Halo EFT for $^{31}$Ne in a spherical formalism

We calculate the electromagnetic properties of the deformed one-neutron halo candidate $^{31}$Ne using Halo Effective Field Theory (Halo EFT). In this framework, $^{31}$Ne is bound via a resonant $P$-wave interaction between the $^{30}$Ne core and the valence neutron. We set up a spherical formalism for $^{31}$Ne in order to calculate the electromagnetic form factors and the E1-breakup strength distribution into the $^{30}$Ne-neutron continuum at leading order in Halo EFT. The associated uncertainties are estimated according to our power counting. In particular, we assume that the deformation of the $^{30}$Ne core enters at next-to-leading order. It can be accounted for by including the $J^P=2^+$ excited state of $^{30}$Ne as an explicit field in the effective Lagrangian.


I. INTRODUCTION
The emergence of halo nuclei is an intriguing aspect of atomic nuclei near the driplines [1][2][3]. They were discovered in the 1980s at radioactive beam facilities and are characterized by an unusually large interaction radius [4]. Nuclear halo states consist of a tightly bound core with a characteristic size ∼ 1/M hi and a cloud of halo nucleons of size ∼ 1/M lo , which is much larger than neighbouring isotopes. The large separation between the momentum scales M hi M lo leads to universal properties, which are independent of the details of the core [5][6][7]. These properties are most pronounced in neutron halos as they are not affected by the long-range Coulomb repulsion between charged particles.
The separation of scales in halo nuclei can formally be exploited using Halo effective field theory (Halo EFT) [8,9] (see Ref. [7] for a recent review). It uses effective degrees of freedom and allows to describe observables in a systematic expansion in M lo /M hi , thus enabling uncertainty estimates based on the expected size of higher order terms in the expansion. For the dynamics of the halo nucleons, the substructure of the core can be considered short-distance physics that is not resolved, although low-lying excited states of the core sometimes have to be included explicitly. One assumes the core to be structureless and treats the nucleus as a few-body system of the core and the valence nucleons. Corrections from the core structure appear at higher orders in the EFT expansion, and can be accounted for in perturbation theory. Since the relevant halo scale M lo is small compared to the pion mass, even the pion exchange interaction between nucleons and/or the nuclear core is not resolved. Thus, halos can be described by an EFT with short-range contact interactions. A new facet compared to few-nucleon systems is the appearance of resonant interactions in higher partial waves [8,9]. However, there are many light halo nuclei where S-wave interactions are dominant.
In heavier halo nuclei in the region Z = 9 -12, the structure of the ground state is believed to be more complicated and deformed halos are expected. Nakamura and collaborators provided the first indications of a halo structure in 31 Ne [10]. Subsequently, they showed that the ground state of 31 Ne has a low one-neutron separation energy and is a deformed P -wave halo [11]. A similar structure was found for 37 Mg [12] which is even heavier. In the case of 31 Ne, they used state of the art shell model calculations to analyze nuclear and electromagnetic 1n-removal reactions on C and Pb targets and found that the weakly-bound P -wave neutron carries only about 30% of the single-particle strength. The first excitation energy of the 30 Ne core is 800 keV above the ground state which has spin and parity quantum numbers J P = 0 + . Meanwhile, the quantum numbers of 31 Ne are J P = 3/2 − with a neutron separation energy of 150 keV.
The possibility that 31 Ne could a be a one-neutron halo was suggested in theoretical work using density-dependent relativistic mean-field theory [13]. Various authors have analyzed the experimental data on Coulomb dissociation of 31 Ne based on this assumption [14,15]. Urata et al. showed that the data can be well reproduced in the particle-rotor model when the quadrupole deformation parameter of the 30 Ne core is around β 2 = 0.2...0.3 [15]. The preferred structure of a P -wave neutron halo with J P = 3/2 − was also obtained using a microscopic G-Matrix calculation [16] and a deformed Woods-Saxon potential for the neutron-core interaction [17]. A theoretical analysis of the ground state quantum numbers 31 Ne based on experimental data on Coulomb breakup and neutron removal reached the same conclusion [18]. Most recently, the Gamow shell model was applied to the Neon isotopes 26 Ne− 31 Ne [19]. This study confirmed the P -wave neutron halo character of 31 Ne and suggested that 29 Ne could also be a neutron halo.
The separation of scales in 31 Ne allows for a controlled, systematic description of its properties using Halo EFT. A first Halo EFT calculation of the electric properties of 31 Ne based on the general framework of [20] was carried out in [21]. In this paper, we present a complete discussion of the electromagnetic structure of 31 Ne, as well as its E1 breakup. We describe 31 Ne as a P -wave 30 Ne-neutron bound state. The deformation of the 30 Ne core enters at next-to-leading order and can be calculated by including the J P = 2 + excited state as an explicit field in the effective Lagrangian. Instead of using the standard Cartesian formulation of the field theory applied in [21], we introduce a spherical basis that is ideally suited for the description of halo nuclei beyond the S-wave. It employs the correct number of field components in a given partial wave and thus does not require any auxilliary conditions, leading to more compact and transparent expressions. Moreover, we also calculate magnetic observables.
In Sec. II, we present the Halo EFT for 31 Ne in the spherical basis, derive the 30 Ne-neutron scattering amplitude, and discuss the corresponding power counting. The electromagnetic (EM) sector is discussed in Sec. III. We incorporate EM interactions and derive the scalar and vector currents and their corresponding form factors. In Sec. IV, we extract the leading moments from the form factors and discuss universal correlations between them. Moreover, we elucidate the implications of the multipole moments with respect to the deformation of 31 Ne and determine the quadrupolar deformation parameter β 2 . The E1 breakup of 31 Ne into the 30 Ne-neutron continuum is analyzed in Sec. V. Finally, we present our conclusions in Sec. VI.

