Dichroic subjettiness ratios to distinguish colour flows in boosted boson tagging

Journal of High Energy Physics, Mar 2017

Abstract N-subjettiness ratios are in wide use for tagging heavy boosted objects, in particular the ratio of 2-subjettiness to 1-subjettiness for tagging boosted electroweak bosons. In this article we introduce a new, dichroic ratio, which uses different regions of a jet to determine the two subjettiness measures, emphasising the hard substructure for the 1-subjettiness and the full colour radiation pattern for the 2-subjettiness. Relative to existing N -subjettiness ratios, the dichroic extension, combined with SoftDrop (pre-)grooming, makes it possible to increase the ultimate signal significance by about 25% (for 2 TeV jets), or to reduce non-perturbative effects by a factor of 2−3 at 50% signal efficiency while maintaining comparable background rejection. We motivate the dichroic approach through the study of Lund diagrams, supplemented with resummed analytical calculations.

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Dichroic subjettiness ratios to distinguish colour flows in boosted boson tagging

Received: December Dichroic subjettiness ratios to distinguish colour ows Gavin P. Salam 0 1 2 4 Lais Schunk 0 2 3 Gregory Soyez 0 2 3 0 Open Access , c The Authors 1 On leave from CNRS, UMR 7589, LPTHE , F-75005, Paris , France 2 CH-1211 , Geneva 23 , Switzerland 3 IPhT, CEA Saclay , CNRS UMR 3681 4 CERN, Theoretical Physics Department N -subjettiness ratios are in wide use for tagging heavy boosted objects, in particular the ratio of 2-subjettiness to 1-subjettiness for tagging boosted electroweak bosons. In this article we introduce a new, dichroic ratio, which uses di erent regions of a jet to determine the two subjettiness measures, emphasising the hard substructure for the 1-subjettiness and the full colour radiation pattern for the 2-subjettiness. Relative to existing N -subjettiness ratios, the dichroic extension, combined with SoftDrop (pre-)grooming, makes it possible to increase the ultimate signal signi cance by about 25% (for 2 TeV jets), or to reduce non-perturbative e ects by a factor of 2 3 at 50% signal e ciency while maintaining comparable background rejection. We motivate the dichroic approach through the study of Lund diagrams, supplemented with resummed analytical calculations. Jets; QCD Phenomenology 1 Introduction Setup and useful tools for discussion A tagger, a groomer and a jet shape A useful graphical representation Radiation constraints (N -subjettiness) Dichroic subjettiness ratios Combining mMDT/SD with N -subjettiness Dichroic subjettiness with SoftDrop (pre-)grooming Performance in Monte-Carlo simulations N -subjettiness and mass distributions with various 21 ratios ( = 2) Signal v. background discrimination and other performance measures Brief comparison with other tools Brief analytic calculations A Dichroic subjettiness ratios for = 1 B Explicit expressions for the analytic results C Example code for dichroic subjettiness ratios Introduction With the increasingly high-energy scales probed by the Large Hadron Collider (LHC), masmomentum much larger than their mass. In this boosted regime, when they decay hadronically, they are reconstructed as single jets that have to be separated from the much more common quark- and gluon-initiated jets. Over the past few years, several techniques relying on jet substructure, i.e. on the internal dynamical properties of jets, have been devised in order to achieve this task. These techniques are now routinely used in LHC analyses and new-physics searches. There are three common families of methods used to separate boosted heavy objects from standard QCD jets: (i) taggers, which impose that a jet contain two hard cores (or three for a top-quark), a situation more common in signal jets than in QCD jets which are dominated by soft-gluon radiation; an increasingly widespread technique for tagging is the modi ed MassDrop tagger (mMDT) [1, 2] and its generalisation, SoftDrop [3], which will be our chosen tools here; (ii) radiation constraints, which constrain soft-gluon radiation inside jets, expected to be larger in QCD jets than in colourless weak-boson decays; a widespread way of applying radiation constraints is to cut on jet shapes, for example the of soft-and-large-angle radiation, often dominated by the Underlying Event and pileup, hence ensuring a better mass resolution. To reach a large discriminating power, it is helpful to combine several of these techniques. Since taggers and groomers share many similarities, one often starts by applying a tagger/groomer and then imposes a cut on the value of a jet shape computed on that tagged/groomed jet. Finally, one selects jets with a (tagged or groomed) mass close-enough to the weak boson mass. In this paper, we introduce the concept of \dichroic" subjettiness ratios for applying radiation constraints. Starting from an object in which two hard prongs have been identi ed (\tagged"), the dichroic variant of subjettiness di ers from standard subjettiness ratio. These two (sub)jets will generally overlap and correspond to di erent degrees of tagging/grooming. The reason for calling this \dichroic" is that the radiation patterns in the two di erent (sub)jets are driven by distinct colour ows.1 In particular we will use a large jet for calculating 2 and a smaller, tagged subjet for 1. Calculating 2 on the large jet provides substantial sensitivity to the di erent colour structures of signal (colour singlet when viewed at large angles) and background (colour triplet for a quark-jet or octet for a gluon-jet). Calculating 1 on the tagged subjet ensures that it is not substantially a ected by the overall colour ow of the large jet, but rather is governed essentially by the invariant gives enhanced performance compared to existing uses of N -subjettiness, which adopt the same (sub)jet for numerator and denominator (see e.g. [7{11] for recent examples). Performance of radiation-based discrimination involves two criteria: the ability to distinguish signals from backgrounds and the robustness of that discrimination, notably its insensitivity to non-perturbative e ects. As discussed already in [12], these two criteria are often in tension, because the region of large-angle soft kinematics on one hand provides substantial discrimination power, but is also the region where the Underlying Event and hadronisation have the largest impact. A point central in our discussion will be the between these aspects. To reduce the tension between discrimination power and perturbative robustness we will show how the dichroic subjettiness ratio can be used in combination not just with tagging but also a separate (pre-)grooming step. Setup and useful tools for discussion Before introducing the dichroic tools in section 3, let us rst discuss the individual building blocks used in our new combination and introduce a simple framework to facilitate the discussion of the underlying physics and expected performance. 1By \colour ow" we mean the ow of colour between two partons in leading-colour sense; this may be the ow of colour from initial to nal-state partons, or between two initial or nal-state partons. We will concentrate on the modi ed MassDrop tagger, used here as a tagger, N -subjettiness as a radiation-constraining jet shape, and SoftDrop as a groomer.2 These are all common choices in the literature, but we believe that our generic strategy can be extended to other combinations if needed. To ease the physics discussion below, let us brie y recall how these methods are de ned. The modi ed MassDrop tagger and SoftDrop both start by reclustering the jet with the Cambridge/Aachen algorithm. They then recursively undo the last step of the clustering, splitting the current jet j into two subjets j1 and j2. The procedures then stop if the splitting is symmetric enough, i.e. if with pti the transverse momentum of the subjet ji, 12 their angular separation in the rapidity-azimuthal angle plane and R the jet radius. If the symmetry condition eq. (2.1) is not met, the procedure is recursively applied to the subjet with the largest pt. Eq. (2.1) that to some extent mMDT and SD have both tagging and grooming properties. When we use mMDT and SD together, the zcut parameter of SD will be renamed cut in order to avoid confusion. N -subjettiness is de ned (in the unnormalised version we use here) as follows: for a given jet, one nds a set of N axes a1; : : : ; aN (see below) and introduces distance iaj = q where the sum runs over all the constituents of the jet, of momentum pti and with an angular is a free parameter and in what follows we will concentrate on the case in Monte-Carlo numerical simulations and considerably simpli es the physical discussions below. However, the techniques introduced in this paper straightforwardly apply to other , including the frequent experimental choice = 1, and we will comment on this in section 4.3 and appendix A. We still need to specify how to choose the N -subjettiness axes. In practice, there are several methods that one can use. Common choices include using exclusive kt axes or using \minimal" axes, i.e. use the set of axes that minimise the N . We will instead consider the case of exclusive axes obtained by declustering the result of a generalised-kt 2The distinction between a \tagger" and a \groomer" is often thin. Here tools which are meant to nd a multi-prong structure in boosted jets are referred to as taggers, while tools which clean the jet of soft-and-large angle (mostly non-perturbative) radiation are called groomers. In other words, a tagger can fail (for objects with no hard substructure) while a groomer will always return a valid groomed jet (possibly with a single constituent). 