Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in GENEVA

Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in GENEVA

Abstract

We extend the lowest-order matching of tree-level matrix elements with parton showers to give a complete description at the next higher perturbative accuracy in at both small and large jet resolutions, which has not been achieved so far. This requires the combination of the higher-order resummation of large Sudakov logarithms at small values of the jet resolution variable with the full next-to-leading-order (NLO) matrix-element corrections at large values. As a by-product, this combination naturally leads to a smooth connection of the NLO calculations for different jet multiplicities. In this paper, we focus on the general construction of our method and discuss its application to and collisions. We present first results of the implementation in the Geneva Monte Carlo framework. We employ -jettiness as the jet resolution variable, combining its next-to-next-to-leading logarithmic resummation with fully exclusive NLO matrix elements, and Pythia 8 as the backend for further parton showering and hadronization. For hadronic collisions, we take Drell-Yan production as an example to apply our construction. For jets, taking from fits to LEP thrust data, together with the Pythia 8 hadronization model, we obtain good agreement with LEP data for a variety of -jet observables.

Keywords:
QCD, Monte Carlo, NLO Computations, Resummation, Jets, Collider Physics

DESY 12-221

November 29, 2012

[-7ex]

1 Introduction

Accurate and reliable theoretical predictions for measurements at collider experiments require the inclusion of QCD effects beyond the lowest perturbative accuracy in the strong coupling . This is especially important in the complex environment of the LHC, which requires precise predictions for a broad spectrum of observables. Higher-order corrections in are important to predict total cross sections and other inclusive observables. Exclusive jet observables, such as jet-vetoed cross sections, require the all-orders resummation of logarithmically enhanced contributions. For many observables, an accurate description across phase space demands a combination of both types of corrections. For experimental analyses to benefit from these advances, it is crucial to provide the best possible theoretical predictions in the context of fully exclusive Monte Carlo event generators.

The goal of modern Monte Carlo programs is to provide a proper description of the physics at every jet resolution scale. This is the motivation for the by-now standard combination of matrix elements with parton showers (ME/PS). Catani:2001cc (); Lonnblad:2001iq () Here, the parton shower provides the correct lowest-order description at small jet resolution scales, where the resummation of large Sudakov logarithms is needed, while at large jet resolution scales the exact tree-level matrix elements are needed to provide the correct lowest-order description. Hence, the ME/PS merging provides theoretical predictions at the formally leading accuracy relative to the lowest meaningful perturbative order. Once one has a consistent matching between these two limits of phase space, the possibility to include exact tree-level matrix elements for several jet multiplicities follows almost automatically by iteration.

Given the necessity of higher-order perturbative corrections to make accurate predictions, it is important to extend the perturbative accuracy of the Monte Carlo description to formal accuracy relative to the lowest order. This requires including the formally next higher-order corrections that are relevant at each scale. At small scales, i.e., small values of the jet resolution variable, this requires improving the leading-logarithm (LL) parton shower resummation with higher-order logarithmic resummation, while at large scales this requires including the fully differential next-to-leading-order (NLO) matrix elements. It is important to realize that typically a large part of phase space, often including the experimentally relevant region, is characterized by intermediate scales, i.e., by a transition from small to large scales. In the end, providing an accurate description of this transition region requires a careful combination of both types of corrections.

Such a Monte Carlo description at relative accuracy across phase space has never been achieved and is the subject of our paper. (We briefly summarize the existing efforts to combine NLO corrections with parton showers in section 1.1 below.) The crucial starting point in our approach is that all perturbative inputs to the Monte Carlo are formulated in terms of well-defined physical jet cross sections Bauer:2008qh (); Bauer:2008qj (). This allows us to systematically increase the perturbative accuracy by incorporating results for the relevant ingredients to the desired order in fixed-order and resummed perturbation theory.

