Factorization and Resummation for Generic Hierarchies between Jets
Abstract
Jets are an important probe to identify the hard interaction of interest at the LHC. They are routinely used in Standard Model precision measurements as well as in searches for new heavy particles, including jet substructure methods. In processes with several jets, one typically encounters hierarchies in the jet transverse momenta and/or dijet invariant masses. Large logarithms of the ratios of these kinematic jet scales in the cross section are at present primarily described by parton showers. We present a general factorization framework called , which is an extension of SoftCollinear Effective Theory (SCET) and allows for a systematic higherorder resummation of such kinematic logarithms for generic jet hierarchies. In additional intermediate soft/collinear modes are used to resolve jets arising from additional soft and/or collinear QCD emissions. The resulting factorized cross sections utilize collinear splitting amplitudes and soft gluon currents and fully capture spin and color correlations. We discuss how to systematically combine the different kinematic regimes to obtain a complete description of the jet phase space. To present its application in a simple context, we use the case of jets. We then discuss in detail the application to jet processes at hadron colliders, considering representative classes of hierarchies from which the general case can be built. This includes in particular multiple hierarchies that are either strongly ordered in angle or energy or not.
DESY 15129
NIKHEF 2015024
January 19, 2016
1 Introduction
A thorough understanding of the production of hadronic jets is crucial to take full advantage of the data from highenergy colliders. Jet processes typically involve hierarchies between the shortdistance scale of the hard scattering (e.g. the jet energies or invariant masses between jets) and the scale at which the individual jets are resolved (e.g. the mass or angular size of a jet), leading to logarithms of the ratio of these scales in the perturbative expansion of the cross section. An accurate description of these effects is obtained by resumming the dominant logarithmic corrections to all orders in perturbation theory.
In multijet events one generically encounters additional hierarchies in the hard kinematics of the jets, namely among the jet energies and/or among the angles between jets. At the LHC, an important class of examples are jet substructure methods to reconstruct boosted heavy objects, which essentially rely on identifying soft or collinear (sub)jets. Another example is cascade decays of heavy new (colored) particles leading to experimental signatures with jets of widely different . There are also cases where additional jets produced by QCD are used to tag or categorize the signal events, a prominent example being the current Higgs measurements. Whenever such kinematic hierarchies arise among QCDinduced jets, in particular in the corresponding background processes, the enhancement of soft and collinear emissions in QCD leads to additional logarithms of the jet kinematics in the cross section. So far, a complete and general factorization framework for multijet processes that allows for a systematic resummation of such kinematic logarithms for generic jet hierarchies has been missing. Current predictions therefore rely on Monte Carlo parton showers and are thus mostly limited to leading logarithmic accuracy.
In this paper we develop such a factorization and resummation framework for generic jet hierarchies in hardscattering processes with large momentum transfer by considering an extension of SoftCollinear Effective Theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001ct (); Bauer:2001yt (); Bauer:2002nz (); Beneke:2002ph () referred to as . Compared to the usual soft and collinear modes in SCET, contains additional intermediate modes that behave as soft modes (with eikonal coupling) with respect to the standard collinear modes but at the same time behave as collinear modes with respect to the overall soft modes. Their precise scaling, which is now simultaneously soft and collinear, depends on the considered measurement or observable (in analogy to how the scaling of the modes in SCET is determined by the considered observable).
In SCET individual hard QCD emissions are resolved as jets, while the effects of soft and collinear emissions on observables are each resolved at a single scale. The intermediate modes in are required to further resolve the additional scales induced by measurements or hierarchies which are not separated in SCET.^{1}^{1}1We stress that this does not imply that SCET describes such effects incorrectly. It does correctly contain these effects at each fixed order but it is not sufficient for resumming the associated additional logarithms. In fact, we will match onto SCET in the limit where the additional hierarchies disappear and the corresponding logarithms are not enhanced. This is precisely analogous to the relation between SCET and fixedorder QCD for the logarithms resummed by SCET. The case we discuss in detail in this paper is the explicit measurement of soft or collinear (sub)jets. Here, also individual soft or collinear emissions are explicitly resolved, and allows us to capture their effects on observables.
Generically, there are two types of intermediate modes that appear which can be distinguished by their origin as follows

Collinearsoft (csoft) modes arise as soft offspring from a collinear sector of a parent SCET.