3Throughout this paper, we assume that the parameter of the mMDT is set to 1. Choosing a small would have an e ect similar to that of a (recursive) N -subjettiness cut, as discussed in [12]. phasespace available for an emission from the jet initial parton at an angle and carrying a momentum fraction z. The diagram shows a given emission (the solid dot) as well as lines with the same momentum fraction, kt and also [15, 16]) and is related to the fact that, since it preserves the ordering in mass, it produces results very close to the much more complex minimal axes.4 2= 1 is expected to be smaller for weak bosons and one imposes a cut 21 < cut as a radiation constraint to distinguish weak bosons from the QCD background. A useful graphical representation To guide our discussion, it is helpful to consider the available phasespace for radiation inside a (QCD) jet in the soft-and-collinear limit and see how the various methods under consideration constrain that phasespace. This is conveniently done using Lund diagrams [20]. Consider an emission at an angle from the jet axis, carrying a fraction z of the transverse momentum of the parent parton. Lund diagrams represent the two-dimensional phases gure 1, a line of constant momentum fraction z corresponds to a diagonal line with log(kt) = constant log(1= ) and a line of a given mass, m2 z 2 in the soft and small In the soft-and-collinear approximation, su cient for the following discussion, each emission comes with a weight d2! = 2 s(kt)CR d log(1= ) d log(kt) ; 4For a generic (WTA) recombination scheme [17{19] for 1 to avoid inconvenient recoil e ects. (left) and the mMDT mass (right). The solid black point corresponds to the emission dominating the jet mass and can be anywhere along the solid red line. It gives the prefactor in the jet mass distribution. The shaded red area corresponds to the vetoed region yielding the Sudakov exponent. respectively for quark and gluon jets. The strong coupling constant, s, is evaluated at a scale equal to the transverse momentum of the emission relative to its emitter. Apart from running-coupling e ects and subleading corrections in the hard-collinear and softlarge-angle regions, this weight is uniform over the Lund plane. In the leading logarithmic approximation, the radiation in a jet is a superposition of independent and strongly-ordered (primary) emissions in that plane, as well as secondary emissions emitted from the primary emissions and which can be represented as extra Lund triangles (leaves) originating from each of the primary emissions, tertiary emissions emitted from secondary ones, etc. . . Leaves will be discussed in more detail below. To illustrate how one can use this pictorial representation to discuss physics processes, Lund diagram is represented in the left panel of gure 2. One rst needs an emission that provides the dominant contribution to the mass of the jet, i.e. an emission such that m2 = z 2pt2 or such that z 2 = the angles in units of the jet radius R and introduced the dimensionless (squared) mass integrated weight for emissions that generate a (normalised) jet mass equal to Rf0ull( ) = ) f=.c. sCR log(1= ); 5The R use below in weights and Sudakov factors stands for \Radiator" and is not to be confused with the jet radius. where for the last equality we have illustrated the structure of the answer in a xed coupling approximation, as indicated by the superscript \f.c.". Modulo corrections induced by the running of the strong coupling, the logarithmic behaviour basically comes from the integration over the solid line of equal mass in the Lund representation. We also need to impose that no emissions occur at larger mass. This induces a Sudakov suppression exp[ Rfull( )] where6 Rfull( ) = d2! (z 2 > ) f=.c. sCR log2(1= ): The double-logarithmic behaviour corresponds to the shaded area in the Lund diagram. In the end, the leading-logarithmic (LL) result for the cross-section can be written as = Rf0ull( ) e Rfull( ): This expression has a simple graphical representation: a prefactor corresponding to the emission setting the mass, the solid line in the Lund diagram, and a Sudakov suppression for larger masses, the shaded area in the Lund diagram. Let us now consider the jet mass distribution after the application of the mMDT. This is represented in the right panel of gure 2. In this case [2], emissions with z < zcut are discarded by the mMDT recursive procedure,7 so that both the prefactor Rm0MDT for having an emission setting the jet mass and the Sudakov exponent RmMDT are restricted to z > zcut and we have (assuming Rm0MDT( ) = RmMDT( ) = = Rm0MDT( ) e RmMDT( ); ) (z > zcut) f=.c. sCR log(1=zcut); Compared to the full mass result, eq. (2.6), the prefactor is smaller but the Sudakov suppression is also less important. In practice, we will therefore have a suppression of the QCD background at intermediate masses but an increase at very small masses. More generally, we see that in order to have a large suppression of the QCD background, we want a method that keeps the prefactor small but gives a large Sudakov suppression. This will be a key element of our dichroic approach. 6Technically, the exponential comes from the fact that, in the region z 2 > , real emissions are vetoed while virtual contributions are present. 7Strictly speaking, only emissions with z < zcut and at an angle larger than the rst emission with z > zcut will be discarded. This has no impact on the discussion of the jet mass since the di erence only introduces a subleading correction. Subleading corrections to the mMDT/SD mass distribution are discussed in ref. [21]. 8The assumption zcut 1 could be lifted (see e.g. ref. [2]). We keep it here mostly for simplicity and to match the double-logarithmic accuracy assumed in our discussion based on Lund diagrams. Next, we consider the signal (electroweak boson) jets in the context of Lund diagrams. For, say, a W boson, the original splitting, W ! qq occurs on a line of constant mass at small z, this splitting will be concentrated close to the large-z end of that constantmass line, with the small-z tail exponentially suppressed (in our logarithmic choice of axes). As a direct consequence, no emissions are possible at larger mass and there will not be any Sudakov factor for that region. For simplicity in our discussion below, we will assume a constant splitting function in z, which would be the case e.g. for a Higgs boson or an unpolarised W boson. For an mMDT zcut condition, this yields a signal e ciency 2zcut. Subsequent emissions from the original qq pair will happen as if they were secondary emissions from these two quarks, i.e. essentially in two separate Lund planes each of them restricted to angles smaller than the separation qq between the two quarks, because of angular ordering. One of those Lund planes (that for the softer of the qq pair) will be represented as a leaf, cf. gure 3. Now that we have discussed how mass distribution and radiation constraints are represented in terms of Lund diagrams, we will use Lund diagrams to discuss more complex substructure methods, leaving corresponding analytic expressions to section 5. Radiation constraints (N -subjettiness) Let us now examine how a cut on N -subjettiness on the full jet a ects the pattern of allowed radiation. Our discussion will be in a context where the full jet has a speci ed mass, denoted through . The constraints imposed by a cut on the N -subjettiness ratio 21 can then again be presented quite straightforwardly in terms of Lund diagrams, at least in the small 21 limit, which is what we will consider in our discussion. Say that we have a rst emission with an angle 1 and momentum fraction z1 that dominates the jet mass, proximation, N (with like scale" z 2 = and imposing a cut 21 < cut is equivalent to vetoing emissions down to a \masscut, for both primary and secondary emissions. This is represented in gure 3 for QCD and signal jets, where the extra constraint on N -subjettiness corresponds to an extra Sudakov factor represented by the blue shaded region. In the background case, the leaf that emerges from the plane corresponds to a region of secondary emissions, while in the signal it corresponds to the region of emissions from the softer of the qq pair. Assuming a background mainly consisting of quark jets, the main parts of the plane in the gures are both associated with a CF colour factor, while the leaf in the background case is associated with a CA colour factor, in contrast with the