VINCIA for hadron colliders
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Abstract
We present the first public implementation of antennabased QCD initial and finalstate showers. The shower kernels are \(2\rightarrow 3\) antenna functions, which capture not only the collinear dynamics but also the leading soft (coherent) singularities of QCD matrix elements. We define the evolution measure to be inversely proportional to the leading poles, hence gluon emissions are evolved in a \(p_\perp \) measure inversely proportional to the eikonal, while processes that only contain a single pole (e.g., \(g\rightarrow q\bar{q}\)) are evolved in virtuality. Nonordered emissions are allowed, suppressed by an additional power of \(1/Q^2\). Recoils and kinematics are governed by exact onshell \(2\rightarrow 3\) phasespace factorisations. This first implementation is limited to massless QCD partons and colourless resonances. Treelevel matrixelement corrections are included for QCD up to \(\mathcal {O}(\alpha _s^4)\) (4 jets), and for Drell–Yan and Higgs production up to \(\mathcal {O}(\alpha _s^3)\) (V / H + 3 jets). The resulting algorithm has been made publicly available in Vincia 2.0.
1 Introduction
The basic differential equations governing renormalisationgroupimproved (resummed) perturbation theory for initialstate partons were derived in the 1970s [1, 2, 3]. The resulting DGLAP^{1} equations remain a cornerstone of highenergy phenomenology, underpinning our understanding of perturbative corrections and scaling in many contexts, in particular the structure of QCD jets, parton distribution functions, and fragmentation functions.
In the context of event generators [4], DGLAP splitting kernels are still at the heart of several presentday parton showers (including, e.g., [5, 6, 7, 8, 9]). Although the DGLAP kernels themselves are derived in the collinear (smallangle) limit of QCD, which is dominated by radiation off a single hard parton, the destructiveinterference effects [10] which dominate for wideangle softgluon emission can also be approximately accounted for in this formalism; either by choosing the shower evolution variable to be a measure of energy times angle [11] or by imposing a veto on nonangularordered emissions [12]. The resulting partonshower algorithms are called coherent. A third alternative, increasingly popular and also adopted in this work, is to replace the partonbased DGLAP picture by socalled colour dipoles [13] (known as antennae in the context of fixedorder subtraction schemes [14, 15, 16, 17, 18]),^{2} which incorporate all singleunresolved (i.e., both soft and collinear) limits explicitly. In the context of shower algorithms, this approach was originally pioneered by the Ariadne program [13, 21] and is now widely used [22, 23, 24, 25, 26, 27, 28, 29, 30]. We note that the word “coherence” is used in different contexts, such as angular ordering. When we use coherence in the context of antenna functions, we define it at the lowest level, as follows: antenna functions sum up the radiation from two sides of the leading\(N_C\) dipole coherently, at the amplitude level; see also Ref. [27].
In addition, shower algorithms rely on several further improvements that go beyond the LO DGLAP picture, including: exact momentum conservation (related to the choice of recoil strategy), colourflow tracing (in the leading\(N_C\) limit, related to coherence at both the perturbative and nonperturbative levels), and higherorderimproved scale choices (including the use of \(\mu _R=p_\perp \) for gluon emissions and the socalled CMW scheme translation which applies in the soft limit [31, 32]). Each of these are associated with ambiguities, with Sect. 2 containing the details of our choices and motivations.
Finally, in the context of initialstate parton showers, the evolution from a high factorisation scale to a low one corresponds to an evolution in spacelike (negative) virtualities, “backwards” towards lower resolution. The correct equations for backwards partonshower evolution were first derived by Sjöstrand [33]; in particular it is essential to multiply the evolution kernels by ratios of parton distribution functions (PDFs), to recover the correct lowscale structure of the incoming beam hadrons. We shall use a generalisation of backwards evolution to the case of simultaneous evolution of the two incominghadron PDFs, similar to that presented in [26].

They are intrinsically coherent, in the sense that the correct eikonal structure is generated for each singleunresolved soft gluon, up to corrections suppressed by at least \(1/N_C^2\). Especially for initial–final antennae, where gluon emission off initial and finalstate legs interfere, has some challenges.^{3} In the final–final case which was already testable with previous Vincia versions, a recent OPAL study of 4jet events [37] found good agreement between Vincia and several recently proposed coherencesensitive observables [38].

They are extremely simple, relying on local and universal \(2\rightarrow 3\) phasespace maps which represent an exact factorisation of the nparticle phase spaces not only in the soft and collinear regions but over all of phase space. This makes for highly tractable analytical expansions on which our accompanying matrixelement correction formalism is based [39]. The pure shower is in some sense merely a skeleton for generating the leading singularities, with corrections for both hard and soft emissions regarded as an intrinsic part of the formalism, restoring the emission patterns to at least LO accuracy up to the matched orders.