Softcollinear modes arise as collinear offspring from a soft sector of a parent SCET.
This distinction is helpful, as it automatically determines the correct Wilsonline structure and interactions of the modes with respect to the other modes present in the final . Both types of modes can be present at different scales and in different directions. There can also be cases where the two types become degenerate.
first appeared in ref. Bauer:2011uc (), where its purely collinear regime described by csoft modes was constructed and used to describe the situation of two energetic jets collinear to each other. In ref. Procura:2014cba (), was used to describe the situation where two resolution variables are measured simultaneously, requiring csoft modes separated from the collinear modes in either virtuality or rapidity depending on the measurements. The purely soft regime of involving softcollinear modes was first considered in ref. Larkoski:2015zka (). There it was shown that this regime is essential for the resummation of nonglobal logarithms by explicitly resolving additional soft subjets (see also ref. Neill:2015nya ()). In ref. Larkoski:2015kga (), the soft and collinear regimes were used to factorize and resum a twoprong jet substructure variable (defined in terms of energycorrelation functions Larkoski:2014gra ()). They also discussed a way to treat the overlap between the two regimes by removing the double counting at the level of the factorized cross section. More recently, a setup was applied in refs. Becher:2015hka (); Chien:2015cka () for the factorization of both global and nonglobal logarithms appearing in jet rates (see e.g. refs. Seymour:1997kj (); Cheung:2009sg (); Ellis:2010rwa (); Banfi:2010pa (); Kelley:2012kj (); vonManteuffel:2013vja ()).
In this paper, we give a general description of for generic jet hierarchies. We first focus on the case of a single hierarchy. We review the purely collinear regime, following ref. Bauer:2011uc (), which we will label as . Furthermore, we present in detail the purely soft regime (labeled ) as well as the overlap between the collinear and soft regimes (labeled ), involving both csoft and softcollinear modes. The corresponding kinematic hierarchies for jets are illustrated in Fig. 1. Standard SCET applies to case (a) where the jets are parametrically equally hard and well separated, , where are the dijet invariant masses and the total centerofmass energy. The collinear regime is shown in case (b), where two jets (labelled 1 and 2) are close to each other. It is characterized by the hierarchy . The soft regime is shown in case (c), where one jet (labelled 1) is less energetic than the others. It is characterized by the hierarchy . Finally, in the soft/collinear overlap regime, shown in case (d), one jet is softer than the others and at the same time closer to one of the hard jets, leading to the hierarchy .
In general, can have multiple soft and collinear regimes (along with the corresponding overlap regimes), which is necessary to describe multiple hierarchies between several jets. We discuss in detail the application of the formalism for a generic jet process at hadron colliders and for a number of different hierarchies. The cases we explicitly consider include

One soft jet.

Two jets collinear to each other, with or without a hierarchy in their energies.

Two jets collinear to each other plus an additional soft jet.

Two soft jets with or without a hierarchy in their energies.

Two soft jets collinear to each other.