CF factor for the signal, and correspondingly represented with a darker shade of blue. We see that we now have a Sudakov suppression for both the signal and the QCD background. Since the vetoed area is larger for the background than for the signal, the former is more suppressed than the latter, implying a gain in discriminating power. Furthermore, since, for a given cut, the vetoed area increases when gets smaller, the discriminating power will also be larger for more boosted jets. to the requirement of a given full jet mass with a cut on the N -subjettiness ratio 21. The red shaded region (present only in the background case) corresponds to the Sudakov vetoed region for the mass, as in gure 2, together with the prefactor for having an emission on the solid red line. The blue shaded region corresponds to the additional veto coming from the cut on N -subjettiness. The dashed/dotted red line for the signal case represents the fact that, for signal jets, small-z con gurations are exponentially suppressed. The region that emerges from the plane is referred to as a \leaf" and in the left-hand diagram represents secondary emissions from emission 1, while in the right-hand diagram it represents emissions from the softer of the two prongs of the decay. Dichroic subjettiness ratios Combining mMDT/SD with N -subjettiness We can now present the main proposal of this paper concerning the dichroic combination of a tagger with a radiation constraint. The discussion below assumes that we use SoftDrop or the modi ed MassDrop tagger as our tagger and a cut on 21 as a radiation constraint, but we believe that the core argument can also be applied to other shapes, for example to energy correlation functions [22{24]. Let us consider a high-pt large-radius (R ' 1) jet on which we have applied an mMDT (or SD) tagger. The original large-radius jet will be called the full jet. The part of the jet that remains after the mMDT/SD tagging procedure will be called the tagged jet, and has an angular size comparable to the angle between the two hard prongs identi ed by the tagger. The N -subjettiness variables 1 and 2 can be evaluated either on the full or the tagged jet and there are three combinations of interest: The rst two options are currently widely used in the literature (see e.g. [7{11] for recent examples). The third, \dichroic", option is a new combination, and is the subject of this paper.9 To understand how these di erent variants work, we will take two approaches. First we will consider what values of 21 arise for di erent kinematic con gurations involving three particles in the jet, i.e. two emissions in the case of QCD jets, and the original two prongs plus one additional emission in the case of signal jets. Then we will use this information to understand how a cut on 21 constrains the radiation inside the jet. During this discussion it will be useful to keep in mind the core di erence between signal and background jets. In the case of the background jets, the whole Lund plane and the leaf can contain emissions, as shown in gure 3(left). In the case of signal jets, emissions are mostly limited to the region shown in blue in gure 3(right), i.e. at angles smaller than the decay opening angle and transverse momenta smaller than the mass. The leaves in the two cases have di erent colour factors, however we will neglect this aspect in our discussion.10 Rather we will concentrate on the di erences that arise at large angle, i.e. from the di erent coherent radiation patterns of coloured versus net colour-neutral objects. We consider the situation where, after the tagger has been applied, the tagged jet mass is dominated by emission \a", i.e. za a2 (in the case of the signal jet this is the softer of the two prongs). The Lund-plane phasespace can then be separated into 3 regions depicted gure 4. Region A (in red) is the region that is constrained to be free of radiation by the fact that the tagger has triggered on emission a. This corresponds to the region where both z 2 > za a2 and eq. (2.1) are satis ed. It is responsible for the Sudakov exponent associated with the tagger, cf. eq. (2.7). Of the remaining phasespace, region B (blue) corresponds to emissions that are contained inside the tagged jet. It is populated in both signal and background cases. It contains not only emissions that satisfy the mMDT/SD condition (z > zcut in the case of mMDT), but also emissions with z 2 < za a2 and < a, due to the Cambridge/Aachen declustering used by mMDT/SD. Region C (green) corresponds to emissions that are in the original full jet, but not in the tagged jet. It is uniformly populated in the background case, while in the signal case it is mostly empty of radiation, except at the left-hand edge (initial-state radiation) and the right-hand edge (leakage of radiation from the colour-singlet qq decay). The emission with the largest z 2 in each of regions B and C will respectively be labelled b and c and we will assume strong ordering between emissions, as in section 2. There are three kinematic cases to consider for the relative z 2 ordering of emissions a, b and c, cf. gure 4. In each case, gure 4 gives the result for each of the 21 variants, for both background and signal. The signal case simply assumes that there are no emissions 9One can be tempted to also consider a fourth option where 1 is computed on the full jet and 2 on the tagged jet. It is straightforward to show, following the same arguments as below, that this is not the best combination, as one might expect intuitively. 10At low pt a signi cant part of 21's discriminating power is arguably associated with the leaf and, for gluon-initiated background jets, with the part of the main Lund plane that is at small angles compared to the decay opening. This is mostly equivalent to quark-gluon discrimination, which is known to be only moderately e ective [22, 25{27] and not to improve signi cantly at high-pt. These e ects are included in the analytic calculations of section 5. of 21 with mMDT/SD (shown speci cally for mMDT or SD with = 0). In each Lund diagram, emission \a" corresponds to the emission that dominates the mMDT/SD jet mass. This de nes three regions: region A (red) is vetoed by mMDT, region B (blue) contains the constituents of the mMDT/SD jet and region C (blue) is the di erence between the mMDT/SD jet and the full jet. Emissions \b" and \c" are respectively in regions B and C, and the three plots correspond to three di erent orderings of zc c2 compared to za a2 and zb b2. The table below the plots shows the corresponding value of 21 for both the QCD background (where all three regions have to be included) and the signal (where only regions A and B are present). For simplicity, \b/a" stands for (zb b2)=(za a2), and so forth. in region C, which is appropriate in a double-logarithmic approximation. The results are expressed as a shorthand, i=j zi i2=zj j2. The case of the signal is particularly simple: since zb b2 < za a2 and there is nothing same, the performance of the signal/background discrimination will be best for the method that gives the largest background 21 result (recall that one enhances signal relative to background by requiring 21 < cut). Let us examine the background separately for each of the three kinematic cases shown in gure 4: 1. For za a2 zb b2=za a2.11 2. For za2 a2 zc c2, all three 21 variants give the same result as for the signal, 11Even if the signal and background have the same value, the di erent colour factor of the leaf, discussed (signal) than for CA colour factors (background). cut on 21. See text for details. in this case. 3. Finally, for zc2 c2 za2 a2 and so may or may not be advantageous. 21 dichroic has a value of za a2=za a2 = 1, which is always larger than the signal and larger than the other two variants. Overall therefore, 2d1ichroic is expected to be the best of the three variants. Alternatively, we can also see the bene t of the dichroic combination by examining directly how emissions are constrained when one applies a given cut on the 21 ratio, similarly to the discussion in section 2.3. We have represented the Lund diagrams relevant for our discussion in gure 5, where we have used the same regions A, B and C as in the above discussion. We start by considering a jet for which we already have applied the mMDT/SD procedure, resulting in a (mMDT/SD) mass dominated by emission \a". This automatically comes with a mMDT/SD prefactor and Sudakov suppression represented by the solid red line and shaded light red area (region A) in gure 5, guaranteeing that there are no emissions at larger mass kept by the mMDT/SD. For 2t1agged, emissions in region B are vetoed down to a mass scale cut while emissions in region C, i.e. outside the mMDT/SD tagged jet, are left unconstrained. This results in the (additional) Sudakov suppression given by the blue area (region B) in gure 5(a). The situation for 2fu1ll is a bit more involved and we have three cases to consider. The rst case is when there is (at least) one emission in region C with z 2 > za a2 = cut = cut and is represented in gure 5(b). Let us then call emission \c" the emission in