There is a close correspondence with the antennasubtraction formalism used in fixedorder calculations [16, 17, 18], which is based on the same subtraction terms and phasespace maps. This property was already utilised in [40] to implement a simple and highly efficient procedure for NLO corrections to gluon emission off a \(q\bar{q}\) antenna. Highly nontrivial fixedorder results which have recently been obtained within the antenna formalism include NNLO calculations for \(Z+\mathrm {jet} \) [41], \(H+\mathrm {jet} \) [42] (for \(m_t\rightarrow \infty \)), \(gg\rightarrow gg\) [43], and leadingcolour \(q\bar{q}\rightarrow t\bar{t}\) [44] production at hadron colliders. While it is (far) beyond the scope of the present work to connect directly with these calculations, their feasibility is encouraging to us, and provides a strong motivation for future developments of the antennashower formalism.
In Sect. 2 we introduce the basic antennashower formalism, including our notation and conventions. We mainly focus on initial–initial and initial–final configurations and summarise final–final configurations only briefly, as a more extensive description is available in [23, 39]. Our conventions for colour flow are specified in Sect. 2.6. These are intended to maximise information on coherence while simultaneously generating a state in which all colour tags obey the indexbased treatment of subleadingcolour correlations proposed in [48, 49]. By assigning these indices after each branching and tracing them through the shower evolution, rather than statistically assigning them at the end of the evolution as was done in [49], we remove the risk of accidentally generating unphysical colour flows.^{4} We therefore believe the procedure proposed here represents an improvement on the one in [49]. The extension of Vincia ’s automated treatment of perturbative shower uncertainties to hadron collisions is documented in Sect. 2.7.
In Sect. 3, we present the extension of the GKS^{5} matrixelementcorrection (MEC) formalism [39] to initialstate partons, starting with the case of a basic process accompanied by one or more jets whose scales are nominally harder than that of the basic process in Sect. 3.1. In Sect. 3.2, we present some basic numerical comparisons between treelevel matrix elements and our shower formalism expanded to the equivalent level (i.e., setting all Sudakov factors and coupling constants to unity), to validate that combinations of \(2\rightarrow 3\) antenna branchings do produce a reasonable agreement with the full nparton matrix elements. We discuss our extension of “smooth ordering” [39] to reach nonordered parts of phase space in Sect. 3.3, again focusing on the initialstate context. Section 3.5 summarises the application of smooth ordering to the specific case of hard jets in QCD processes. In Sect. 3.6 we extend and document Vincia ’s existing use of MadGraph 4 [50] matrix elements.
The set of numerical parameters which define the default “tune” of Vincia 2.0 is documented in Sect. 4, including our preferred convention choice for \(\alpha _s\), the most important parameter of any shower algorithm. A set of comparisons to a selection of salient experimentally measured distributions for hadronic Z decays, Drell–Yan, and QCD jet production are included to document and validate the performance of the shower algorithm with these parameters.
Finally, in Sect. 5, we summarise and give an outlook. Additional material, as referred to in the text, is collected in the Appendices.
2 Vincia ’s Antenna showers
A QCD antenna represents a colourconnected parton pair which undergoes a (coherent) \(2\rightarrow 3\) branching process [13, 14, 15, 16, 51]. In contrast to conventional shower models (including both DGLAP and Catani–Seymour dipole ones) which single out one parton as the “emitter” with one (or more) other partons acting as “recoiler(s)”, the antenna formalism treats the two prebranching “parent” partons as a single entity, with a single radiation kernel (an antenna function) driving the amount of radiation and a single “kinematics map” governing the exact relation between the prebranching and postbranching momenta. Formally, the antenna function represents the approximate (to leading order in the vanishing invariant(s)) factorisation between the pre and postbranching squared amplitudes, while the kinematics map encapsulates the exact onshell factorisation of the \((n+1)\)parton phase space into the nparton one and the \((2\rightarrow 3)\) antenna phase space.
Note that for branching processes involving flavour changes of the parent partons, such as \(g\rightarrow q\bar{q}\), a distinction between “emitter” and “recoiler” and thus a treatment independent of the above description is possible. However, this is not compulsory and we are therefore still using the same \((2\rightarrow 3)\) antenna phasespace and kinematics map as in the case of gluon emission. Moreover, applying a \(2\rightarrow 3\) branching amounts to using the lowest number of involved partons which admit an onshell to onshell mapping.
In this section we briefly review the notation and conventions that will be used throughout this paper (Sect. 2.1), followed by definitions for all of the phasespace convolutions or factorisations, respectively, antenna functions, and evolution variables on which Vincia ’s treatment of initial–initial, initial–final, and final–final configurations are based (Sects. 2.2, 2.3, and 2.4). The expressions for final–final configurations are unchanged relative to those in [23, 39], with the default antenna functions chosen to be those of [52] averaged over helicities. Some further details on the explicit kinematics constructions are collected in Appendix A. The explicit form of the showergeneration algorithm is presented in Sect. 2.5. Finally, we round off in Sect. 2.8 with comments on some features of earlier incarnations of Vincia which have not (yet) been made available in Vincia 2.0.
2.1 Notation and conventions
We use the following notation for labelling partons: capital letters for prebranching (parent) and lowercase letters for postbranching (daughter) partons. We label incoming partons with the first letters of the alphabet, a, b, and outgoing ones with i, j, k. Thus, for example, a branching occurring in an initial–final antenna (a colour antenna spanned between an initial–state parton and a final–state one) would be labelled \( AK \rightarrow a jk\). This is consistent with the conventions used in the most recent Vincia papers [29, 39].^{6} The recoiler or recoiling system will be denoted by R and r respectively (compared with \(R'\) and R in [29]).
The evolution variable, which we denote t, is evaluated on the postbranching partons, hence, e.g., \(t_{\mathrm {FF}}=t(s_{ij},s_{jk})\). It serves as a dynamic factorisation scale for the shower, separating resolved from unresolved regions. As such, it must vanish for singular configurations. Generally, we define the evolution variable for each branching type to vanish with the same power of the momentum invariants as the leading poles of the corresponding antenna functions, see below. The complementary phasespace variable will be denoted \(\zeta \).
The sum in Eq. (5) runs over all possible \((n+1)\)parton states that can be created from the nparton state, and will be implicit from here on. \(\text {d}\Phi _{\text {ant}}\) is the antenna phase space, providing a mapping from two to three onshell partons while preserving energy and momentum. The specific form for the two configurations, initial–initial and initial–final, are defined below, along with the specific forms of the evolution variable.
2.2 Initial–initial configurations
2.3 Initial–final configurations
In traditional (DGLAPbased) partonshower formulations, the radiation emitted by a colour line flowing from the initial to the final state is handled by two separate algorithms, one for ISR and one for FSR. Coherence can still be imposed by letting these algorithms share information on the angles between colourconnected partons and limiting radiation to the corresponding coherent radiation cones. But even so, several subtleties can arise in the context of specific processes or corners of phase space. Examples of problems encountered in the literature involving Pythia’s \(p_\perp \)ordered showers include how radiation in dipoles stretched to the beam remnant is treated [59], whether the combined ISR+FSR evolution is interleaved or not [60] and whether/how coherence is imposed on the first emission [36].
In the context of antenna showers, the radiation off initial–final (IF) colour flows is generated by IF antennae, which are coherent ab initio. We therefore expect the treatment of wideangle radiation to be more reliable and plagued by fewer subtleties. The main issue one faces instead is technical. Denoting the pre and postbranching partons participating in an IF branching by \(AK\rightarrow akj\), the choice of kinematics map specifying the global orientation of the akj system with respect to the AK one is equivalent to specifying the Lorentz transformation that connects the prebranching frame, in which A is incoming along the z axis with momentum fraction \(x_A\), to the postbranching one, in which a is incoming along the z axis with momentum fraction \(x_a\). For a general choice of kinematics map, this can result in boosted angles entering in the relation between \(x_a\) and the branching invariants, producing highly nontrivial expressions, and the phasespace boundaries can likewise become very complicated. To retain a simple structure for this first implementation, and since we anyway intend our shower as a baseline to be improved upon with matrixelement corrections, the algorithm we present in this paper is based on the simplest possible kinematics map, in which momentum is conserved locally within the antenna, \(p_a  p_j  p_k = p_A  p_K\). This implies that the momentum of the hard system, R, is left unchanged, meaning IF branchings doe not produce a transverse recoil in the hard system. This is indicated by the unchanged momenta of the other incoming parton B and the finalstate R, cf. the illustrations in Fig. 3.

Even in cases where there is only one original II antenna (as e.g., in Drell–Yan), it is not true that recoil can only be generated by the first emission. In particular, if the first branching is a (sea) quark evolving backwards to a gluon, that gluon will participate in a new II antenna, which will generate added recoil according to the above prescription for the II case. For cases with more than one II antenna (e.g., \(gg\rightarrow H\)), the number of possible \(p_\perp \) kicks of course increases accordingly.

