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%\sectionauthor{Cluster effective field theory calculation of electromagnetic breakup reactions with the Lorentz integral transform method}[]{Ylenia}{Capitani}[]\label{sec:capitani}

The results presented in this contribution are the outcome of applying an Effective Field Theory (EFT) approach to the study of a three-body nuclear cluster system, where $\alpha$-particles are considered as low-energy degrees of freedom alongside nucleons.
%
In this exposition, I will emphasize some of the typical discussions that arise when dealing with effective potentials---power counting, the system's response to cutoff variations, and the inclusion of a three-body force---while also allowing room for comparison with experimental data.
%
The formalism presented below provides a valid starting point for generalization to cluster systems with more than three components, where interesting new aspects may emerge, since different characteristic energy scales and additional many-body forces could come into play.
As mentioned in the Introduction, this transition from few- to many-body cluster systems offers an interesting prospect.

Theoretical studies of electromagnetic breakup reactions such as
$\gamma + {}^{12}\mathrm{C} \to \alpha + \alpha + \alpha$ and
$\gamma + {}^{9}\mathrm{Be} \to \alpha + \alpha + n$
are important from the perspective of nuclear astrophysics, since the inverse processes provide pathways for Carbon nucleosynthesis.
In particular, the reaction involving ${}^{9}$Be,
i.e.~${}^{4}$He$(\alpha n,\gamma){}^{9}$Be,
plays a role when an environment rich in neutrons becomes available~\cite{OBERHUMMER2001269,SUMIYOSHI2002467}.

To date, we have carried out calculations of the ${}^{9}$Be
photodisintegration cross section~\cite{YC9Be} using a fully three-body \textsl{ab initio}
approach and restricting to the low-energy regime of astrophysical
relevance.
For this process, the cross section can be expressed in terms of a response function $R(\omega)$, whose explicit form is given by
\begin{equation}
	R(\omega) = \sum_{f}\hspace{-14pt}\int\,
	| \langle
	\psi_f|\hat{O}_{\lambda}(\omega)|\psi_0
	\rangle |^2
	\delta(E_f-E_0-\omega)\,,\label{eq:response}
\end{equation}
where $\omega$ is the energy of the real photon with polarizations $\lambda = \pm1$, and $E_0$ ($E_f$) is the energy of the initial ground (final) state.
To avoid calculating the continuum wave functions $\psi_f$, the Lorentz integral transform method~\cite{Efros_2007} is employed.
This technique involves two key steps: first, an integral transform of $R(\omega)$ is evaluated, where the kernel is chosen to be of Lorentzian form;
then, the response function is determined by an inversion procedure.
In this way, the initial problem is reduced to a purely bound-state-like problem, whose inputs are the transition operator $\hat{O}_\lambda$, chosen here as the dipole operator, and the ground-state wave function $\psi_0$.
The solution to the bound-state problem is obtained through a variational procedure, in which a Non-Symmetrized Hyperspherical Harmonics (NSHH)~\cite{PhysRevC.83.024001,Deflorian2013} basis constructed in momentum space is employed.

The ${}^{9}$Be nuclear system exhibits interesting features that provide a suitable ground for developing an EFT~\cite{RevModPhys.92.025004}. The shallow three-body $\alpha\alpha n$ binding of $-1.573$ MeV sets an energy scale that is very well separated from that of the $\alpha$-particle (its proton separation energy is around $20$ MeV).
Therefore, it is legitimate to study ${}^{9}$Be as an effective three-body cluster system.
Furthermore, this nucleus has a Borromean structure: the three-body system is bound, although the two-body substructures---$\alpha n$ and $\alpha\alpha$---are not.