An essential aspect of any higher-order prediction is a reliable estimate of its perturbative uncertainty due to neglected higher-order corrections. To the extent that parton shower Monte Carlos provide perturbative predictions, they should be held to the same standards. An important benefit in our approach is that we have explicit control of the perturbative uncertainties and are able to estimate reliable fixed-order and resummation uncertainties. As a result, in Geneva each event comes with an estimate of its perturbative uncertainty; i.e., Geneva provides event-by-event theory uncertainties.1

In our approach, the Monte Carlo not only benefits from the resummation, but in turn also provides important benefits to analytic resummed predictions. For one, it greatly facilitates the comparison with experimental data, as it allows easy application of arbitrary kinematic selection cuts, which can often be tedious to take into account in analytic predictions. More importantly, resummed predictions require nonperturbative inputs which can be treated as power corrections at intermediate scales but become corrections at very small scales. Here, these are provided “on-the-fly” by the nonperturbative hadronization model. In essence, we are able to combine the precision and theoretical control offered by higher-order resummed predictions with the versatility and flexibility offered by fully exclusive Monte Carlos.

In this paper, we focus on the theoretical construction. We leave a discussion of the implementation details of the Geneva Monte Carlo framework to a dedicated publication.2 We will however highlight some of the main technical issues we had to overcome and discuss some implementation details in the application sections. In the remainder of this section, we briefly summarize the existing efforts to include NLO corrections in parton shower Monte Carlos and give a short overview of our basic construction. In section 2, we discuss in detail the requirements to obtain full accuracy as well as our method to achieve it. In section 3, we discuss the application to jets, where we combine next-to-next-to-leading logarithmic (NNLL) resummation with NLO matrix elements, and present results from the implementation in Geneva together with a comparison to LEP measurements. In section 4, we discuss the application to hadronic collisions and show first results for Drell-Yan production, jets, obtained with Geneva. We conclude in section 5.

1.1 Previous Approaches Combining NLO Corrections with Parton Showers

Over the past decade, many steps have been taken to include NLO corrections into Monte Carlo programs Collins:2000gd (); Collins:2000qd (); Potter:2001ej (); Dobbs:2001dq (); Frixione:2002ik (); Frixione:2003ei (); Kramer:2003jk (); Soper:2003ya (); Nagy:2005aa (); Kramer:2005hw (); Nason:2004rx (); Frixione:2007vw (); Alioli:2010xd (); Torrielli:2010aw (); Hoche:2010pf (); Frixione:2010ra (); Frederix:2011zi (); Platzer:2011bc (). By now, the MC@NLO Frixione:2002ik (); Frixione:2003ei () and Powheg Nason:2004rx (); Frixione:2007vw (); Alioli:2010xd () methods are routinely able to consistently combine the fixed NLO calculation of an inclusive jet cross section for a given jet multiplicity with additional parton showering. These methods have also been extended to include the full tree-level matrix elements for additional jet multiplicities Bauer:2008qh (); Hamilton:2010wh (); Hoche:2010kg (); Giele:2011cb ().

Recently, efforts have been made to extend these approaches in order to combine NLO matrix elements for several jet multiplicities with parton showers Lavesson:2008ah (); Alioli:2011nr (); Hoeche:2012yf (); Gehrmann:2012yg (); Frederix:2012ps (); Platzer:2012bs (); Lonnblad:2012ix (). We will discuss some issues faced by some of these approaches in section 2.1.5. Here, we would like to stress that including several NLO matrix elements by itself does not provide a full extension of the lowest-order ME/PS matching to relative perturbative accuracy, since the fixed NLO corrections only suffice to increase the perturbative accuracy in the region of large jet resolution scales. To the same extent that the inclusion of the LL Sudakov factors in the ME/PS merging are needed to get meaningful results at intermediate and small jet scales, higher-order resummation is necessary to improve the perturbative accuracy in this region.

In our approach, the full information from NLO matrix elements for several jet multiplicities is automatically included as follows: For a given Born process with partons, a small jet scale corresponds to the exclusive -jet region, and here the -parton virtual NLO corrections are incorporated in conjunction with the higher-order resummation; in fact, they are a natural ingredient of it. On the other hand, a large jet scale corresponds to the inclusive -jet region with additional hard emissions. Here, the -parton virtual NLO corrections are included in the usual way by the fixed NLO calculation for jets.