In Vincia matrixelement corrections (MECs) are regarded as an integral component of the evolution. Up to the first several orders (typically three powers of \(\alpha _s\)) we therefore expect to be able to apply MECs which will change the relative weighting of branching events in phase space, emphasising those regions which would have benefited most from large recoils and deemphasising complementary ones. Matrixelement corrections will ensure that the emission pattern is correctly described with fixedorder precision. The allorders resummation of nonLL configurations (e.g., configurations with balancing soft emissions), is, however, not formally improved, meaning a residual effect of the recoil strategy remains. Note that the MECs will nonetheless attribute a sensible lowestorder weight to hard configurations that are usually out of reach of strongly ordered parton showers.

As already pointed out above and illustrated by [29, Figs. 3, 4], the IF radiation patterns remain coherent, in the sense that large colour opening angles are a prerequisite for wideangle radiation. This is a nontrivial and important property of the antennashower formalism, which is preserved independently of the recoil strategy.
In the following, we describe the phasespace convolution, antenna functions and resulting noemission probability used for initial–final evolution.
2.4 Final–final configurations
We denote the pre and postbranching partons participating in a final–final branching by \(IK\rightarrow ijk\), with no recoils outside the antenna. In the following, we specify the phasespace factorisation, antenna functions, evolution variables and the resulting noemission probability. More extensive descriptions of Vincia ’s finalstate antennashower formalism can be found in [23, 39].
2.5 The shower generator
We now illustrate how the shower algorithm generates branchings, starting from trial branchings generated according to a simplified version of the noemission probability in Eq. (5). For definiteness we consider the specific example of initial–initial antennae, initial–final ones being handled in much the same way, with a PDF ratio that only involves one of the beams, and final–final ones not involving any PDF ratios at all. The full antennashower evolution (II+IF+FF) is combined with Pythia’s \(p_\perp \)ordered multiplepartoninteractions (MPI) model, in a common interleaved sequence of evolution steps [8].
 Instead of the physical antenna functions, a, we use simpler (trial) overestimates, \(\hat{a}(s_{aj},s_{jb},s_{AB})\). For instance the trial antenna function for gluon emission off an initialstate quark–antiquark pair is chosen to be$$\begin{aligned} \hat{a}_{q\bar{q}\,g}^\text {II}=2\,\frac{s_{ab}^2}{s_{AB}s_{aj}s_{jb}}. \end{aligned}$$(43)
 Instead of the PDF ratio, \(R_\text {pdf}\), we use the overestimatewhere \(t_\text {min}\) is the lower limit of the range of evolution variable under consideration and \(\alpha \) a parameter, whose value is, wherever possible, chosen differently, depending on the type of branching, to give a good performance.$$\begin{aligned} \hat{R}_\text {pdf} = \left( \frac{x_A}{x_a}\frac{x_B}{x_b}\right) ^\alpha \frac{f_a(x_A,t_\text {min})}{f_A(x_A,t_\text {min})} \frac{f_b(x_B,t_\text {min})}{f_B(x_B,t_\text {min})}, \end{aligned}$$(44)

In cases where the physical \(\zeta \) boundaries depend on the evolution variable t, we allow trial branchings to be generated in a larger hull encompassing the physical phase space, with \(\zeta \) boundaries that only depend on the t integration limits.
2.6 Colour coherence and colour indices
When assigning colour indices to represent colour flow after a branching, we adopt a set of conventions that are designed to approximately capture correlations between partons that are not LCconnected, based on the arguments presented in [49]. Specifically, we let the last digit of the ”Les Houches (LH) colour tag” [63, 64] run between 1 and 9, and refer to this digit as the ”colour index”. LCconnected partons have matching LH colour tags and therefore also matching colour/anticolour indices, while colours that are in a relative octet state are assigned nonidentical colour/anticolour indices. Hence the last digit of a gluon colour tag will never have the same value as that of its anticolour tag. This does not change the LC structure of the cascade; if using only the LH tags themselves to decide between which partons string pieces should be formed, the extra information is effectively just ignored. It does, however, open for the possibility of allowing strings to form between nonLCconnected partons that ”accidentally” end up with matching indices, in a way that at least statistically gives a more faithful representation of the full SU(3) group weights than the strictLC one [49].
In Figs. 4 and 5 we illustrate our approach, and the ambiguity it addresses. For definiteness, and for simplicity, we consider the specific case of \(Z\rightarrow q gg\bar{q}\), but the arguments are general. The two diagrams in Fig. 4 show the outgoing partons, produced by a Z boson decaying at the point denoted by \(\bullet \). Both axes correspond to spatial dimensions, hence time is indicated roughly by the radial distance from the Z decay point. Examples of the colour indices defined above are indicated by subscripts, hence e.g., \(g_{13}\) denotes a gluon carrying anticolour index 1 and colour index 3. Due to our selection rule, the type of assignment represented by Fig. 4a is always selected when \(m_{gg}\) is small, \(s_{gg}< s_{qg}\), while the one represented by Fig. 4b is selected when \(s_{qg}<s_{gg}\) (when the second emission occurs in the \(qg\) antenna, and completely analogously when it occurs in the \(g\bar{q}\) one). The subleadingcolour ambiguity illustrated by Fig. 5 can only occur for the latter type of assignment, hence will be absent in our treatment for collinear \(g\rightarrow gg\) branchings (where the flow represented by Fig. 4a dominates), in agreement with the collinearly branching gluon having to be an octet. We regard this as an improvement on the treatment in [49], in which there was no mechanism to prevent collinear gluons from ending up in an overall singlet state; see also the remarks accompanying [49, Fig. 15].
As a last point, we remark that this new assignment of colour tags is currently left without impact, but is implemented in order to enable future studies, such as colour reconnection within Vincia .
2.7 Uncertainty estimations
Traditionally, shower uncertainties are evaluated by systematic up/down variations of each model parameter, which mandates the generation of multiple event sets, one for each variation. To avoid this timeconsuming procedure, Vincia instead generates a vector of variation weights for each event [39], where each of the weights corresponds to varying a different parameter. A separate publication details the formal proof of the validity of the method [65], which we have here extended to cover both the initial and finalstate showers in Vincia . (Note added in proof: during the publication of this manuscript, two further papers appeared reporting similar implementations in Herwig and Sherpa, see [66, 67].) In this section, we only give a brief overview of the implementation, referring to [39, 65] for details and illustrations. Technical specifications for how to switch the uncertainty bands on and off in the code, and how to access them, are provided in Vincia ’s HTML User Reference [46].

Vincia ’s default settings, with default antenna functions, scale choices and colour factors.

Variation of the renormalisation scale. Using \(\alpha _s(t/k_\mu )\) and \(\alpha _s(t\,k_\mu )\), with a userspecifiable value of the additional scaling factor \(k_\mu \).

Variation of the antenna functions. Using antenna sets with large and small nonsingular terms, representing unknown (but finite) processdependent LO matrixelement terms. Note that these are cancelled by LO MECs (up to the matched orders).

