Inferring nuclear structure from heavy isobar collisions using Trajectum

We wish to thank the referee for their careful reading of the manuscript. We believe we have addressed their questions below. In addition to the referee's questions, we found that in the left panels of Fig.~6, the ratio was incorrectly labelled as ZrZr/RuRu. We have corrected this to RuRu/ZrZr.

The referee writes:

This manuscript studied how to use measurements in ultra-relativistic heavy-ion collisions to probe the nuclear structure of the colliding nuclei. The authors performed high statistics numerical simulations for Ru+Ru and Zr+Zr collisions at the top RHIC energy with the \emph{Trajectum} framework. They studied how particle yield, mean transverse momentum, and anisotropic flow coefficients depend on different nuclear structure configurations parameterized by five sets of Woods-Saxon parameters. The paper was written clearly and contained important physics insights for the RHIC isobar program. This study also builds connections between low-energy nuclear structures and high-energy relativistic heavy-ion collisions. I would recommend it for publication once the authors clarify the following questions.

To build a connection between the structure of nuclei and high-energy heavy-ion collisions, the authors should explain the underlying assumptions for how the produced initial-state energy density profile in the heavy-ion collision is related to the nucleus' structure. For example, will different energy deposition models weaken the sensitivity of the Woods Saxon deformation parameters on heavy-ion observables?

Our response:

In the ratio between Ru and Zr, dependence on model parameters usually cancels to a large degree. We show an explicit example of this in Fig.~9, where we show that changing the viscosities changes $v_2$ and $v_3$ for both Ru and Zr, but in the ratio this dependence cancels to within statistical uncertainties. Given that computing isobars is statistically demanding, we did not check explicitly whether varying parameters related to the initial energy deposition has an effect on the sensitivity of the heavy-ion observables on Woods-Saxon deformation parameters, but any such effects are similarly expected to cancel out when taking the ratio between Ru and Zr.

We have added the sentence ``More generally, it is expected that the dependence of observables on other model dependencies such as $d_{\rm min}$ in the initial state or other pre-hydrodynamic parameters mostly cancel when taking a ratio of observables from the two isobars.'' on page 8 to make this clear from the text.

The referee writes:

The Woods-Saxon parameters listed in Table 1 assumed the nucleon were point-like objects. However, in the Trento initial condition model, the nucleons are assumed to have finite sizes. Did the authors correct the Woods-Saxon parameters for finite nucleon sizes, as discussed in Phys.~Rev.~C 79, 064904 (2009)?

Our response:

Indeed one can make Woods-Saxon parameters which either describe the charge or baryon number density, or describe the point density of the nucleons. As the referee points these Woods-Saxon parameters are only approximately equal. In principle, our calculation requires the parameters for the positions of the nucleons. The parameters for case 1 and 2, however, come from relatively old references that likely do not include the effects described in Phys.~Rev.~C 79, 064904 (2009). Cases 3 till 5 are more modern and we think that they describe the point densities of the nucleons.

However, similar to the point made above the small difference in the Woods-Saxon parameters affects Ru and Zr equally and hence in the ratio these differences cancel. Since we were quite clear that our study is not a precision attempt at describing Ru and Zr separately we decided not to further comment on the charge versus point density in this paper. If the referee is interested we have a more specific discussion in 2206.13522 about this, but we did not feel it relevant enough for isobars to expand on this in the current work.

The referee writes:

Did the authors consider the short-range hard-core repulsion between nucleons in their nuclear configurations? Would these short-range correlations affect the observable ratios between the two isobar collisions?

Our response:

The Trento model incorporates a minimal distance requirement for the placement of the nucleons inside the nucleus, where we require nucleons to be at least $d_\text{min}$ apart. In Bayesian analyses we generally find little dependence on $d_\text{min}$, and it has little effect on observables. As mentioned in our reply to the referee's first question, especially in the isobar ratio any dependence is expected to largely cancel out.

We have added the following on page 3 to make this clear in the text: ``As in [26], Trento also includes a hard-core repulsion implemented through a minimal inter-nucleon distance $d_\text{min}$.''

As in our response.