region C with the largest z 2, which thus comes with a Sudakov suppression imposing that there are no other emissions in region C with z 2 > zc c2. Emission \c" will dominate 1 so that the cut on 21 will come with an extra suppression factor in region C extending from depicted in gure 5(b). The second case is when the emission in region C with the largest gure 5(b), is entirely forbidden because it would give a value of 21 always larger than cut. The third case is when there are no emissions in region C with . This directly comes with a Sudakov suppression in region C vetoing emission further vetoes emissions with in both regions B and C. These two vetoes combine to vetoing all emission down to cut as represented in gure 5(c). If instead we use our new 21 dichroic variable, we are always in the situation of gure 5(c), where we veto all emissions down to a mass scale cut in both regions B and C. This new version therefore comes with the strongest Sudakov suppression, i.e. of the three 21 variables it is the one that, for background jets, is least likely to have a small 21 value. Given that the three 21 variants behave similarly to each other for signal, the signal-tobackground discrimination should be improved for the dichroic variant. With our dichroic method, we actually recover the same overall Sudakov suppression as the one we had when measuring the full jet mass and cutting on the full N -subjettiness (see section 2.3 and gure 3(left)). The gain of our new method (3.1c) compared to this full N -subjettiness case comes from the fact that the prefactor associated with the jet mass is now subject to the constraint imposed by the tagger. If we take for example the case of the mMDT, this prefactor would be largely suppressed for the background | going from s log(1= ) for full N -subjettiness to s log(1=zcut) for the dichroic method | while the signal would only be suppressed by a much smaller factor 2zcut. Additionally, measuring the tagged jet mass instead of the full jet mass signi cantly reduces ISR and non-perturbative e ects which would otherwise a ect the resolution of the signal mass peak za a2=zc c2 which is (see also [28, 29]). Finally, we note that the gain in performance is expected to increase for larger boosts due to region C getting bigger (double-logarithmically in ). Dichroic subjettiness with SoftDrop (pre-)grooming Since 2d1ichroic uses 2 computed on the full jet, including all the soft radiation at large angles, we can expect this observable to be quite sensitive to poorly-controlled non-perturbative e ects | hadronisation and the Underlying Event | and to pileup. The standard strategy to mitigate these e ects is to kill two birds with one stone and to use mMDT (or SD) both as a two-prong tagger and as a groomer, and impose the 21 constraint on the result. This is equivalent to the 21 with the drawback and loss of performance described in the previous section. tagged variant discussed ( gure 5(a)), We show here how we can achieve a background rejection that is larger than for 2t1agged and more robust with respect to non-perturbative e ects than 21 dichroic. Conceptually, the idea is that the tagger and groomer achieve two di erent tasks: the tagger selects a twoprong structure in the jet, imposing a rather hard constraint on the soft radiation in order to do so, leading to a small R0 prefactor for the jet mass. This is not quite what we want from tained from our new combination including grooming: rst groom the jet, e.g. with SoftDrop (SD). We then compute both the jet mass and 1 on the tagged jet (here using the mMDT), yielding the solid red line prefactor and the shaded red region (A) for the Sudakov exponent. We then impose a cut on the 21 ratio with 2 computed on the SD jet, leading to the extra shaded blue and green regions (B and C) for the Sudakov exponent. a groomer, which should get rid of the soft-and-large-angle radiation while retaining enough of the jet substructure to have some discriminating power when using radiation constraints. This suggests the following picture: we rst apply a \gentle" grooming procedure to the jet, like a SoftDrop procedure with a positive value of . This is meant to clean the jet of the unwanted soft junk12 while retaining as much as possible the information about the perturbative radiation in the jet. We can then carry on with the dichroic method presented in the previous section, i.e. use a more aggressive tagger, like mMDT,13 to compute the jet mass and 1 and compute 2 on the SD (pre-)groomed jet: This is depicted in gure 6, where regions A and B are the same as in the previous section, but now region C indicates the region where emissions are kept by the groomer but rejected by the tagger. Similarly, we can introduce 2(SD jet) 1(mMDT jet) 2(SD jet) 1(SD jet) Note that we will always choose our mMDT-tagging and SD-grooming parameters such that the tagged jet is the same whether tagging is performed before or after grooming. For mMDT-tagging with parameter zcut and SD-grooming with parameters cut and , this implies cut Using the same arguments as in section 3.1, we can show straightforwardly that this method will have a larger rejection than with the other two variants where one would be computing the jet mass on the mMDT-tagged jet and the 21 ratio either on the mMDTtagged jet, 2t1a;gggreodomed tagged, or on the SD-groomed jet, 21;groomed, owing to a larger full 21 Sudakov suppression of the background, for a similar signal e ciency. 12In the presence of pileup, one should still apply a pileup subtraction procedure [30], like area-median subtraction [31, 32], charged-track-based techniques [33{35], the constituent subtractor [36], SoftKiller [37] or PUPPI [38]. This can be done straightforwardly with SoftDrop and mMDT. 13Or SD with a smaller value of than used in the grooming. Compared to the other possible situation where both the jet mass and the 21 ratio are computed on the SD-groomed jet, the dichroic variant would have a smaller R0 prefactor, associated with mMDT instead of SD. This again leads to a larger background rejection. Because of the initial grooming step, the groomed dichroic subjettiness ration is expected to be less discriminating than the ungroomed version introduced in section 3.1. Indeed, the associated Sudakov exponent is smaller since we have amputated part of the soft-large-angle region. One should however expect that this groomed variant will be less sensitive to non-perturbative e ects. Overall, there is therefore a trade-o between effectiveness, in terms of achieving the largest suppression of the QCD background for a given signal e ciency, and perturbative robustness, in terms of limiting the sensitivity to poorly-controlled non-perturbative e ects. Performance in Monte-Carlo simulations Let us now investigate the e ectiveness and robustness of dichroic subjettiness ratios in Monte-Carlo simulations, using Pythia 8.186 [39], at a centre-of-mass energy of determining signal and background e ciencies we keep all jets above a given pt cut.14 We cut on the 21 ratio. Whenever a SoftDrop (SD) grooming procedure is included, we use cut = 0:05 and respectively for the parameters of mMDT and SD). Jet reconstruction and manipulation are performed with FastJet 3.2.0 [14, 40] and fjcontrib 1.024 [41]. N -subjettiness and mass distributions with various 21 ratios ( = 2) We start by examining the 21 distribution. This is plotted in gure 7 for both QCD jets (solid lines) in dijet events and W jets (dashed lines) in W W events. jets above 2 TeV and always apply SoftDrop grooming. In practice, we use parton-level events, and impose a cut on the reconstructed jet mass (SD-groomed or mMDT-tagged) 60 < m < 100 GeV. We consider four cases: the 2fu1;lglroomed distribution when we cut on the SD-groomed mass and the 21 dichroic 21;groomed distributions when we cut on the mMDT-tagged mass. As expected, the distributions for signal (W ) jets are peaked at smaller values of 21 than the corresponding distribution for background (QCD) jets. Figure 7 shows that all the signal distributions, and in particular the three options where one measures the mMDT-tagged jet mass, are very similar. This is in agreement with our discussion in the previous section. Comparatively the background distributions look rather di erent. The case where everything is computed from the mMDT-tagged jet (the solid blue curve) peaks at smaller values of 21 as expected from its smaller Sudakov 14All jets in the signal sample above that cut are considered to be signal-like, even if they came from initial-state radiation; however such initial-state jets will have been relatively rare in our sample and so should not a ect our nal conclusions. d /σd 0.6 tagger: mMDT(zcut=0.1) pt > 2 TeV (anti-kt, R = 1) at parton level, including SoftDrop grooming. The dashed lines, in red for the SD-groomed jet and in blue for the mMDT-tagged jet, are the mass distributions with no constraint on N -subjettiness. The solid lines have an additional cut 21 < 0:3 with di erent combinations of jets used for the computation of the jet mass, 1 and 2 as indicated in the legend, our dichroic combination