Contents

MAN/HEP/2011/16


Non–global logs and clustering impact on jet mass with a jet veto distribution

Kamel Khelifa–Kerfa111Kamel.Khelifa@hep.manchester.ac.uk

School of Physics & Astronomy, University of Manchester,

Oxford Road, Manchester, M13 9PL, U.K.



Abstract

There has recently been much interest in analytical computations of jet mass distributions with and without vetos on additional jet activity [1, 2, 3, 4, 5, 6]. An important issue affecting such calculations, particularly at next–to–leading logarithmic (NLL) accuracy, is that of non–global logarithms as well as logarithms induced by jet definition, as we pointed out in an earlier work [3]. In this paper, we extend our previous calculations by independently deriving the full jet–radius analytical form of non–global logarithms, in the anti– jet algorithm. Employing the small–jet radius approximation, we also compute, at fixed–order, the effect of jet clustering on both and colour channels. Our findings for the channel confirm earlier analytical calculations of non–global logarithms in soft–collinear effective theory [5]. Moreover, all of our results, as well as those of [3], are compared to the output of the numerical program EVENT2. We find good agreement between analytical and numerical results both with and without final state clustering.

1 Introduction

Event and jet shape variables have long served as excellent tools for testing QCD and improving the understanding of its properties (for a review, see [7]). Event/jet shape distributions have been used to extract some prominent parameters in QCD including the strong coupling and the quark–gluon colour ratio [8]. Due to the fact that shape variables are, by construction, linear in momentum, they exhibit a strong sensitivity to non–perturbative (NP) effects [7, 9]. They have thus been exploited to gain a better analytical insight into this QCD domain [7, 10]. Furthermore, jet shapes have been used not only to study the jet structure of hadronic final states, including jet multiplicities, jet rates and jet profiles (Ref. [11] and references therein), but also the subjet structure, or substructure, of the jets themselves (for a recent example, see [1]). The latter subject has received significant attention in recent years, particularly in the area of boosted objects with the aim to separate the decay products of Beyond Standard Model (BSM) particles from QCD background at LHC (for a review, see [12]).

Although shape variables are, by construction, Infrared and Collinear (IRC) safe, fixed–order perturbative (PT) calculations break down in regions of phase space where the shape variable is small. These regions correspond to gluon emissions that are soft and/or collinear to hard legs and lead to the appearance of large logs that spoil the PT expansion of the shape distribution [11] (and references therein). While measured shape distributions have a peak near small values of the shape variable and then go to zero, fixed–order analytical distributions diverge. To deal away with these divergences and successfully reproduce the experimentally–seen behaviour, one ought to either perform an all–orders resummation of the large logs, matched to fixed–order result, or rely on Monte Carlo event generators. We are concerned, in the present paper, with the resummation method as it paves the way for a better understanding of QCD dynamics including the process of multiple gluon radiation. The general form of resummed distributions for observables that have the property of exponentiation can be cast as [11]

(1.1)

where , is an expansion in with constant coefficients that can be inferred from fixed–order calculations and collects terms that are proportional to powers of the shape variable . The function resums all the leading logs (LL) , while resums the next–to–leading logs (NLL) and so on.

There are two types of jet shape observables 222from the point of view of our calculations in this paper.: global and non–global [13]. Global observables are shape variables that are sufficiently inclusive over the whole final state phase space. The resummation of such variables, e.g, thrust, heavy jet mass and broadening, up to NLL accuracy have long been performed [14, 15]. The resultant resummed distributions were then matched with NLO fixed–order results for a better agreement with measurements over a wide range of values of the shape variable [14, 16]. In the recent past, the NNLL NLO distribution has been obtained for energy–energy correlation [17], as well as NNLL NNLO [18, 19] for the thrust distribution [20], both in annihilation processes in QCD. Within the framework of Soft and Collinear Effective Theory (SCET) [21, 22], the NLL resummation for various event/jet variables have been performed [23, 24] and used, after matching to NNLO, for a precise determination of the coupling constant . The extracted value is consistent with the world average with significant improvements in the scale uncertainty.

