# Combining states without scale hierarchies with ordered parton showers

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## Abstract

We present a parameter-free scheme to combine fixed-order multi-jet results with parton-shower evolution. The scheme produces jet cross sections with leading-order accuracy in the complete phase space of multiple emissions, resumming large logarithms when appropriate, while not arbitrarily enforcing ordering on momentum configurations beyond the reach of the parton-shower evolution equation. This requires the development of a matrix-element correction scheme for complex phase-spaces including ordering conditions as well as a systematic scale-setting procedure for unordered phase-space points. The resulting algorithm does not require a merging-scale parameter. We implement the new method in the Vincia framework and compare to LHC data.

## 1 Introduction

High-energy physics in the era of the Large Hadron Collider relies on accurate calculations of Standard-Model scattering signatures—both to determine backgrounds when directly searching for new physics and to allow for setting indirect bounds by comparing measurements to precision calculations. Since measurements at the LHC are typically sensitive to the formation and evolution of jets, much attention has been devoted to calculating QCD corrections. This has led to exquisite dedicated high-precision calculations, and to the development of general schemes to overcome the limited applicability of individual fixed-order QCD calculations by combining multiple calculations into a single consistent result. To this end, modern General Purpose Event Generators [1, 2, 3, 4] include a variety of complex matching [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and merging [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] schemes.

A unified Standard-Model prediction that is applicable for precision measurements and new-physics searches alike must naturally include particle configurations that probe very different aspects of the calculation. The optimal perturbative description of very different particle (and momentum) configurations can consequently vary significantly within one measurement, so that care must be taken to avoid applying specialized arguments outside of their region of validity. Otherwise, the accuracy of the calculation is in jeopardy and its uncertainty might be underestimated. For example, applying QCD reasoning to events without large hierarchies in the hardness of jets can lead to problematic effects [36].

Standard-model calculations at the LHC can somewhat artificially be categorized as focussing on momentum configurations with or without large scale (hardness) hierarchies between jets. Fixed-order QCD calculations are often appropriate for the latter, while the former require a resummation of large perturbative enhancements by means of evolution equations. Both approaches have complementary strengths and should be combined for a state-of-the-art calculation. It is crucial to avoid bias when constructing a single calculation that describes very different limits.

*n*hard, well-separated partons with fixed-order matrix elements while retaining the jet evolution given by the parton shower. We enforce strict requirements on the new scheme to improve on previous ideas:

- 1.
The introduction of new parameters into the calculation is avoided. This is especially important when the correlation with existing parameters is not obvious.

- 2.
The method should provide a uniform accuracy over the complete phase space for one particle multiplicity. For now, this means that the rate of

*n*jets should be given with leading-order accuracy in QCD, irrespectively of the hardness of jets. - 3.
The method should be largely agnostic to parton-shower-inspired arguments when configurations without large scale hierarchies are discussed.

*n*jets will be correctly described with fixed-order accuracy, without the need for explicit subtractions of negative weight, even if the rate for \(n+1\) jets is also corrected with matrix elements. We will describe how the new method allows to achieve leading-order accuracy in QCD for multi-parton configurations. This establishes a baseline for future developments beyond leading-order QCD.

The new scheme relies on applying leading-order matrix-element corrections in phase-space regions that are accessible by a sequence of splittings ordered in a parton-shower evolution variable, supplemented with fixed-order results for configurations that cannot be reached by any such sequence. We will use the misnomer “shower configurations” for such states, and call states which cannot be reached by an ordered sequence of shower emissions “non-shower states”.

A very brief introduction to the parton-shower formalism and the notation is established in Sect. 2. The new method to iteratively correct parton showers with matrix elements is described in detail in Sect. 3. The combination of this scheme of matrix-element corrections for ordered parton-shower evolution with non-shower states is discussed in Sect. 4. An executive summary of the algorithm is given in Sect. 5, followed by a discussion of the impact of combining parton-shower-like and non-shower phase-space regions at parton level. Then results and data comparisons are presented in Sect. 6 before we summarize and give an outlook in Sect. 7. Additional details about the smoothly ordered showers and “GKS” matrix-element corrections previously used in Vincia are collected in Appendix A, while a thorough validation of new matrix-element corrections for ordered parton-shower evolution is given in Appendix B.