II. HALO EFT FOR NEON-31
A. Lagrangian: strong sector We describe 31 Ne as a shallow P -wave bound state of the 30 Ne core and the valence neutron. Our effective Lagrangian includes a bosonic field c with J P = 0 + for the 30 Ne core and a J P = 1/2 + spinor field n α with α ∈ {−1/2, 1/2} for the neutron. Moreover, a J P = 3/2 − dimer field π β with β ∈ {−3/2, −1/2, 1/2, 3/2} captures the physics of 31 Ne and the core-neutron continuum. The corresponding Lagrangian is given by where M nc ≡ m n + m c denotes the kinetic mass of the core-neutron system, while η 1 ≡ ±1 is a sign to be determined from matching to scattering observables. Moreover, denotes the core-neutron reduced mass. The coefficient C

B. Full Dimer Propagator
For convenience, we use the power divergence subtraction scheme from Refs. [22] and [23] with renormalization scale µ. In order to determine the full dimer propagator, we dress the bare propagator with dimer self-energies. We end up with the Dyson equation which is depicted diagramatically in Fig. 1. This geometric series represents the exact solution of the core-neutron problem.
The dimer self-energy is diagonal in the spin indices of the incoming and outgoing dimer fields and reads with Since the Σ β β and D 0 (p 0 , p) are diagonal in the spin indices, the full dimer propagator is also diagonal and reads iD β β (p 0 , p) = iD(p 0 , p)δ β β , where the scalar full propagator is given by The full dimer propagator must have a simple pole at the energy p 0 = p 2 2Mnc −B 1 with B 1 = γ 2 1 /(2m R ) the one-neutron separation energy of 31 Ne, whereas γ 1 > 0 is the corresponding binding momentum. In order to calculate 31 Ne observables, we need the wave function renormalization constant defined by Figure 2. Pictoral representation of the neutron-core scattering amplitude including spin indices. The thick line denotes the full dimer propagator, the single solid line represents the neutron field, and the dashed line represents the core field.
which yields

C. Scattering Amplitude and Matching
The P -wave neutron-core scattering amplitude in the J = 3/2-channel is obtained by attaching external core and neutron lines to the full dimer propagator from Eq. (6), see Fig. 2. In the center-of-mass frame with E = p 2 /(2m R ) = p 2 /(2m R ) and p = |p| = |p |, it reads where σ is the three-dimensional vector with the Pauli matrices as its components. The corresponding result for a J = 1/2 state is given in Appendix B. Comparing Eq. (9) to the general form of the amplitude in terms of the effective range parameters, we obtain the matching conditions Since the parameter r 1 has to be negative for causal scattering [24,25], the sign η 1 is determined to be η 1 = +1 according to Eq. (11). With these matching conditions, the wave function renormalization constant reads Z π is the residue of the bound state pole at the energy p 0 = p 2 /(2M nc ) − γ 2 1 /(2m R ) in the full dimer propagator where γ 1 is a real positive solution to the equation

