# Finite-size effects in heavy halo nuclei from effective field theory

## Abstract

Halo/Cluster Effective Field Theory describes halo/cluster nuclei in an expansion in the small ratio of the size of the core(s) to the size of the system. Even in the point-particle limit, neutron-halo nuclei have a finite charge radius, because their center of mass does not coincide with their center of charge. This point-particle contribution decreases as \(1/A_\mathrm {c}\), where \(A_\mathrm {c}\) is the mass number of the core, and diminishes in importance compared to other effects, e.g., the size of the core to which the neutrons are bound. Here we propose that for heavy cores the EFT expansion should account for the small factors of \(1/A_\mathrm {c}\). As a specific example, we discuss the implications of this organizational scheme for the inclusion of finite-size effects in expressions for the charge radii of halo nuclei. We show in particular that a short-range operator could be the dominant effect in the charge radius of one-neutron halos bound by a P-wave interaction. The point-particle contribution remains the leading piece of the charge radius for one-proton halos, and so Halo EFT has more predictive power in that case.

## 1 Introduction

Theoretical models of many-body systems usually treat the constituent particles as having no internal structure. This point-particle approximation is also used in cluster models, e.g., for the description of halo systems [1, 2, 3, 4], even though one might encounter situations for which the core is rather large. Finite-size effects are included a posteriori, but can become significant for certain observables. As an example, the total charge radius is usually calculated by simply adding in quadrature [5] the charge radii of the constituents and the calculated point-particle radius; see e.g. the calculation of charge radii for neutron-rich helium isotopes in the Gamow Shell Model [6]. Instead, it would be useful to construct a framework in which finite-size effects can be included systematically.

Constituent-size effects can be accounted for in effective field theories (EFTs), where they appear through derivative interactions. For example, the nucleon charge radius (and, more generally, nucleon form factors) can be calculated [7] in Chiral Perturbation Theory (ChPT) in an expansion in powers of \(k_{\pi }/M_{\mathrm{QCD}}\), where \(k_{\pi }\sim 150~\mathrm {MeV}\) is a momentum scale associated with the lightest carrier of the nuclear force, the pion, and \(M_{\mathrm{QCD}}\sim 1\) GeV is the characteristic mass scale of QCD. The relevant pion parameters are its mass and decay constant. Chiral EFT [8] is a generalization of ChPT to a typical nucleus, for which the binding energy per nucleon is \(B/A\sim k_\pi ^2/M_{\mathrm{QCD}}\) and the radius \(R\sim A^{1/3}/k_\pi \). The nuclear charge radius includes the sum of the nucleons’ radii plus many-body effects generated by internucleon interactions and currents [9]. We would like to have a similar framework for clusterized systems.

Clusterized systems, with much smaller energies and larger radii, are additionally characterized by scales beyond the pion scales. These nuclei can be viewed as a collection of valence nucleons orbiting around either no core (few-nucleon systems), one core (halo nuclei) or many cores (cluster nuclei). The cores themselves frequently—but not always—have properties of typical nuclei. The generic existence of such systems can be understood as a consequence of a fine-tuning in QCD, which introduces a lighter momentum scale \(\aleph \sim 30\) MeV [10, 11]. For such loosely bound systems we can devise EFTs that exploit the separation of scales without involving pions explicitly. In these EFTs one considers processes with typical momenta \(k_\mathrm {lo}\), such that \(k_\mathrm {lo}\ll k_\mathrm {hi}\), where \(k_\mathrm {hi}\lesssim k_\pi \) is a high-momentum scale. One then develops an expansion for observables in powers of \(k_\mathrm {lo}/k_\mathrm {hi}\). The very lightest nuclei are dilute systems with no core, where the dominant (two- and three-nucleon) interactions are S-wave. The corresponding EFT is Pionless EFT, for which power counting is relatively well understood [8].

