (2008a) to one-loop order, including the summation of the next-to-leading order logarithms using renormalization group (RG) evolution in effective field theories. Our analysis of the jet cross section at this order includes both invariant masses in the peak region and the tail region above the peak, and the final results are analytic up to integration over the soft-function model.

For massless jets there has been a lot of work done on the program of resumming logs in event shape variables Catani et al. (1991a, 1993, a); Burby and Glover (2001); Krauss and Rodrigo (2003); Gardi and Magnea (2003); Gardi and Rathsman (2001); Korchemsky and Marchesini (1993); Dokshitzer et al. (1998); Catani and Webber (1998); Berger (2002); Berger et al. (2002); Banfi et al. (2001, 2002); Trott (2007). In this paper we do not use the traditional approach to resummation, but rather an approach that sums the same large logs based on the renormalization of operators in effective field theories, including HQET and SCET Bauer et al. (2001b, a). The effective theory resummation technique has the advantage of being free of Landau-pole singularities Manohar (2003); Becher et al. (2007a), since it only depends on the evaluation of anomalous dimensions at perturbative scales. This technique can also be extended in a straightforward manner to arbitrary orders, in the resummation Neubert (2005); Becher et al. (2007b). A recent application of the SCET technique is the resummation for thrust in to massless jets at NLL order Schwartz (2007).

In our log-summation there is an important distinction between large logs which affect the overall cross section normalization, and large logs that change the shape of the distribution in . In predicting the normalization in the dijet region we must sum up a series of double Sudakov logarithms that occur for and for . However, it turns out that the same is not true for logs affecting the shape of the invariant mass spectrum. As we discuss in detail, the form of the spectrum is protected from large logs below the scale until we reach the fundamental low energy scale governing the dynamics of either the soft or jet functions. This conclusion is not affected by the mass threshold at , and is valid to all orders in perturbation theory (ie. for both leading and subleading series of logarithms). In order for this cancellation to occur it is important that the invariant mass definition includes soft radiation at wide angles. The hemisphere mass definition of and , as well as other definitions which associate wide angle soft radiation to both and , are in this category. In the effective field theory this protection against the appearance of shape changing large logs is described by a set of “consistency conditions”. From our analysis we find that the only shape changing large logs occur between the low energy scale where logs in the jet functions are minimized, and a perturbative low energy scale where logs in the soft function are minimized. Here is the hadronic scale where the interactions are non-perturbative. As indicated there are two scales appearing in each of these functions, and the question of which dominates depends on the size of these parameters.

The program of this paper is as follows. In Sec. II.1 we review the formulation of the factorization theorem for the invariant mass cross-section from Ref. Fleming et al. (2008a). In Sec. II.2 we show that the finite lifetime effects can be treated as a convolution of jet functions for stable top quarks with a Breit-Wigner, and we describe models for that are consistent in the presence of perturbative corrections. In Sec II.3 we discuss the structure of large logarithms and present the factorization with log resummation. In section III we discuss the connection between renormalization and the resummation of large logs in SCET and HQET, derive the consistency conditions, and summarize results for the NLL renormalization group evolution. Results for the matching, running, and matrix elements in SCET including the soft hemisphere function are given in Sec. IV. In Sec. V we give matching, running, and matrix element results in HQET. A short-distance jet-mass scheme is discussed in detail in Sec. VI, including its relation to other schemes at one-loop. In Sec. VII we discuss the non-perturbative soft function model and the scheme for the gap parameter we use in our numerical analysis. An analysis of the one-loop cross-section with NLL log-summation is given in Sec. VIII for both the peak and tail regions. Conclusions are given in section IX. Additional computational details are given in the Appendices A–F.

## Ii Formalism

### ii.1 Invariant Mass Cross-Section

In this section we review the main definitions of effective theory objects needed for our calculations of terms in the factorization theorem in Eq. (I). Further details can be found in Ref. Fleming et al. (2008a). Starting from QCD, the two-jet cross section can be written as

(6) |

where , and , and is the leptonic tensor including vector and axial vector contributions from photon and boson exchange. This result is valid to all orders in the QCD coupling but lowest order in the electroweak interactions. The superscript on the summation symbol denotes a restriction on the sum over final states to the kinematic situation given in Eq. (1). The QCD top quark currents are , where and . In Ref. Fleming et al. (2008a) we started with Eq. (6) and derived the factorization theorem for the double differential invariant mass distribution in the peak region in Eq. (I). There the factor is the tree-level Born cross-section,

(7) |

where and . Equation (I) can be easily generalized to include the angular distribution in where is the angle between the top jet direction and the momentum:

(8) |

where

(9) |

The remaining functions in Eq. (I) include , a hard-function
that encodes quark-gluon interactions at the production scale , which
encodes perturbative effects^{7}

(10) |

where contains short-distance dynamics, while describes all scales that are longer distance than . After making a field redefinition Bauer et al. (2002b) the SCET production current at leading order in is given by

(11) |

where we have collinear fields and Wilson lines defined in the jet-fields and , as well as soft -Wilson lines and mass-mode -Wilson lines to be discussed below. Here the indicates that the fields are at coordinate ; recall that this dependence carries information about residual momenta at scales . The dependence on larger momenta is encoded in the labels of the collinear fields Bauer and Stewart (2001). For example, forces the total minus-label-momentum of to be . In terms of defined in Eq. (10), the hard-function appearing in Eq. (I) is simply

(12) |

and after including RG-evolution we have where is the evolution kernel discussed below in sections II.3 and III.