being plotted using a solid black line. The cross section used for normalisation, is that for jets above the pt cut. 21 distributions for jets in dijet (solid lines) and W W (dashed lines) events again imposing pt > 2 TeV and including SoftDrop grooming. Di erent colours correspond to di erent combinations of jets used for the computation of the jet mass, 1 and 2 as indicated in the legend, our new dichroic combination being plotted in black. We have selected jets with a mass is between 60 and 100 GeV. The crosssection used for normalisation, , is de ned after the jet pt and mass cut, so that all curves integrate to one. mass distribution ⎯ dijets ⎯ Pythia8 tagged mass [mMDT(zcut=0.1)] suppression, related to the fact that this combination puts no constraints on large-angle emissions (region C in the previous section). Furthermore, the dichroic combination, the solid black curve in gure 7, is expected to have the largest suppression and is indeed peaked at larger 21 values, translating into a larger discrimination against signal jets. Note that the 21 distribution for the dichroic combination also shows a peak for 12 > 1 that we have not discussed in our earlier argumentation. This comes from events with multiple emissions in region C and will be discussed brie y in our analytic calculations in section 5. Results for the mass distribution obtained for background (QCD dijets) jets at parton level (without UE) are presented in gure 8. As in gure 7, SoftDrop grooming has always been applied prior to any additional tagging or N -subjettiness cut. Again, we can identify most of the features discussed in section 3. First of all, if we compare the mMDT-tagged mass (dashed blue curve) to the SD-groomed jet mass (dashed red curve) we see that the latter is smaller than the former at small masses, owing to the larger Sudakov factor RSD > RmMDT, but larger at intermediate masses, due to the larger prefactor RS0D > Rm0MDT. Then, we can consider the e ect of the additional constraint on the 21 ratio, taken here as 21 < 0:3 for illustrative purpose. If we compute 21 on the same jet as for the mass ( 2fu1;lglroomed in solid red and 21 tagged in solid blue for the SD-groomed and mMDT-tagged jets respectively), we see that the cut reduces the background, that the reduction increases for smaller masses and that the reduction is larger for the SD-groomed jet than for the mMDT-tagged jet. This last point is a re ection of the fact, that the Sudakov suppression associated with the N -subjettiness cut is larger when both the mass and 21 are computed on the SD-groomed jet ( gure 3(left)) than when both the mass and 21 are computed on the mMDT-tagged jet ( gure 5(left)). Then, when measuring the mMDT-tagged jet mass, one sees that computing 21 on the SD-groomed jet ( 2fu1;lglroomed, the solid green curve in gure 8) shows a larger suppression than computing 21 on the mMDT-tagged jet, although the di erence is reduced at very small masses. Finally, if we consider our new, dichroic case, eq. (3.2) ( 2d1ic;ghrroooicmed, the solid black curve), we see a larger suppression than in all other cases, as expected from our earlier arguments. Signal v. background discrimination and other performance measures To further test the performance of our new method, we have also studied ROC (receiver operating characteristic) curves, shown in gure 9 for parton-level simulations and in gure 10 for hadron-level events including hadronisation and the Underlying Event. In all cases, we impose the constraint that the (full, tagged or groomed) mass is between 60 and 100 GeV. E ciencies are given relative to the inclusive cross-section for having jets above our pt cut. Let us rst discuss the result of parton-level simulations, gure 9, where the dichroic ratio is again represented by the black curves. Without grooming (the left-hand plot in the gure), our method shows a substantial improvement compared to all other combinations considered, outperforming them by almost 30% in background rejection at a signal e ciency of 50% and by more than a factor of 2 at a signal e ciency of 40%. After SoftDrop grooming (right-hand plot), the dichroic method, i.e. computing the jet mass and 1 on the mMDTtagged jet and 2 on the SD-groomed jet, still shows an improvement, albeit less impressive than what is observed using the full jet to compute 2 If instead we consider the results at hadron level, including both the perturbative parton shower as well as non-perturbative e ects, gure 10, we see that the dichroic subjettiness ratio still does a better job than the other variants but the gain is smaller. For example, measuring the mMDT-tagged mass with a cut on the groomed dichroic ratio, dichroic 21;groomed, the optimal choice in gure 10, is only slightly better than the next best choice where one measures the SD-groomed mass and imposes a constraint on 2fu1;lglroomed. This is because in going from parton to hadron level, the groomed- 2fu1;lglroomed curve has moved down more than the tagged- 2d1ic;ghrroooicmed curve, i.e. the former is getting a signi cantly larger boost in its discriminating power from non-perturbative e ects.15 This is potentially prob15That there should be larger non-perturbative e ects in the groomed- 2fu1;llgroomed can be understood as follows: because groomed accepts a larger fraction of signal events in a given mass window than tagged, to reach the same nal e ciency the 21 cut must be pushed closer to the non-perturbative region. ROC curves ⎯ parton level ⎯ ungroomed ROC curves ⎯ parton level ⎯ groomed √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV tagger: mMDT(zcut=0.1) √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV tagger: mMDT(zcut=0.1) parton level. The left plot is obtained starting from the full jet, while for the right plot, a SoftDrop grooming step has been applied. The ROC curves are obtained by varying the cut on the 21 ratio. ROC curves ⎯ hadron level ⎯ ungroomed ROC curves ⎯ hadron level ⎯ groomed √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV tagger: mMDT(zcut=0.1) √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV tagger: mMDT(zcut=0.1) lematic, because one does not necessarily want signal-to-background discrimination power for a multi-TeV object to be substantially driven by the physics that takes place at a scale of 1 GeV, physics that cannot, with today's techniques, be predicted from rst principles. Additionally, phenomena happening on a scale of 1 GeV are di cult to measure reliably. performance for various pt cuts ε /√ 6 ( :e 5 isg 3 l ing 2 S 1 tagger: mMDT(zcut=0.1) 60 < m < 100 GeV 60 < m < 100 GeV B ε 4 ce 3 n isg 2 tagger: mMDT(zcut=0.1) (de ned as the ratio between the background \fake" tagging rate at hadron and parton level). Di erent curves correspond to di erent combinations indicated in the legend. For the solid curves, a SoftDrop ( curves. In the left-hand plot, we impose a 2 TeV pt cut on the initial jet. The symbols on each curve then correspond to a signal e ciency (computed at hadron level) ranging from 0.05 upwards the right-hand extremity explicitly labelled. In the right-hand plot, the signal e ciency (computed at hadron level) is xed to be 0.5 and the pt cut on the jet is varied between 500 GeV and 3 TeV (in steps of 500 GeV, labelled explicitly for the groomed dichroic ratio), with the large symbol on each line corresponding to a 3 TeV cut. It would be interesting to investigate non-perturbative e ects in greater depth, both analytically, e.g. following the approach used in [2], or by studying their dependence across di erent Monte-Carlo generators and associated tunes. However, for the purpose of this article, we limit ourselves to using the results from Pythia 8. In evaluating the overall performance of di erent 21 combinations we will consider both the signal signi cance and the size of non-perturbative e ects. We will use the following alternative to ROC curves. For a given method and pt cut, we rst determine the 21 cut required to obtain a desired signal e ciency (at hadron level). For that value of the 21 cut, we can compute the signal signi cance, de ned as S=p B (computed at hadron level) which is a measure of the discriminating power of the method; we then estimate non-perturbative e ects as the ratio between the background e ciency at hadron level divided by the background e ciency at parton level, which is a measure of robustness against non-perturbative e ects. We will show results for a range of di erent signal-e ciency choices and jet pt cuts. In gure 11, which highlights the key performance features of the dichroic method, we plot the signal signi cance versus the non-perturbative e ects for di erent methods. In the left-hand panel, the curves correspond to a range of 21 cuts for jets with pt > 2 TeV. The points on the curves correspond to di erent signal e ciencies (starting from 0.05, in steps on the curves correspond to di erent pt cuts, with the 21 cut adjusted (as a function of pt) so as to ensure a constant signal e ciency of 0:5. In both plots, the 21 cut is determined so as to achieve the expected signal e ciency at hadron level and the same