\(\alpha _s\)suppressed counterparts of the finiteterm variations above^{8} which are not cancelled by (LO) MECs.

Variation of the colour factors. All gluon emissions use the colour factor of either \(C_A=3\) or \(2C_F=8/3\).
 Modified \(P_\text {imp}\) factor,$$\begin{aligned} P_\text {imp}' = \frac{\hat{t}^2}{\hat{t}^2+t^2}. \end{aligned}$$(51)
2.8 Limitations

Sector showers [28]: a variant of the antennashower formalism in which a single term is responsible for generating all contributions to each phasespace point. It has some interesting and unique properties including being onetoone invertible and producing fewer (one) term at each order of GKS matrixelement corrections leading to the numerically fastest matching algorithm we are aware of (see [28]), at the price of requiring more complicated antenna functions with more complicated phasespace boundaries. For the initialstate extension of Vincia we have so far focussed on the technically simpler case of “global” (as opposed to sector) antennae.

Oneloop matrixelement corrections. The specific case of oneloop corrections for hadronic Z decays up to and including 3 jets was studied in detail by HLS [40]. The extension of this method to hadronic initial states, and a more systematic approach to oneloop corrections in Vincia in general, will be a major goal of future efforts.

Helicity dependence [52]. The shower and matrixelementcorrection algorithms described in this paper pertain to unpolarised partons. Although this is fully consistent with the unpolarised nature of the initialstate partons obtained from conventional parton distribution functions (PDFs), we note that an extension to a helicitydependent formalism could nonetheless be a relatively simple future development. Moreover, we expect this would provide useful speed gains for the GKS matrixelement correction algorithm equivalent to those observed for the finalstate algorithm [52].