Halo/Cluster EFT, here labeled HEFT, was proposed as an EFT for systems with one [12, 13] or more [14] cores and valence nucleons [15, 16]. (See Ref. [17] for a recent review.) HEFT power counting is a generalization of the power counting for Pionless EFT allowing for dominant interactions in waves with non-vanishing angular momentum and for a breakdown scale \(k_\mathrm {hi}\) estimated from the first excitation of the core and/or its size. In the first cases considered, \(^{5,6}\)He [12, 13, 18, 19, 20], there is an alpha-particle core, and the neutron–alpha (*n*–\(\alpha \)) interaction is mostly of P-wave nature, generating a near-threshold \(^{5}\)He resonance. The \(\alpha \)–\(\alpha \) interaction, in turn, is obtained [14] from \(\alpha \)–\(\alpha \) scattering and the lowest \(^8\)Be state. HEFT has since been extended to heavier cores and to proton halo systems [17]. In most of these cases the high scale \(k_\mathrm {hi}\) in HEFT is associated with the size of the core, i.e. \(k_\mathrm {hi} \sim 1/R_\mathrm {c}\). This means HEFT is an expansion in powers of \(R_\mathrm {c}/R_\mathrm {halo}\), where \(R_\mathrm {halo}\) is the unnaturally large size of the halo system.

The different contributions to the charge radius of a halo nucleus can then be organized in the HEFT expansion and the size of the effect due to the finite size of the core (and of the nucleon) estimated. We do that and thereby derive—for both S- and P-wave one-neutron halos—the charge-radius formula that is frequently used in nuclear theory. However, we also point out that there is, in principle, another expansion parameter present when HEFT is applied to systems with a relatively large number, \(A_\mathrm {c}\), of core nucleons. To leading order in \(1/A_\mathrm {c}\) the core is static, its recoil being small compared to nucleon recoil. Consequently the center of mass of a neutron halo coincides with its center of charge. Thus, whereas for light halos (e.g. \({}^6\)He) the difference between these two generates an important contribution to the charge radius [21], for heavier systems the corresponding effect goes to zero. Correctly assessing the impact of the finite size of the constituents on the charge radius requires keeping track of factors of \(1/A_\mathrm {c}\).

Moreover, the charge radii of halo nuclei are affected by a short-range operator, which is subleading in \(R_{\mathrm {c}}/R_\mathrm {halo}\) but leading in \(1/A_\mathrm {c}\). We show that for one-neutron halos bound by a P-wave interaction (e.g. the excited state of \({}^{11}\)Be) this effect may be as important as the long-distance contributions to the halo’s charge radius that have been previously computed in HEFT [22]. Similar considerations also apply to the form factors of two-neutron halos such as those discussed in Ref. [23]. They are not, however, as pressing for proton halos, where a finite charge radius will be generated by the photon coupling to the valence proton(s) even if the core is infinitely heavy.

This exemplifies the importance of keeping track—to the extent possible—of factors of \(1/A_c\) in observables, rather than just counting powers of \(k_\mathrm {lo}\). This is quite similar to the need to distinguish between relativistic corrections that are suppressed by powers of the inverse nucleon mass, \(1/m_\mathrm {N}\), and other corrections that only carry powers of \(1/k_\mathrm {hi}\) [24, 25, 26]. The significant difference between these two scales produces a hierarchy between effects that scale with the same power of \(k_\mathrm {lo}\). Only once that hierarchy has been identified can the power counting be formulated in an efficient manner.

We isolate the \(A_\mathrm {c}\) dependence that enters the charge radius through kinematic effects, i.e., because the nucleon–core mass ratio, \(m_\mathrm {N}/m_\mathrm {c}\), is small. In contrast, we assume that all Lagrangian coefficients (LECs) scale with a power of \(k_\mathrm {hi}\) that is solely determined by the naive engineering dimension of the operator they multiply, i.e., we use naive dimensional analysis with respect to \(k_\mathrm {hi}\) and do not attach any additional \(A_\mathrm {c}\) dependence to the LECs. It is true that \(k_\mathrm {hi}\) is also \(A_\mathrm {c}\) dependent, because \(k_\mathrm {hi}\) will generically be of order the inverse core size, \(1/R_\mathrm {c}\), and \(R_\mathrm {c}\) can be taken to be \(\propto A_\mathrm {c}^{1/3}\). But any additional accounting of the \(A_\mathrm {c}\) dependence of short-distance physics in HEFT beyond this would require a more microscopic understanding of the \(A_\mathrm {c}\) dependence of all the cofficients in the EFT. This could be achieved by matching HEFT to a microscopic calculation, but it could be argued that such matching goes beyond the EFT philosophy of writing down a theory that is independent of the short-distance physics. In contrast, the kinematic effects we identify here are universal, in the sense that they occur irrespective of the nature of the forces between the halo nucleons and the core(s).