The two-body potentials that we use are derived from a Halo/Cluster EFT~\cite{Hammer_2017,PhysRevC.90.044004_Ji}. They are naturally born in momentum space and depend on the initial (final) relative momentum $p$ ($p'$).
Each $\ell$-component of the partial-wave expansion is represented as a series of contact interaction terms
\begin{align}
	&V_\ell(p,p') =
	p^{\ell} p'^{\ell}
	g(p;\Lambda^{(\ell)}) g(p';\Lambda^{(\ell)})
	\sum_{i,j=0}^{1}
	p^{2i} \lambda^{(\ell)}_{ij} p'^{2j}
	\,,
	&&\lambda^{(\ell)}=
	\begin{pmatrix}
		\lambda^{(\ell)}_0&\lambda^{(\ell)}_1\\
		\lambda^{(\ell)}_1&0
	\end{pmatrix}\,,\label{eq:partial}
\end{align}
parametrized by specific Low Energy Constants (LECs), $\lambda^{(\ell)}_{0},\lambda^{(\ell)}_{1}$.
A cutoff parameter $\Lambda^{(\ell)}$ is also introduced through the Gaussian function $g(p;\Lambda^{(\ell)})=e^{-(p/\Lambda^{(\ell)})^{2m}}$ ($m$ positive integer), which regularizes the effective potential~(\ref{eq:partial}).
Values of $\Lambda^{(\ell)}$ are kept finite, and restricted to those allowed by the Wigner bound~\cite{Wigner_PhysRev.98.145}.

The LECs are determined by relating them to scattering observables.
For the $\alpha n$ system, the calculation procedure can be briefly sketched as follows.
First, the Lippmann-Schwinger equation is solved by using the effective potential~(\ref{eq:partial}), together with an ansatz for the $T$-matrix that follows a similar partial-wave expansion.
Next, the resulting on-shell expression for the $T$-matrix is expanded for low momenta and compared term by term to the familiar Effective Range Expansion (ERE),
$k^{2\ell} / T_{\ell,\alpha n}(E)
\propto \left(%
1/a_{\ell} - r_{\ell} k^{2}/2 + i k^{2 \ell+1}
\right)$.
Finally, for a fixed cutoff $\Lambda^{(\ell)}$, the LECs are calculated
using the experimental values of the scattering length (or volume), $a_\ell$, and the effective range, $r_{\ell}$.
The procedure is quite similar for the case of the $\alpha\alpha$ system when considering the Coulomb-distorted strong term of the
$T$-matrix and the Coulomb-modified ERE.

In the low-energy regime of interest, the $\alpha n$ and $\alpha \alpha$ interactions to be taken into account are the $\ell_j=P_{3/2}$ and $S_0$, respectively. Both waves exhibit a resonant behaviour at low energy and a large scattering length
($a^{\textrm{exp}}_1=-62.951$~fm$^3$,
$r^{\textrm{exp}}_1=-0.8819$~fm$^{-1}$~\cite{ARNDT1973429},
$a^{\textrm{exp}}_0=-1920$~fm,
$r^{\textrm{exp}}_1=1.099$~fm~\cite{HIGA2008171,AFZRevModPhys.41.247}).
%
To capture the shallow $\alpha n$ $P$-wave resonance, the naive power counting for the scattering parameters, formulated solely in terms of the proper power of the high-momentum scale $M_{hi}$, should be modified. Here the one proposed by Bedaque, \textsl{et al.}~\cite{BEDAQUE2003159} is adopted, which reads $a_1 \sim 1/(M_{lo}^2 M_{hi})$, $r_1 \sim M_{hi}$, with $M_{lo}$ the scale set by the existing resonance.
It follows that, at low energy, both $a_1$ and $r_1$ terms give a Leading Order (LO) contribution.
The $S$-wave shallow resonance existing at low energy in the $\alpha\alpha$ system is supported by the interplay between the short-range strong potential and the long-range Coulomb interaction. In this case, we use the power counting by Higa, \textsl{et al.}~\cite{HIGA2008171},
$a_0\sim M_{hi}^2 / M_{lo}^3$, $r_0\sim 1/(3k_C) \sim 1/ M_{hi}$,
where $k_C \propto \alpha_{\textrm{em}}m_\alpha $, for which $a_0$ and $r_0$ must be retained in LO calculations.
This justifies truncating the effective potential as in Eq.~(\ref{eq:partial}).