D. Power Counting
Following Ref. [9], we assume only one combination of coupling constants to be fine-tuned, namely ∆ 1 /g 2 1 . This is sufficient in order to produce a shallow P -wave bound state. With this choice, the scattering volume a 1 is enhanced by whereas the P -wave effective momentum r 1 scales like where M lo and M hi denote the typical low-and high-momentum scales of the system, respectively. The low-momentum scale is given by the binding momentum of the shallow P -wave bound state. In the case of 31 Ne with a binding energy of B 1 = 150 keV [11], this yields The high-momentum scale can be approximated by the breakdown scale of the theory. Since we do not include the J P = 2 + state of the 30 Ne core explicitly, M hi can be estimated by the associated momentum scale of the excitation energy E ex = 792 keV [26]. The corresponding value is given by M hi ≈ √ 2m R E ex ≈ 40 MeV. At this momentum scale, the deformation of the 30 Ne core due to this excited state starts to play a role.
Given this power counting scheme, the equation for the pole position, Eq. (13), and the wave function renormalization, Eq. (12), can be expanded at leading order in M lo /M hi to yield The quantity g 2 1 Z LO π is proportional to the absolute value squared of the EFT wave function at the bound state pole. Thus, Z π must be positive. As a consequence, r 1 must be negative. At NLO, the wave function renormalization is given by the expression from Eq. (12), Thus, only if |r 1 | > 3γ 1 holds, we end up with a normalizable state with positive residue. This requirement is consistent with the hierarchy M lo M hi which forms the basis of our power counting.
For a shallow P -wave state, we have at least two effective range expansion parameters, a 1 and r 1 , which have to be fixed by observables. Until now, we only know the neutron separation energy of 31 Ne from experiment, which is not enough in order to fix both effective range expansion parameters. Therefore, we will estimate the P -wave effective momentum r 1 in an interval around the breakdown scale, according to Eq. (15) and use the neutron separation energy to determine the scattering volume a 1 . Based on these assumptions we can calculate other observables accessible in our theory, such as the electromagnetic current and the corresponding multipole moments as well as the associated radii.
Taking everything together, we estimate Values of r 1 in this interval are consistent with unitarity and the estimated breakdown scale of our theory associated with the J P = 2 + excited state of the core.

III. ELECTROMAGNETIC SECTOR
We now go on to include electromagnetic interactions in the effective theory. Moreover, we derive the corresponding form factors using spherical coordinates. We present results for the form factors of 31 Ne and provide general expressions for form factors of arbitrary multipolarity L.
In the first step, electromagnetic interactions are included via minimal substitution meaning that the usual derivative ∂ µ in the Lagrangian in Eq. (1) is replaced by the covariant derivative D µ containing the charge operatorq, the elementary charge e > 0 and the photon field A µ = (A 0 , A). In the second step, all possible gauge-invariant operators involving the electric field E and also the magnetic field B have to be considered within our power counting scheme. It turns out that only gauge-invariant operators proportional to the magnetic field B are contributing at LO whereas operators involving the electric field E contribute at higher orders.