1.2 Brief Overview of Our Construction

The starting point of our approach is the separation of the inclusive -jet cross section into an exclusive -jet region and an inclusive -jet region,

(1)

Here is a suitable resolution variable, which is a function of , and denotes the fully differential cross section for a given . In ME/PS merging, this role is played by the variable that determines the merging scale. However, in our case the parameter is not a jet-merging cut but instead serves as an infrared cutoff for the calculation of and ideally is taken as small as possible.

In the -jet region at small (both above and below ), logarithms of become large and must be resummed to maintain consistent perturbative accuracy to some order in . On the other hand, in the -jet region at large , a fixed-order expansion in will suffice. To consistently match the resummed and fixed-order calculations, we use the following prescription for the jet cross sections entering in eq. (1):

(2)

The superscript “resum” indicates an analytically resummed calculation and “FO” indicates a fixed-order calculation or expansion. This construction properly reproduces the fixed-order calculation at large , the resummed calculation at small , and smoothly interpolates between them.

It is straightforward to extend our formulation to combine higher jet multiplicities at NLO with higher-order resummation, as we will show. This is done by replacing in eq. (1.2) with an inclusive -jet cross section separated into the exclusive -jet and inclusive -jet cross sections and iteratively applying eq. (1.2).

The key ingredients in our approach are the higher-order resummation of the jet resolution variable, the fully differential fixed-order calculation, and the parton shower and hadronization. While each of these components is known, there is a sensitive interplay of constraints between them that must be satisfied to achieve a consistent combination. This is precisely what is accomplished in the Geneva framework and is the focus of this paper.

2 General Construction

In this section, we derive our theoretical construction in a process-independent manner. We start in section 2.1 with a slightly simplified setup, considering the singly differential spectrum in the jet resolution variable. We use this to discuss in detail the perturbative structure and the accuracy in the different phase space regions. In section 2.2, we discuss the extension to the fully differential case and how to combine the fixed-order expansion and resummation in this situation. In section 2.3, we further generalize these results to include several jet multiplicities by iteration. Finally, in section 2.4, we discuss the Monte Carlo implementation and how to attach parton showering and hadronization.

2.1 What Resummation Can Do for Monte Carlo

Basic Setup

The basic idea of Monte Carlo integration is to randomly generate points in phase space (“events”) that are distributed according to some differential (probability) distribution. By summing over all points that satisfy certain selection criteria, we are able to perform arbitrary integrals of the distribution. In our case, that distribution is the fully differential cross section, allowing one to compute arbitrary observables. For simplicity, we will first focus on the singly differential cross section in some phase space resolution (or jet resolution) variable of dimension one. The precise definition of is not important at the moment, so we keep it generic for now. We use the convention that the limit corresponds to Born kinematics, i.e., the tree-level cross section is . We also require that is an IR-safe observable, such that the differential cross section can in principle be well defined to all orders in perturbation theory and for contains no IR divergences.

To give an example, for our application to jets in section 3, we will use 2-jettiness , where is the usual thrust Farhi:1977sg (). Alternatives include other -jet event shapes. For Drell-Yan in section 4, we will use beam thrust Stewart:2009yx (). An alternative would be the of the leading jet. If the Born cross section we are interested in has signal jets,3 then could be -jettiness or the largest of any additional jet. The important point is that we can think of as a resolution variable which determines the scale of additional emissions in the phase space, such that for there are no emissions above the scale . For later convenience, we also define the dimensionless equivalent of as

(3)

Here, is the relevant hard-interaction scale in the Born process, e.g., for jets or for Drell-Yan . In terms of , the limit corresponds to the exclusive limit close to Born kinematics. For , there are additional emissions at the hard scale , which means we are far away from Born kinematics and we should switch the description to consider the corresponding Born process with one additional hard jet.

