An Optimal Observable for Color Singlet Identification

We have addressed the requested changes as follows:

1 - Changed

2 - Changed

3 - Changed

4 - Added explanation and references

5 - Changed

6 - Added after equation (6) "where we have exploited the fact that our analysis is blind to both the color and the flavor of the final-state partons."

7 - Changed

8 - Inserted the color singlet figure now. Updated the text to mention the new figure.

9 - We acknowledge that our notation was indeed confusing. Eq.(1) and Eq.(2) do not represent the full matrix element square for signal and background, respectively. Rather, we have exploited the collinear limit of the two subjets we are tagging and factorized their contribution from the complete expression. The remaining part of the matrix element squares, which include both initial-state contributions such as parton distribution functions, as well as eventual extra jets, contributes with an overall factor which is not relevant, within our approximation, for the construction of the observable. We have changed the sentence just before Eq.(1) to be:
"The matrix element square describing the decay of a boosted color-singlet state into two hard partons plus a single low-energy gluon is".
Furthermore, we have rewritten the paragraph before Eq.(8).

10 - Added more lines and a contour plot for pull. Updated the caption and in-text paragraph describing figure 3 appropriately. Added a reference to the new pull angle contours plot in the paragraph about relationship to the pull angle.

11 - Added "Such jet shape may be Sudakov safe~\cite{Larkoski:2013paa,Larkoski:2014wba,Larkoski:2015lea}, although it is not clear what its natural safe companion observable would be~\footnote{In a recent study~\cite{Komiske:2020qhg}, a generalized defintion of IRC safety (which includes Sudakov safety) based on continuity properties on the space of events was proposed. In that context, $N$-(sub)jettiness~\cite{Stewart:2010tn,Thaler:2010tr} plays the role of a universal safe companion.}"

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14 - This is simply a description of the algorithm as implemented and attached to the arXiv submission, which makes a minimal flavour-tag requirement on the large-R jet before the relatively expensive computation of matched, b-tagged subjets used to apply a tighter requirement. The "preselection" role of this cut has now been clarified in the text.

15 - The first two comments made here are addressed below in point 16. Addressing the 3rd comment, we appreciate the suggestion for alternative dipole-subjet selection via maximum invariant mass. We have implemented and tested this alternative scheme, and no change is seen for the bb subjet selection (unsurprisingly, as there is rarely a 3rd tagged subjet to give ambiguity), but more substantial changes are seen for the H->gg and Z->qq analyses. These are not significant enough to change the thrust of the results, but give a slight improvement in ROC performance for dipole-based measures in H->gg, and a more ambiguous shift of the ROC curves for Z->qq: we have hence chosen to make the max-mass scheme our default and updated the paper and plots accordingly, with a note about the impact of using an alternative leading-pTs scheme.

16 - We believe that the additional explanation added in order to address the Referee's point 9) partly answers this objection too. Furthermore, we have modified the sentence as follows:
"However, contrary to our naive expectation, the background is now also strongly peaked with a similar shape. This is mostly likely due to a breakdown of our factorized "production$\times$decay" approximation, which for the background heavily relies on the collinear limit of the tagged subjets. "

17 - We have clarified that the observable O' is a monotonic variant of the jet color ring for "fixed kinematic of the two tagged subjets a and b."