At hadron colliders, what one often measures instead is jets, which only occupy patches of the phase space. The corresponding jet shape variables are thus non–inclusive, or non–global, and the resummation becomes highly non–trivial even at NLL level. Consider, for example, measuring the normalised invariant mass, , of a subset of high– jets in multijet events. A veto is applied on final state soft activity to keep the jet multiplicity fixed. Jets are only defined through a jet algorithm, which generally depends on some parameters such as the jet size  [25]. We are thus faced with a multi–scale (, hard scale, veto, jet size) problem where potentially large logs in the ratios of these scales appear. In addition to the Sudakov leading logs, , coming from independent primary gluon emissions, there are large subleading non–global logs (NGLs) of the form , where and are two different scales, coming from secondary 333These are emissions that are not radiated off primary hard legs. correlated gluon emissions.

We argued in [3] that in the narrow well–separated jets limit, the non–global structure of the distribution, at hadronic colliders, becomes much like that of hemisphere jet mass [13]. This is mainly due to the fact that non–global logs arise predominantly near the boundaries of individual jets. We had therefore considered dijet events where only one of the jets is measured while the other is left unmeasured. We found, in the anti– algorithm [26], NGLs in the ratio as well as where and are the veto and hard scale respectively. These logs were completely missed out in [1, 2]. The resummation of these NGLs to all–orders had been approximated to that of the hemisphere mass [13] up to terms vanishing as powers of . Furthermore, we pointed out, by explicitly computing the jet mass (without jet veto) distribution under clustering, that different jet definitions differ at NLL due to clustering–induced large logs. Here we compute these logs, which we refer to as clustering logs (CLs), for the jet mass with a jet veto distribution.

Within the same context of multijet events, Kelley et al. [4] (version ) proposed that if one measures the masses of the two highest–energy jets, instead of a single highest–energy jet as done in [3], then the resulting distribution is free from NGLs. This is clearly not correct since the latter shape observable, which we shall refer to, following [27], as threshold thrust 444This name is more appropriate at hadron colliders where at threshold the final state jets are back–to–back and there is no beam remnant [27]., is still non–global. To clearly see this consider, for example, the following gluonic configuration in dijet events at . A gluon is emitted by hard eikonal legs into the interjet energy region, . then emits a softer gluon into, say the quark jet region. This configuration then contributes to the quark jet mass. The corresponding virtual correction, whereby gluon is virtual, does not, however, contribute to the quark jet mass. Hence, upon adding the two contributions one is left with a real–virtual mis–cancellation resulting in logarithmic enhancement of the jet mass distribution. The latter is what we refer to as NGLs. The other, antiquark, jet receives identical enhancement. Thus the sum of the invariant masses of the two jets does indeed contain NGLs contribution. The latter is actually twice that of the single jet mass found in [3].

Moreover, the authors of [4] (version ) claimed that the anti– [26] and Cambridge–Aachen (C–A) [28] jet algorithms only differ at NNLL for the threshold thrust 555This claim has been removed from version .. From our calculation in [3] for the jet mass, which is not -with respect to clustering- much different from the threshold thrust, we know that the latter statement is incorrect. Nonetheless, an explicit proof will be presented below. Now, what is interesting in [4] and triggers the current work, is that the total differential threshold thrust distribution computed in the C–A algorithm and which contains neither NGLs nor CLs contributions, seemed to somehow agree well with next–to–leading (NLO) program EVENT2 [29].

In this paper we shall shed some light on the result of [4] by considering the individual colour, and , contributions to the total differential distribution as well as the effect of C–A clustering. We show that at both NGLs and CLs are present and that the above agreement with EVENT2 is, on one side merely accidental 666As we shall see in sec. 5, while individual colour contributions do not agree with EVENT2 their sum does, but only in the shape variable range and for the jet–radius considered in [4]. Outside the latter range or for other smaller jet–radii they do not agree., and on the other side due to the fact that the interval of the threshold thrust considered in [4] does not correspond to the asymptotic region where large logs are expected to dominate. The current work may be regarded as an extension to [3]. It includes: (a) computing the full dependence of the leading NGLs coefficient in the anti–, (b) computing the small approximation of the latter as well as the leading CLs coefficient in the C–A algorithm and (c) checking our findings, as well as those of [3], against EVENT2. It turns out, from the latter comparison, that the above approximation is actually valid for quite large values of .