## 2 Parton showers and matrix-element corrections

*O*(in the notation of [41]),

*n*-particle momentum configuration \(\Phi _n\). Though not explicitly stated, any

*n*-particle state contains an arbitrarily complicated Born state, \(\,{\Phi _{n}} \equiv {\Phi _{\text {B}+n}}\). The first term in Eq. (1) encodes the contribution from no resolvable shower emissions, while the second piece includes one or more emissions. The parton flavors \(\vec {a}'\) of the \((n+1)\)-particle momentum configuration \(\Phi _{n+1}\) include the resolved emission and the partons \(\vec {a}\), with momenta changed according to the recoil prescription of the parton shower and flavor changes where applicable. The generating functional obeys the evolution equation

*x*the momentum fraction of the incoming parton in \(\Phi _n\). The first term in Eq. (2) corresponds to evolution by initial-state radiation, while the second term represents final-state radiation. Backward evolution [39] for initial-state radiation introduces a ratio of parton distribution functions (PDFs)

*f*in the first term. The quality of the shower real-radiation pattern is governed by the unregularized, dimensionful splitting kernels \(P_{ij}\equiv P_{ij}({{{\Phi _{n+1}}/{\Phi _{n}}}}).\)

^{1}For brevity, we will suppress the indices of the splitting functions. The shower will produce an accurate real-emission pattern if the sum of all products of splitting probabilities and transition probabilities \(\left| \mathcal {M}{({{{\Phi _{n}}}})}\right| ^2\) is a good approximation of the full real-emission probability \(\left| \mathcal {M}{({{{\Phi _{n+1}}}})}\right| ^2\). For a transition from an

*n*-particle to an \((n+1)\)-particle state, this can be achieved by the (symbolic) replacement

The impact of ME corrections is largest for hard, well-separated jets, as splitting kernels do not approximate the full fixed-order matrix element well for configurations with hard, well-separated jets. Thus, the most significant improvement of ME corrections can be obtained when correcting the *n* hardest splittings in the shower cascade. In practise, this means that hardness-ordered parton showers allow for simpler MEC schemes [44, 45, 46], which in particular do not require knowledge of high-multiplicity matrix elements as a function of evolution variables.^{2} Instead, it is sufficient that the parton shower generates complete, physical intermediate momenta \( {\Phi _{n}}\) that can be used to evaluate \(\left| \mathcal {M}{({{{\Phi _{n}}}})}\right| ^2\) numerically. Thus, we will limit our discussion to hardness-ordered shower programs. This will allow for a level of process-independence, and make the *iteration* of ME corrections possible.

The key technical difficulty for a consistent application of ME corrections is the construction of the sum over parton-shower paths in the denominator of the correction factor \(\mathcal {R}({{{\Phi _{n+1}}}})\). Since parton showers are formulated as Markov processes, neither the weight nor the existence of each term in the sum is known a priori when the splitting governed by \(P({{{\Phi _{n+1}}/{\Phi _{n}}}})\) is generated, and all terms have to be reconstructed explicitly.

## 3 Matrix-element corrections for ordered parton showers

The formalism of ME corrections for ordered parton showers (MOPS) is close in spirit to the idea of the iterative MEC approach of [37, 38].^{3} These previous ideas rely on a history-independent parton shower that is able to fill the complete available phase space. This necessitates abandoning parton-shower ordering, i.e. the property that ensures the resummation of large logarithms in ratios of evolution scales. Sensible resummation properties then rely on the introduction of auxiliary functions. Furthermore, configurations with hard well-separated jets might contain poorly understood higher-order corrections. It is thus sensible to limit ME corrections for the parton shower to phase-space regions reachable by an ordered sequence of branchings. This means that we need to rederive appropriate MEC factors \(\mathcal {R}({{{\Phi _{n}}}})\) that correctly encode the presence of complicated phase-space constraints due to ordering—making the resulting method substantially different from previous attempts.

To not overcomplicate the derivation of the MOPS formalism, we drop all coupling- and PDF factors in this section. These pieces are evaluated exactly as in an uncorrected parton shower (the probability of a splitting at evolution scale *t* includes a factor \(\alpha _s(t)/2\pi \), splittings involving initial legs induce ratios of PDFs \(f(\frac{x}{z},t)/f(x,t)\), cf. Eq. (2)), and do not enter in the MEC factors. Similarly, Sudakov factors are not explicitly written out when demonstrating the MOPS method. The MOPS procedure is applied during the Sudakov veto-algorithm as a redefinition of the splitting kernels, meaning that both the (real) emission probability and the no-emission probabilities are ME corrected. This ensures the unitarity of the method, i.e. that corrections to higher parton multiplicities vanish in observables that are only sensitive to a lower multiplicity.