To describe the differential spectrum, we want the Monte Carlo to generate events at specific values of , which are distributed according to the differential cross section . The total cross section is then simply given by summing over all events,

(4)

The essential problem every Monte Carlo generator faces is that in perturbation theory the differential cross section contains IR divergences from real emissions for , which only cancel against the corresponding virtual IR divergences upon integration over the region. As a result, the perturbative spectrum for can only be defined in a distributional sense in terms of plus and delta distributions [see eq. (8) below]. To deal with this, we have to introduce a small cutoff and define the cumulant of the spectrum as

(5)

In the Monte Carlo, the total cross section is then obtained by combining the cumulant and spectrum as

(6)

In practice, this is implemented by generating two distinct types of events: (i) events that have and relative weights given by , and (ii) events that have nonzero values and relative weights given by . The first type of events have Born kinematics and represents the tree-level and virtual corrections together with the corresponding real emissions integrated below . The second type of events contains one or more partons in the final state, since the real-emission corrections determine the shape of the spectrum for nonzero . We now have two basic conditions:

  1. From a numerical point of view, we want the value of to be as small as possible, so as to describe as much differential information as possible. In practice, our ability to reliably compute the cumulant in perturbation theory sets a lower limit on the possible value of few times .

  2. Since is an unphysical parameter, we want the dependence on it to drop out (to the order we are working at). In practice, this is guaranteed by including the corresponding dominant higher-order corrections in the cumulant and spectrum.

Perturbative Expansion and Order Counting

In perturbation theory, the differential cross section in and the cumulant in have the general form

(7)

where we have distinguished “singular” and “nonsingular” contributions. For , the singular terms in scale like , while the nonsingular terms in contain at most integrable singularities. For the cumulant, this means that contains all terms in enhanced by logarithms , while .

The singular part of the spectrum is given by

(8)

where denotes the Born cross section, and we denote the usual plus distributions as

(9)

They encode the cancellation between real and virtual IR divergences. The corresponding singular contribution to the cumulant cross section integrated up to is

(10)

At , only the coefficients with contribute, so has logarithms up to , while has logarithms up to , where we use the abbreviations

(11)

The expansion of the coefficients and in the singular contributions can be written as

(12)

Similarly, the expansion of the nonsingular contributions can be written as

(13)

Using eqs. (12) and (13), the spectrum and cumulant up to are given by

(14)

Note that the constant term in the singular corrections, which contains the finite virtual corrections to the Born process, only appears in the cumulant.

Figure 1: Illustration of the different parametric regions in the jet resolution.

We now distinguish three parametrically different regions in , which are illustrated in figure 1:

  • Resummation (“peak”) region : In this limit, the logarithms in the singular contributions are large, such that parametrically one has to count4

    (15)

    This means one has to resum the towers of logarithms in the spectrum and in the cumulant in eq. (2.1.2) to all orders in to obtain a meaningful perturbative approximation at some order. At the same time, the nonsingular corrections can be regarded as power suppressed, since they are of relative .

  • Fixed-order (“tail”) region : In this limit, the logarithms are not enhanced, and a fixed-order expansion in is applicable. The singular and nonsingular contributions are equally important and both must be included at the same order in . In particular, there are typically large cancellations between these for , so it is actually crucial not to resum the singular contributions in this region, since otherwise this cancellation would be spoiled.

  • Transition region: The transition between the resummation and fixed-order regions.

There are of course no strict boundaries between the different regions. This is why it is important to have a proper description not just in the two limits but also in the transition region, which connects the resummation and fixed-order regions. In fact, in practice the experimentally relevant region is often somewhere in the transition region, where both types of perturbative corrections can be important.

Lowest Perturbative Accuracy

For the Monte Carlo to provide a proper description at all values of , it has to include at least the lowest-order terms relevant for each region. Keeping only these, and dropping all other terms, the spectrum and the cumulant are given by

(16)

where the functions and are given in terms of the coefficients in eq. (12) as

(17)

The and resum the leading-logarithmic series in the cross section, which we denote as LL. The functions and resum the next-to-leading-logarithmic series in the cross section, which we denote as NLL.