We obtain the SCET two-jet cross section by replacing the QCD current in Eq. (6) with the SCET current. The resulting expression can be factorized as discussed in Ref. Fleming et al. (2008a)

(13) | ||||

This result can be used to compute the cross-section in the ultra-tail region, where . Due to the large suppression this region is not interesting experimentally, however we will still discuss formal aspects of Eq. (13) in detail because it is an important step towards deriving the peak region factorization theorem, and is also important for making the analogy with massless event shapes. The soft function in Eq. (13) is the same as the soft function in Eq. (I), up to perturbative effects due to top-quark vacuum polarization graphs denoted by the extra argument . It can be either derived by using eikonal Ward identities Collins et al. (1988) or properties of the coupling of usoft gluons to collinear particles in SCET Bauer et al. (2004). For the case of hemisphere invariant masses it is where

(14) |

The same function appears in event shapes for massless two-jet production, and besides the and dependence, Eq. (13) is analogous to the factorization theorem for massless dijets Bauer et al. (2003); Korchemsky and Sterman (1995); Korchemsky and Tafat (2000). In Eq. (II.1) , , , are color indices, , and the soft Wilson lines are

(15) |

with for the antitriplet representation, where . In Eq. (II.1) is defined as the soft momentum components from the state that are included in the experimental determination of (and for ). We also have operators and that project out the soft momentum components and

(16) |

For hemisphere masses the operator is defined to project out the total plus-momentum of soft particles in hemisphere- (and the minus-momentum in hemisphere-). In Ref. Fleming et al. (2008a) it was shown that does not depend on the top quark width, and when we pass below the top quark mass scale is only modified by a perturbative prefactor,

(17) |

The matching coefficient is induced by the coupling of gluons to top-vacuum polarization bubbles at zero-momentum. This result applies at any order in , but at NLL order .

In Eq. (13) the mass-mode function contains virtual perturbative corrections due to gluons and quarks with momenta , and is given by

(18) |

The definition of these mass-mode -Wilson lines is identical to those in Eq. (II.1), except that they involve gluon fields which couple to massive top-quarks for any momentum, and which have zero-bin subtractions to avoid double counting the momentum region accounted for by the gluons. This implies that only gets contributions from graphs with a top-vacuum polarization bubble Kniehl (1990); Burgers (1985); Hoang et al. (1995) coupling to the gluons. At NLL order the function , but is relevant at NNLL order and beyond when considering virtual top loops. Note that due to the invariant mass constraint , the quarks never appear in the final state. This is important for the validity of Eq. (13).

Matrix elements of top quark collinear fields in SCET give the jet functions for the top quark jet, and for the antitop jet,

(19) |

These jet functions and depend on both the mass and width of the top quarks. The matrix elements of collinear fields are defined with the zero-bin subtractions Manohar and Stewart (2006), which avoids double counting the soft region.

For predictions in the peak region, the and functions should be factorized further by integrating out the top quark mass. This is accomplished by matching onto jet functions in HQET with boosted heavy quarks. The relevant Feynman rules are given in Appendix B. The jet function matching takes the simple form Fleming et al. (2008a)

(20) |

The HQET jet functions and also depend on the residual mass term that fixes the mass definition in HQET. They are defined by

(21) |

where the are vacuum matrix elements of T-products of HQET operators

(22) |

Here for we have , while for we have . The gluons in and and HQET fields are only sensitive to fluctuations below and are built of gluons describing low energy fluctuations down to in the top and antitop rest frames respectively. In Ref. Fleming et al. (2008a) these gluons were called ultracollinear. We emphasize that to make the matching consistent, the collinear gluons in Eq. (II.1) have zero-bin subtractions for the same region as those in the SCET jet functions. These subtractions ensure that the jet-functions do not double-count the soft region encoded in , and are critical for ensuring that the functions are IR-finite, as we discuss further in Appendix A. In Eq. (II.1) the Wilson lines are

(23) |

with analogous formulas for and in terms of . Note that if the Wilson lines and were absent, then would just define the HQET heavy quark/antiquark propagators Manohar and Wise (2000). The Wilson lines, let’s say for , encode the color dynamics of gluons that are soft in the top rest frame and come from the highly boosted antitop quark, and they render this vacuum matrix element into a gauge-invariant physical object. The analogous situation with top and antitop switched applies for the vacuum matrix element . For the SCET jet functions, the Wilson lines appearing in Eqs. (11) and (II.1) have the analogous physical interpretation where the top quark mass has not yet been integrated out.