Here we focus on the charge radius of single-neutron halo nuclei, where the point contribution is suppressed by \(1/A_\mathrm {c}\) for the reason described above. But the presence of the heavy-core propagator is ubiquitous, so similar effects will affect other observables as well. For example, one expects the Born–Oppenheimer approximation to emerge in systems with multiple heavy cores and/or valence nucleons.

Our paper is structured as follows: In Sect. 2 we discuss the interplay between the various scales that are involved in an EFT for a halo system with a heavy core. We also discuss the low-energy scattering parameters for the nucleon–core system and introduce the charge radius in terms of the momentum expansion of the low-energy charge form factor. In Sect. 3 we derive the observable charge radius for S- and P-wave one-neutron-halo states. The power counting is exemplified by considering the charge radius for selected halo states. We summarize our findings in Sect. 4. An appendix discusses the corresponding results for proton halos, where considering factors of \(1/A_\mathrm {c}\) does not lead to any change in the hierarchy of the various physical mechanisms that contribute to the charge radius.

## 2 Power counting

Once the relevant degrees of freedom are chosen, a model consists of a specific set of interactions among them. In contrast, with an EFT one considers the most general dynamics consistent with the known symmetries. It is crucial to organize the corresponding infinity of contributions to any observable according to their size (“power counting”).

We are interested in a clusterized system where the size \(R_\mathrm {halo}\sim 1/k_{\mathrm{lo}}\) of the system is sufficiently large that the constituents can be taken as point-like in a first approximation. This system might be probed with particles (photons, electrons, neutrinos, nucleons) of wavelength \(\sim 1/k_{\mathrm{lo}}\) that cannot resolve the inner structure of the constituents. For simplicity we consider a few valence nucleons orbiting around a single core of radius \(R_\mathrm {c}\) consisting of \(A_\mathrm {c}\gg 1\) nucleons. The arguments below can be generalized straightforwardly to multiple-core systems.

In addition to these kinematic factors of \(1/A_\mathrm {c}\), there may be dependence on \(A_\mathrm {c}\) coming through the interaction coefficients, or “low-energy constants” (LECs). As a trivial example, electromagnetic interactions add up constructively for protons and the corresponding LECs in general depend on the core charge \(Z_\mathrm {c}=A_\mathrm {c} - N_\mathrm {c}\). It is not clear how to deal with this quantity a priori. In neutron halos, \(Z_\mathrm {c}\) can be significantly smaller than \(A_\mathrm {c}/2\), but this is not necessarily so for proton halos. We will keep factors of \(Z_\mathrm {c}\) explicit and deal with them on a case-by-case basis.

Likewise, the LECs for strong interactions might in specific cases represent some constructive or destructive interference in the interactions of the valence nucleon with the core nucleons. One way to determine the \(A_\mathrm {c}\) dependence of these LECs is by matching HEFT to the ab initio solution of the same system with a more fundamental EFT [27, 28, 29], in a region where both EFTs are valid. Another way is to look at systematic trends in LECs fitted to data for different cores. In either case a manifestation of strong \(A_\mathrm {c}\) dependence would be a particularly large or small LEC value with respect to the expected power of \(k_\mathrm {hi}\). Since there is no clear case at this point, below we limit ourselves to the kinematical factors arising from the core mass, although the counting of factors of \(1/A_{\mathrm{c}}\) could be improved later if needed.

### 2.1 Nucleon–core scattering

EFTs incorporate from the start the coupling to the continuum, so that most calculations of halo structure, including form factors, are intimately connected with nucleon–core scattering. A discussion of nucleon–core interactions is therefore necessary for the calculation of form factors, and we briefly review previous work on the subject here.

*T*matrix has a pole at \(k=i\gamma _0\) with

^{1}. The mildest assumption [13] is that the effective range, just as for S-waves, is not fine tuned and directly reflects the breakdown scale,

*S*-matrix poles of non-zero energy require a single fine tuning,

Just as for S-waves, the assumption that no further powers of \(1/A_\mathrm {c}\) appear in the ERE parameters implies that the only change in power counting when treating \(1/A_\mathrm {c}\) as small is the extra expansion (3).