Given the two-body effective potentials mentioned above and the Coulomb interaction as input,
it is interesting to discuss how the calculated ${}^9$Be ground-state energy $E_0$ depends on the two-body cutoffs $\Lambda^P_{\alpha n}$ and
$\Lambda^S_{\alpha \alpha}$ (see Fig.~\ref{fig:gs}).
\begin{figure}[tbp!]
	\centering
	\includegraphics[width=\textwidth]{capitani_y_fig1}
	\caption{\label{fig:gs}%
		 Ground-state energy of ${}^9$Be ($\alpha \alpha n$) as a function of the two-body cutoff $\Lambda_{\alpha n}^P$ at fixed $\Lambda_{\alpha \alpha}^S$ (left panel) and as a function of $\Lambda_{\alpha \alpha}^S$ at fixed $\Lambda_{\alpha n}^P$ (right panel). The horizontal dotted line indicates the experimental value of -1.573 MeV~\cite{TILLEY2004155}.}
\end{figure}
A variation of $\Lambda^S_{\alpha \alpha}$ results in a narrower range ($\approx 0.7$ MeV) for $E_0$ with respect to the range ($\approx 1.8$ MeV) obtained by varying $\Lambda^P_{\alpha n}$.
A possible explanation of the stronger $\Lambda^P_{\alpha n}$-dependence could be given in terms of a residual discrete scale invariance when two resonant pairwise $P$-wave interactions are present in a three-body system~\cite{EFIMOV,kievsky,Pwaveefimov}.
The findings of Refs.~\cite{PhysRevC.90.044004_Ji,Li:2023hwe}, in which the $^{6}$He nucleus is studied within Cluster EFT as an $\alpha nn$ system with resonant pairwise $\alpha n$ $P$-wave interactions, may support this interpretation.
The weaker dependence of $E_0$ on $\Lambda^S_{\alpha \alpha}$ may be seen as a consequence of the presence of the repulsive Coulomb interaction.

Due to the cutoff-dependent behaviour of the calculated $E_0$, we introduce in the potential model a state-dependent three-body force,
$V_3(Q,Q')
=\lambda_3 f(Q;\Lambda_3) f(Q';\Lambda_3)$,
with regulator $f(Q;\Lambda_3)=e^{-(Q/\Lambda_3)^2}$, where $Q^2$ is a mass-weighted combination of the squared relative momenta of the three different particle pairs.
In such a situation, for fixed values of the two-body cutoffs
$\Lambda^P_{\alpha n},\Lambda^S_{\alpha \alpha}$ and
for each choice of the three-body cutoff $\Lambda_3$,
the new LEC $\lambda_3$ is tuned to reproduce a three-body observable, here $E^{\textrm{exp}}_0=-1.573$ MeV~\cite{TILLEY2004155}.

In principle, two other $\alpha n$ partial waves should be explored at low energy: the $S_{1/2}$ and the broadly resonant $P_{1/2}$.
The latter gives a higher-order effect, as also dictated by the adopted power counting.
The $\alpha n$ $S_{1/2}$ interaction, whose $a_0$ term is formally of LO, makes a small contribution of the order of keV to the calculated ${}^9$Be ground-state energy.
On the other hand, it produces a LO effect in the positive low-energy regime of the cross section calculations.
Consequently, in addition to the aforementioned $\alpha n$ $P_{3/2}$ and $\alpha \alpha$ $S_{0}$
interactions, the $\alpha n$ $S_{1/2}$ potential is also retained.
%
The inclusion of the $\alpha n$ $S_{1/2}$ effective interaction gives rise to deep two-body bound states, as well as unphysical three-body bound states.
As within the adopted integral transform approach it is preferable to avoid the presence of these spurious states, a projection potential term~\cite{KUKULIN1978330} constructed with the $S_{1/2}$ deep bound-state wave function is used: $V_{\Gamma}(p,p') = \Gamma \phi^{*}(p)\phi(p')$.
This does not introduce any new parameters to the model. In principle, one should take $\Gamma\to\infty$, but a sufficiently large value of $\Gamma$ can be used, ensuring $\Gamma$-independent results both in bound-state and cross section calculations.