Fullfledged fermion mass effects [68]. Our treatment of mass effects for initialstate partons is so far limited to one parallelling the simplest treatment in conventional PDFs, the “zeromassvariableflavournumber (ZMVF) scheme”. In this scheme, heavyquark PDFs are set to zero below the corresponding mass threshold(s) and are radiatively generated above them by \(g\rightarrow Q\bar{Q}\) splittings, with \(m_Q\) formally set to zero in those splittings and for the subsequent heavyquark evolution. Thus, in Vincia 2.0, all partons are assigned massless kinematics, but \(g\rightarrow Q\bar{Q}\) splittings are switched off (also in the final state) below the physical mass thresholds. This only gives a very rough approximation of mass effects [69, 70] but at least avoids generating unphysical singularities. Beyond the strict ZMVF scheme, optionally and for finalstate branchings only, we allow for a set of universal antenna mass corrections to be applied and/or for tighter phasespace constraints to be imposed, with the latter obtained from the wouldbe massive phasespace boundaries. We note that a mixed treatment similar to the one currently employed by Pythia, with massive/massless kinematics for outgoing/incoming partons, respectively, would not be straightforward to adopt in Vincia as it would be inconsistent with the application of onshell matrixelement corrections.
 The socalled “Ariadne factor” [21] for gluon splitting antennaewith \(S_N\) the invariant mass squared of the colour neighbour on the other side of the splitting gluon and \(s_P\) the invariant mass squared of the parent (splitting) antenna is limited to its original purpose, that of improving the description of 4jet observables in Z decay, and is not applied outside that context.$$\begin{aligned} P_\text {Ari} = \frac{2s_N}{s_N+s_P}, \end{aligned}$$(52)
3 Matrixelement corrections
In this section we focus on the MEC formalism in Vincia and discuss our strategy for reaching the nonordered parts of phase space, both with respect to the factorisation scale in the case of the first branching and with respect to previous branching scales.
Note that in this paper all matrix elements are generated with MadGraph [50, 71]. The output is suitably modified to extract the leadingcolour matrix element, i.e. to not sum over colour permutations, but pick the (diagonal) entry in MadGraph’s colour matrix that corresponds to the colour order of interest. All plots shown in this paper are based on leadingcolour matrix elements.
3.1 Hard jets in nonQCD processes
In this section we describe our formalism to combine events which are accompanied by at least one very hard jet, with the ones which are not. We emphasise that the considerations are general and apply to any processes that do not exhibit QCD jets at the Born level.
Since the ISR shower formally corresponds to a “backwards” evolution of the PDFs [33], the factorisation scale represents the natural upper bound (starting scale) for the initialstate shower evolution. This implies that any phasespace points with \(t > t_\text {fac}\) will not be populated by the shower, potentially leaving a “dead zone” for hight emissions. In principle, the freedom in choosing the evolution variable can be exploited to define t in such a way that the entire physical phase space becomes associated with scales \(t<t_\text {fac}\) [30], including points with physical \(p_\perp ^2\gg t_\text {fac}\). Here, however, we wish to maintain a close correspondence between the evolution variable and the physical (kinematic) \(p_\perp \), requiring the development of a different strategy.
The inclusive cross section obtained from Eq. (57) does not reduce to the zeroparton Born cross section, the changes being only due to hard emissions which have not been incorporated in the first term in (57). This differs from cross section changes in CKKWinspired merging prescriptions [76, 77], which arise from real–virtual mismatches at the merging scale,^{9} or from the definition of the inclusive cross section in unitarised merging schemes [78, 79]. In the latter, the inclusive cross section is almost entirely given by the first term in (57), and only changed by ”incomplete” states which cannot be associated with valid partonshower histories. The definition of what is deemed an ”incomplete state” is not conventional and thus may depend on the details of a particular implementation. Note, however, that [78, 79, 80] do not advocate including the factors ”\(\Delta _0\)” when reweighting ”incomplete” states. This could lead to interesting differences in observables relying on very boosted Zboson momenta.
We note that, although the described method of adding hard jets in nonQCD processes is the default choice in Vincia , we include the possibility to perform an ordinary shower, starting off the factorisation scale \(t_\text {fac}\). This is the recommended option when combining Vincia ’s shower with external matching and merging schemes.
3.2 Strong ordering compared with treelevel matrix elements
One could force the dead zone to disappear by simply removing the ordering condition and starting the shower at the phasespace maximum for each antenna. However, as can be seen from Fig. 9, this would highly overcount the matrix element in the unordered region, again parallelling the observations for the equivalent case of finalstate radiation in [39]. The strongordering condition is clearly a better approximation to QCD, even if it does not fill all of phase space. To improve the shower, we will therefore need to allow the shower to access the whole phase space while suppressing the overcounting in the unordered region.
3.3 Smooth ordering compared with treelevel matrix elements
A further point that must be addressed in the context of the ordering criterion is that our matrixelementcorrection formalism, discussed below, requires a Markovian (historyindependent) definition of the \(\hat{t}\) variable in the \(P_\mathrm {imp}\) factor in Eq. (68). Rather than using the scale of the preceding branching directly (which depends on the shower path and hence would be historydependent), we therefore compute this scale in a Markovian way as follows: Given a nparton state we determine the values of the evolution variable corresponding to all branchings the shower could have performed to get from any \((n1)\) to the given nparton state. The reference scale \(\hat{t}\) is then taken as the minimum of those scales. The dead zone, equivalent to the unordered region, is now populated by allowing branchings of a restricted set of antennae to govern the full relevant phase space. Such antennae are called unordered, while other antennae are called ordered. It is in principle permissible to treat all antennae in an event as unordered. To mimic the structure of effective \(2{\rightarrow }4\) and higher branchings, we, however, only tag those antennae which are connected to partons that partook in the branching that gave rise to the chosen value for \(\hat{t}\) as unordered. Branchings of ordered antennae may then contribute below the scale \(\hat{t}\).