cut is used for parton-level results. To avoid the proliferation of curves, the result for the ungroomed full- 2fu1ll is not shown since it is obvious from the ROC curves in gures 9 and 10(left) that it is extremely sensitive to non-perturbative e ects. In both plots, we see that the dichroic method comes with larger discriminating power with a relatively limited sensitivity to non-perturbative e ects, provided one rst applies a grooming step. Without the grooming step, one observes a much larger sensitivity to non-perturbative e ects, as one might expect.16 It also appears that the performance gain increases when the boost, i.e. the jet pt, increases. This was also expected from our arguments in section 3. Finally, compared to the common setups in the literature, namely with modi ed MassDrop tagging with a cut on 21 applied either on the mMDT ( tag- 2t1agged, the dot-dashed blue curve) or on the full jet ( tag- 2fu1ll, the dashed green curve), our dichroic method with grooming (solid black) gives up to a factor of two improvement in signal signi cance, with comparable non-perturbative e ects. Considering other combinations that have not been widely used experimentally, 2fu1;lglroomed with either a groomed ( groom, still remains the best, with an optimal signi cance that is about 25% larger, and smaller non-perturbative corrections for any given signal signi cance. As a nal check, we have studied the dependence of the signal e ciency on the 21 cut, as shown in gure 12. Comparing the left and right-hand plots, it appears clearly that applying SoftDrop grooming helps to reduce non-perturbative e ects which otherwise signi cantly lower the signal e ciency. It is also interesting to notice that without grooming, the signal e ciency obtained with our dichroic method (the dashed black curve on the left plot of gure 12) only reaches its plateau for cuts on 21 larger than 1 already at parton level. This can likely be attributed to initial-state radiation in the jet at angles larger than the decay angle of the W boson. These e ects are strongly reduced by SoftDrop grooming (see also the discussion in section 5). In the end, a more complete study would include variations of the SD parameters and of the cuts on the mass. A brief investigation of the SD parameters shows that our choice domain under study. However, in view of the good signal e ciency reached when computing both the jet mass and 21 with SoftDrop, it might also be interesting to investigate our dichroic combination where we also use SoftDrop for the tagger instead of the mMDT. An extensive analytic study foreseen in a follow-up paper [42] would allow for a systematic study of these e ects. Such an analytic understanding could also be of use in the context of building decorrelated taggers [10]. 16It can also be shown that grooming largely reduces the impact of initial-state radiation as well (see also [12]). tagger: mMDT(zcut=0.1) considered in gures 9 and 10. Solid curves correspond to hadron-level results while dashed curves are obtained at parton level. The left plot is obtained starting from the full jet, while for the right plot, a SoftDrop grooming has been applied. Brief comparison with other tools To complete our Monte Carlo studies, in gure 13 we compare the performance of 2d1ic;ghrroooicmed with various other tools: mMDT tagging alone, SoftDrop grooming alone ( = 2 as above), and also the Ym variant [29] of Y-splitter [43], combined either with SoftDrop (pre-)grooming or with trimming [44], as described in detail in ref. [29] (see also ref. [28]). Whereas in the analogous gure 11, all curves involved the same signal e ciency, here this is no longer the case. Accordingly e ciencies are reported versus pt in table 1. Let us start by examining the pure mMDT result: as known already from [2] it provides mild tagging, it has small non-perturbative corrections and only modest dependence on pt. SoftDrop ( perturbative corrections.17 These two tools have the highest signal e ciencies, of about 63% and 76% respectively at 2 TeV. Next we examine combinations that involve Ym-splitter. Recall that this tool undoes y = min(pt21; pt22) This cut is similar in its e ect to zcut in mMDT. When used in conjunction with SD (pre-)grooming, the highest-mass emission that passes the SD cut is also the one that is unclustered by Ym-splitter and so it is required to pass the ycut condition. As a result, the constraint in the Lund plane turns out, at the leading-log level, to be identical to section 7 of ref. [3]). √ /εS 6 icfi 4 g lis 3 Ym-split.(ycut=0.11) + trim.(ftrim=0.1,Rtrim=0.3) mMDT(zcut=0.1) 60 < m < 100 GeV tagger: mMDT(zcut=0.1) √ /εS 6 icfi 4 g lis 3 60 < m < 100 GeV from 500 GeV to 3 TeV in steps of 500 GeV, as in gure 11(right). The 3 TeV point is always = 2) with a range of other tools, including Ym-splitter (left) and not explicitly labelled, it is equal to 2. Note that the default signal-e ciency working point for the dichroic subjettiness ratios is 0:4 here rather than the 0:5 chosen in gure 11. The signal e ciencies for other cases are given in table 1. Ym-splitter+trimming SoftDrop+Ym-splitter all other variants dichroic 21;groomed < 1 jet pt cut [GeV] that obtained with dichroic 21;groomed and the condition cut = 1, with a Sudakov suppression vetoing all emission down to a mass scale in the SD-groomed jet, and a small prefactor s ln(1=ycut). This is re ected in gure 13, where one sees that the dichroic 21;groomed < 1 curve (black open diamonds) is remarkably similar to the SD+Ym-splitter curve (red open Where the dichroic 21;groomed variable has an advantage is that one can now further adjust the choice cut, whereas with SD+Ym-splitter that freedom is not available. Of the various Ym combination considered in ref. [29], the one that gave the best signal-to-background discrimination was Ym with trimming, shown as red solid squares in gure 13. Overall it performs less well than the mMDT plus 2d1ic;ghrroooicmed combination with xed to 0:4, even though is has a broadly similar signal e ciency. Another point to discuss concerns the choice of in the N -subjettiness de nition, eq. (2.2). Many experimental uses of N -subjettiness ratios have concentrated on the choice = 2. A discussion of the = 1 case is given in appendix A, including comparisons of dichroic and normal variants. Dichroic always perform best also for only show dichroic results. An argument often given for the choice of = 1 is that it is less sensitive to nonperturbative e ects. Figure 13 (right) shows groomed ( lled symbols, solid lines) and ungroomed (open symbols, dashed lines) results for = 1 (squares and triangles) and = 2 (circles). For the standard E-scheme four-vector recombination (triangles), or the exclusive-kt axes with the winner-takes-all (WTA) recombination scheme (squares) [17{19]. In both the SD-groomed and ungroomed cases, the non-perturbative corrections are somewhat smaller for = 1 (except in the WTA groomed case). In the ungroomed case, = 1 also leads to better signal-discrimination. However once SD-grooming is included the signal discrimination is best for the e ects for the SD-groomed performance in dichroic = 2) performance is very similar to the 21;groomed( = 1, S = 0:4) gure 13(right). Therefore, it is the SD-groomed, = 2, dichroic ratio that appears to give the best overall performance. There are a number of other variables that one might also consider, notably energycorrelation functions (ECFs) [22{24]. In particular we expect that dichroic ratios may be of use also for the most recent set of ECFs discussed in ref. [24], a number of which are designed to have similarities to N -subjettiness. Their study is, however, beyond the scope of this work. Brief analytic calculations In this section, we consider brief analytic calculations relating to the observables we have presented so far. Our main goal here is to illustrate that the discussion from section 3 | where we used Lund diagrams to motivate dichroic subjettiness ratios | does indeed capture the qualitative picture observed in Monte-Carlo simulations. To that aim, it is su cient to use leading-logarithmic accuracy, where we control double logarithms, i.e. 1. Note that, recently, several jet substructure methods have been understood at higher accuracy, see e.g. [21, 45], and we intend to provide a more precise calculation in future work [42]. { 22 { block T , eq. (5.1), used to write all Sudakov exponents. Centre: representation of the full jet Sudakov Rfull( ; cut; z), eq. (5.3a), including secondary emissions. Right: representation of the full jet Sudakov RSD( ; cut; z), eq. (5.3c), including secondary emissions. For both the centre and right plots, the dot indicated by z corresponds to the emission dominating the jet mass and we will integrate over allowed values of its momentum fraction z. In practice, we will express everything in terms of the following fundamental block (cf. gure 14(left)): T ( ; zcut; CR) = where angles are normalised to the jet radius R and we use the 1-loop running-coupling s(z ptR) = s=(1 + 2 s 0 log z ) with s(ptR) and 0 = (11CA 4nf TR)=(12 ). Explicit expressions for T are given