Three jets collinear to each other with or without a hierarchy in the angles between them.
These cases contain the nontrivial features and essential building blocks that are needed to describe arbitrary hierarchies. In particular, we show how spin and color correlations are captured in the associated factorization theorems.
Each regime requires a different mode setup in , so technically corresponds to a different effective field theory. We explain how they are appropriately combined and matched to the corresponding SCET in the nonhierarchical limit. This yields a complete description of the jet phase space that accounts for all possible kinematic hierarchies.
We will consider an exclusive jet cross section and require that the jets can always be distinguished from each other by imposing the parametric relation . We assume that the corresponding jet resolution variable(s) that enforce this constraint do not exhibit any hierarchies among themselves, such that there are no parametrically large nonglobal logarithms from soft emissions. For definiteness and simplicity, we consider jettiness Stewart:2010tn () as our overall jet resolution variable. In refs. Stewart:2015waa (); Thaler:2015xaa (), it was shown that jettiness can be promoted into an exclusive cone jet algorithm, and with a suitable choice of jettiness measure the resulting jets are practically identical to anti jets. We stress though that the general setup for the treatment of kinematic hierarchies is largely independent of the specific choice of jet resolution variable and jet algorithm. In the application to jet substructure the setup can get more complicated when subjets get sensitive to the jet boundary Larkoski:2015zka (); Larkoski:2015kga (). For earlier analytic work on jet hierarchies in jets see Ref. Catani:1992tm ().
The remainder of this paper is organized as follows: In sec. 2 we describe the basic regimes (, , and ) and the structure of the resulting factorization theorems for jets that resum the corresponding kinematic logarithms. In sec. 3, we present a detailed discussion with explicit perturbative results for the case of jets, which is simple enough that the single hierarchies shown in fig. 1 are sufficient to exhaust all kinematic limits. We subsequently discuss stepbystep the generalizations required to treat a generic LHC process jets plus additional nonhadronic final states. Specifically, collinear initialstate radiation, spin and color correlations for a single kinematic hierarchy are addressed in sec. 4. In sec. 5 we discuss the various cases with multiple hierarchies outlined above. We conclude in sec. 6.
2 Overview of the effective field theory setup
In this section, we discuss the general factorization framework for each regime of , considering for simplicity jets. We start in sec. 2.1 with reviewing the standard case without additional hierarchies, which also serves to establish our notation. The purely collinear, purely soft, and soft/collinear regimes are discussed in secs. 2.2, 2.3, and 2.4. For now we only consider kinematic configurations with one hierarchy. The general case will be discussed in sec. 5 in the context of jets. In sec. 2.5 we show how to combine the resulting factorization theorems from the different kinematic regions. We first explicitly consider a jet resolution observable, and we outline the modifications required for a measurement in sec. 2.6.
2.1 Standard SCET: equally separated and energetic jets
We first discuss the hard kinematics for processes with jets. The total momentum of the th jet is given by
(1) 
Here, the massless reference (label) momentum contains the large component of the jet momentum. That is, corresponds to the jet energy and we take the unit vector to point along the direction of the jet. The residual momentum then only has small components of .
mode  