For example, consider the case of a gluon emission being associated with the smallest value of the evolution variable. In this case the gluon as well as the two partons playing the role of the parent antenna that emitted the gluon, are marked for unordering and therefore all antennae in which these three partons participate are allowed to restart the evolution at their phasespace limits. This limited unordering reflects that no genuinely new region of phase space would be opened up by allowing partons/antennae completely unrelated to the “last branching” to be unordered, as these will already have explored their full accessible phase spaces during the prior evolution.
Similarly we repeat the twodimensional histograms for the smoothly ordered antenna shower in Fig. 12 without and in Fig. 13 with the cut on \(m_{\perp \,Z}^2\). As expected, we obtain an improved description as compared to both the strong and unordered showers, Figs. 8 and 9 respectively. Due to the form of the improvement factor in Eq. (68) we get a factor of 0.5 at the green line, around where the scales of the two branchings coincide, leading to a better description already of this region. Once again these plots show that the shower undercounts the region where the Z boson is very soft and should have been generated with a weak shower, representing a path that is not available in Vincia yet. The strongly unordered region remains somewhat overcounted, though by less than a factor 2, far better and with narrower distributions than was the case for the fully unordered shower, Fig. 9.
An extended set of plots, including Higgs production processes, can be found in Appendix B.
3.4 Smooth ordering vs. strong ordering
This section presents a comparison of strong and smooth ordering, first in terms of their analytical leadinglogarithmic structures, and then using jet clustering scales, investigating the processes \(e^+e^\rightarrow \) jets as well as \(pp\rightarrow Z+\)jets. The analyses are adapted from the code used in [30], originally written by Höche. In order to focus on the shower properties we present partonlevel distributions, with MECs switched off, a fixed strong coupling with \(\alpha _s(m_Z)=0.13\), and a very low cutoff, \(10^{3}~\text {GeV}\) for \(e^+e^\rightarrow \) jets and \(10^{2}~\text {GeV}\) for \(pp\rightarrow Z+\)jets. To furthermore put the magnitude of the differences between smooth and strong ordering into perspective, an \(\alpha _s(m_Z)\)variation band for the strongly ordered result is included in Figs. 14 and 15.
3.4.1 Leading logarithms
3.4.2 Hadronic Z decays
To increase the available phase space we used a heavy Z with \(m_Z=1000~\text {GeV}\) which decays hadronically. In Fig. 14 we present the partonlevel result for four successive jet resolution measures, \(y_{m\,m+1}\) (with \(m\in \{2,3,4,5\}\)), and their ratios \(y_{m\,m+1}/y_{m1\,m}\), using the Durham jet algorithm. Jet resolution scales exhibit a Sudakov suppression for low values, and exhibit fixedorder behaviour for large values. We note that in realistic calculations (and in experimental data), lowscale values are typically strongly affected by hadronisation corrections, which are absent here since we are at parton level, with a fixed \(\alpha _s\). We also exclude values of \(y_{m\,m+1}\) corresponding to scales below the shower cutoff. Small values of the ratios \(y_{m\,m+1}/y_{m1\,m}\) highlight the modelling in the region of large scale separation, i.e. where effects of resummation become relevant. Large values of \(y_{m\,m+1}/y_{m1\,m}\) are associated with the region of validity of fixedorder calculations.
In the distributions of the jet resolution scales themselves we observe moderate differences between the different ordering modes, up to \(\mathcal {O}(20~\%)\). Smooth ordering generates more events with larger \(y_{m\,m+1}\) separation and, consequently, fewer events with small separation, compared to strong ordering.
3.4.3 Drell–Yan
The partonlevel results for \(Z+\)jets events are presented in Fig. 15: four successive jet resolution measures, \(d_{m\,m+1}\) (with \(m\in \{0,3\}\)), and their ratios \(d_{m\,m+1}/d_{m1\,m}\), using the longitudinally invariant \(k_\perp \) jet algorithm with \(R=0.4\). As before, jet resolution scales show a fixedorder behaviour for large values, a Sudakov suppression and potentially large nonperturbative corrections for low values. The ratios \(y_{m\,m+1}/y_{m1\,m}\) are used to more clearly reveal the successive scale hierarchies.
The observations for both, the jet resolution scales, and their rations, are qualitatively similar to the \(e^+e^\rightarrow \) jets case, though quantitatively the effects here are larger. We notice the same turnover when going from \(d_{12}/d_{01}\) to \(d_{34}/d_{23}\) we saw for Z decays, with the explanation being very similar to the case before. Smooth ordering will allow additional phasespace regions to be filled with harder emission (cf. Fig. 10). Due to the unitarity of the partonshower algorithm, this naively means that fewer soft emissions occur. This is counteracted by the Markovian restart scale, which means that the smoothly ordered shower yields softer emissions from “ordered” antennae. At low multiplicity, the former dominates, as all antennae are allowed to fill their available phase space, while at higher multiplicity, the latter drives the differences. Figure 15 shows trends in \(d_{01}\) and \(d_{12}\) similar to the ones visible in Figs. 10 and 20 of [87]. Note again that the additional, compensating effect of the Markovian restarting scale starts playing an important role for higher multiplicities.
3.5 Hard jets in QCD processes
We already discussed our strategy to include hard branchings in nonQCD processes in Sect. 3.1. For processes with QCD jets in the final state we apply a different formalism, as the Born process already comes with a QCD scale. The first branching is allowed to populate all of phase space; however, the region with scales above the factorisation scale, \(t>t_\text {fac}\), is treated with smooth ordering, as described in Sect. 3.3. In Fig. 16 we show the PStoME ratios for \(gg\rightarrow ggg\) and \(q\bar{q}\rightarrow ggg\) where the factorisation scale is chosen to be the transverse momentum of the finalstate partons in the Born \(2\rightarrow 2\) process. We show a comparison of strong ordering, i.e. not including \(t>t_\text {fac}\), smooth ordering with \(\hat{t}=t_\text {fac}\) in the \(P_\text {imp}\) factor, and no ordering, which corresponds to adding an event sample with \(t>t_\text {fac}\). The plots indicate that the smooth ordering is preferred over adding hard jets as a separate event sample. Note that the asymmetric distribution of the PStoME ratio for \(gg\rightarrow ggg\) is the result of combining the distributions of different colour flows.
One could imagine applying the same treatment to nonQCD processes as well. However, this is not done in Vincia as the factorisation scale in these processes is not a QCD scale and therefore not suited to enter the \(P_\text {imp}\) factor.
3.6 Matrixelement corrections with MadGraph 4
In this section we review the GKS procedure for iterative matrixelement corrections (MECs) [39]. To first order, the formalism is equivalent to that by Bengtsson and Sjöstrand in Refs. [5, 12], and to the approach used for real corrections in Powheg [88, 89]. In the context of finalstate showers, the approach was generalised to multiple emissions in [39] where it was successfully used to include MECs through \(\mathcal {O}(\alpha _s^4)\) for hadronic Z decays. A generalisation at the oneloop level has also been developed [40], though so far limited to \(\mathcal {O}(\alpha _s^2)\). Here, we focus on treelevel corrections only.