in appendix B and are mostly taken For the QCD background, we nd, for cut < 1: RmMDT( ; cut; z) ; (5.2b) where zSD( ) = max ( c2ut)1=(2+ ); and (cf. gure 14(middle,right)) Rfull( ; ; z) = T0( RmMDT( ; ; z) = Rfull( ; ; z) RSD( ; ; z) = Rfull( ; ; z) =z; CR); ; cut( =z)( +1)=2; CR): Note that the full and mMDT jet mass Sudakov introduced respectively in eq. (2.5) and eq. (2.7) (and used below) can be written as Rfull( ) = Rfull( ; 1; \any z") ; RmMDT( ) = RmMDT( ; 1; \any z") : In the above expressions, z corresponds to the momentum fraction of the emission dominating the jet mass (emission \a" in gures 4 and 5). Compared to the simple R0 factor that we had in section 2.2, we keep the z integration explicit since the secondary emissions, the CA terms, depend explicitly on z. In all cases, the integration over z runs over the region kinematically allowed by the tagger de ning the jet mass. The Sudakov exponent in these expressions is then essentially given by the jet on which we compute 2 While we only target leading-logarithmic accuracy, our results also include the singlelogarithmic contributions coming from hard collinear splittings, which are often phenomenologically important. They appear as the bi factors in eqs. (5.2) and (5.3), where we have introduced bi = exp(Bi) with Bq = 3=4 and Bg = 4nf TR)=(12CA). These contributions can e ectively be taken into account by limiting all z integrations to bi for primary emissions and bg for secondary emissions. Finally, as expected, if one takes the limit ! 1 of the SD results, one recovers the full results. Also, the limit ! 0 of (5.2c), reduces to (5.2b). So far, we have not yet discussed the case where is computed from the mMDT-tagged jet and 21 from the full jet. This is more involved due to the two separate kinematic con gurations involved (see gure 4(b-c)). In the end, we nd (assuming = cut c Z zcut d c Z zcut dzc s(pzc cptR)CR Rout,full( c; cut; zc) ; (5.5) and a similar expression with \full" replaced by \SD" for the case where 21 is calculated Rout,SD( c; ; zc) = Rout,full( c; ; zc) p czc ; cut( c=zc)( +1)=2; CR): The con gurations contributing to the last two lines of eq. (5.5) come from jets with at least one emission in region C (discarded by mMDT) with c result in an extra contribution to the mass distribution, which would then be larger than what we obtain with our dichroic combination (eq. (5.2d) or, equivalently, the of eq. (5.5)). When using the dichroic combination, these con gurations would all have = cut. They combination leads to: Z zcut d c Z zcut dzc s(pzc cptR)CR cu=t>1 Z bi dz s(pz ptR)CR e Rout,full( c; cut= c;zc) Rout,full( ) = T0( ; zcut; CR) Rout,SD( ) = Rout,full( ) This result splits into 2 contributions corresponding to the two terms in the round bracket on the second line of (5.7): the rst term comes from con gurations where there is no emission in region C with z 2 > cut, and it corresponds to values of 2d1ichroic < 1 (this is manifest, because in eq. (5.7), given for cut > 1, it has no dependence on cut). For the second contribution, the part corresponding to values of 2d1ichroic 1, there is an emission cut. To guarantee 21 < cut, we then need to veto emissions (both primary and secondary) with z 2 > Note that this second contribution itself includes two sub-contributions: the case where emission \c" is the only emission in region C with z 2 > , yielding a contribution to the 21 distribution proportional to ( 21 (recall that 2full is set by the second hardest emission overall, which makes it equal to tagged); and a second sub-contribution where, in addition to emission \c", there is at least one additional emission with cut > z 2 > , yielding a continuum with 21 > 1 in the 21 distribution (see gure 7 as well as the right plot of gure 15 below). One can calculate 1) contribution to the 21 distribution by taking the di erence between (5.7) and (5.2d) for cut ! 1 which gives Z bi dz s(pz ptR)CR Z zcut d c Z zcut dzc s(pzc cptR)CR in round brackets in eq. (5.7). In practice the -function contribution gets smeared out to values of 12 > 1 through the e ect of multiple emissions. Note that it is relatively straightforward to check that the limit cut ! 1 in eq. (5.5), From the equations above, the 21 distribution, for a given jet mass, can be obtained by taking the derivative with respect to cut and normalising by the jet mass distribution without any cut on 21. Background e ciencies can also be obtained straightforwardly by integrating any of the above mass distributions over the allowed mass window. 18Note that the di erence between the Sudakov suppression in the two contributions comes from secondary For signal jets, we assume that if the jet mass is not within some reasonable window around the boson mass, then the jet is discarded. We then nd the following signal e ciency S = fISR dz psig(z) exp with zmin = , zSD( ) or zcut depending on whether the mass is computed on the full jet, the SD-groomed jet or the mMDT-tagged jet, respectively. The 21 distribution for a given jet mass can be obtained by taking the derivative of S with respect to cut (and normalising appropriately). In eq. (5.10) the Sudakov exponent is given by Rsig( ; ; z) = hT0(pz(1 z) =zbi; CR) T0(pz =(1 z) ; pz =(1 z) =z; pz =(1 z) =zbi; CR) z)bi; CR) ; sn ln2n , though we also valid for small . Here we target double-logarithmic accuracy, include a set of nite-z and hard-splitting corrections that were found to be numerically important in ref. [12] (cf. eq. (A.24)). These represent only a subset of next-to-leading logarithmic terms. Note that for z 1) the term on the fourth (second) line is zero because of the last of the -functions in eq. (5.1), while the term on the third ( rst) line corresponds to the leaf in gure 3(right). For simplicity, in our numerical results we will use psig(z) = 1 in eq. (5.10).19 Eq. (5.10) also includes a factor fISR that accounts for the e ect of initial-state radiation (ISR). Such e ects are present both for signal and background jets and are generically single-logarithmic. As such they are subleading compared to the double-logarithms that Nevertheless, if we consider signal jets and examine the limit of large pt with M , cut, etc. all xed, then because of the absence of double logarithms of , single-logarithmic ISR e ects ( s ln )n can be numerically dominant [28]. Physically, they are associated with the requirement that ISR should not substantially modify the mass of the signal jet. The correction involves ( s ln )n terms, only when the mass is determined on the full jet and the factor fISR then takes the form fISR = exp = 0 s(pt) log 19For the W W process under consideration, correlations between the incoming quarks and the nal quarks after the decay of the two W bosons have been calculated in [46] and could in principle be used to compute psig(z). This would however be speci c to the W W process considered here just as an example. We therefore use the \splitting function" of an unpolarised W boson. This simpli cation does not a ect signi cantly any of the results presented here. mass distribution ⎯ dijets ⎯ analytic tagged mass [mMDT(zcut=0.1)] d /σd 0.6 tagger: mMDT(zcut=0.1) (dijets) has been represented with nite width and scaled down by a factor of 5. where a non-global contribution (formally of the same logarithmic order) is ignored for simplicity. In the above formula, M is size of the mass window in which signal jets are accepted, and a full treatment of all single-logarithmic corrections would need to account also for logarithms of precise phenomenological applications. The nite O ( s) component associated with highpt emissions could be obtained e.g. using POWHEG [47{49], aMC@NLO [50] or at NNLO using MATRIX [51] or MCFM [52{54]. We can now compare our analytic predictions with the Monte-Carlo results from the previous section. We use the previous section, and freeze the coupling for scales below fr = ~frptR, which we set to 1 GeV. We start with the QCD mass distributions, shown on the left plot of gure 15, to be compared to the Monte-Carlo results presented in gure 8. Globally, we see that our analytic calculation captures correctly the main patterns discussed earlier. We note however that the analytic distributions, especially those involving the full jet mass, are less peaked than the Monte-Carlo ones. This is likely due to subleading logarithmic corrections, like multiple-emission corrections which would e ectively increase the Sudakov exponent. The 21 distributions for both QCD jets and signal (W ) jets are shown in the right plot of gure 15, to be compared with gure 7. The ordering between the di erent curves is well captured by our analytic expressions. Di erences related to the over-simplicity of our leading-logarithmic approximation are larger than what was seen for the mass distribution. First, our analytic calculations are non-zero when 21 ! 