In the resummation region at , the LL terms in the spectrum scale as (relative to the overall scaling) and provide the lowest level of approximation. The NLL terms scale as , and one can argue about whether they are needed as well in order to get a meaningful lowest-order prediction. Formally, they are necessary to obtain the spectrum at , which one might consider the natural leading-order scaling of the spectrum (or equivalently if one does not want to rely on the enhancement of the LL series). Experience shows that the NLL terms are indeed numerically important. For example, in analytic resummations, one rarely gets a sensible prediction without going at least to NLL. Similarly, to obtain sensible predictions from a parton shower, it is almost mandatory to include important physical effects such as momentum conservation in the parton splitting and the choice of scale Catani:1990rr (). In the cumulant, the LL series in scales as and must be included. The NLL series in scales as and, for consistency, should be included in the cumulant if it is included in the spectrum.

In the fixed-order region at , the lowest meaningful order in the spectrum is given by the complete terms, requiring one to include the and terms, which are part of the and functions, as well as the nonsingular corrections . Since we take to be small, the cumulant is always in the resummation region. Hence, its nonsingular corrections [see eq. (2.1.2)] are suppressed by and can be safely neglected.

The leading level of accuracy in eq. (2.1.3) closely corresponds to what is achieved in the standard ME/PS matching. In this case, the LL resummation is provided by the parton shower Sudakov factors (either generated by the shower or multiplied by hand), where the jet resolution variable corresponds to the shower evolution variable, since that is the variable for which the shower directly resums the correct LL series. The LL series has a well-known and simple exponential structure,

(18)

such that

(19)

The resummation exponent at LL is given by the integral over the leading term in the spectrum. This is precisely what the standard parton shower veto algorithm exploits to generate the resummation exponent. The analogous structure does not hold at NLL, which is why the parton shower cannot resum the NLL series by exponentiating the integral of the term. As already mentioned, in practice, parton showers include important partial NLL effects, so practically this provides a numerically close approximation to the correct NLL series. The nonsingular corrections in the spectrum, , are obtained by including the full tree-level matrix element for one additional emission. Since the full matrix element also includes the and terms, this requires a proper matching procedure to avoid double counting these terms. At LL, a simple way to do this is to multiply the full fixed-order result from the matrix element with the shower’s LL resummation exponent,

(20)

which corresponds to the CKKW-L Catani:2001cc (); Lonnblad:2001iq (); Krauss:2002up (); Lavesson:2005xu () procedure. The reason this gives the spectrum correctly at LL is the simple structure in eq. (19), where the LL exponent multiplies the term in the spectrum.5. At large , the exponent in eq. (20) can be expanded as , so eq. (20) gives the correct leading fixed-order result.

Compared to eq. (2.1.3), the NLO matching performed in MC@NLO and Powheg amounts to adding to the cumulant the singular constant, containing the virtual corrections, as well as the nonsingular contributions . Assuming the same set of NLL terms are included in the cumulant and spectrum, this achieves that inclusive quantities that are integrated over a large range of , such as the total cross section, are correctly reproduced at fixed NLO, which provides them with accuracy. In these approaches, the goal is not to improve the perturbative accuracy of the spectrum (or the cumulant at small ), which has the same leading accuracy as in eq. (20).

Next-To-Lowest Perturbative Accuracy

We now want to improve the Monte Carlo description in eq. (6) from the lowest-order accuracy, given by eq. (2.1.3), to the next-to-lowest perturbative accuracy in . This requires us to include the appropriate higher-order corrections in each region, which gives

(21)

where we denote the series of logarithms resummed by the functions and by NLL and the series resummed by and by NNLL. They can again be written in terms of the coefficients in eq. (12) as

(22)

In the resummation region, the NLL series in the spectrum scales as and thus provides the correction to the LL series in . Similarly, the NNLL series scales as providing the correction to the NLL series in . They can again be obtained by performing the standard resummation in the exponent of the cross section to NLL and NNLL respectively. (Here, NLL refers to those parts of the full NNLL resummation that arise from the combination of the one-loop matching corrections with the NLL resummation, see section 3.1.1 and table 2.)