While the current paper was in preparation, a paper by Hornig et al. [5] appeared in arXiv which studied NGLs in various jet algorithms, including anti– and C–A, within SCET. On the same day, Kelley et al. published version of [4] in which they realised that this distribution is not actually free of NGLs and computed the corresponding coefficient in the anti– algorithm. Our findings on NGLs, which were independently derived using a different approach to both papers, confirm the results of both SCET groups. Clustering effects on primary emission sector are unique to this paper.

The organisation of this paper is as follows. In sec. 2 we compute the full logarithmic part of the LO threshold thrust distribution. We then consider, in sec. 3, the fixed–order NLO distribution in the eikonal limit and compute the NGLs coefficient, in both anti– and C–A jet algorithms. In the same section we derive an expression for the CLs’ first term as well. Note that our calculations for the C–A algorithm are performed in the small limit. Sec. 4 is devoted to LL resummation of our jet shape including an exponentiation of the NGLs’ and CLs’ fixed–order terms. The latter exponentiation suffices for our purpose in this paper, which is to compare the analytical distribution with EVENT2 at NLO. It also provides a rough estimate of the size and impact of NGLs and CLs on the total resummed distribution. In appendix C, the corresponding resummation in SCET [4, 27, 30] is presented. Numerical distributions of the threshold thrust obtained using the program EVENT2 are compared against analytical results and the findings discussed in sec. 5. In light of this discussion, we draw our main conclusions in sec. 6.

2 Fixed–order calculations:

After briefly reviewing the definition of the threshold thrust observable, or simply the jet mass with a jet veto, presented in [4, 27], a general formula for sequential recombination jet algorithms is presented. We then move on to compute the LO integrated distribution of this shape variable. At this order, all jet algorithms are identical. Note that partons (quarks and gluons) are assumed on–mass shell throughout.

2.1 Observable and jet algorithms definitions

Consider annihilation into multijet events. First, cluster events into jets of size (radius) with a jet algorithm. After clustering, label the momenta of the two hardest jets and and the energy of the third hardest jet . The threshold thrust is then given by the sum of the two leading jets’ masses after events with are vetoed [4],

(2.1)

and are the jet mass fractions for the two leading jets respectively. We have shown in [3] that the single jet mass fraction, , is a non–global shape variable. Thus must obviously be a non–global variable too.

A general form of sequential recombination algorithms at hadron colliders is presented in [25]. The adopted version for machines may be summarised as follows [25]: Starting with a list of final state pseudojets with momenta  777 may be the momenta of individual particles or each may be the total momentum of the particles whose paths are contained in a small cell of solid angle about the interaction point, as recorded in individual towers of a hadron calorimeter., energies and angles w.r.t. c.m frame, define the distances

(2.2)

where can be any (positive or negative) continuous number. At a given stage of clustering, if the smallest distance is then and are recombined together. Otherwise if the smallest distance is then is declared as a jet and removed from the list of pseudojets. Repeat until no pseudojets are left. The recombination scheme we adopt here is the –scheme, in which pairs are recombined by adding up their –momenta. Two pseudojets, and , are merged together if

(2.3)

The anti–, C–A and algorithms correspond, respectively, to and in eq. (2.2). We shall only consider the first two algorithms, anti– and C–A in this paper. Calculations for the inclusive are identical to those for the C–A algorithm as shown in [3]. With regard to notation, the jet–radius in [4], which we shall denote , is given in terms of by

(2.4)

Here we work with instead of .

To verify that the definition (2.1) is just the thrust in the threshold (dijet) limit, hence the name, we begin with the general formula of the thrust,

(2.5)

where the sum is over all final state –momenta and the maximum is over directions (unit vectors) . In the threshold limit, enforced by applying a veto on soft activity, annihilates into two back–to–back jets and the thrust axis, the maximum , coincides with jet directions. At LO, an emission of a single gluon, , that is collinear to, and hence clustered with say, , produces the following contribution to the thrust

(2.6)

where is the energy of the hard leg , the gluon’s energy and we have discarded an term. Recalling that the first two terms in the RHS of eq. (2.6) are just the mass fractions and , respectively, at LO and neglecting the third term (quadratic in ) one concludes that

(2.7)

This relation can straightforwardly be shown to hold to all–orders.

