# Sequential Flows by Irrelevant Operators

### Submission summary

 Authors (as Contributors): Christian Ferko
Submission information
Date submitted: 2022-11-07 02:27
Submitted by: Ferko, Christian
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
Approach: Theoretical

### Abstract

We explore whether one can $T \overline{T}$ deform a collection of theories that are already $T \overline{T}$-deformed. This allows us to define classes of irrelevant deformations that know about subsystems. In some basic cases, we explore the spectrum that results from this procedure and we provide numerical evidence in favor of modular invariance. We also study the flow of the classical Lagrangian for free bosons and free fermions under successive deformations. Some of the models found by sequentially flowing are likely to have interesting holographic interpretations.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2206.04787v3 on 12 December 2022

Resubmission 2206.04787v2 on 7 November 2022
Submission 2206.04787v1 on 1 September 2022

## Reports on this Submission

### Report

I would like to thank the authors for their kind reply.
I believe they addressed the points I raised in a satisfactory way, with possibly the exception of point 1.

I still find the concept of a leading deformation in the context of this paper quite confusing.
The considered setup is obtained by applying two distinct and consecutive finite deformations and cannot be understood as being generated by a single irrelevant operator.
In fact, the deformed theory is controlled by three parameters, namely $\{λ_1,λ_2, λ_3\}$, and there is no point along the flow where these are all small. One can insist on studying such a regime, but it is probably just misleading. For instance, I don't see how the operator in Eq. (1.7) should be in any way related to the theory with finite $λ_3 = -λ_1 = -λ_2$, yet the authors state that "[...] the procedure of sequentially deforming that we described would seem to define some theory, whose leading order deformation is (1.7)."

### Requested changes

1 - The authors should clarify the paragraph mentioned above.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Christian Ferko  on 2022-12-09  [id 3116]

(in reply to Report 2 on 2022-12-05)

We would like to thank the referee for his or her comments. We have gone ahead and revised the paragraph for improved clarity concerning this point.

### Report

I would like to thank the authors for their kind reply and clarifications in the paper.

Unfortunately, I believe that my remark about equations (3.22), (3.24) and (3.8) was misunderstood. The point was not to compare (3.8) to (3.22), but rather to explain why the authors obtained a different bound on $r$ from (3.24) than they did from (3.22). The reason is that to obtain (3.24), they used the same methodology as in (3.8), but I believe equation (3.8) is wrong.

To see this explicitly, one can check (3.8) in a special case, namely $\lambda_2 = \lambda_1$ and $\beta = \alpha$. The approximation (3.8) than reduces to $$\mathcal{E}_{n, m} \sim \frac{2 \sqrt{\alpha \lambda_1} - R}{\lambda_1 + \lambda_3} \ .$$ However, we can obtain the exact result for $\lambda_2 = \lambda_1$ and $\beta = \alpha$ by starting from (3.6): $$\mathcal{E}_{n, m} = \frac{R + \lambda_3 \mathcal{E}_{n, m}}{\lambda_1} \left( \sqrt{1 + \frac{4\lambda_1 \alpha_n}{(R + \lambda_3 \mathcal{E}_{n,m})^2}} - 1 \right) \ ,$$ which can be solved to give $$\mathcal{E}_{n, m} = \frac{-R + \sqrt{R^2 + 4 \alpha \lambda_1 + 8 \alpha \lambda_3}}{\lambda_1 + 2 \lambda_3} \ .$$ This result is exact and does not agree with the approximation from (3.8). In particular, the numerators between the prediction from (3.8) and the exact result differ with a factor of 2 in front of $\lambda_3$. I believe this is the same factor of 2 that distinguishes the bound $r < 1$ obtained in (3.24) from the actual bound $r < 1/2$ from (3.22).

What went wrong is that an implicit equation for $\mathcal{E}_{n,m}$ was used after the large $\alpha$ approximation (3.7), but $\mathcal{E}_{n,m}$ itself is of order $\sqrt{\alpha}$. Presumably one can obtain a correct version of (3.8) by taking into account the next order in (3.7).

Unless the authors disagree with this argument, I think it best to address this issue before proceeding with the publication of the paper.

### Requested changes

Address the issue with (3.8) and (3.24).

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Christian Ferko  on 2022-12-09  [id 3117]

(in reply to Report 1 on 2022-11-21)

We thank the referee for his or her comments. The issue the referee raised is now clear to us. The resolution is interesting, and we have revised the discussion around (3.7) - (3.8). It turns out that the formulae we originally listed were correct but only in a perturbative expansion in $\lambda_3$ around $0$. The discrepancy with the later formula goes away when one performs this expansion to leading order in $\lambda_3$. We have also revised any subsequent discussion that relied on those results.