This example highlights the core features of the MOPS method. The recursive structure of the correction factor represents a crucial difference to the GKS method (see Appendix A). At first sight, it seems counter-intuitive that knowledge of ordered *and* unordered paths is required to correctly calculate the correction factor for a phase-space point that has been generated by an ordered sequence of splittings. However, the necessity becomes clear when the weight of intermediate states is taken into account.

To obtain a uniform accuracy over the complete *n*-parton phase space, states beyond the reach of the parton shower have to be included. We discuss the treatment of these non-shower states in the next section, and present the general formula for the MOPS factor in Sect. 5.

## 4 Completing the calculation with non-shower configurations

The MOPS formalism discussed in Sect. 3 only covers the parton-shower phase space characterized by an ordered sequence of splitting scales (\(t_{\text {fac}}>t_1>t_2\dots \)). As a consequence, a prescription for the missing phase space is required. The precise definition of these regions depends on the parton shower itself, the starting scale, definition of the evolution variable, and recoil strategy. Configurations can either be forbidden by restricting the first emission to scales below the factorization scale, cuts on lowest-multiplicity phase space points, or by the ordering property of the shower.

^{4}Uniform (leading-order) accuracy then is obtained across the complete emission phase space by also applying a ME corrected shower when adding soft-collinear shower radiation to non-shower states. This will, if performed naively, introduce overlap between (the shower off) different non-shower states. Three steps are required to avoid the overlap:

- 1.
Non-shower events are defined as unordered if no ordered path exists, i.e. if different paths to the same ME state are present, the event is only considered unordered if none of the paths can be reproduced with an ordered sequence of branchings scales.

- 2.
Potential overlap between non-shower states with different parton multiplicities has to be removed, e.g. a maximally unordered \(+2\)-particle state may also be produced as a shower emission off a maximally unordered \(+1\)-particle state. The explanation how this overlap is identified and removed in the higher-multiplicity states, is deferred to the end of Sect. 5, since it is helpful to first discuss how non-shower states are showered.

- 3.
States produced by ordered parton showers overlap with soft-collinear radiation attached to non-shower states if the “history” of a phase-space point contains both ordered and unordered paths. Therefore, both have to be ME corrected with correction factors taking into account both possibilities of population.

To obtain a flexible scale-setting prescription, we borrow the idea of constructing all possible event histories from the CKKW-L [26, 27, 28] The aim of the procedure is twofold: define dynamical scales by exploiting the information about the phase-space points with the help of the weight and “substructure” of multi-jet matrix elements, while further ensuring a smooth inclusion of non-shower states with shower-accessible events.

*n*-particle state. To obtain the correct (leading-order) result when summing over all possible splittings \(\Phi '_{n}\rightarrow {\Phi _{n+1}}\), we can apply the corrective factor

- (a)
if one splitting dominates over all other splittings, then a natural scale to capture the dynamics is strongly correlated with the relative jet separation of the dominant splitting,

- (b)
if no splitting dominates, i.e. all splittings contribute democratically, there should be no strong preference for a scale, and a weighted average of jet separations seems appropriate.

An expression for the effective scale could also have been obtained by including PDF ratios in Eq. (13), which would mean that the choice of effective scale captured dynamics of underlying “hadronic” cross sections. We do not implement such a scale-setting prescription since we believe that the scale setting should be based on perturbative parton-level quantities.

Note that the scale-setting mechanism in Eq. (14) allows for \(t_{\text {fac}}< t^\mathrm {\,eff}\) if the scales entering the calculation are sufficiently large. An example of such a configuration are non-shower states with multiple hard (and possibly balancing) jets without \(p_\perp \) hierarchy. In this case, using a scale defined for the lowest-multiplicity process can result in pathologies [50]. It is desirable that \(t^\mathrm {\,eff}\) is not bounded by \(t_{\text {fac}}\), the factorization scale assigned to a fictitious lowest-multiplicity process. Instead, \(t^\mathrm {\,eff}\) should provide a more “natural” scale for this genuine multi-jet configuration. Furthermore, \(t^\mathrm {\,eff}\) is bound to remain in the perturbative region, since we only include non-shower phase space points for which clustering scales (as defined by the shower evolution variable) are above the parton-shower cut-off.