2.2 LO distribution

In [3] we computed the LO distribution of the jet mass fraction, , in the small () limit using the matrix–element squared in the eikonal approximation. In this section, we use the full QCD matrix–element to restore the complete dependence of the singular part of the distribution. The general expression for the integrated and normalised distribution, or equivalently the shape fraction, is given by

(2.8)

where is the total hadrons cross–section. The perturbative expansion of the shape fraction in terms of QCD coupling may be cast in the form

(2.9)

where refers to the Born contribution and is equal to . The derivation of the first order correction, , to the Born approximation is presented in appendix A. The final result reads

(2.10)

where we have used eq. (A.2) to change the normalisation in eq. (2.8) from to . The reason for this change is that the matrix–element we have used in EVENT2 is normalised to the Born cross–section 888Note that there are three sets of matrix–elements included in the program, of which only one is not normalised to the Born cross–section.. The only difference between the two normalisations at is in the one–loop constant. If we normalised to we would have found instead of . The function is given by

(2.11)

Notice that eqs. (2.10) and (2.11) are identical to eqs.  and  of [4] v and the sum of the parts of eqs.  and in [5] provided that the jet radius in the latter, which we refer to as , is related to by: . It is worthwhile to note that in the limit the distribution (2.10) reduces to the well known thrust distribution [31] with upper limit . For the threshold thrust distribution includes, in addition to thrust distribution, the interjet energy flow distribution [32] too,

(2.12)

Here the interjet region (rapidity gap), referred to in literature as , is defined by the edges of the jets. Specifically, it is related to the jet–radius by

(2.13)

The important features of the distribution that are of concern to the present paper are actually contained in the second order correction term , which we address in the next section.

3 Fixed–order calculations:

We begin this section by recalling the formula of the matrix–element squared for the annihilation into two gluons, in the eikonal approximation. Let us first define the final state partons’ –momenta as

(3.1)

where the angles are w.r.t. direction (which lies along the z–axis) and we assume the energies to be strongly ordered: . This is so that one can straightforwardly extract the leading NGLs. Contributions from gluons with energies of the same order, , are subleading and hence beyond our control. The recoil effects are negligible in the former regime and are thus ignored throughout. The eikonal amplitude reads [11],

(3.2)

where and stand for primary and secondary emission amplitudes respectively. If we define the antenna function then the latter amplitudes are given by

(3.3)

and

(3.4)

For completeness, the two–parton phase space is given by

(3.5)

It is worth noting that the primary emission, , contribution to the distribution is only fully accounted for by the single–gluon exponentiation in the anti– algorithm case. If the final state is clustered with a jet algorithm other than the latter, integration over the modified phase space, due to clustering, leads to (see below) new logarithmic terms that escape the naive single–gluon exponentiation. On the other hand, the secondary amplitude contribution is completely missing from the latter Sudakov exponentiation in both algorithms.

First we outline the full structure of the distribution up to NLL level in the anti– including the computation of the NGLs coefficient. After that, we investigate the effects of final state partons’ clustering on both primary and secondary emissions. The C–A algorithm is taken as a case study to illustrate the main points. Calculations where the final state is clustered with other jet algorithms should proceed in an analogous way to the C–A case.

3.1 distribution in the anti– algorithm

The anti– jet algorithm works, in the soft limit, like a perfect cone. That is, a soft gluon is clustered to a hard parton if it is within an angular distance , from the axis defined by the momentum of the latter. This feature of the algorithm greatly simplifies both fixed–order and resummation calculations. Considering all possible angular distances between and we compute below the corresponding contributions to primary and secondary pieces of the distribution. Note that we use LL and NLL to refer to leading and next–to–leading logs of (and not ) in the exponent of the resummed distribution (discussed in sec. 4).

