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For systems characterized by large scattering lengths and short effective ranges, the unitarity limit, where the scattering length diverges and the effective range vanishes, can serve as a natural starting point for a perturbative expansion. Short-Range Effective Field Theory (SREFT), which involves only contact interactions, offers a systematic framework for organizing such calculations. While the standard framework of expansion around the unitarity limit within SREFT (or Pionless EFT in nuclear physics) is well established \cite{Konig:2015aka, Konig:2016utl, Konig:2016iny, Konig:2019xxk}, several open questions remain. Here, I examine some of these issues in the two-, three-, and four-body sectors.

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\textbf{Power counting at high orders in the two-body sector}
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In the two-body sector, a non-derivative two-body interaction appears at LO in SREFT, with its low-energy constant (LEC) $C_0$ tuned to reproduce the unitarity limit. This results in scale invariance. Deviations from unitarity, including the scattering length and effective range corrections, start to appear at next-to-leading order (NLO). The LO two-body interaction is then corrected to account for the large scattering length, while the finite effective range is introduced through a two-derivative contact interaction with a new LEC $C_2$.

At even higher order, the shape parameter correction appears. It can be incorporated by introducing a four-derivative two-body interaction with a LEC $C_4$. Although its effect is recognized as two orders higher compared to the effective range correction, this $C_4$ term is already introduced at next-to-next-to-leading order (N$^2$LO) in the conventional power counting \cite{Kaplan:1998we, vanKolck:1998bw, Chen:1999tn}. In our forthcoming work \cite{Wu:pionlessNN}, we show that the inclusion of the $C_4$ term at N$^2$LO is unnecessary if the shape correction is not enhanced. It comes in naturally at next-to-next-to-next-to-leading order (N$^3$LO), consistent with the natural size of the shape parameter. A direct N$^2$LO calculation without the $C_4$ term generates a residual regulator-dependent contribution that has the same energy dependence and the same size as the shape parameter correction. Therefore, we do not need to worry about it at N$^2$LO. Including $C_4$ at N$^3$LO is sufficient to remove such a regulator-dependent contribution.

With a finite regulator, such as a momentum cutoff, residual regulator dependence is generally unavoidable. Can it provide any useful information? It turns out that the residual regulator dependence together with naturalness (see Ref. \cite{vanKolck:2020plz} for a review) can provide a guide for the order subleading parameters come in if no further fine tuning exists, as has been emphasized in the context of Chiral EFT in Ref. \cite{Long:2012ve}. An example is the residual regulator-dependent contribution associated with the $C_4$ term discussed above.

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\textbf{General form of the three-body limit cycle}
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To renormalize the three- and more-body systems, a three-body force characterized by a single dimensionful parameter, $\Lambda_\star$, is required at LO \cite{Bedaque:1998kg, Bedaque:1998km}. In the unitarity limit, this gives rise to the Efimov effect \cite{Efimov:1970zz}, where the three-body spectrum exhibits an infinite geometric tower of bound states, known as Efimov trimers. This is a manifestation of discrete scale invariance (DSI). Correspondingly, a limit cycle emerges in the renormalization-group (RG) flow of the three-body interaction strength \cite{Bedaque:1998kg, Bedaque:1998km}. While the analytic form of the three-body renormalization relation has been established for a sharp cutoff regulator based on the Skorniakov--Ter-Martirosian (STM) equation~\cite{Skorniakov:1957kgi}, its universality for more general regulators and other formalism remains underexplored. Through a comprehensive analyses of both the STM equation and the Faddeev equation \cite{Faddeev:1960su}, we establish the universal functional form of three-body renormalization relation for general separable regulators, showing that the running of three-body LEC at the two-body unitarity limit follows a real M\"{o}bius transformation that involves only three parameters. For further details, see Ref. \cite{Chen:2025iqp}. Several interesting questions about the three-body limit cycle still remain.

\begin{itemize}
    \item When DSI is weakly broken by a large scattering length, how will the limit cycle change for general regulators? Numerical results with the scattering length $a_0$ treated non-perturbatively indicate that deviations from DSI diminish as either the cutoff $\Lambda$ or the scattering length increases, implying that the correction is suppressed by inverse powers of $a_0\Lambda$, consistent with the analysis in Ref. \cite{Braaten:2011sz} for a sharp cutoff regulator. A perturbative analysis along the direction of Ref. \cite{Ji:2011qg} may provide a quantitative estimate of its NLO effect.
    \item What if the regulator is local and non-separable? It is well-known that the three-body problem with short-range interactions is relevant to the singular $1/r^2$ potential \cite{Braaten:2004rn}. It is shown that, when the long-range $1/r^2$ potential is cut off at short distance by a square well, the corresponding limit cycle is multi-valued~\cite{Beane:2000wh, Bawin:2003dm}. Our preliminary results for the three-body problem with non-separable regulators are consistent with this, though the structure is more complex and warrants further study.
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\textbf{Four-boson system expanded around the unitarity limit}
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For four- and more-body systems, a four-body force is required for renormalization at NLO when the range correction is included \cite{Bazak:2018qnu}. The strength of this four-body force needs to be fixed by a four-body datum, thereby introducing a new four-body scale.  Together with the scattering length and effective range, this four-body scale modifies the universal LO correlations among the properties of few-body systems. The impact of these new scales on the correlations between the four-body excited- and ground-state properties has been investigated in Ref. \cite{Wu:2023mhg}.