### 2.2 Charge form factor

*f*is defined in Eq. (4). Indeed, the loop for neutron-halo systems was calculated by Hammer and Phillips [22] with the leading-order (LO) results:

In summary, these power-counting arguments make explicit that the point-like contribution for one-nucleon halos involves a kinematical suppression factor \(1/A_\mathrm {c}^2\) for neutron halos. But this has the consequence that, for P-wave one-neutron halos, a short-range operator enters at the same order as the finite-size core contribution. The existence of such additional short-range operators will have negative influence on the predictive power of LO calculations.

## 3 The charge-radius formula

*c*with S- and P-wave interactions is given by

*f*was defined in Eq. (4). The field \(A_0\) is the zeroth component of the photon four-vector field. Here \(\tau _3\) is the third isospin Pauli matrix, and we have defined the charge number of the core \(Z_\mathrm {c}\). Note that \(Z_\mathrm {h}\) is the charge (\(Z_\mathrm {c}\) or \(Z_\mathrm {c}+1\)) of the nucleon–core system. The charge radius of the nucleon (core) field is \(\rho _\mathrm {N}\) (\(\rho _\mathrm {c}\)), while \(\rho _\sigma \) and \(\rho _\pi \) are additional short-range parameters with sizes given by Eq. (19), which will be discussed below.

This Lagrangian includes all operators that will contribute to the charge radius of the halo system up to NLO. Higher-order terms, such as the one that leads to the shape parameter in the ERE and terms that do not contribute to the charge form factor, are denoted by the ellipsis. The coefficients of the \(\nabla ^2A_0\) terms in Eq. (21) encode the finite size and composite nature of the core. We assign the same scaling to them as Hammer and Phillips, who used naive dimensional analysis to argue that \(\rho _\mathrm{c}^2, \rho _\sigma ^2, \rho _\pi ^2\) are all \(\sim R_\mathrm {c}^2\) ^{2}. However, whereas Ref. [22] argued that this scaling rendered these effects of higher order, here we contend that they can provide the main contribution to the charge radius for heavy-core systems.

### 3.1 S-wave neutron halos

Here we compute the expectation value of the zeroth component of the electromagnetic current, \(\langle J^0\rangle \), which appears in Eq. (13), for an S-wave one-neutron halo. The long-distance contributions to this quantity are well known, cf. Refs. [22, 25, 30], where the diagrams in Fig. 1a, b were evaluated (although only for \(A_{\mathrm {c}}=1\) in Refs. [25, 30]). Here we include diagrams Fig. 1d–f as well, and so account for finite-size effects and the leading short-range, two-body operator.

Orders of the various contributions to the charge radius of S-wave neutron halos listed in Eq. (30). In each column effects of a particular order in the usual HEFT expansion parameter \(k_\mathrm {lo}/k_\mathrm {hi}\) appear. Meanwhile the rows organize contributions due to additional small factors: inverse powers of the number of core nucleons (\(A_\mathrm {c}\)) and protons (\(Z_\mathrm {c}\))

\(\mathcal{O}(k_\mathrm {lo}^{-2})\) | \(\mathcal{O}((k_\mathrm {lo}k_\mathrm {hi})^{-1})\) | \(\mathcal{O}(k_\mathrm {hi}^{-2})\) | \(\mathcal{O}(k_\mathrm {lo}k_\mathrm {hi}^{-3})\) | |
---|---|---|---|---|

\(\mathcal{O}(1)\) | — | — | \(\rho _\mathrm {c}^2\) | \(\gamma _0r_0(\rho _\mathrm {c}^2-\rho _\sigma ^2)\) |

\(\mathcal{O}(A_\mathrm {c}^{-3/2}Z_\mathrm {c}^{-1})\) | — | — | \(\rho _\mathrm {n}^2/Z_\mathrm {c}\) | \(\gamma _0r_0\, \rho _\mathrm {n}^2/Z_\mathrm {c}\) |

\(\mathcal{O}(A_\mathrm {c}^{-2})\) | \(r_\mathrm {pt,LO}^2\) | \(\gamma _0r_0\, r_\mathrm {pt,LO}^2\) | \(\cdots \) | \(\cdots \) |

- 1.
A change in the size of the core when it is placed in the bound state with the neutron.

