# Living on the walls of super-QCD

### Submission summary

 As Contributors: Francesco Benini · Matteo Bertolini Arxiv Link: https://arxiv.org/abs/1812.04645v3 Date accepted: 2019-04-01 Date submitted: 2019-03-19 Submitted by: Benini, Francesco Submitted to: SciPost Physics Domain(s): Theoretical Subject area: High-Energy Physics - Theory

### Abstract

We study BPS domain walls in four-dimensional $\mathcal{N}=1$ massive SQCD with gauge group $SU(N)$ and $F<N$ flavors. We propose a class of three-dimensional Chern-Simons-matter theories to describe the effective dynamics on the walls. Our proposal passes several checks, including the exact matching between its vacua and the solutions to the four-dimensional BPS domain wall equations, that we solve in the small mass regime. As the flavor mass is varied, domain walls undergo a second-order phase transition, where multiple vacua coalesce into a single one. For special values of the parameters, the phase transition exhibits supersymmetry enhancement. Our proposal includes and extends previous results in the literature, providing a complete picture of BPS domain walls for $F<N$ massive SQCD. A similar picture holds also for SQCD with gauge group $Sp(N)$ and $F < N+1$ flavors.

### Ontology / Topics

See full Ontology or Topics database.

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.

### List of changes

- 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.

### Submission & Refereeing History

Resubmission 1812.04645v3 on 19 March 2019
Submission 1812.04645v2 on 17 January 2019

## Reports on this Submission

### Anonymous Report 1 on 2019-3-24 Invited Report

• Cite as: Anonymous, Report on arXiv:1812.04645v3, delivered 2019-03-24, doi: 10.21468/SciPost.Report.889

### Strengths

1) The paper contains an independent analysis from 3d and 4d perspectives that provides non-trivial cross-checks

2) It supplements recent results on 3d N=1 IR dualities with a natural 4d embedding

3) Clarity of presentation

### Report

I would like to thank the authors for the detailed explanations. They provide a satisfactory response to my question.

As I noted in my first review, this is a very good paper with several interesting results and non-trivial cross-checks. It makes progress in an exciting subject. I am happy to recommend this paper for publication in its current version.

• validity: high
• significance: top
• originality: high
• clarity: top
• formatting: perfect
• grammar: perfect