3.1.1 term

The LL contribution to the distribution comes from diagrams corresponding to two–jet final states. That is diagrams where both real gluons, and , are clustered with the hard partons and . Diagrams where one of the two gluons is in the interjet region, and hence not clustered with either hard parton, contribute at NLL level. Other gluonic configurations lead to contributions that are beyond our NLL control and thus not considered. The part of the distribution may be found by expanding the exponential of the LO result (2.10). The full expression including the running coupling at two–loop in the will be presented in sec. 4. For the sake of comparison to the clustering case, we only report here the the LL term, which reads

(3.6)

Next we consider the derivation of the contribution to the distribution including the full jet–radius dependence.

3.1.2 term and NGLs

In the anti– algorithm the non–global logarithmic contribution to the distribution is simply the sum of that of the single jet mass fraction, , with a jet veto distribution studied in [3] 999Here we go beyond the small approximation assumed in [3].. This is in line with the near–edge nature of non–global enhancements. In two–jet events, the well separated 101010such that the jet–radius is much smaller than the jets’ separation; , where is the angle between jets and . jets receive the latter enhancements independently of each other.

Figure 1: Schematic representation of gluonic arrangement giving rise to NGLs. We have only shown the NGLs contributions to the –jet. Identical contributions apply to the –jet.

Possible final state gluonic arrangements relevant to NGLs at second order are depicted in fig. 1. The all–orders resummed NGLs distribution may be written in the form [13]

(3.7)

with being the evolution parameter defined in terms of the coupling by

(3.8)

where the exact form of the upper and lower limits, and , depend on the gluonic configuration and the second line in (3.8) assumes a fixed coupling. To make contact with interjet energy flow calculations [33, 34], we work in this particular section with hadronic variables instead of variables . The pseudo–rapidity and transverse momentum (both measured w.r.t. incoming beam direction) are related, respectively, to the angle and energy by 111111Otherwise, one can redefine the partons’ –momenta in terms of and and use the antenna function expressions of and to rewrite them in terms of the hadronic variables.

(3.9)

Using the secondary emissions eikonal amplitude (3.4) in terms of the new variables, the NGLs coefficient reads

(3.10)

where the phase space measure, , is of the general form given in eq. (3.5) with the integrals included in the definition of  (3.8) and new restrictions coming from the jet shape definition. For configuration in fig. 1, it reads

(3.11)

where the interjet (gap) region, is given in eq. (2.13). Due to boost invariance of rapidity variables the latter region has been centred at . Moreover, the factor in (3.11) accounts for the –jet contribution. Since neither the integrand nor the integral measure in eq. (3.10) depends explicitly on the azimuthal angles (, ), we use our freedom to set , average over and then perform the rapidity integration. The resultant expression for in configuration at the limit reads,

(3.12)

An identical expression was found for the NGLs’ coefficient in the interjet energy flow distribution [33] 121212Our jet–radius, , is given in terms the parameter , used in [33], by the relation: .. The fact that is the same for and interjet energy flow distributions means that the NGLs’ coefficient only depends on the geometry of the phase space and not on the observable itself. This is of course only true in the limit where the jet shape variable goes to zero. The difference between the jet shape variables amounts only to a difference in the logarithm’s argument.

It should be understood that there are –function constraints on and resulting from rapidity integrations not explicitly shown in eq. (3.12). Performing the remaining trivial integrals yields

(3.13)

where a factor of has been absorbed in  (3.12).

Now consider configuration in fig. 1. Adding up the corresponding virtual correction, one obtains the following phase space constraint

(3.14)

The phase space measure is analogous to in (3.11) with and the two –functions in (3.11) replaced by those in eq. (3.14). The limits on are then . If we impose the constraint given in eq. (3.13), i.e, , then the lower limit becomes . The NGLs coefficient thus reads

(3.15)

where we have averaged the eikonal amplitude over and moved s’ –functions onto the integral of the evolution parameter , which is given at by

(3.16)

The contribution is then beyond our NLL accuracy. In fact, vanishes in the limit as can be seen from eq. (3.15).

The last contribution to NGLs at comes from configuration in fig. 1. Upon the addition of the virtual correction, one is left with the constraint

(3.17)

The corresponding NGLs coefficient and evolution parameter read

(3.18)
(3.19)

Since we have assumed strong ordering, , then the lower limit of in (3.18) is . Consequently the coefficient vanishes in the limit . For this reason, this configuration will not be considered.

