Interleaved Resonance Decays and Electroweak Radiation in the Vincia Parton Shower

We are grateful to the reviewers for their careful reading of the manuscript and their detailed comments. We reply to them here point-by-point. They are also added as replies to the previous version in pdf format.

Referee 1:

1. We agree that the enumerated list describing the interleaved algorithm could be improved. We have significantly expanded it and have tied the steps more explicitly to the illustration in figure 1 (now figure 2). To keep the list itself relatively concise, some of the content (such as the scale definitions under point 2) were moved up into the body text in the introduction to section 2, which also allowed to comment more extensively on those choices, cf our response to point 3 of referee 2.

2. We have expanded the discussion on the resolution measure, and the relevance of the parameter R.
   In short, the exact form of the resolution measure is not of importance as long as it correctly separates the singular regions of phase space associated with each shower. For internal consistency, we have chosen to stay as close as possible to the more limited implementation in Pythia's default showers.

3. We have added two figures showing the computation time penalties for enabling interleaved resonance decays (fig 3) and the EW shower (fig 15). Regarding the point of validation of the individual EW splitting kernels, these have been validated to give identical collinear limits against the branching kernels of 2002.09248, which were calculated separately. We have added a note of this in the text. Furthermore, the tests shown in, for instance, figure 10 and 11 (now figs 12 and 13) serve as an indirect validation, as the EW shower would not be able to produce the correct spectra without the correct splitting kernels. This is also reflected in the text.

4. A check against a theoretical prediction is shown in figure 10 (now figure 12). We had struggled to format this figure, and accept that its message was not conveyed clearly. We have now revised it by emphasising the exact ME curve (which serves as theory baseline) by using a thick black line for that. We also dashed the lines that represent the individual components of the full Vincia result. Finally, to reduce number of curves, we also dropped the Pythia result, since that is anyway unrelated to the validation of the Vincia treatment as such. We hope that the figure now shows much more clearly that, if one does not enable the overlap veto, the sum of the QCD and the EW path overcount the exact matrix element by a large amount (left pane). On the other hand, with the veto enabled, the shower lines up quite well with the exact matrix element (right pane).
   We have tried to clarify this further in the text.

5. This should have been the jet radius R of the anti-kT algorithm. We have adjusted the figure accordingly.

Referee 2: 1. We agree that both references should be included. arXiv:2108.10817 was released on the same day as this manuscript, but arXiv:1403.4788 should have been included in the first place, which we apologize for. We have added both.

1. The point about running widths leading to potential gauge violations is of course completely correct. We would like to keep our remarks about it relatively brief, since this aspect is not by itself a main point of our paper. In the introduction on p4, we have therefore sought to change the relevant paragraph while trying not to lengthen it, as follows:

   ...as well as options for allowing partial widths (and hence relative branching fractions) to vary with $Q^2$. The latter allows, for example, to account for kinematic thresholds and effects of running couplings across a reasonable range around the pole, but cannot be pushed too far, especially in the electroweak sector where masses and couplings are not independent of each other.

One further point to clarify is that the masses of resonances produced by the EW shower are also distributed according to Breit-Wigner distributions with running widths. As is already explained in Appendix C, the gauge violations that lead to dangerous high-energy behaviour are taken care of in the same way as in the calculation of the EW antenna functions. Furthermore, Figure 2 (4 in the new version) shows that at high invariant mass, for the high-energy resonances that the EW shower tends to produce, the shower dominates over the Breit-Wigner distribution, meaning that the exact treatment of running width effects are of little consequence in that region. We have verified that this is true for all SM resonances. A note of this has been added to the resonance matching section.

1. In Vincia, the default cutoff scale for final-state evolution is 0.75 GeV, so the given definitions of Q_res are typically above the shower cutoff, for top, Z, and W resonances in the SM (which all have widths larger than 1 GeV). This means that, when one of the dynamical scale choices is selected, most of these resonances will still decay before the cutoff is reached. (Although the Q_res$ distributions are peaked at zero, the integrated probabilities below the shower cutoff are still less than 50%). We added a new figure 1 to highlight this, which shows the surviving fraction of t, Z, and W resonances as functions of the shower scale, for the three different dynamic-scale options. The region of typical shower cutoff values (0.5 - 1 GeV) is also illustrated.

The reasoning for using a measure of offshellness is given in the introductory paragraph to section 2, specifically strong ordering of propagator virtualities, which dictates which diagrams are enhanced and which are suppressed. The default choice, eq.(4), explicitly represents the denominator structure of eq.(3); the alternatives are mainly provided to make it possible to study the sensitivity to variations on this choice. We have added the following two paragraphs to the beginning of section 2 to elaborate on this:

The desire to connect with the strong-ordering criterion in the rest
of the perturbative evolution, as the principle that should dictate the
leading amplitude structures, leads us to prefer a dynamical
scale choice for 
resonance decays, whereby resonances that are highly off shell will
persist over shorter intervals in the evolution than ones that are almost
on shell. We note that this has the consequence 
that the on-shell tail will be resolvable by soft photons or gluons,
albeit suppressed by the survival fraction. To illustrate this,
fig.~1 shows the survivial fractions (denoted $\Delta_R$)
as functions of 
evolution scale, for $t$, $Z$, and $W$ resonances, for three different
options for dynamical scale choices, all of which are roughly
motivated by the propagator structure:
\begin{eqnarray}
i)~~~Q_\mathrm{RES}^2 & = & (m - m_0)^2~, \

Listed in the author comments.