INTEGER MCMODE selects between Leading Colour (0), Vincia Colour (1), and Full Colour (2), as defined below,

INTEGER ICOL selects which colour ordering is desired for MCMODE=0,1,

DOUBLE PRECISION P1(0:3,NEXTERNAL) the momenta of the particles (in this example NEXTERNAL =6),

INTEGER HEL1(4) holding up to 4 helicity configurations to be summed over, sufficient to average over an unpolarised initial 2parton state or decaying vector boson, with specified finalstate helicities. The enumeration of helicity configurations follows MadGraph’s normal helicitycounting convention.

The requested matrix element squared is saved in the doubleprecision ANS variable, which in Vincia always has only a single element.
Note that, though we show all matrixelement comparisons with leading colour, the conclusions do not change when replacing leading with full colour.
Interference between different Bornlevel processes: In previous versions of Vincia the interference contributions from different Bornlevel processes were ignored; e.g., the interference between \(Z\rightarrow d\bar{d}(g\rightarrow u\bar{u})\) and \(Z\rightarrow u\bar{u}(g\rightarrow d\bar{d})\) contributing to \(Z\rightarrow d\bar{d}u\bar{u}\) was not included. As those interferences can become fairly large and are already present for the first branching, e.g., \(qg\rightarrow qgg\) can arise from \(gg\rightarrow gg\) or \(qg\rightarrow qg\) Bornlevel processes, we developed a more general formalism capable of handling these cases. Yet more interesting and illustrative are the interferences between \(gg\rightarrow H\) and \(Q\bar{Q}\rightarrow H\) Born processes, which both contribute to \(Qg \rightarrow QH\) (with Q a heavy quark) but involve completely different types and orders of couplings. For this special case of Higgs production and decay we provide an option to allow/disallow such interferences.
4 Preliminary results and tuning
4.1 The strong coupling
All components of Vincia (i.e., both matrixelement corrections and showers) use a single reference value for strong coupling constant, with the default value \(\alpha _s^{\overline{\mathrm {MS}}}(M_Z) = 0.118\), in agreement with the current world average [53, 93]. By default, we use twoloop running expressions, with the number of active flavours changing at each quarkmass threshold (including at \(m_t\)), though options for oneloop running or even fixed \(\alpha _s\) values are provided as well. The inclusion of threeloop running effects is not relevant at the present (LO+LL) level of precision of the shower. In the infrared, the behaviour of \(\alpha _s\) is regulated by allowing to evaluate it at a slightly displaced scale, \(\alpha _s(\mu ) \rightarrow \alpha _s(\mu + \mu _0)\) and by imposing an upper bound \(\alpha _s < \alpha _s^\mathrm {max} \). The set of default parameter values are:
+ Vincia:alphaSvalue = 0.118 ! Default alphaS(mZ) MSbar +
+ Vincia:alphaSorder = 2 ! Default is twoloop running +
+ Vincia:alphaSmuFreeze = 0.4 ! mu0 scale in alphaS argument, in GeV +
+ Vincia:alphaSmax = 1.2 ! max numerical value of alphaS +
4.2 Vincia 2.0 default tune
Two main tools were used to perform the analyses: Vincia ’s own ROOTbased analysis tool, VinciaRoot [39], and Rivet [95]. For the hadroncollider distributions, we compare Vincia 2.0 with Pythia 8.2. For the \(e^+e^\rightarrow \mathrm {hadrons} \) analyses, we also include Vincia 1.2, since this version included NLO corrections to \(e^+e^\rightarrow 3\ \mathrm {jets} \) which have not yet been migrated to Vincia 2.0. Note, however, that even without the NLO corrections the two Vincia versions are not exactly identical due to a slightly revised definition of the smoothordering criterion, to make it truly Markovian.
We note that these tunings were done manually (by “eye”), rather than by automated minimisation of \(\chi ^2\) or equivalent measures. The latter is not as straightforward as it may sound, due to correlations between measurements and the influence of regions of low theoretical accuracy. These issues can be at least partially addressed by combining global knowledge and experience to (subjectively) choose binwise weighting factors. Nevertheless, manual and automated approaches may be considered complementary, with the former certainly competitive for the purpose of determining a set of “reasonable default values”, which is our principal aim here.
4.2.1 Hadronic Z decays
The finalstate showering and hadronisation parameters are constrained using hadronic Z decays, mainly from the LEP experiments. In the context of Vincia 2.0, the rates of perturbative finalstate branchings depend on the effective renormalisation scheme and scale choice, cf. Eq. (83), for which we have chosen the default values:
+ Vincia:CMWtypeFF = 2 ! CMW rescaling for FF antennae +
+ Vincia:alphaSkMuF = 0.6 ! muR prefactor for gluon emissions +
+ Vincia:alphaSkMuSplitF = 0.5 ! muR prefactor for gluon splittings +
Figure 18 shows the eventshape observables^{13} that were used as the primary tuning constraints, compared with lightflavour tagged data from the L3 experiment [96]. In the main (top) plot panes, experimental data is represented by black square symbols, with 1\(\sigma \) and 2\(\sigma \) uncertainties represented by black vertical error bars and lightgrey extensions, respectively. In the ratio panes, the inner (green) bands indicate the 1\(\sigma \) uncertainties on the data; outer (yellow) bands represent 2 \(\sigma \).
Note that, since Vincia 2.0 does not incorporate the NLO corrections to \(Z \rightarrow 3\) jets internally (unlike Vincia 1.2 [40]), we have chosen to allow the default tune to undershoot the reference data slightly in regions dominated by hard, resolved 3jet events. This hopefully produces a more universal global tuning which should also be appropriate for use with the NLO merging strategies that are available within Pythia, notably UNLOPS [97].
+ Vincia:cutoffScaleFF = 0.9 ! Cutoff value in GeV for FF antennae +
+ StringZ:aLund = 0.5 ! Lund a parameter +
+ StringZ:bLund = 1.15 ! Lund b parameter +
+ StringZ:aExtraDiquark= 1.12 ! (extra for diquarks) +
+ StringPT:sigma = 0.295 ! Soft pT in string breaks +
The corresponding full set of default parameter values are:
+ ! * String breakup flavour parameters +
+ StringFlav:probStoUD = 0.21 ! StrangenesstoUD ratio +
+ StringFlav:mesonUDvector = 0.45 ! Lightflavour vector suppression +
+ StringFlav:mesonSvector = 0.555 ! Strange vectormeson suppression +
+ StringFlav:mesonCvector = 1.03 ! Charm vectormeson suppression +
+ StringFlav:mesonBvector = 2.2 ! Bottom vectormeson suppression +
+ StringFlav:probQQtoQ = 0.077 ! Diquark rate (for baryon production) +
+ StringFlav:probSQtoQQ = 1.0 ! Optional Strange diquark suppression +
+ StringFlav:probQQ1toQQ0 = 0.027 ! Vector diquark suppression +
+ StringFlav:etaSup = 0.53 ! Eta suppression +
+ StringFlav:etaPrimeSup = 0.105 ! Eta’ suppression +
+ StringFlav:decupletSup = 1.0 ! Optional Spin3/2 Baryon Suppression +
+ StringFlav:popcornSpair = 0.9 ! Popcorn +
+ StringFlav:popcornSmeson = 0.5 ! Popcorn +
+ StringZ:rFactC = 1.60 ! Bowler parameter for c quarks +
+ StringZ:rFactB = 1.1 ! Bowler parameter for b quarks +
+ StringZ:useNonstandardB = true ! Special treatment for b quarks +
+ StringZ:aNonstandardB = 0.82 ! a parameter for b quarks +
+ StringZ:bNonstandardB = 1.4 ! b parameter for b quarks +
4.2.2 Drell–Yan
In Figs. 22, 23 and 24 we show a set of observables in Drell–Yan events with ATLAS data from [105] and [106] and CMS data from [92] and [107]. We show predictions of default Vincia 2.0 in red, Vincia 2.0 wimpy (representing an ordinary shower, starting at the factorisation scale, i.e. no hard jets, no MECs, and strong ordering) in green, and Pythia 8.2 in blue. The Vincia 2.001 results correspond to the following default parameter choices:
#+ Perturbative shower parameters +
+ Vincia:CMWtypeII = 2 ! CMW rescaling of Lambda for II antennae +
+ Vincia:CMWtypeIF = 2 ! CMW rescaling of Lambda for IF antennae +
+ Vincia:alphaSkMuI = 0.75 ! Renormalisationscale prefactor for ISR +
+ ! emissions +
+ Vincia:alphaSkMuSplitI = 0.7 ! " for g>qq splittings +
+ Vincia:alphaSkMuConv = 0.7 ! " for ISR conversions +
# + Shower IR cutoff and primordial kT +
+ Vincia:cutoffScaleII = 1.0 ! Cutoff value (in GeV) for II antennae +
+ Vincia:cutoffScaleIF = 0.9 ! Cutoff value (in GeV) for IF antennae +
+ BeamRemnants:primordialKThard = 1.05 ! Primordial kT for hard interactions +
+ BeamRemnants:primordialKTsoft = 0.7 ! Primordial kT for soft interactions +
Figure 22 shows angular correlations and the transversemomentum spectrum of the Drell–Yan lepton pair. As one would expect the spectrum of Vincia 2.0 wimpy dies out at the Z mass. The prediction of default Vincia 2.0 shows too much activity in the hard tail of the spectrum which is caused by the reweighting of the event sample that includes high\(p_\perp \) jets, see Sect. 3.1. The tuning of the renormalisationscale prefactors was chosen to produce as good a compromise as possible between the regions above and below \(p_\perp \sim m_Z/2\).
4.2.3 Underlying event
Although softinclusive QCD physics is not the main focus of this version of Vincia , it is nonetheless relevant to verify that a reasonable description of the underlying event (UE) is obtained. We rely on the basic multipartoninteraction (MPI) modelling of Pythia 8 [8, 34, 108] including its default colourreconnection (CR) model, with parameters reoptimised for use with Vincia ’s initial and finalstate showers.
The MPI and CR parameter choices for the default Vincia 2.001 tune are as follows:
+ ! UE/MPI tuning parameters +
+ SigmaProcess:alphaSvalue = 0.118 +
+ SigmaProcess:alphaSorder = 2 +
+ MultiPartonInteractions:alphaSvalue = 0.119 +
+ MultiPartonInteractions:alphaSorder = 2 +
+ MultiPartonInteractions:pT0ref = 2.00 +
+ MultiPartonInteractions:expPow = 1.75 +
+ MultiPartonInteractions:ecmPow = 0.21 +
+ ! Parameters for PYTHIA 8’s baseline CR model +
+ ColourReconnection:reconnect = on +
+ ColourReconnection:range = 1.75 +