- 2.
Pieces of the ab initio wave function not in the core + neutron piece of the Hilbert space, e.g., those due to excited states of the core.

As a concrete example we consider the S-wave ground state of the one-neutron halo \({}^{11}\mathrm{Be}\), whose form factor and photodisintegration were investigated in Ref. [22]. The neutron separation energy is \(B_{\mathrm {s}0}=0.502~\mathrm {MeV}\) [33], corresponding to \(k_\mathrm {lo}\sim \gamma _0 \simeq 30~\mathrm {MeV}\) through Eq. (9). Using the charge radius of the \({}^{10}\mathrm{Be}\) core, \(\rho _\mathrm {c}=2.357(18)~\mathrm {fm}\) [34], as an estimate for its size, the breakdown scale is \(k_\mathrm {hi}\sim 1/R_\mathrm {c}\simeq 80~\mathrm {MeV}\). This is also the momentum \(\sqrt{2m_\mathrm {R}E_\mathrm {ex}}\simeq 80~\mathrm {MeV}\sim k_\mathrm {hi}\) corresponding to the first excitation of the core at \(E_\mathrm {ex}=3.368~\mathrm {MeV}\) [35], so there is no need to include a field for this state. These scales then give us the expansion parameter \(k_\mathrm {lo}/k_\mathrm {hi}\approx 0.4\). This means that \(r_\mathrm {pt,LO}^2\) is numerically of the same size as \({\mathcal {O}}((k_{\mathrm{lo}}/k_{\mathrm{hi}})^3)\) corrections. Since \(Z_\mathrm {c}\sim A_\mathrm {c} k_\mathrm {lo}/k_\mathrm {hi}\), the neutron-radius contributions are suppressed by more than five powers of \(k_\mathrm {lo}/k_\mathrm {hi}\) compared to \(\rho _\mathrm {c}\). At LO there is a charge-radius prediction, but it is trivial since it is just the charge radius of the \({}^{10}\mathrm{Be}\) core, \(\rho _\mathrm {c}^2\simeq 5.56(4)~\mathrm {fm}^2\). This does, though, explain most of the measured value of \(r_\mathrm {{}^{11}Be}^2 \simeq 6.07(8)~\mathrm {fm}^2\) (\(r_\mathrm {{}^{11}Be}=2.463(16)\) fm [34]). Estimating \(\gamma _0r_0\) from the EFT expansion parameter \(\sim 0.4\), we find that the point-charge contribution to \(r_\mathrm {ch}^2\), i.e., the first term of Eq. (29), is \(r_\mathrm {pt,LO}^2 / \left( 1-\gamma _0r_0 \right) \simeq 0.3~\mathrm {fm}^2\). This explains more than half of the difference \(r_\mathrm {{}^{11}Be}^2 - r_\mathrm {{}^{10}Be}^2\). The rest must come from the short-distance effect \(\rho _\sigma ^2\): the experimental value for \(r_\mathrm {{}^{11}Be}\) can be reproduced if the short-range parameter is given by \(\rho _\sigma ^2\approx 5.1~\mathrm {fm}^2\), which is of the expected order of magnitude, \(1/k_\mathrm {hi}\sim 2.5~\mathrm {fm}\). This supports the power counting presented here. We thus see that \(\rho _\sigma ^2\) must be a little smaller than \(\rho _\mathrm {c}^2\) in order to explain the data, although the errors on the atomic measurements of the \({}^{10}\mathrm{Be}\) and \({}^{11}\mathrm{Be}\) radii make it difficult to extract a precise value for \(\rho _\mathrm {c}^2 - \rho _\sigma ^2\).

### 3.2 P-wave neutron halos

An important aspect of the S-wave halo system is that all the charge form-factor diagrams we considered are finite. For P waves this is not the case. The increased singularity of the P-wave interaction can be seen already in the need for the effective-range term (10) at LO to allow proper renormalization of nucleon–core scattering. As before we will consider operators up to second order in the photon momentum \(\varvec{Q}\) and will show that, if the charge-radius contributions of the constituents are to be considered explicitly, we need an additional short-range operator to renormalize the halo charge radius.

