Living on the walls of super-QCD

We thank the reviewers for their close reading of the paper and useful remarks. We edited the paper to address the concerns of the second reviewer. For the benefit of the reader, we include here our response to the second reviewer's comments.

• Let us now comment on the weakness highlighted by the referee. The 3d low-energy theory we propose to describe the domain walls has a quartic superpotential interaction and is not UV complete. It should be thought of as an effective theory, for values of the mass close to the phase transition (notice that, in any case, the domain wall can have a 3d description only below some scale, set by the SQCD scale and the 4d quark mass). A possible UV completion is in terms of a similar theory with no quartic superpotential interactions, and with a bare mass close to the value corresponding to the phase transition. The analysis of the vacuum structure of this theory has been done by Choi et al. [JHEP 1810 (2018) 105], taking into account perturbative effects at large values of the fields. Their results agree with the semiclassical analysis we did using the effective description. We think this gives strong support to the claim that the analysis of vacua is complete.

• Regarding the question raised by the reviewer, it would surely be interesting to perform a more complete analysis that includes the adjoint multiplet $\Phi$. In particular, one may worry that the theory with the adjoint has more vacua (in which $\Phi$ gets a VEV) than the low-energy theory with the adjoint integrated out. We do not have a rigorous answer, but we believe that various arguments support the belief that no other vacua exist.

1) Close to the phase transition and in the vacuum that hosts the CFT, our analysis is complete because, no matter how small is the mass of the adjoint, we can always integrate out the adjoint as long as we look at lower energy scales. This will be true even if we move a little bit away from the CFT by a mass deformation, as long as this is smaller than the adjoint mass.

2) If we make the mass of the 3d quarks larger in our effective description, we cannot really use the effective description anymore. In particular, we could use the UV complete theory with no quartic superpotential interactions and with the adjoint field -- whose bare mass is set to zero. For large positive values of the quark bare (i.e. UV) mass, the 3d quarks can be integrated out, leaving a theory with the adjoint. This theory has been carefully analyzed by Bashmakov et al. [JHEP 1807 (2018) 123], with the conclusion that, for zero bare adjoint mass, no other vacua exist.

3) It does not make sense to take the bare mass of the 3d quarks to be large and negative in the UV complete description, because such a regime does not pertain to the physics of the domain walls. In particular, when that mass (in the description with no quartic superpotential) is zero, some vacua run to infinity and the Witten index jumps. However, we know from 4d arguments that the Witten index on the domain walls cannot jump. Indeed, probably that mass vanishes when the 4d mass vanishes, but in this case there are no 4d vacua nor domain walls, and so that value cannot really be reached.

4) Therefore, the regime in which the bare mass in the UV complete description is between zero and the positive value corresponding to the phase transition, is the one where in principle there could be other vacua once the adjoint is taken into account. We agree that a more complete analysis would be required to settle the issue. However, notice that the potential extra vacua should contribute a total of zero to the Witten index, because the analysis for large positive values of the mass was reliable, and the corresponding Witten index is already saturated by the vacua we found. This observation makes the existence of extra vacua unlikely (in our opinion), but of course not impossible.

We hope this addresses the reviewer’s concern.

- Eq. (2.12) had indeed a missing \epsilon. Thanks for spotting it out. We corrected the equation.

- We have changed the first line after eqn. (5.1) for the sake of clarity.

- We have corrected a typo in the second line of page 26: W(Phi) -> W(\tilde M)

- We have added a footnote (now number 10) at page 12 to take into account the second reviewer's comment.