1. This region is however not under control within our strongly-ordered approximation. Similarly, the kink observed 0:5 is not physical. It comes from the onset of the secondary-emission contribu ROC curves ⎯ analytic ⎯ ungroomed ROC curves ⎯ analytic ⎯ groomed anti-kt(R=1), pt>2 TeV, 60<m<100 GeV 0.1 tagger: mMDT(zcut=0.1) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV combination is given by the black curves in the right plot of gure 15. The dijet case clearly has a contribution proportional to ( 21 1) (cf. eq. (5.9)) (scaled down by a factor of 5 for clarity), which is not observed in the Monte-Carlo results. In practice, additional emissions at smaller z 2 would also contribute to 21, and they would transform the ( 21 contribution into a Sudakov peak at 21 & 1, which is visible on the Monte-Carlo simulations. We are currently working on a better analytic calculation, lifting the assumption that emissions are strongly ordered in z 2 [42]. Finally, let us turn to the ROC curves, plotted in gure 16. We again see that they reproduce the main qualitative features observed in section 4. There are however quantitative di erences between our analytic results and the Monte-Carlo simulations. For example, our calculation over-estimates the signal e ciencies. A more quantitative description would require a more precise analytic treatment including subleading corrections, beyond the strong-ordering approximation, and xed-order corrections for signal e ciencies. Conclusion In this paper we have examined the interplay between boosted-object tagging algorithms, mMDT or SoftDrop, and radiation constraints, notably as imposed through N -subjettiness numerator is evaluated on the full jet, while the denominator is evaluated on the set of constituents left after the tagging stage. The name \dichroic" comes from the fact that the large-angle colour ow, present in backgrounds but not signals, gets directed exclusively to the numerator and not the denominator. It is this feature that leads to an enhanced signi cance in distinguishing (colour-singlet) signals from (colour-triplet or octet) backgrounds, notably compared to current widely used N -subjettiness ratios. As well as considering signal-signi cance, it is important to keep non-perturbative e ects under control: a method that is overly reliant on non-perturbative physics for its discrimination power is one for which signal-e ciency and background-rejection estimates may be highly model-dependent, and correspondingly uncertain. It is also likely to be subject to large detector e ects. We have found that the combination of a light grooming step based on SoftDrop ( 2SD= 1tagged is e ective in maintaining good signal-to-background signi cance while substantially limiting nonperturbative e ects. The overall behaviour of our dichroic 21 variable, with grooming, was illustrated in gure 7: the 21 distribution for signal jets is left largely unmodi ed by the change to a dichroic variant (black dashed curve versus any of the other dashed curves), whereas the distribution for background jets is shifted to substantially higher values of 21 (black solid curve versus any of the other solid curves), increasing the ability to distinguish signal and Figures 11 and 13 provide a summary of the signal-signi cance (vertical axis) and non-perturbative corrections (horizontal axis) for a range of boosted-object identi cation methods. The points along the lines correspond to di erent signal-e ciency working points ( gure 11(left)) or pt cuts (the other plots). One sees that 2d1ic;ghrroooicmed with = 2, in black, provides the best signal signi cance of any of the methods and that, for a given signal signi cance, it tends to limit the size of non-perturbative e ects relative to other methods. In addition to the Lund-plane based arguments given in section 3 and the Monte Carlo studies of section 4, we have also outlined the analytic leading-logarithmic structure of di erent combinations of taggers and 21 ratios. As well as bringing insight into the behaviour of di erent taggers, such calculations provide a basis for the future design of \decorrelated" [10] combinations of taggers and dichroic radiation constraints, providing background rejection that is independent of the tagged jet mass and thus straightforward to use in the context of data-driven background estimates. Acknowledgments GS thanks CERN for hospitality while part of this work was being nalised. GS's work is supported in part by the French Agence Nationale de la Recherche, under grant ANR-15CE31-0016. GPS and GS are both supported in part by ERC Advanced Grant Higgs@LHC (No. 321133). LS and GS are also grateful to the Erwin-Schrodinger-Institute for Mathematics and Physics for hospitality when parts of this work was being completed. Dichroic subjettiness ratios for = 1 In section 3, we have argued in favour of the dichroic subjettiness ratios using N -subjettiness = 1, for which the dichroic variant can also be considered. Note that for = 1, we have de ned the N -subjettiness axes through an exclusive-kt declustering. This can be done either using the standard Escheme four-vector recombination or the winner-takes-all (WTA) recombination scheme. For simplicity, we will focus on E-scheme results here. A brief comparison between the two axis choices is shown in gure 13(right). √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV 0.1 tagger: mMDT(zcut=0.1) √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV 0.1 tagger: mMDT(zcut=0.1) √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV 0.1 tagger: mMDT(zcut=0.1) √s=13 TeV, Pythia(8.186) anti-kt(R=1), pt>2 TeV, 60<m<100 GeV 0.1 tagger: mMDT(zcut=0.1) = 1 (dashed lines) and = 2 (solid lines). The same 4 variants as in gures 9 and 10 are included. The left (right) column corresponds to full (SD-groomed) jets. The top (bottom) row corresponds to parton-level (hadron-level) events. Figure 17 shows ROC curves similar to those presented in gures 9 and 10, this time including results for = 1 as dashed lines. We can make several observations based on these plots. First, as for = 2, we see that the dichroic ratio also outperforms the other combination for = 1. The performance gain is however smaller, especially with SD grooming. In terms of the sensitivity to non-perturbative e ects, we see that N -subjettiness ratios to non-perturbative e ects might have been anticipated since the corresponding kt cut is less a ected by soft-and-large-angle emissions than for = 2. A consequence of this observation is that grooming is less critical when using a cut on N -subjettiness ratios performance gain compared to the other approaches, cf. the bottom-left plot of gure 17. Finally, we can argue that = 1. To be fair, the comparison should be made between on the bottom-right plot of gure 17) and 2d1ichroic for dichroic 21;groomed for = 2 (the solid black line = 1 (the dashed black line on the bottom-left plot) which both show good signal signi cance and limited non-perturbative corrections. This comparison shows a somewhat larger background rejection in the = 2 case for typical signal e ciencies in the 0:2 0:6 range, as also seen in gure 13. Explicit expressions for the analytic results For completeness, we give the result of the building block used for all the analytic calculations in section 5, see eq. (5.1). We work with a one-loop running coupling (with 5 active avours), appropriate at our T ( ; 0; CR) L<=Lfr L0<=Lfr<L L0=>Lfr (L Lfr)(L+Lfr 2L0) with W (x) = x log(x) and L = log(1= ); L0 = log(1= 0); Lfr = log(1=~fr); L = L0 + (1 + )L = 2 s 0L; 0 = 2 s 0L0; fr = 2 s 0Lfr; = 2 s 0L: Example code for dichroic subjettiness ratios In this last appendix, we brie y indicate how dichroic subjettiness ratios can be implemented using tools available in FastJet and fjcontrib. In particular, we make use of the RecursiveTools contrib (for ModifiedMassDropTagger and SoftDrop) and of the Nsubjettiness contrib. First, besides standard FastJet headers needed for jet clustering, one needs to include the following headers: #include <fastjet/contrib/ModifiedMassDropTagger.hh> // mMDT tagger #include <fastjet/contrib/SoftDrop.hh> #include <fastjet/contrib/Nsubjettiness.hh> // optional SD grooming // tau1 and tau2 Then, one should declare the basic objects needed for tagging, computing 1 and 2, and, optionally, grooming: // the tagger [here mMDT with a z cut] // Note: by default, this automatically reclusters with Cambridge/Aachen double zcut = 0.1; fastjet::contrib::ModifiedMassDropTagger mmdt_tagger(zcut); // (optional) groomer [here SoftDrop] // Note: by default, this automatically reclusters with Cambridge/Aachen double beta = 2.0; double zetacut = 0.05; fastjet::contrib::SoftDrop sd_pre_groomer(beta, zetacut); // (for theoretical reasons it is preferred to use an unnormalised measure) double beta_tau = 2.0; fastjet::contrib::UnnormalizedMeasure measure(beta_tau); fastjet::contrib::GenKT_Axes axes_gkt(1.0/beta_tau); fastjet::contrib::Nsubjettiness tau1(1, axes_gkt, measure); fastjet::contrib::Nsubjettiness tau2(2, axes_gkt, measure); Note that all parameters here are given as examples and have not been optimised. 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Gavin P. Salam, Lais Schunk, Gregory Soyez. Dichroic subjettiness ratios to distinguish colour flows in boosted boson tagging, Journal of High Energy Physics, 2017, 22, DOI: 10.1007/JHEP03(2017)022