In the fixed-order region, increasing the perturbative accuracy by requires the complete corrections, including the nonsingular corrections. Similarly, for the cumulant, and resum the NLL and NNLL series of logarithms, which scale as and , respectively, and provide the improvement over the LL and NLL series in and . In addition, going to the next higher order in the cumulant requires including the full singular constant ,6 as well as the nonsingular corrections , which both scale as .

It is instructive to see where the information from the virtual NLO matrix elements enters in eq. (2.1.4). As already mentioned, the virtual NLO corrections to the Born process are given by . In addition, by multiplying the LL series it contributes part of and . Hence, consistently combining the virtual corrections with the resummation requires one to go to at least NLL. The virtual NLO corrections with one extra emission (plus the integral over the two-emission tree-level matrix element) yield the full corrections in the spectrum, i.e., both the singular terms as well as the nonsingular terms in eq. (2.1.2). Adding these corrections again requires one to avoid double counting the singular terms that are already included in the resummation. In analytic resummation, it is well known how to do this, namely by simply adding the nonsingular corrections. These are obtained by taking the difference of the full NLO corrections and the singular NLO corrections, where the latter are given by expanding the resummed result to fixed order. Since this construction involves the virtual contribution to both the Born process and the process with one extra emission, we see that going consistently to higher order in both the resummation and fixed-order regions naturally leads to a combination of the information from two successive NLO matrix elements.

Merging NLO Matrix Elements with Parton Shower Resummation Only

We stress that, for a description at the next-higher perturbative accuracy across the whole range in , it is not sufficient to include the fixed NLO corrections to the spectrum and take care of the double counting with the parton shower resummation. This only provides the proper NLO description in the fixed-order region at large . In the transition and resummation regions, a proper higher-order description necessitates higher-order resummation. Of course, this is not a problem if the only goal is to improve the fixed-order region at large , as is the case for example in a recent MC@NLO publication Frederix:2012ps ().

However, including the fixed NLO corrections outside the fixed-order region, as is done in Sherpa’s recent NLO merging Hoeche:2012yf (); Gehrmann:2012yg (), can actually make things worse in two respects: First, numerically this will typically force the spectrum to shift toward the fixed-order result and away from the resummed one. Since this can shift the spectrum in the wrong direction, it can potentially make the result less accurate.7 At the same time, the perturbative uncertainties from fixed-order scale variation decrease, which only aggravates this problem. Multiplying the NLO corrections to the spectrum with LL parton shower Sudakov factors (see, e.g., ref. Hamilton:2012np ()) can mitigate this to some extent but does not solve the problem. The only consistent way to include the fixed NLO corrections to the spectrum outside the fixed-order region, and in particular obtain reliable perturbative uncertainties, is to properly combine them with a higher-order resummation.

Second, this explicitly spoils the formal accuracy of the inclusive cross section. To see this, consider adding the fixed NLO corrections to the lowest-order spectrum and cumulant in eq. (2.1.3), properly taking care of the double counting at , which gives

(23)

Using these expressions yields for the inclusive cross section

(24)

While the first two terms give the correct NLO inclusive cross section, the terms induced by the fixed NLO corrections in the spectrum formally scale as and and therefore spoil the formal perturbative accuracy for the inclusive cross section and in fact for any inclusive observable. This directly contradicts the claim in refs. Hoeche:2012yf (); Gehrmann:2012yg () that this description maintains the higher-order accuracy of the underlying matrix elements in their respective phase space range. It only preserves the fixed terms, which in the context of combining fixed-order corrections with a logarithmic resummation is necessary but not sufficient to preserve the higher perturbative accuracy.

This problem cannot be avoided by multiplying the corrections in the spectrum with the LL parton shower Sudakov factors, since this does not provide the proper NLL and NNLL series. Note also that we have already assumed in eq. (2.1.5) that the full NLL series is included in the spectrum and cumulant. In general, the parton shower cannot provide this, which means there will be even terms induced in eq. (2.1.5).