A. Scalar Current
First, we calculate the matrix element of the zeroth component of the electromagnetic current of 31 Ne. Therefore, we consider the amplitude with an irreducible vertex for an A 0 photon with four momentum (0, q) coupling to the 30 Ne-n P -wave bound state with initial momentum p and final momentum p . Thus, we have q = p − p and define q = |q|. The initial and final states are characterized by their momenta and projections of the spin, denoted by |π β (p) and |π β (p ) , respectively. The LO contributions to this amplitude are depicted in Fig. 3. Since 31 Ne has a total spin of 3/2, there are four possible projections for each the initial and final state. Hence, the tensors connecting initial and final state projections are 4 × 4 matrices in spin space.
The scalar electromagnetic transition amplitude can be written as where e q is the unit vector of q, q c is the charge of the core in terms of the elementary charge e and µ Q is the quadrupole moment. Moreover, the electric monopole and quadrupole form factors are denoted by G E0 (q) and G E2 (q), respectively. Since the multipole moments are explicitly factored out of the form factors in Eq. (21), G E0 (q) and G E2 (q) are normalized to one in the limit of vanishing photon momentum by construction. The tensorsT 00 3/2 andT 2M 3/2 are normalized 4 × 4 polarization matrices [27]. For general J, they are (2J + 1) × (2J + 1) matrices given by They are normalized such that they have a coefficient of 1 for maximal projections. Consequently, the multipole moments are defined for maximal projections as it is usually done by convention. The subscript J indicates the spin of the considered two-particle bound state, while L stands for the angular momentum of the photon. The possible contributions for L result from coupling the two P -wave spherical harmonics appearing in the right diagram of Fig. 3. Furthermore, M denotes the projection of the angular momentum L. This implicit angular momentum coupling can yield contributions of the photon multipolarities L = 0, 1, 2 in the electromagnetic transition amplitude, Eq. (21). Due to parity conservation, however, only even numbers of L contribute so that we are left with L ∈ {0, 2}. Hence, as we can read off Eq. (21), the electric monopole and quadrupole form factors with their corresponding multipole moments appear for J = 3/2, but no dipole form factor. 1 The LO results for the electric form factors read Since gauge invariance ensures charge conservation, the normalization lim q→0 G E0 (q) = 1 is automatically fulfilled and hence serves as a consistency check. The normalization condition + (0, q) Inserting this quadrupole moment in Eq. (24), we obtain Note that the result for G E0 (q), Eq. (23), is the same for a spin 1/2 dimer but appears with the appropriate J = 3/2 polarization matrix in Eq. (21).

B. Vector Current
Next, we investigate the vector electromagnetic current of 31 Ne. For this purpose, we consider the amplitude with an irreducible vertex for an A k photon with four momentum (0, q) coupling to the 30 Ne-n P -wave bound state. The corresponding diagrams are depicted in Fig. 4. The two diagrams on the right, which are crossed out, can be shown to vanish by parity conservation such that only the two diagrams on the left contribute. Furthermore, we have to take into account local gauge invariant contributions from the magnetic coupling to the spins of the corresponding fields. Assuming that both the anomalous magnetic moment of the neutron κ n and the magnetic moment of the dimer L M scale naturally, they contribute at LO. As a matter of fact, the counterterm L M is necessary for renormalization purposes already at LO. The diagrams are shown in Fig. 5 and the corresponding magnetic interaction vertices are given by where κ n denotes the anomalous magnetic moment of the neutron and µ N is the nuclear magneton. (See Ref. [28] for a discussion of the J P = 1/2 + case.) L M is a counterterm required for renormalization. Furthermore, B = (∇ × A) is the magnetic field of the photon while S J is a three-dimensional vector with spin matrices as its components. These Figure 5. Diagrams contributing to the irreducible vertex for a B k photon coupling to the 30 Ne-n where (S J ) m and B m are the corresponding components in spherical coordinates. The components of the three spin matrices are given by [27] [(S J )) m ] σ σ = J(J + 1)C Jσ (Jσ)(1m) .
Therefore, the matrix element for maximal projection is always multiplied by B 0 . The vector electromagnetic transition amplitude can be written as where µ D denotes the magnetic dipole moment, µ O is the magnetic octupole moment, whereas G M1 and G M3 are the corresponding form factors, respectively. As for the scalar current, the multipole moments are explicitly factored out in Eq. (32) and therefore the form factors are normalized to one in the limit of vanishing photon momentum by construction.
Obviously, the physics of the vector electromagnetic current is richer than that of the scalar current. Not only the electric monopole and quadrupole form factor appear in Eq. (32) but also the magnetic contributions. In our case, or rather for a spin-3/2 particle, in addition to the magnetic dipole moment there is also a magnetic octupole moment. Due to the integral over the two P -wave spherical harmonics and over the spherical harmonic from the vector photon in the second diagram of Fig. 4, we now have an implicit coupling of three angular momenta. The corresponding photon multipolarities are L = 0, 1, 2, 3. However, parity conservation restricts the possible values for the magnetic contributions to L ∈ {1, 3}. The contributions of L = 1 and L = 3 manifest in Eq. (32) through terms proportional to Y *