+ ! VINCIA is not compatible with perturbative diffraction +
+ Diffraction:mMinPert = 1000000.0 +
4.2.4 QCD jets
As our final set of validation checks, we consider the following observables in hardQCD events: azimuthal dijet decorrelations, jet cross sections, and jet shapes. A technical aspect is that, due to the steeply falling nature of the jet \(p_\perp \) spectrum, we use weighted events for all MC results in this section. The basic \(2\rightarrow 2\) QCD process at the scale \(\hat{p}_\perp \) is oversampled by an amount of \((\hat{p}_\perp /10)^4\), while the compensating event weight is \((10/\hat{p}_\perp )^4\). This allows one to fill the lowcrosssection tails of the distributions with a reasonable amount of events. Note, however, that, for observables that are not identical to the biasing variable (which are all observables since no onetoone measurement of the partonic \(\hat{p}_\perp \) is possible), rare events with large weights can then produce “spurious” peaks or dips in distributions, accompanied by large error bars. Such features are to be expected in some of the distributions we show below; removing them would require generating substantially more events. While these features appear in the predictions of Vincia , they are not present in Pythia’s distributions. The reason is as follows: The aforementioned event weight becomes large for small values of \(\hat{p}_\perp \). As this value serves as the starting scale in Pythia’s shower, the event will not produce any high\(p_\perp \) jets. In Vincia , however, the full phase space for the first emission is explored with the suppression factor \(P_\text {imp}\) which is necessary for the application of MECs. In the rare cases, where Vincia produces a jet with \(p_{\perp \,j}\gg \hat{p}_\perp \), the large event weight becomes visible in distributions which require high\(p_\perp \) jets.
A second technical aspect is that, as shown in Fig. 16, the PStoME ratios for QCD processes result in rather broad distributions already for the firstorder correction with gluon emission only. This complicates including MECs for QCD processes, as violations in the Sudakov veto algorithm for generating emission and noemission probabilities in the shower become more likely. By default, we neglect such violations. It is, however, possible for the user to check the effect of taking the violations into account properly via the procedure outlined in Ref. [111], which has been included in Vincia .
In Fig. 26 we show the predictions of Vincia 2.0 and Pythia 8.2 for dijet azimuthal decorrelations for different ranges of the jet transverse momentum and compare to ATLAS data from [112]. While we observe no glaring discrepancies with the data—the general trends of the distributions are well reproduced by both Vincia and Pythia—there still appears to be some room for improvement, in particular with Vincia undershooting the precisely measured data points around \(\Delta \phi \sim 0.9\) in the lower two \(p_\perp ^\text {max}\) bins by about 10–20 %.
Figures 27 and 28 show the transversemomentum and jet mass spectra for different ranges of the jet rapidity and compare the MC predictions to CMS data from [113] and [114] respectively. We note that, whereas Pythia lies systematically above the data here, the lower default \(\alpha _s\) value chosen in Vincia causes the Vincia normalisations to be substantially lower, even to the point of undershooting the measurements. This is not surprising given that the inclusivejet cross section in Pythia/Vincia is calculated at LO. The tails of the distributions unfortunately suffer from rather large weightfluctuation effects, as was discussed above; nonetheless we note that the bins for which a reasonable statistical precision is obtained are generally closer to the data than the Pythia reference comparison.
Finally, in Fig. 29 we show the differential jet shape variable \(\rho (r)\) and its cumulative integral \(\Psi (r)\) for different ranges of the jet transverse momentum, compared with ATLAS data from [115]. This validates that the FSR broadening of QCD jets is in reasonable agreement with the experimental measurements, though we note that Vincia ’s distributions may be slightly too narrow, which we again regard as being consistent with the LL nature of Vincia ’s antenna functions and analogous to the slightly too narrow thrust distribution we allowed in the \(e^+e^\) event shapes. As far as a first default set of parameters goes, we are satisfied with this level of tuning, with future directions being informed both by lessons from combinations with external matrixelement matching and merging schemes and by attempts to integrate NLO antennafunction corrections into the shower itself, e.g. in the spirit of [40].
5 Summary and conclusions
We presented the first publicly available antenna shower for initial and final state in Vincia 2.0, with focus on antenna functions and kinematic maps for initialstate radiation. Vincia 2.0 includes two different methods to explore the full phase space for the first emission, depending on the hard process at hand, without the disadvantages of a “power shower”. The full phase space of subsequent emissions is populated in a Markovian way. We compare explicitly to treelevel matrix elements for \(pp\rightarrow Z/H\,jj(j)\) and \(pp\rightarrow jjj\) to check the validity of our approximations.
We extended the iterative MEC approach to the initial state and include MECs for QCD up to \(\mathcal {O}(\alpha _s^4)\) (4 jets), and for Drell–Yan and Higgs production up to \(\mathcal {O}(\alpha _s^3)\) (V / H + 3 jets). This is the first time MECs beyond one leg have been applied to hadron collisions. However, this implementation was not without its complications; the large phase space available for initialstate branchings implies that “unordered” emissions account for a larger fraction of the full phase space than was the case for FSR, and the MEC factors are less well behaved and therefore more difficult / less efficient to implement, compared to pure finalstate MECs. We also saw in Sect. 4.2 that biased event samples result in larger weight fluctuations for Vincia than in the case of pure Pythia, presumably due to unordered emissions in Vincia allowing a larger range of corrections to each event. In the context of future developments of Vincia , these aspects will therefore merit further consideration.
We presented first validation results with Vincia 2.0 for the main benchmark processes for FSR and ISR, including hadronic Z decays, Drell–Yan, and QCD jets. We observe good agreement with experimental data from the LEP/SLD and LHC experiments.
The development of a more highly automated interface to MadGraph 5 is among the main development targets for the near future. The feasibility of an interface to Njet2 [116] is also being explored.
Footnotes
 1.
Dokshitzer–Gribov–Lipatov–Altarelli–Parisi. We mourn the recent passing of Guido Altarelli (1941–2015), a founder of this field and a great inspirer.
 2.
 3.
Older partonshower models often treat initialstate (ISR) and finalstate (FSR) evolution in disjoint sequences. In this case, it is challenging to ensure that FSR evolution from the enlarged and changed parton ensemble after ISR evolution recovers the coherent features. Implementations of a combined simultaneous evolution chain for ISR and FSR may also be challenging. The current \(p_\perp \)ordered showers in Pythia 8 [8, 34, 35] do, for example, not account for the coherence structure of the hardest gluon emission in \(t\bar{t}\) events [36]. In contrast, we suspect that due to the fact that physical output states (parsed through hadronisation) are only constructed at the end of the evolution, the angularordered algorithms of Herwig [6] and Herwig++ [7] produce coherent sequences of emission angles in IF configurations correctly. This assessment relies on the assumption that the algorithms ensure that the angular constraints on finalstate emission variables are unchanged by the ISR shower evolution, and vice versa.
 4.
 5.
Giele–Kosower–Skands [39].
 6.
The earliest Vincia paper on final–final antennae [23] used an alternative convention: \(\hat{a} + \hat{b} \rightarrow a + r + b\).
 7.
 8.
Up to and including Vincia 2.001, these variations were erroneously applied by multiplying or dividing the antenna functions by \((1+\alpha _s)\), which is degenerate with the renormalisationscale variations.
 9.
The value of merging scales is typically well below \(t_\text {fac}\).
 10.
This is trivial for \(q\bar{q}\rightarrow Zgg\) as the corresponding antenna function already is the ratio of the LO matrix elements.
 11.
For initial–initial antennae, replace m in the phasespace limit on the rapidity integral in Eq. (75) by \(\sqrt{s}=\sqrt{s_{AB}/(x_A x_B)}\), assuming \(x_Ax_B \ll 1\). For initial–final antennae, replace it by \(\sqrt{s_{AK}/x_A}\) assuming \(x_A\ll 1\).
 12.
In MadGraph 4 the colour matrix for amplitudes with multiple quark pairs is more complicated and required a decomposition by hand to separate the leading from the subleadingcolour parts, as is now done automatically by MadGraph 5.
 13.
For definitions, see e.g. [96].
Notes
Acknowledgments
We thank the HepForge project, www.hepforge.org [117] for providing the hosting and repository services used for developing and maintaining the Vincia code. HepForge is funded via the UK STFC and hosted at the Durham IPPP. We thank Johannes Bellm for useful discussions on [87]. This work was supported in part by the ARC Centre of Excellence for Particle Physics at the Terascale. SP is supported by the US Department of Energy under contract DEAC0276SF00515. The work of MR was supported by the European Research Council under Advanced Investigator Grant ERC–AdG–228301 and ERC Advanced Grant No. 320651, “HEPGAME”. PS is the recipient of an Australian Research Council Future Fellowship, FT130100744: “Virtual Colliders: highaccuracy models for high energy physics”.
Supplementary material
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