In Sect. 6 we will show that the scale setting outlined in this section results in a very good description of LHC data.

## 5 The complete algorithm

Since non-shower states are included without a hard cut-off (e.g. a merging scale), the effective scale \(t^\mathrm {\,eff}\) may differ significantly from the factorization scale \(t_{\text {fac}}\). In this case, we further attach Sudakov factors by means of trial showering [26, 27] to the non-shower states to include a sensible suppression due to the resummation of large logarithms of \(t_{\text {fac}}/t^\mathrm {\,eff}\). This is relatively straight-forward for \(+2\)-particle states—a Sudakov factor \(\Delta (t_{\text {fac}},t_{2}^\mathrm {\,eff})\) is applied to ensure a sensible result if the \({\vec p}_\perp \) of the combined Born\(+2\)-parton system is small. For higher-multiplicity non-shower states, more scale hierarchies arise, and a more detailed scheme is necessary to cover all relevant cases. However, only two types of scale hierarchies can remain after removing the overlap between *n*-particle non-shower events and states that are produced by showering lower-multiplicity non-shower configurations: the ordering \(t_{\text {fac}}> t_{n}^\mathrm {\,eff}\), or the ordering \(t_{\text {fac}}> t_{n-1}^\mathrm {eff} > t_{n}\).^{5} The hierarchy \(t_{\text {fac}}> t_{n}^\mathrm {\,eff}\) is again ameliorated by applying a single Sudakov factor \(\Delta (t_{\text {fac}},t_{n}^\mathrm {\,eff})\) to produce a sensible result for small \({\vec p}_\perp \) of the combined Born\(+n\)-parton system. If instead a hierarchy \(t_{\text {fac}}> t_{n-1}^\mathrm {\,eff} > t_{n}\) can be constructed, then a product of Sudakov factors \(\Delta (t_{\text {fac}},t_{n-1}^\mathrm {\,eff}) \Delta (t_{n-1}^\mathrm {\,eff},t_{n})\) is appropriate. This guarantees a uniform weighting of \(+n\)-particle events arising from either \(+n\)-particle non-shower states or showered \(+(n-1)\)-particle configurations. Note that the Sudakov factors \(\Delta (t_{\text {fac}},t^ \mathrm {\,eff})\) are unity if \(t_{\text {fac}}< t^\mathrm {\,eff}\).

The information about the different types of scale hierarchies are also used to remove the overlap between non-shower states with different parton multiplicities. States with scale hierarchies of the type \(t_{n-m}^\mathrm {\,eff}> t_{n-(m-1)}> \cdots > t_{n}\) are removed for \(m\ge 2\). For states that contain the hierarchy \(t_{n-1}^ \mathrm {\,eff} > t_{n}\), the event is removed if the clustered \(+(n-1)\)-particle state is itself an unordered state. Events without scale hierarchies that could have resulted from showering lower-multiplicity states are kept; that includes all \(+2\)-particle states with unordered scales \(t_{2}>t_{1}\) and \(+1\)-particle states with \(t_{1}>t_{\text {fac}}\). For the interested reader we include further methodological instructions in Appendix C.

## 6 Results

In this section, we present results obtained with the new method, including both the MOPS factor and the non-shower states (called “MOPS \(+\) unordered” in the following). A detailed validation can be found in Appendix B. The analyses are performed with Rivet [51]. We begin this section by studying the effect of the new method on jet separations, before moving to comparisons to LHC data. In both cases, we juxtapose the results with the GKS ME corrections implemented in Vincia. The GKS MECs scheme includes emissions above the factorization scale \(t_{\text {fac}}\) (see Appendix A2 for how those are generated) as does the MOPS \(+\) unordered method by adding non-shower \(+1\)-particle states. Emissions with scales \(t_{1}>t_{\text {fac}}\) would not naturally be present in the pure or MOPS corrected shower, where Born states are showered beginning at \(t_{\text {fac}}\). For the following results we add \(+1\)-particle states with scales \(t_{1}>t_{\text {fac}}\) explicitly to the pure and MOPS corrected shower, and shower these states using \(t_{1}\) as shower starting scale. This decreases the significance of including non-shower states w.r.t comparing to a strictly ordered shower evolution, but should avoid using an “overly conservative” shower setup when comparing to default Vincia.