1(M +k)
and Y * 3(M +k) , respectively. In contrast, the electric contributions to the vector current are apparent through the term proportional to (p + p) k .
A closer look at the vector current in Eq. (32) suggests a generalized structure including arbitrary high multipole electric and magnetic form factors that can be found in Appendix C.
The LO results for the magnetic form factors for a J = 3/2 dimer read with A c denoting the mass number of the core while a, b and c are functions given by For a J = 1/2 dimer, only the magnetic dipole form factor is observable. The corresponding functions a, b and c slightly differ from the ones given in Eqs. (35) to (37) and can be found in Appendix B. The first term in the numerator of Eq. (33) proportional to L M (µ) is a contribution due to the direct magnetic moment coupling to the spin of the dimer field. The second term is a contribution due to the magnetic moment of the neutron proportional to κ n . Finally, the origin of the third contribution proportional to yq c /A c lies in the finite angular momentum of the charged core which induces a magnetic dipole moment and therefore contributes to the magnetic dipole form factor. Applying the normalization conditions at the real photon point, lim q→0 G M1 (q) = 1 and lim q→0 G M3 (q) = 1, the magnetic dipole and octupole moments are read off as Note that the magnetic dipole moment cannot be predicted in Halo EFT. Instead the counterterm L M (µ) is matched to the magnetic dipole moment using Eq. (38). The contribution of the dimer to the magnetic moment, L M (µ), thus is resolution dependent. Its scale dependence is governed by the renormalization group equation This is the exact same expression as in Eq. (27) except for the substitution y → (1 − y) in the dependence on the mass factor from Eq. (25).

IV. BOUND STATE OBSERVABLES AND THEIR CORRELATIONS
The electromagnetic form factors from the previous section can be expanded for low three-momentum transfer q 2 in order to extract the corresponding radii: where r 2 (E/M)L denotes the expectation value of the electric/magnetic radius squared with multipolarity L, respectively. Below, we give general expressions for a halo nucleus with a J P = 0 + core and a halo neutron in a J P = 3/2 − P -wave state as well as explicit numbers for 31 Ne based on the assumptions discussed in Sec. II.

A. Results in the Electric Sector
The electric monopole and quadrupole radii at LO are Given these expressions, we can establish universal correlations with other observables. Considering the result for the quadrupole moment in Eq. (26), we find the correlation where the numbers in parentheses give the EFT uncertainties of 40%. Furthermore, we find a correlation between the quadrupole radius squared r 2 E2 and the neutron separation energy S n given by It is depicted in the right panel of Fig. 6, where the red cross indicates our 31 Ne result for S n = 0.15 MeV given by

B. Results in the Magnetic Sector
The LO result for the magnetic octupole radius squared reads This correlation between the squared octupole radius and the neutron separation energy is similar to the correlation in Eq. (48). The corresponding value for the octupole radius of 31 Ne is Given the octupole moment in Eq. (39) and estimating r 1 as before yields the following result for the octupole moment of 31 Ne Since the magnetic dipole moment contains the counterterm proportional to L M (µ), it is not possible to predict its value in Halo EFT. However, as discussed above, the full q 2dependence of µ D G M1 (q) is predicted. In particular, the "renormalized magnetic radius" defined as µ D r 2 M1 is independent of L M (µ). For a J = 3/2 state, we have Using this relation, it is either possible to predict r 2 M1 once the magnetic dipole moment is determined experimentally or vice versa. The corresponding result for J = 1/2 can again be found in Appendix B.