Pragmatically, the inclusive cross section can be restored to formal accuracy by either explicitly including the corresponding corrections in the cumulant to cancel these terms, where numerical methods to do so have been described very recently in refs. Lonnblad:2012ng (); Platzer:2012bs (); Lonnblad:2012ix (), or alternatively by explicitly restricting the fixed NLO corrections in the spectrum to the fixed-order region at large , such that the induced terms in the total cross section are not logarithmically enhanced and are formally . This is essentially the approach taken in ref. Frederix:2012ps (). However, neither of these approaches improves the perturbative accuracy in the spectrum outside the fixed-order region.

2.2 What Monte Carlo Can Do for Resummation

For being the resolution variable between and more than jets, we showed in the previous subsection that combining the NLO matrix-element corrections for and partons at the level of the singly differential spectrum is equivalent to combining the NNLL resummation of the singular contributions with the higher-order nonsingular contributions. Our goal now is to extend this singly differential description to the fully differential case, in order to use the full -parton and -parton information of the matrix elements. We will use the notation (N)LO or (N)LO to indicate up to which fixed order in the -parton or -parton matrix elements are included.

To start with, it is straightforward to generalize the jet resolution spectrum and its cumulant to include the full dependence on the -body Born phase space,

(25)

such that eq. (6) becomes

(26)

Here, is the inclusive -jet cross section. The discussion in section 2.1 can be precisely repeated in this case, since the perturbative structure of the differential spectrum with respect to is precisely the same as in eqs. (7) and (8). Namely, we can write it as the sum of singular and nonsingular contributions,

(27)

The nonsingular contributions are general functions of and , but as before are integrable in for . The singular contributions have the structure

(28)

where is now the fully differential Born cross section. Since the singular contributions arise from the cancellation of virtual and real IR singularities, which only know about , their dependence naturally factorizes from the kinematics of the underlying hard process. This is what allows the resummation of the singular terms to higher orders for a given point in . At LL, the entire dependence is that of the Born cross section. At higher logarithmic orders, this is not the case anymore, since the coefficients can have nontrivial dependence. In addition, the precise definition of also becomes important. Depending on its definition, the higher-order singular coefficients can depend on clustering effects or other types of nonglobal logarithms Dasgupta:2001sh (); Dasgupta:2002bw (); Delenda:2006nf (); Hornig:2011tg (); Kelley:2012kj (); Kelley:2012zs (), which can be difficult to resum to high enough order with currently available methods. Therefore, it is important to choose a resolution variable with simple resummation properties. An example is -jettiness, for which the complete NNLL resummation for arbitrary is known Stewart:2010tn (); Jouttenus:2011wh (). For the purpose of our discussion below, we will assume that a resummed result for the spectrum and its cumulant in eq. (26) at sufficiently high order is available to us.

We can think of the cumulant in eq. (26) as the exclusive -jet cross section with no additional emissions (jets) above the scale , while the spectrum for is the corresponding inclusive -jet cross section. While the cumulant is differential in and thus already as differential as it can be, the spectrum contains a projection from the full phase space down to . To also be fully differential in the -jet phase space, we can generalize eq. (26) to

(29)

where denotes the fully differential spectrum for a given . We explicitly denote the dependence on and to clearly distinguish the spectrum from the cumulant. We have also used the shorthand notation

(30)

where denotes a projection from an -body phase space point to an -body phase space point. This projection defines what we mean by jets at higher orders in perturbation theory. Note that beyond LO, both the cumulant and spectrum must be well-defined jet cross sections; i.e., they require a specific IR-safe projection from to for both and . We will see below where this definition enters. Using eq. (29) at the next-higher perturbative accuracy requires us to combine the higher-order resummation in for the cumulant and spectrum with the fully exclusive -jet and -jet fixed-order calculations at NLO and NLO. To achieve this, we have to construct appropriate expressions for the cumulant and the spectrum , which we do in the next two subsections.