We conclude that in the regime , the only non–vanishing contribution to the NGLs comes from the phase space configuration . Other configurations, and , vanish in the limit . Hence

(3.20)

In fig. 4 we plot as a function of the jet–radius . At the asymptotic limit (or equivalently ) saturates at . This value (or rather half of it) is used as an approximation to in [3]. From eq. (3.12), we can see that the correction to such an approximation is less than for jet–radii smaller than , which is equivalent to . Furthermore, Eq. (3.12) confirms the claim made in the same paper that NGLs do not get eliminated when the jet–radius approaches zero. One may naively expects that when the jet size shrinks down to () there is no room for gluon to be emitted into. This means that becomes inclusive and hence vanishes. To the contrary, reaches its maximum in this limit.

Few important points to note:

  • If we choose to order the energy scales in the –functions of (3.13) and (3.16) the opposite way, i.e, then configuration becomes leading, in NGLs, while the contribution from configuration vanishes. That is in eq. (3.16) becomes

    (3.21)

    and in eq. (3.12). We do not consider this regime here though.

  • If, on the other hand, we do not restrict ourselves to any particular ordering of the scales, as it is done in Refs. [5] and [4], then both configurations and would contribute to the leading NGLs. Adding up , in (3.13), and , in (3.21), the –functions sum up to unity and one recovers the result reported in the above mentioned references. Notice that it is a straightforward exercise to show that eq. (3.12) is equal to given in eq.  of [5] in the case where ( given in sec. 2). Moreover, the coefficient given in eq.  of [4] v is related to by .

  • Setting the cut–off scale in , eq. (3.13), and , eq. (3.21), would diminish NGLs coming from both configurations and and the threshold thrust becomes essentially a global observable. This is unlike the observation made in the study of the single jet with a jet veto distribution [3] where the above choice of kills the NGLs near the measured jet but introduces other equally significant NGLs near the unmeasured jet.

In the next subsection we recompute both and contributions to the distribution under the C–A clustering condition. For the term, we only focus on configuration and do not attempt to address the subleading contributions coming from configurations and .

3.2 distribution in the C–A algorithm

The definition of the C–A algorithm is given in eq. (2.2) with . Unlike the anti– algorithm, which successively merges soft gluons with the nearest hard parton, the C–A algorithm proceeds by successively clustering soft gluons amongst themselves. Consequently, a soft parton may in many occasions be dragged into (away from) a jet region and hence contributing (not contributing) to the invariant mass of the latter. The jet mass, and hence , distribution is then modified. It is these modifications, due to soft–gluons self–clustering, that we shall address below.

Any clustering–induced contribution to the distribution will only arise from phase space configurations where the two soft gluons, and , are initially (that is, before applying the clustering) in different regions of phase space. Configurations where both gluons are within the same jet region, gluon is in one of the two jet regions and gluon is in the other or both gluons are within the interjet region are not altered by clustering and calculations of the corresponding contributions will yield identical results to the anti– algorithm. We can therefore write the distribution in the C–A algorithm, at , as

(3.22)

It is the last term in eq. (3.22) that we compute in the present subsection. Starting at configurations with two gluons in two different regions, the jet algorithm either:

  1. recombines the two soft gluons into a single parent gluon if the clustering condition (2.3) is satisfied. The latter parent gluon will either be in one of the two jet regions or out of both of them (and hence in the interjet region).

  2. or leaves the two gluons unclustered, if the clustering condition is not satisfied. This case is then identical to the anti– one but with a more restricted phase space. This restriction comes from the fact that for the two gluons to survive the clustering they need to be sufficiently far apart. Quantitatively, their angular separation should satisfy the relation

    (3.23)

Below, we examine the contributions from configurations (A) and (B) to the and colour pieces of the distributions. All calculations are performed in the small approximation using the variables .

3.2.1 term

Figure 2: A schematic representation of a three–jet final state after applying the C–A algorithm on real emission along with virtual correction diagrams. The two gluons are clustered in the E–Scheme (see sec. 2). Identical diagrams hold for the –jet.