collinear ()  
ultrasoft 
To describe the degrees of freedom of the effective field theory, it is convenient to use lightcone coordinates,
(2) 
where , and contains the components perpendicular to and . The subscript will be dropped if it is obvious which lightcone coordinates we are referring to.
For definiteness, we consider jettiness Stewart:2010tn () as the jet resolution observable, defined as
(3) 
We use a geometric measure with , where the parameter controls the size of the th jet region and can in principle depend on the hard jet kinematics. It roughly corresponds to the typical jet radius and we consider it as . The minimization assigns particles to the jet they are closest to, and we denote the contribution to from the th jet region by . Note that is equal to the jet invariant mass up to power corrections.
The SCET description applies in the exclusive jet limit where all jets are sufficiently narrow and there are no additional jets from additional hard emissions. This limit corresponds to taking . Formally, we work at leading order in the power expansion in , where we use to denote the typical (average) jet mass. Due to the singular structure of QCD, jets typically have masses much smaller than their energy. Hence, in practice most of the events naturally have .
We stress that our discussion of the kinematic jet hierachies largely decouples from the precise choice of , and in principle any jet resolution observable which constrains (more precisely, any type variable) can be utilized. Furthermore, the precise jet algorithm that is used to find the actual jet momenta , which then determine the , is also not relevant to our discussion. One option is to promote eq. (3) itself to a jet algorithm by further minimizing the value of over all possible jet directions Stewart:2010tn (). This is the basis of the recently introduced XCone jet algorithm Stewart:2015waa (); Thaler:2015xaa (). Any other jet algorithm that yields the same jet directions up to power corrections can be used, which includes the usual type clustering algorithms.
We denote the large pairwise invariant mass between two jets with
(4) 
We order the jets such that
(5) 
and we define
(6) 
So, is the smallest dijet invariant mass, and measures the softness of jet 1. For jets, is just the intermediate dijet invariant mass.
The situation where all jets are equally energetic and well separated corresponds to and and therefore . It is described by the usual SCET framework, since all dijet invariant masses are of the same order so there are no additional hierarchies between physical scales. In contrast, the regimes illustrated in fig. 1 and discussed in the following subsections are characterized by ( regime), ( regime), and ( regime).
The degrees of freedom in consist of collinear modes for every jet direction and ultrasoft (usoft) modes interacting with these. The parametric scaling of these modes is summarized in table 1. The collinear modes for the different jet directions cannot interact with each other in the effective theory, while the interactions with the usoft modes decouple at leading power in via the BPS field redefinition Bauer:2001yt (). This leads to the following SCET Lagrangian for jet production
(7) 
The Lagrangians and describe the dynamics of the collinear and usoft sectors, respectively, and only contain interactions among the fields within each sector. Their explicit expressions can be found in refs. Bauer:2000yr (); Bauer:2001ct (); Bauer:2001yt (). The hardscattering Lagrangian consists of leadingpower SCET operators, built from collinear fields and usoft Wilson lines, and their Wilson coefficients. It arises from matching the hardscattering processes in QCD onto SCET, where fluctuations with a virtuality above the scale are integrated out.
The factorization theorem for the differential cross section following from eq. (7) has the following structure Fleming:2007qr (); Bauer:2008dt (); Ellis:2010rwa (); Stewart:2010tn ()
(8) 
The Wilson coefficients arise from and encode the shortdistance physics of the hardscattering process. They determine the hard function . The jet functions incorporate the dynamics of the collinear radiation that leads to the formation of jets, which takes place at the scale . Finally, the cross talk between the jets via usoft radiation is described by the soft function at the scale . Here, is a vector and and are matrices in the color space of the external hard partons. The jet functions are scalars in color space, i.e. color diagonal, and can therefore be pulled outside the color trace. The precise form of the jet and soft functions and the structure of the convolution between them is determined by the jet resolution variable. Since each function in the cross section eq. (8) only involves a single scale, the logarithms of can be systematically resummed by evaluating each function at its natural scale and evolving them to a common scale using their renormalization group evolution (RGE).
2.2 regime: two collinear jets
We now consider the kinematic situation where the first two jets come close to each other, but remain energetic, i.e.,
(9) 
Thus, all of the dijet invariant masses remain equally large except for . This additional hierarchy introduces large logarithms of in the hard and soft functions in eq. (8). The theory that resums these logarithms (which we now regard as the regime of ) was introduced in ref. Bauer:2011uc ().^{2}^{2}2The refactorization of the hard sector was already discussed earlier in refs. Bauer:2006mk (); Baumgart:2010qf (). We briefly recall it here and refer to ref. Bauer:2011uc () for a detailed derivation. It was applied in refs. Larkoski:2015kga (); Larkoski:2015uaa () in the context of jet substructure.
The relevant modes in the regime are given in table 2. Due to the measurement of there are additional collinearsoft (csoft) modes. Compared to the usoft modes, they have a higher angular resolution allowing them to resolve the two nearby jets separated by the angle of order . Hence, they interact with the usoft modes as collinear modes with lightcone direction . At the same time, they interact with the two nearby jets 1 and 2 (the collinear and collinear sectors) as soft modes. In particular, at their own collinearity scale the directions and belong to the same equivalence class as . The requirement that their plus component is constrained by the jet resolution measurement implies which then fully determines their scaling as given in table 2.
To disentangle all physical scales, we perform the twostep matching shown in Fig. 2. We first match QCD onto standard SCET with collinear sectors with corresponding invariant mass fluctuations and an associated usoft sector at the scale . At this point, the two nearby jets are not separately resolved yet and contained in the collinear sector. After decoupling the collinear and usoft modes, this theory is matched onto . For the collinear sectors of jets 3 to as well as for the usoft sector only the virtuality scale is lowered to and , respectively. The collinear sector of the parent SCET with scaling is matched onto the two collinear sectors for jets 1 and 2 and the csoft mode. This step involves nontrivial matching coefficients, related to the collinear splitting amplitudes. They appear when matching the hardscattering Lagrangian of the parent SCET onto the final of . As shown in ref. Bauer:2011uc (), the interactions between the two collinear modes and the csoft modes can be decoupled via a further BPS field redefinition. This leads to the leadingpower Lagrangian, which has again no interactions between different sectors,
(10) 
Here, is the Lagrangian for the csoft modes and is identical to the Lagrangian for collinear modes , except for the different scaling of the label momenta and associated scaling of the csoft gauge fields. It is important that the csoft fields are defined with a zerobin subtraction Manohar:2006nz () to avoid double counting with the usoft fields in analogy to the collinear fields. In addition, the and collinear modes are now defined with an appropriate zerobin subtraction with respect to both csoft and usoft modes.
mode  