In the final factorization theorem in Eq. (I) we have the functions, as well as the matching condition for the mass fluctuations,

(24) |

In bHQET all dynamic effects associated with the top-quark mass appear in , and there are no mass-mode quarks or gluons in this theory. Since and encode finite matching corrections at the scale due to top-vacuum polarization, we have , and so these factors drop out from our NLL analysis. Therefore in later sections we simply use . This coefficient becomes sensitive to the ratio through its anomalous dimensions which depends on a logarithm of . Including the RG-summation of these logarithms gives the coefficient appearing in the factorization theorem, where is the bHQET current evolution factor discussed below in sections II.3 and III. Note that in principle and the factors in Eq. (24) can also have dependence at NNLL. For related discussions see Refs. Becher and Melnikov (2007); Chiu et al. (2007).

Alternatively, the matching coefficient of SCET and HQET jet functions given by in Eq. (24) can be determined from currents,

(25) |

where the boosted HQET current is

(26) |

with

(27) |

The soft Wilson lines in this current are the same as those used in the SCET soft function. The only distinction is that soft gluons in bHQET no longer couple to massive top-bubbles.

Due to the large width of the top quarks the jet functions can be computed in perturbation theory. At tree level they are Breit-Wigner distributions

(28) |

where we have adopted a normalization such that

(29) |

The Wilson coefficients in the factorization theorem in Eq. (I) are also normalized to unity at tree level, and .

### ii.2 Factorization of Lifetime Effects and Soft Function Models

The leading order bHQET Lagrangian is

(30) |

In light-cone coordinates, , we have and
and gluons/residual momenta scaling as and . Unlike standard HQET, the
ultracollinear gluon fields in bHQET are defined with zero-bin
subtractions Manohar and Stewart (2006) for the soft region. In
Eq. (30) is a Wilson coefficient obtained by
matching to the full theory and is equal to the top quark total width. This is
true to leading order in electroweak interactions, to and
in the power counting, and to all orders in
.^{8}

(31) |

is the residual mass term that fixes the top quark mass definition that is used in the HQET computations. It needs to be consistent with the bHQET power counting Fleming et al. (2008a),

(32) |

can be computed perturbatively, and is UV- and IR-finite. Note that the way in which Eq. (30) will be used is to compute a jet-function where the width smears over a set of states of invariant mass . Thus, for our analysis there are no corrections to Eq. (30), just corrections of .

In Eq. (21) the jet functions are expressed in terms of the imaginary part of vacuum matrix elements in Eq. (II.1). From it is straightforward to see that can be obtained from the imaginary part of the vacuum matrix element for (fictitious) stable top quarks by shifting the energy variable ,

(33) |

Here we defined results for stable top quarks, namely the jet function , and a vacuum matrix element . They are related by

(34) |

and we will refer to as the stable jet function in what follows. The result in Eq. (II.2) is in complete analogy to the relation between the production rate of top quark pairs in the nonrelativistic threshold region, , where the leading order finite lifetime effects can be implemented by the shift prior to taking the imaginary part of the forward scattering matrix element Fadin and Khoze (1987).

To separate the different physical effects in the cross section it is convenient to derive a factorization theorem for the leading order finite lifetime effects to all orders in . To do so we define the function

(35) |

It is analytic everywhere in the complex -plane, except along the positive real axis, , where the vacuum matrix elements , defined using Eq. (II.1) with , has a cut for intermediate states having invariant masses larger than the top quark mass. Using the residue theorem for a contour that envelops the cut, it is then straightforward to derive the dispersion relation

(36) |

where is any point in the complex plane not on the positive real axis. With the choice and a change of variable , this dispersion relation can be brought into the form

(37) |

Note that the upper limit of the integration can be replaced
by since the stable jet function only has support for
positive values of its energy variable.
Equation (37) states that the bHQET jet
functions for the physical unstable top quark can be written as a
convolution of the stable jet functions with a Breit-Wigner function
of the width . Thus the leading order finite lifetime
effects can be factorized from the jet function.^{9}

Eq. (37) reflects the fact that the top quark width acts as an infrared cutoff for the jet function through smearing over a Breit-Wigner function Poggio et al. (1976). In the factorization theorem in Eq. (I) additional smearing is provided by the convolution with the soft function, where the width of the distribution is of order the hadronic scale . Eq. (37) allows us to group both types of smearing into a common infrared function , with the following modified version of the factorization theorem,

(38) | ||||