Consider the gluonic configuration in (A) where the harder gluon is in the interjet region and the softer gluon is in the –jet region. We account for the –jet region through multiplying the final result by a factor of two. Applying the C–A algorithm (2.2), the smallest distance is . Hence gluon pulls gluon out of the –jet region and form a third jet, as depicted in fig. 2. The latter is then vetoed to have energy less than . The corresponding clustering angular function, in the small angles limit, reads

Adding up the corresponding virtual corrections, where one or both of the gluons are virtual, one obtains the following constraint on the phase space

(3.25)

Since we are working in the strong energy–ordered regime, , only the last –function survives. The new contribution to the piece of the distribution is then given by

(3.26)

This result is identical to 131313It is actually twice that found in [3] for a single jet mass (without a jet veto) distribution. The reason for this is that the clustering requirement only affects the distribution to which the softest gluon contributes. Which is in both cases the jet mass distribution.

Figure 3: A schematic representation of a two–jet final state after applying the C–A algorithm on real emission along with virtual correction diagrams. The two gluons are clustered in the E–Scheme (see sec. 2). Identical diagrams hold for the left (–) jet.

The second possible configuration that corresponds to case (A) is where gluon is in, say, the –jet region and the softer gluon is in the interjet region. If the two gluons are clustered, i.e, gluon pulls in gluon , then upon adding real emission and virtual correction diagrams, depicted in fig. 3, one obtains the following phase space constraint

(3.27)

where we have assumed small angles limit and employed the LL accurate approximation

(3.28)

Given the fact that and and must be close to each other to be clustered, i.e, they should satisfy condition (3.23), then the first two –functions in eq. (3.27) are substantially suppressed and one is only left with the veto on . Applying the C–A algorithm one obtains an identical clustering function to eq. (3.2.1). Hence the CLs’ coefficient for this configuration is equal to given in eq. (3.26). That is . The evolution parameter does however change. It is now given, at , by

(3.29)

This contribution is then beyond our NLL control. Note that the CLs contribution in eq. (3.29) is equal to what one would find for interjet energy flow distribution provided that the rapidity gap is defined through eq. (2.13).

Let us now turn to case (B) where the two gluons are not merged together. If gluon is in the interjet region and gluon is in one of the two jet regions then the corresponding phase space constraint reads

(3.30)

The limits on –integral are then given by: . Imposing the constraint , it is straightforward to see that the above constraint yields NNLL contribution and thus beyond our control. Similarly, the configuration where gluon is in the jet region and gluon is in the interjet region yields subleading logs.

Hence the piece of the clustering–induced correction term , in eq. (3.22), up to NLL, reads

(3.31)

Next we compute the piece of .

3.2.2 term

Consider the gluonic configuration depicted in fig. 1. Applying the C–A clustering algorithm on the latter yields two possibilities. Namely the two gluons are either clustered or not. The former case completely cancels against virtual corrections and thus does not contribute to NGLs. It is when the two gluons survive the clustering, the latter case, that a real–virtual mismatch takes place and NGLs are induced. The corresponding evolution parameter is equal to of the anti– case, eq. (3.20). The clustering condition is simply one minus that in eq. (3.2.1). The NGLs’ coefficient can then be written, using the eikonal amplitude (3.4), as

(3.32)

where is given in eq. (3.20) and

(3.33)

We can perform the –integral analytically and then resort to numerical methods to evaluate the remaining and integrals. The result, in terms of the jet–radius , is depicted in Fig. 4.

Figure 4: Non–global coefficient in the anti– and C–A algorithms.

saturates at around , i.e, a reduction of about in . This is due to the fact that for the two gluons to survive clustering they need to be sufficiently far apart (). The dominant contribution to comes, however, from the region of phase space where the gluons are sufficiently close. This corresponds to the collinear region of the matrix–element; . Hence the further apart the two gluons get from each other, the less (collinear) singular the matrix becomes and thus the smaller the value of NGLs coefficient.

Note that the C–A coefficient , where is given by eq.  in [5], at least in the small jet–radius region. Noticeably, the two results coincide at both limits and (equivalently and in [5]). In fact, the coefficient