collinear ()  
collinearsoft ()  
ultrasoft 
The factorization theorem for the differential cross section following from eq. (10) has the structure Bauer:2011uc ()
(11) 
Compared to eq. (8), the hard coefficient got factorized into for hard external partons at the scale (arising from the first matching step in fig. 2) and a collinear splitting coefficient describing the splitting of the collinear sector into the  and collinear sectors at the scale (arising from the second matching step in fig. 2). The corresponding hard functions are and . The soft function got factorized into a usoft function at the scale that only resolves the wellseparated jets, and a csoft function that describes the csoft radiation between the two nearby jets at the scale . Note that and have a trivial color structure, since they are related to a collinear splitting for which the relevant color space is one dimensional. In other words, in the collinear limit the full parton color space separates into the subspace for partons and the subspace for the collinear splitting.
2.3 regime: one soft jet
Next, we consider the kinematic situation that the first jet becomes less energetic, while all jets remain well separated from each other, i.e.,
(12) 
In this case, all dijet invariant masses involving the first soft jet are all of the same order . This additional hierarchy leads to large logarithms of in eq. (8), appearing this time only in the hard function. There are no large logarithms in the soft function as it only depends on the angles between the jet directions, which do not exhibit any hierarchy. Hence, the appropriate EFT setup, which we identify with the regime of , only refactorizes the hard function. This type of setup was also considered in refs. Larkoski:2015zka (); Larkoski:2015kga () to calculate energycorrelation functions describing jet substructure. Note however, that their conjectured factorization theorem for the general jet case does not correctly account for color correlations.
The relevant modes in the regime are given in table 3. In addition to the usual collinear modes for the energetic jet sectors and the overall usoft modes, we have a softcollinear mode with momentum scaling that is responsible for the collinear dynamics of the soft jet. Its overall scaling is fixed by the kinematic constraint and the constraint imposed by the measurement of the jet resolution variable requiring that .^{3}^{3}3Here it is important that we are using a jet resolution variable like jettiness, which fixes the size of small lightcone component . Since we still have , this softcollinear mode cannot couple to any of the other wellseparated collinear modes. Hence, it is just a collinear mode with a smaller energy and consequently a smaller invariant mass, .
To match onto the regime, we perform the twostep matching shown in fig. 3. We first match QCD onto standard SCET with collinear sectors at the scale and a corresponding usoft sector at the scale . At this point, the soft jet is still unresolved and contained in the usoft sector. After decoupling the collinear and usoft modes, we match this theory onto . The virtuality of the collinear sectors is simply lowered to . The decoupled usoft sector with momentum scaling is matched onto the softcollinear mode for the now resolved jet 1 and the usoft sector at the lower scale . This involves nontrivial matching coefficients related to the soft gluon current, which appear when matching the hardscattering Lagrangian from the parent SCET onto the of . The softcollinear and usoft sectors can be decoupled via a second BPS field redefinition in the softcollinear sector. Since the parent usoft sector is equivalent to full QCD at a lower scale, this decoupling proceeds completely analogous to the usual matching from QCD to SCET. The final leadingpower Lagrangian has again all sectors completely decoupled,
(13) 
The Lagrangian for the softcollinear mode is given by the usual collinear Lagrangian, but with a different power counting for the label momenta.
mode  