C. Nuclear Deformation
The appearance of higher multipole moments such as the electric quadrupole as well as the magnetic octupole moment indicates that 31 Ne is not a spherically symmetric nucleus. Following Ref. [29], we assume a quadrupolar deformed shape with a sharp edge at radius where R 0 is the equilibrium radius, meaning the radius if the nucleus would be spherically symmetric. The additional term β 2 Y 20 (θ, φ) accounts for the quadrupolar deformation where β 2 is called the deformation parameter. Having defined this surface radius and using β 2 1, we can relate it to the spectroscopic quadrupole moment via [29,30] µ Q (3/2) = 1 5 In the second line of Eq. (55) we used r 2 E0 = (3/5)R 2 0 . As a result, we find a linear correlation between the quadrupole moment and the mean squared electric monopole radius. This is exactly the same correlation we found in our Halo EFT calculation and hence equating the proportionality factors allows us to determine the deformation parameter of 31 Ne to be β 2 = 0.53. This value is similar to β 2 = 0.41 found in an antisymmetrized molecular dynamics calculation with the Gogny D1S interaction [16]. A deformation parameter of β 2 ≈ 0.4 was also obtained in Ref. [17] from the analysis of parallel momentum distribution of the charged fragment in the breakup of 31 Ne. These values indicate a significant deformation due to the non-vanishing quadrupole moment. However, we note that our prediction is solely determined by the dynamics of the electrically charged core. The deformation of the core itself is not included here. In this sense, our predictions are relative to the core. Once the intrinsic properties of the core are experimentally determined, they can be included in our theory. In particular, intrinsic deformation properties of the core such as its quadrupole moment due to the J P = 2 + excited state can be described explicitly in Halo EFT by including a corresponding field in the effective Lagrangian. This would allow us to predict the deformation properties due to both the intrinsic core properties and the dynamics of the halo nucleus. Indeed, it is expected to find a quadrupolar deformation of the 30 Ne core. Urata et al. [15] showed that the deformation parameter of the 30 Ne core is around β 2 = 0.2...0.3, while Minomo et al. [16] found β 2 = 0.39. Therefore, the total quadrupolar deformation of 31 Ne is ultimately composed of both deformation effects. Figure 7. E1 Breakup of 31 Ne into the continuum consisting of the core and neutron. We use the same notation as in Fig. 2.

V. E1 BREAKUP: 31 NE INTO 30 NE AND A NEUTRON
In Fig. 7, we show the LO diagram contributing to the E1 breakup of 31 Ne. The photon transfers an angular momentum of 1 onto the two-body system consisting of the core and neutron. Since this two-body system is bound in a P -wave, the possible final angular momenta in the continuum are 0 and 2, corresponding to an S-and a D-wave, respectively.
The scalar transition amplitude in momentum space is given by where |ψ α β represents the bound state of 31 Ne (see Appendix D for explicit expressions), the mass ratio y is defined in Eq. (25), p is the relative momentum between the core and the neutron, while k is the photon momentum. Without loss of generality, we choose the photon to be traveling in theẑ-direction. We insert in Eq. (57) an identity operator in configuration space and express e iykz by its plane wave expansion where j L (x) is a spherical Bessel function. In the low-energy limit, we use j L (ykr) ≈ (ykr) L /(2L + 1)!!. The scalar transition amplitude then reads whereρ with Z This means, that the photon in Fig. 7 transfers all possible angular momenta. For a specific angular momentum transfer, the amplitude reads The matrix element relevant for the calculation of the EL breakup is given by [31] M (EL; 0) = d 3 r p|ρ L (r) |ψ α β r L Y L0 (e r ) , Since we are interested in the E1 breakup, we set L = 1 in Eq. (63). Moreover, we plug in the bound state wave function in configuration space given by with C π denoting the asymptotic normalization constant (ANC) and u(r) the radial wave function. They read This yields The product of the spherical harmonics in Eq. (67) can be expressed as an irreducible sum of spherical harmonics with L = 0 and L = 2. This allows us to extract the two relevant matrix elements for a transition into either an S-wave or a D-wave.
The P → S transition amplitude reads We use the plane wave expansion of e −ipr , integrate over dΩ and couple the angular momentum of L = 0 and the spin of the neutron to a two-particle continuum with total spin quantum numbers (J β ) to find M J β β (E1; 0; P → S) = ieZ (1) eff C π √ 4πY 00 (e p ) drj 0 (pr)u(r)r 2 C We proceed similarly for the P → D transition amplitude and find Performing the radial integral and sum (average) over final (initial) spins and multiply our results with a factor of 3 to make up for the fact that we chose the photon to propagate in theẑ-direction, we finally get The differential E1 transition strength is given by [32] Using E rel = p 2 /(2m R ), the differential E1 transition strength as a function of the relative energy E rel between the core and neutron reads The final results for both transitions read We show the corresponding curves in Fig. 8 for a P -wave effective momentum r 1 = −100 MeV. The total differential B(E1) transition strength is given in blue while the blue shaded band represents an 40% estimate of the EFT uncertainty at LO from our power counting. As expected, the S-wave contribution in the continuum dominates at low energies while the D-wave contribution takes over around E rel = 0.25 MeV. In Fig. 9, we show the differential cross section for the E1 breakup of 31 Ne. The bare cross section corresponding to an infinite energy resolution of the detector is given by where E rel = E γ − S n + k 2 /(2M nc ) with S n the neutron separation energy of 31 Ne and the expression for the virtual photon flux N E1 is taken from Ref. [33]. The corresponding plot can be found in Fig. 10. Since no experimental results have been published to date, we have folded the predicted bare cross section with a hypothetical Gaussian energy resolution,  experimental data. The resolution averaged cross section is obtained from and shown in Fig. 9. The dark shaded band gives the uncertainty from r 1 estimated as r 1 ∈ [−150, −50] MeV, while the light shaded bands also includes the general EFT uncertainty at LO of 40%. The absolute height of the peak has a large EFT uncertainty while the shape of the curve is rather robust. However, it is strongly influencend by the assumed detector resolution parameter σ and larger values of σ will further smear the peak.