### 6.1 Theory comparisons

Hadron-level results for hadronic *Z* decays and Drell–Yan events are presented in Fig. 2. The results have the expected behavior: at low jet resolution, parton-shower effects dominate, while non-shower states contribute mainly to large jet scales. Hence, the MOPS factor is dominating the observable at low scales. At LEP, shower states remain a dominant contribution even when modeling well-separated jets, and the effect of non-shower states remains at below \(10\%\) per bin. Results at the LHC are in stark contrast to this. There, the influence of shower configurations decreases substantially for large jet resolution, and non-shower phase-space regions become increasingly important. The uncertainty from varying the effective scale is not significant at LEP, and should thus not be considered a realistic uncertainty estimate. At LHC, the variation of \(t^\mathrm {\,eff}~(=t_{\text {fac}}=t_{\text {ren}}=t_{\text {start}})\) is larger, and increases for high jet resolution, as expected from varying scales in a tree-level fixed-order variation. At low resolution, we observe a small increase in the scale uncertainty, which stems from the interplay of very large \(\alpha _s\) values with the Sudakov factors that are applied to non-shower states.

By comparing with previous ideas below, we hope to understand the short-comings and benefits of our MOPS \(+\) unordered prescription. In Fig. 3 we compare the results of Vincia2.2 without corrections, with the MOPS correction, MOPS \(+\) unordered scheme, and Vincia2.0.01 with smooth ordering for the GKS ME corrected orders.

The MOPS correction for purely evolution-induced events is small for all jet resolutions. Differences are mostly at the level of 1–5%, illustrating that the uncorrected shower already describes the matrix elements well in phase-space regions reachable by showering. As discussed above, the jet resolution scales exhibit a Sudakov suppression for small values. In the Sudakov region, the corrected predictions should not deviate greatly from the “plain” shower result. This is indeed the case for both the MOPS \(+\) unordered and the GKS MECs method for very small resolution scales. The GKS MECs method generates more events with larger \(d_{m\,m+1}\) separation. Due to the unitarity of the shower, this leads to a depletion of events with small separation compared to the pure shower. The behavior is consistent with the findings in [38], where differences between strong and smooth ordering have been investigated. The impact of non-shower states in the MOPS \(+\) unordered scheme remains noticeable close to the peak of the distribution, although the modeling of the Sudakov region approaches the uncorrected shower more quickly than for the GKS MECs method. This means that the handling of non-shower states with large scale hierarchies (cf. end of Sect. 5) is important in our approach. Merging approaches commonly discard non-shower states with separation below a certain (merging) scale.

In conclusion, we believe that the MOPS \(+\) unordered scheme has desirable features, and that the choices in the method lead to the expected behavior.

### 6.2 Comparisons to data

^{6}Soft-physics parameters are kept at their current Vincia default values. The default Vincia2.0.01 tune [38] corresponds to different \(\alpha _s \) settings. While this results in a slightly better data description, it does not alter the general observations and conclusions of this section.

In Fig. 4 we confront the results of Vincia2.2 without corrections, with the MOPS correction, MOPS \(+\) unordered scheme, and Vincia2.0.01 with GKS ME corrections with ATLAS [52, 53] and CMS [54, 55] measurements.

As already seen in Sect. 6.1, the effect of the MOPS correction factor is small for all observables. This is of benefit for the description of the Drell–Yan \(p_\perp \) spectrum (upper left panel of Fig. 4), for which the plain shower already offers a sensible data description. The quality of the description also remains unchanged for the MOPS \(+\) unordered scheme. The other observables in Fig. 4 test the existence of hard, well-separated emissions in the tails of the distributions and are thus poorly modeled with the parton shower alone. We find a very good data description with the MOPS \(+\) unordered scheme. In particular, the quality of the data description in our scheme relies crucially on the treatment of non-shower states. The scale-setting mechanism presented in Sect. 4 produces promising results, with the naive central scale choice close to the data, but with a large, leading-order-like uncertainty due to scale variations. We anticipate that the width of the band will decrease when performing a next-to-leading-order calculation with a similar scale choice. The uncertainty due to scale variations is largest in phase-space regions most sensitive to non-shower contributions. For the \(S_\perp \) and leading jet \(p_\perp \) distributions, the results of the GKS MECs approach touch the uncertainty bands attributed to non-shower events at low values, but are outside of the band in regions influenced by multiple hard jets. Both of these observables are much improved in the MOPS \(+\) unordered method, compared to the uncorrected shower. For the angle between the *Z*-boson and the hardest jet we observe a satisfactory data description for both our new method and Vincia2.0.01.