collinear ()  
softcollinear ()  
ultrasoft 
The factorization theorem following from eq. (13) has the structure
(14) 
Compared to eq. (8), the hard coefficient got factorized into for hard external partons at the scale (arising from the first matching step in fig. 3) and a soft splitting coefficient describing the splitting of the parent usoft sector in SCET into the softcollinear and the usoft sector in at the scale (arising from the second matching step in fig. 3). The is a matrix in color space that promotes the hard coefficient from the parton color space to the full parton color space in which the soft function acts. Note that at leading power in the soft jet is initiated by a gluon, , since only gluon emissions are enhanced in the soft limit, and the natural scale for its jet function is .
2.4 regime: one soft jet collinear to another jet
We now consider the kinematic situation where the first two jets come close to each other and at the same time the first jet becomes soft. The remaining jets stay equally separated and energetic, i.e.,
(15) 
Hence, this case is characterized by the combination of the collinear and soft hierarchies in the dijet invariant masses, , while all remaining . Treating this case in either the or regimes with the corresponding generic scales would leave large logarithms of either or in the hard and/or soft functions. The resummation of both types of large logarithms is achieved in the regime of , which has not been discussed in the literature before. This EFT setup combines the expansion in the softness of jet 1 and the angle between jets 1 and 2, and is the theory connecting the and regimes. As we will see below, this kinematic situation can effectively be described within the regime by an appropriate choice of resummation scales in the hard sector that takes into account the softness of jet 1. This has been exploited in ref. Larkoski:2015kga (). It is nevertheless important to explicitly consider the regime in order to fully separate all scales and to show that all logarithms are resummed correctly in this way. This also shows that this kinematic situation cannot be described within the regime, which lacks the required refactorization of the soft sector. The regime is also useful to account for the overlap between the and regimes, see sec. 2.5, and to be able to handle more complicated overlapping hierarchies.
The relevant modes in the regime are summarized in table 4. Besides the usual collinear modes with the labels and the usoft modes, we have a softcollinear mode in the direction that describes the collinear dynamics of the soft jet, and a csoft mode that is responsible for the cross talk between the two nearby jets 1 and 2. As for the soft case, the scaling of the softcollinear mode is determined by and . And as for the collinear case, to be able to resolve the two nearby jets 1 and 2, the csoft mode is boosted in the lightcone direction with angular resolution scale . The constraint from the jet resolution measurement, , then fixes its scaling.
mode  