The idea of expanding around the unitarity limit has been successfully applied to nuclear few-body systems \cite{Konig:2015aka, Konig:2016utl, Konig:2016iny, Konig:2019xxk} within the framework of Pionless EFT. When only scattering-length corrections are included, this expansion yields good description of the ground-state binding energies and radii of three- and four-nucleon systems. However, a systematic treatment of effective-range corrections and the four-body force at NLO in calculations of four-body radii and excited-state energies has remained underexplored. Studies of the four-body radii up to NLO within Short-Range (or Pionless in nuclear physics) EFT, incorporating both the effective range and the four-body force, have only recently been carried out in some talks \cite{Konig:2024INT, Wu:2025ECT} by solving the Faddeev--Yakubovsky (FY) equations \cite{Yakubovsky:1967} and in Ref. \cite{Mondal:2025wml} using the Stochastic Variational Method \cite{Suzuki:2002stochastic}. Besides, the calculations for the four-body systems in Refs. \cite{Konig:2016utl, Konig:2019xxk, Mondal:2025wml} are restricted to relatively small cutoffs, where deep trimers do not appear. As pointed out in Ref. \cite{Lin:2023zqw}, the cutoff dependence is more complicated at larger cutoffs, necessitating relatively large cutoffs to achieve converged results.

As cutoff increases, more trimers appear. These states need to be removed in four-body calculations; otherwise, the four-body states become unstable, and decay channels must be taken into account, significantly complicating the problem. The pseudo-potential method, which is often used to exclude spurious two-body bound states \cite{Lehman:1982zz, Nogga:2005hy, Song:2016ale}, can be extended to subtract these unwanted deep trimers. In our calculation, we implemented this subtraction technique in solving the FY equations.

Another issue that may arise when computing the four-body state at large cutoffs is the presence of ``exceptional cutoffs'', at which the LEC diverges and the divergence also propagates to observables. They were first identified in two-nucleon problems when higher order interactions are treated in perturbation theory within chiral EFT \cite{Gasparyan:2022isg}. Their existence has since also been found in the three-nucleon system \cite{Thim:2025vhe}. In our four-boson calculations \cite{Wu:fourbody}, we observe similar behaviors in the four-body LEC and observables when the four-body force is included at NLO together with the effective range correction. This phenomenon occurs in both the perturbative FY formalism \cite{Konig:2019xxk}, used to calculate four-body radii with a separable super-Gaussian regulator, and in the diagrammatic approach \cite{Brodsky:2006d, Lin:2023zqw}, employed to study the binding energy of the four-body excited state with a sharp momentum cutoff. It should be emphasized that this is a regulator-dependent issue: whether ``exceptional cutoffs'' appear, and where they occur, depends on the chosen representation of short-range physics, i.e., regulators. One way to address this issue is to employ, or design, regulators that avoid these exceptional cutoffs; examples of such efforts are provided in Refs.~\cite{Yang:2024yqv, PavonValderrama:2025zzk, PavonValderrama:2025azr}. Alternatively, one can adjust the strategy for fixing the LECs to remain consistent within EFT truncation errors \cite{Peng:2024aiz}.

Although one might regard it as a regulator-dependent artifact, the existence of ``exceptional cutoffs'' motivates further consideration of perturbative renormalizability of EFT \cite{Hammer:2019poc}. Several questions deserve further thought:
\begin{itemize}
    \item In the context of perturbative renormalizability, is it necessary to require a well-defined limit as the regulator (cutoff) increases?
    \item If ``exceptional cutoffs'' are present, do they reduce or impose additional constraints on the convergence radius of the calculation relative to the EFT breakdown scale as suggested in Ref. \cite{PavonValderrama:2025zzk}? Can we still obtain reliable results and limits by staying away from these cutoff values?
    \item Is renormalization group invariance violated if the presence of exceptional cutoffs depends on the specific choice of regulators?
    \item How to properly choose the representation of short-range physics to avoid the appearance of ``exceptional cutoffs''?
\end{itemize}

By staying away from these ``exceptional cutoffs'', our numerical results display good convergence for both the four-body ground state radius and the excited binding energy \cite{Wu:fourbody}. We confirm that a four-body counterterm becomes essential when range corrections are included perturbatively at NLO. Furthermore, by systematically including subleading corrections, we tested the convergence behavior of expansion around the unitarity limit and assessed its predictive power in the four-body sector. Our calculations demonstrate very good convergence of this expansion up to NLO. The LO predictions capture the dominant universal features near unitarity, while the NLO corrections---which incorporate the large scattering length, finite-range effects, as well as a four-body force required by renormalization---provides systematically controlled improvements. In particular, including these corrections significantly improves the description of both the ground state radius and excited-state energy. Although subleading, they are quantitatively important for providing accurate predictions in real physical systems.


\begin{acknowledgments}
 I gratefully acknowledge the hospitality of the ECT*, where part of this work was carried out. I also thank the organizers of the ECT* workshop ``Universality in strongly-interacting systems: from QCD to atoms'' for providing the opportunity to present and develop this work. I am indebted to Prof. U. van Kolck for his valuable suggestions on the content of this proceeding. I also wish to thank my collaborators---Dr. Jaber Balal Habashi, Dr. Mario S\'anchez S\'anchez, Langxuan Chen (PhD student), Dr. Pengfei Zhang, Dr. Xincheng Lin, and Dr. Sebastian K\"{o}nig---whose joint projects form the foundation of this work.
\end{acknowledgments}

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