In this renormalization scheme the effect beyond the “standard” charge-radius formula depends on the extent to which the dicluster counterterm differs from the core radius. In contrast to the S-wave case, here the difference \(\rho _\pi ^2 - \rho _{\mathrm {c}}^2 -\rho _{\mathrm {n}}^2 / Z_{\mathrm {c}}\) must go to zero as the regulator is taken to infinity, in order to yield a finite \({\bar{\rho }}_\pi ^2\). It is important to consider what would happen if we were to include the finite-size contributions, but not the \(\rho _\pi ^2\) short-range operator. In Eq. (37) we see that the constituent charge radii enter with a prefactor that corresponds to a divergent integral. Since the parameters \(\rho _\mathrm {c}^2\) and \(\rho _\mathrm {n}^2\) are observables—these are the charge radii of the core and the neutron—they cannot absorb this divergence. The only parameter available for this purpose is the \(\rho _\pi ^2\). As such it is not possible to add the finite-size contributions without also including the short-range operator. Formally, the scaling of \(\rho _\pi ^2\) is \(\rho _\pi ^2 \sim {\bar{\rho }}_\pi ^2 \sim R_\mathrm {c}^2\) for renormalization scales such that \(L_1 \sim k_\mathrm {hi}\). However, the crucial difference between \(\rho _\pi ^2\) and \({\bar{\rho }}_\pi ^2\) is that the latter is an observable, while the former absorbs a divergence and so is scheme- and regulator-dependent.

## 4 Conclusion

HEFT offers a systematic approach to make model-independent predictions of low-energy observables. In this paper we have discussed a new power-counting scheme for systems with a heavy-core nucleus, and we have derived the finite-size contributions to charge radii of one-nucleon halos. HEFT in general is restricted by appearances of short-range operators at rather low orders. With the heavy-core power counting, these restrictions are even enhanced for some systems and observables. For one-neutron halos where the core is much heavier than the neutron, the point-particle result for the charge radius is demoted from leading to subleading order since the core recoil due to the photon interaction is very small. In contrast, in the case of an S-wave system, the LO charge radius is given by the finite-size contributions of the constituents. For a P-wave one-neutron halo the heavy-core version of HEFT is non-predictive at LO, since the LO charge radius includes an undetermined short-range operator ^{3}.

Note, however, that not all systems are made less predictive in the heavy-core power counting. For proton halos there are no issues for the charge-radius results due to the core being heavy (as shown in the appendix). This is due to the fact that the photon also couples to the proton field, which has a larger recoil than the core field. Furthermore, the expectation for the future is that more cluster data will become available and that this data can then be used to fix the parameters of the corresponding HEFT.

While we considered in detail the case of one-nucleon halo charge radii, the suppression of some contributions by factors of the inverse of the number of core nucleons is not restricted to this class of observables. The suppression for radii can be traced to the small recoil of the core or, equivalently, to the fact that the heavy-core propagator is static at leading order. Similar effects will in principle be present in any calculation at the loop level, where the propagator appears, for example the structure (energies, form factors, etc.) of two-nucleon halos or two-core systems. We leave the investigation of these additional implications of heavy cores to the future.

## Footnotes

- 1.
Note that the P-wave scattering length and effective range have dimensions of volume and momentum, respectively.

- 2.
For \(\rho _\pi ^2\), which undergoes renormalization, this estimate is only valid for a renormalization scale of order \(k_\mathrm {hi}\); see below.

- 3.
As we were finalizing this manuscript we found that a similar conclusion has been reached by Elkamhawy and Hammer [36].

## Notes

### Acknowledgements

Open access funding provided by Chalmers University of Technology. DRP and UvK acknowledge the hospitality of Chalmers University of Technology where this research was initiated. DRP thanks W. Elkamhawy for useful discussion. ER and CF were supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 240603, by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT, IG2012-5158), and by the Swedish Research Council (dnr. 2010-4078). The work of DRP was supported by the US Department of Energy under contract DE-FG02-93ER-40756 and by the ExtreMe Matter Institute EMMI at the GSI Helmholtzzentrum für Schwerionenphysik, Darmstadt, Germany. UvK’s research was supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-FG02-04ER41338, and by the European Union Research and Innovation program Horizon 2020 under grant agreement no. 654002.

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