collinear ()  
softcollinear ()  
collinearsoft ()  
ultrasoft 
We now perform the threestep matching procedure shown in fig. 4. We first match QCD onto SCET with collinear modes and usoft modes with virtuality scales and , respectively. Next, we match onto an intermediate with standard collinear and usoft modes at the lower virtuality scales and , respectively, and a softcollinear sector in the direction at the lower scale with momentum scaling , which can resolve the angular size of the collinear sector. As before, the collinear, softcollinear, and usoft sectors are decoupled by appropriate BPS field redefinitions. At this point, the soft jet is not yet resolved and still contained in the softcollinear sector. This means that there is no nontrivial hard matching coefficient in this step, and as we will see in sec. 3, the matching of the operators in the hardscattering Lagrangian happens entirely at the level of soft Wilson lines. This also means that one could in principle directly construct this and match onto it from QCD (see e.g. refs. Bauer:2011uc (); Procura:2014cba ()).
In the last step in fig. 4, we match the intermediate with collinear sectors onto the final theory. Here, the virtualities of the collinear and usoft modes are simply lowered, with the collinear mode now being refined to the final collinear mode. The parent decoupled softcollinear sector is matched onto the final softcollinear mode for the now resolved jet 1, and the final csoft mode in the direction. (Hence, taking into account its full ancestry, the final csoft mode here could be referred to as a softcollinearsoft mode.) The corresponding matching coefficients are now related to the soft limit of the collinear splitting amplitudes or equivalently the collinear limit of the soft gluon current. Analogous to the regime, the csoft and softcollinear modes are decoupled by a BPS field redefinition. Note that the consistency of fig. 4 can be verified by taking the limit or for which it reduces to the matching for the and regimes, respectively.
The final fully decoupled leadingpower Lagrangian is given by
(16) 
where both and are collinear Lagrangians with the appropriate scaling of their label momenta.
The factorization theorem resulting from eq. (16) has the structure
(17) 
As in eqs. (2.2) and (14), the hard coefficient describes the production of hard partons at the scale . The coefficient now describes the softcollinear splitting at the scale . Compared to the regime in eq. (2.2), corresponds to the soft limit of the collinear splitting coefficient , whose scale got lowered from . Similarly, the scale of the csoft function is now lowered to . Compared to regime in eq. (14), corresponds to taking the collinear limit of the soft splitting coefficient , lowering its scale from and making it color diagonal. In addition, the soft sector got refactorized as in the regime leading to at the scale . As in the regime, the soft jet 1 is always initiated by a gluon with the natural scale for its jet function being .
2.5 Combining all regimes
We now discuss how to combine the factorization theorems for the different regimes as well as the nonhierarchical SCET limit to obtain a complete description for any . This will be generalized to the full jet phase space with arbitrary hierarchies among the in sec. 4. The goal is to be able to resum all logarithms of any ratios of and at the same time to reproduce the correct fixedorder result whenever there are no longer large hierarchies.
Each of the factorization theorems in eqs. (8), (2.2), (14), and (2.4) has been power expanded in the ratio of scales whose logarithms are being resummed. They thus receive power corrections in the corresponding scale ratio, which become in the nonhierarchical limit where that scale ratio is no longer small. To obtain a smooth and complete description, we basically need to add to the resummed result in a given kinematic region the relevant missing nonlogarithmic (“nonsingular”) corrections at fixed order, such that we reproduce the full fixedorder result everywhere. In addition, to ensure a smooth transition across different kinematic regions it is also important to smoothly turn off the resummation in any nonhierarchical limit. This can be achieved through a suitable choice of resummation profile scales Ligeti:2008ac (); Abbate:2010xh ().
A Venn diagram of the fixedorder content of the different theories is shown in fig. 5, from which the nonsingular corrections can be directly read off. The basic idea is to start from the inner most hierarchical (most expanded) case and go outwards step by step matching to the fixedorder content of the next less hierarchical (less expanded) case until we reach the outermost full QCD result. For jets this procedure will be discussed in some detail in sec. 3.5.
We start from the result which resums all kinematic logarithms in the limit and add nonsingular corrections to match it to the and results, which yields the combined cross section,
(18) 
The notation indicates that the hierarchy specified in brackets is not resummed but taken at fixed order. For example, for the logarithms of are resummed, while the logarithms of are not resummed, and are instead expanded to the same fixed order as they are present in . Hence, in the logarithms of are still resummed, while the fixedorder corrections that are singular in cancel between the two terms, such that is a power correction in .
Having obtained , we can further add the nonsingular corrections from SCET in the limit and eventually the nonsingular corrections from full QCD relevant in the limit ,
(19) 
Note that in our approach, the overlap between the and regimes is automatically taken care off via the separate regime. In ref. Larkoski:2015kga () this overlap is removed manually by subtracting it at the level of the factorized cross section, which yields technically the same result.
2.6 observables
Here we briefly discuss for type jet resolution variables, which constrain the transverse momenta within the jets rather than their invariant mass or small lightcone momenta. A simple example is jettiness with the broadening measure,
(20) 
where denotes the component perpendicular to the direction of the th jet. Other examples are the XCone measures with angular exponent Stewart:2015waa (). Measures of this type have been utilized for jet substructure studies using subjettiness Thaler:2010tr (); Thaler:2011gf (). These observables are in principle sensitive to the precise definition of the jet axes due to the fact that the recoil from soft emissions cannot be neglected. To keep the factorization theorem simple, one can employ the recoilinsensitive broadening axes Larkoski:2014uqa ().
The distinct feature of type observables is that all modes in the effective theory, i.e. collinear, soft, csoft and softcollinear, have the same virtuality. This directly follows from the fact that the measurement of constraints their components, which sets the scale of their virtuality . The scaling of the relevant modes in the different regimes is summarized in tables 5, 6, 7, and 8. The different modes are now parametrically separated in rapidity rather than virtuality, and the corresponding logarithms can be summed by using the rapidity renormalization group evolution Chiu:2011qc (); Chiu:2012ir ().
In the regime, there are again csoft modes mediating between the two nearby jets 1 and 2. As before, their scaling is determined by the requirement that they have a resolution angle and the measurement constraint . In the regime, the scaling of the softcollinear modes is fixed by the facts that and . Finally, the regime again combines the features of the and regimes.
The structure of the corresponding factorization theorems is analogous to those in eqs. (8), (2.2), (14), and (2.4). The essential difference is that the convolutions between soft and jet functions are now in transverse momentum variables, and involve the resummation of rapidity logarithms. Since the matching steps are insensitive to the details of the jet measurement, all the arising Wilson coefficients , , , , and are the same as for a type observable. The factorized cross sections for the different regimes can be combined to describe the complete phase space by accounting for the nonsingular corrections as discussed in sec. 2.5.
mode  