In Fig.~\ref{fig:cs}
\begin{figure}[tbp!]
	\centering
	\includegraphics[width=\textwidth]{capitani_y_fig2}
	\caption{\label{fig:cs}%
		Calculated photodisintegration cross section of ${}^9$Be as a function of the photon energy $\omega$, compared with different experimental data sets from Refs.~\cite{Utsunomiya15_PhysRevC.92.064323,Arnold_PhysRevC.85.044605,Burda_PhysRevC.82.015808,Utsunomiya00_PhysRevC.63.018801}.
		The solid line represents the total cross section, which is the sum of three distinct contributions: $1/2^+$ (dashed), $3/2^+$ (dotted) and $5/2^+$ (dash-dotted) resonances.
		The inset provides a detailed view of the near-threshold $1/2^+$ resonance.
		The error bands indicate the considered uncertainties (see the text for more details).}
\end{figure}
we present the final results for the ${}^9$Be photodisintegration cross section, showing overall consistency with the experimental data available from the literature~\cite{Utsunomiya15_PhysRevC.92.064323,Arnold_PhysRevC.85.044605,Utsunomiya00_PhysRevC.63.018801,Burda_PhysRevC.82.015808}.
The different contributions analysed are those related to the electric dipole transitions proceeding from the ${}^{9}$Be ground state, with total angular momentum and parity $J^\pi_0=3/2^+$, to the final states $J^\pi_f=1/2^+,3/2^+,5/2^+$.
While the $1/2^+$ and $5/2^+$ resonances appear as two distinct peaks at energies $\approx 1.7$ MeV and $\approx 3$ MeV, the $3/2^+$ resonance located at around $5$ MeV is much broader.
Although each of the three-body LECs $\lambda_3^{1/2^+}$, $\lambda_3^{3/2^+}$ and $\lambda_3^{5/2^+}$ has been tuned to obtain the correct energy position of the resonance, the heights and widths of the different contributions remain theoretical predictions.
The inset of Fig.~\ref{fig:cs} illustrates in detail the calculated $1/2^+$ contribution, which has a smaller width than the distribution of the experimental points.
We have used a value of $400$ MeV for $\Lambda_3$, which is consistent with the correlation relation between $\Lambda_3$ and the chosen two-body
cutoffs. In the $1/2^+$ case, for example, the differences induced by a cutoff variation from 300 MeV to 400 MeV are at most of the order of 10$\%$ and 20$\%$ in the peak height and width, respectively.
Two kinds of errors have been considered in this study: the uncertainty arising from the inversion procedure used to derive $R(\omega)$, and the error resulting from the HH model space convergence. The former is very small and barely noticeable at the plot scale, whereas the latter is particularly visible in the $5/2^+$ case, giving rise to a clear band.
The absence of higher-order terms in the effective potential of Eq.~(\ref{eq:partial}) could explain the discrepancy with the experiment in the energy region above 3 MeV.
In this regard, the inclusion of the $\alpha n$ $P_{3/2}$ shape parameter term could be significant, and this is the subject of ongoing research.

We have also obtained interesting results regarding the contribution to the cross section from nuclear current terms beyond the one-body term.
Due to the Siegert theorem~\cite{Siegert1937}, which ensures Gauge invariance, this contribution is already included in the presented results.
When the one-body convection current is employed in place of the dipole operator, the resulting cross section shows a slightly more pronounced $1/2^+$ peak. In contrast, the characteristic bump at around 3 MeV due to the $5/2^+$ resonance is completely absent, signalling a considerable effect of the many-body currents.
This is most likely due to the $\alpha n$ $P_{3/2}$ interaction, which has a strong momentum dependence, and enters at the lowest order of the EFT. Further details on this study can be found in Ref.~\cite{YC9Be}.

In conclusion, a number of extensions of the present formalism could be addressed, where the Halo/Cluster EFT potentials introduced above can be used to study other nuclear systems and reactions.
So far, the low-energy cross section relative to the ${}^{12}$C photodisintegration reaction, going from the $J^\pi_n=2^+_1$ bound state to the $0^+_2$ Hoyle state in the continuum, has been analysed using the quadrupole operator.
The preliminary results, including studies of the ${}^{12}$C bound states, are promising~\cite{Capitani}. Further research can therefore be conducted on these lines, especially focussing on deriving the $D$-wave term of the effective $\alpha\alpha$ potential.
When considering an extension of this formalism to study nuclear cluster systems with more than three components,
one could also examine ${}^{16}$O ($\alpha\alpha\alpha\alpha$) and, possibly, the relevant astrophysical reactions  involving this nucleus.
The inclusion of a $nn$ effective interaction would make other nuclear systems and reactions accessible for studies at low energies.
One example is provided in Ref.~\cite{Pinilla} with ${}^6$He ($\alpha n n$). Investigating ${}^{10}$Be as an $\alpha\alpha n n$ system would also be of interest.

\begin{acknowledgements}
Y.C.~acknowledges the financial support of
the European Union-Next Generation EU, Mission 4 Component 1,
CUP F53D23001360006, for the PRIN 2022 project
``Exploiting separation of scales in nuclear structure and dynamics''.
This research was conducted in collaboration with E.~Filandri, C.~Ji, W.~Leidemann and G.~Orlandini.
\end{acknowledgements}

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