## 7 Conclusions

We have presented an algorithm to obtain fixed-order accurate predictions combined with all-order parton-shower evolution that produces finite and non-overlapping results without introducing a merging scale. The new algorithm requires the introduction of a sophisticated matrix-element correction method for evolution-induced configurations. States beyond the reach of the parton shower are included with a systematic scale-setting procedure. This smoothly combines non-shower configurations and states produced in the ordered parton-shower evolution. The algorithm does not depend on specific properties of the parton shower and allows for arbitrary dead zones (which may be required by resummation considerations). The new fixed-order \(+\) parton-shower scheme has been implemented in the Vincia parton shower and will be made publicly available upon the Vincia2.2 release.

The effect of including ME corrections for ordered parton-shower splittings is minor compared to the uncorrected shower. This means that the method does not deteriorate the shower resummation, and gives us confidence that the improvement does not interfere with other improvement strategies [57, 58, 59]. The main improvements stem from a careful treatment of contributions from phase-space regions that are not accessible by ordered parton showers. Such contributions are included with a sophisticated scale-setting prescription. For hadronic initial we find the scale setting to have a sizable influence on observables, since large parts of phase space are not shower accessible. We presented comparisons to data for the \(pp\rightarrow Z+\)jets process and found the results of our new algorithm to be in good agreement with the data.

## Footnotes

- 1.
- 2.
A scheme to correct the hardest emission in angular-ordered showers has been discussed in [47]. This scheme requires to apply the same correction repeatedly, to guarantee that the single hardest emission is corrected to leading-order accuracy. Although promising from the resummation standpoint, it is, however, not obvious how this scheme could be used to correct the

*n*hardest emissions. - 3.
A short review of the GKS approach is given in Appendix A.

- 4.
An arbitrary shower will not correctly describe

*all*sub-leading logs in its evolution variable, so that non-shower configurations may still contain (sub-leading) logarithmic divergences. One famous example of such configurations are the unordered, balancing soft-gluon emissions leading to Parisi–Petronzio scaling in \(p_\perp \) distributions [48, 49]. To avoid such divergences in practice, we only include non-shower phase-space points if each scale at which partons could be recombined (as defined by the shower evolution variable) is above the parton-shower cut-off \(t_{\text {cut}}\approx 1\,\mathrm{{GeV}}\). - 5.
Consider a non-shower (unordered) \(+4\)-particle state. After computing effective scales, it is possible that a scale hierarchy \(t_{\text {fac}}> t_{2}^\mathrm {\,eff}> t_{3} > t_{4}\) exists. Such a configuration can be obtained in several ways showering lower-multiplicity non-shower states. (a) If the reconstructed underlying \(+2\)-particle state is not shower-like (i.e. unordered), then the \(+4\)-particle state with the above hierarchy can be produced by adding two ordered shower emissions to the \(+2\)-particle state. Thus, the state is included by showering a non-shower \(+2\)-particle state. (b) If the reconstructed \(+2\)-particle state can be reached by an ordered sequence of emissions, and furthermore \(t_{3} > t_{4}\) then the “unordering” stems from the \(+2\)-particle to \(+3\)-particle transition. Thus, the \(+4\)-particle configuration can be reached by adding one ordered shower emission to a non-shower \(+3\)-particle state. In conclusion, the states with this more complex scale hierarchy should not be included through a non-shower \(+4\)-particle input, since this would result in over-counting.

- 6.
Different \(k_\mu \) values for different branching types invalidate the interpretation of the effective scale as a single parton-shower starting scale for subsequent showering.

## Notes

### Acknowledgements

We would like to thank Stefan Höche and Peter Skands for helpful discussions, and Peter Skands for comments on the manuscript. This work was supported by the Australian Research Council and by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. NF thanks the SLAC and Fermilab theory divisions for hospitality during the course of this work.

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