collinear ()  
soft 
mode  

collinear ()  
collinearcsoft ()  
soft 
mode  

collinear ()  
softcollinear ()  
soft 
mode  

collinear ()  
softcollinear ()  
collinearcsoft ()  
soft 
3 jets
In this section, we discuss in detail all kinematic regimes for jets, considering each hierarchy in turn. The jets are again ordered according to the kinematics such that
(21) 
As jet resolution variable we use 3jettiness as defined in eq. (3). For simplicity, in this section we use the geometric measure with so and
(22) 
In the exclusive jet limit (or more precisely at leading order in the power expansion in ), we can uniquely associate each jet with one of the partons in the underlying hard partonic scattering process, denoted as
(23) 
Since we label the jets by their kinematic ordering rather than their flavor, we use to denote the partonic channel, which in the present case can be any permutation of where stands for any quark flavor.
By evaluating all functions in the factorization theorems below at their natural scales and RG evolving them to the common arbitrary scale , all kinematic logarithms of , , and in their respective regimes as well as the logarithms of are resummed. The perturbative ingredients required for the resummation to NNLL are fully known. We give the oneloop results for the additional ingredients below. The required common RGE solutions and anomalous dimensions can be found for example in the appendices of Refs. Ligeti:2008ac (); Stewart:2010qs (); Berger:2010xi (); Bauer:2011uc (), and are not reproduced here.
3.1 Standard SCET regime:
We first review the notation and conventions for SCET helicity operators and the matching from QCD. We then discuss the factorization theorem for the standard SCET case where all three jets are equally energetic and well separated.
3.1.1 Helicity operators and matching to SCET
We start by briefly discussing SCET helicity operators Stewart:2012yh (); Moult:2015aoa (); Kolodrubetz:2016uim (), which are convenient for carrying out the matching from QCD onto SCET. In particular, they make it straightforward to construct the complete operator basis in SCET with multiple collinear sectors. We summarize the necessary definitions and some basic properties here, and refer for details to ref. Moult:2015aoa (). A summary of the common SCET notation and conventions is given in app. A.
Collinear quark and gluon jet fields in the collinear sector with specified helicity are defined as
(24) 
which involve the polarization vectors and spinors for massless onshell momenta
(25) 
Since fermions always come in pairs, we can use currents with fixed helicity as basic building blocks of helicity operators,
(26) 
The leptonic vector current is defined analogously but does not contain any QCD Wilson lines. The normalization of the fermion currents and gluon fields are chosen such that the treelevel Feynman rules for the corresponding final state give delta functions of the label momenta ,