Matched Cumulant

We start by discussing the cumulant in eq. (29). Since the resummation is naturally differential in the of the underlying Born process, we can combine the resummed result with the fixed-order one by adding the fixed-order nonsingular contributions to it,

(31)

The first term contains the resummed contributions, while the difference of the two terms in square brackets provides the remaining nonsingular corrections that have not already been included in the resummation. The NLO fixed-order result is given by

(32)

where and are the -parton and -parton tree-level (Born) contributions, is the -parton one-loop virtual correction, and we abbreviated

(33)

Here, implements the definition of . The NLO result also depends on the projection from to , i.e., the precise NLO definition of . However, this dependence only appears in the nonsingular corrections. For a given definition of , the singular NLO corrections do not depend on how the remaining phase space is projected onto , since they arise from the IR limit in which all (IR-safe) definitions agree. In eq. (31), the singular contributions inside the full fixed-order cumulant, are canceled by the NLO expansion of the resummed result at NLL or higher, leaving only the nonsingular fixed-order contributions in square brackets.

Matched Spectrum

To properly combine the higher-order resummation in with the fully differential -jet fixed-order calculation, the inclusive -jet spectrum in eq. (29) has to fulfill two basic matching conditions,

Condition 1: (34)
Condition 2: (35)

The first condition states that integrating the fully differential spectrum over the additional radiative phase space has to reproduce the correct spectrum in including the desired resummation and fixed-order nonsingular corrections, such that eq. (29) reproduces eq. (26). The second condition states that the fixed-order expansion of the fully differential spectrum has to reproduce the full -jet fixed-order calculation, where at NLO,

(36)

Here, and are the -parton and -parton tree-level (Born) contributions, and is the -parton one-loop virtual correction. Integrating over in the last term now requires a projection from to ,

(37)

analogous to eq. (30), which now defines precisely what we mean by jets at NLO.

In principle, there is some freedom to construct an expression for that satisfies both conditions to the order one is working. Our master formula to combine the resummed spectrum with the fully differential is given by

(38)

Expanding the right-hand side to a given fixed order, we can see immediately that Condition 2 is satisfied by construction. Imposing Condition 1 yields the consistency (or “matching”) condition

(39)

If the resummed result already has the nonsingular contributions at the desired fixed order added in, then the term in brackets is by construction equal to unity for any value of . Otherwise, the expansion of the resummed result reproduces the singular terms of the full fixed-order result, leaving the nonsingular fixed-order contributions, such that we get

(40)

Here, denotes the pure resummed result only containing the resummation of the singular contributions. Hence, eq. (38) not only multiplies in the additional dependence on at fixed order, but if needed also adds the nonsingular corrections to the spectrum multiplied by the higher-order resummation factor. (Note that for the expansion of the resummed result to indeed reproduce all the singular terms at the desired fixed order, the resummation has to be carried out to sufficiently high order, which we have already seen in section 2.1.)

To apply Condition 1, we have to integrate eq. (36) using the projection onto and in eq. (33). Therefore, to get the correct spectrum at NLO, the projection in eq. (37) has to satisfy

(41)

i.e., it has to preserve the value of when constructing the projected point. Usually, the simplest way to handle this would be to use the left-hand side to define . However, in our case, eq. (41) provides a very nontrivial condition on the projection since is already defined by our choice of jet resolution variable, which in particular has to be resummable. This turns out to be a nontrivial technical challenge one has to overcome to be able to satisfy Condition 1. We will see where this enters in section 3.1.2 and section 4.1.2.

Note that to ensure that the resummation factor in square brackets in eq. (38) is well behaved in the fixed-order region at large , it is important to turn off the resummation such that the ratio of the resummed spectrum and its expansion becomes up to higher fixed-order corrections. In principle, the fixed-order result in the denominator can also become negative at very small values of . This is not a problem in practice, since this region is explicitly avoided by imposing the cut .

Perturbative Accuracy and Order Counting

The appropriate order counting in the resummation and fixed-order regions is precisely the same as in section 2.1, so there is no need to repeat it here. Applying eq. (38) at the very lowest order, namely LL resummation with LO fixed-order corrections, we get

(42)

where