VI. CONCLUSION
We have investigated the electromagnetic properties of 31 Ne using Halo EFT. Instead of using standard Cartesian coordinates, we have introduced a spherical basis that is ideally suited for the description of halo nuclei beyond the S-wave. It uses the correct number of degrees of freedom by construction and therefore leads to more compact and simplified expressions. We expect it to be useful in future studies of halo nuclei with higher angular momentum.
In our study of the electric properties, we found that our numerical predictions are fairly small. We calculated the charge radius as r 2 E0 1/2 ∈ [0.20(08), 0.35 (14)] fm and the quadrupole moment as µ Q ∈ [0.06(02), 0.17(07)] fm 2 , where the numbers in square brackets give the results for r 1 = −150 MeV and −50 MeV, respectively, while the number in parentheses indicates the error from higher orders in the EFT expansion. The corresponding quadrupole radius is independent of r 1 and predicted to be r 2 E2 1/2 = 0.30 (12) fm. These values are rather small because at leading order they are solely determined by the motion of the electrically charged core around the center of mass. Since the 30 Ne core is almost as heavy as the total system, 31 Ne, this yields small predictions. Therefore, we expect internal electric properties of the core to be important, at least in the electric sector. Such effects can be included at NLO by treating excited states of the core as explicit fields or via counterterms. See Ref. [34] for a discussion of this issue in the case of the charge radius.
Once more experimental data is available, the treatment of the first excited state of the 30 Ne core as an explicit degree of freedom within Halo EFT becomes feasible. This would lead to more precise predictions in the electric sector and potentially to a smaller expansion parameter. Alternatively, one could include the excited states of the core by describing the halo nucleus as a neutron coupled to a rotor similar to the work of Refs. [35,36].
Nevertheless, in the magnetic sector the main contribution to observables arises from the motion of the valence neutron around the center of mass. This means that corrections due to the internal core properties at NLO should be negligible. As a matter of fact, in the magnetic case our numerical predictions are much larger, r 2 M3 1/2 = 9.0(3.6) fm and µ O ∈ [−14(6), −5(2)] µ N fm 2 . The large octupole radius reveals the size of the halo system. Unfortunately, the magnetic dipole moment cannot be predicted since it depends on the counterterm L M already at LO.
In general, the non-vanishing higher multipole moments with multipolarity L > 1 indicate that 31 Ne is not a spherically symmetric nucleus. We extracted the β 2 -deformation parameter from the linear correlation between the quadrupole moment and the charge radius and found β 2 = 0.53. This value indicates a significant deformation due to the quadrupole moment. However, we note that this prediction is solely determined by the dynamics of the electrically charged core whereas the deformation of the core itself is not included here. In this sense, our prediction is relative to the core. Intrinsic deformation properties of the core can be described explicitly in Halo EFT by introducing corresponding fields in the effective Lagrangian. Once more experimental data of the intrinsic deformation properties of the core such as its quadrupole moment are available, they can be included in Halo EFT to calculate the total deformation properties due to both the intrinsic core properties and the dynamics of the halo nucleus.
Moreover, we have derived the differential B(E1) transition strength as a function of the relative energy E rel between the 30 Ne core and the neutron. This transition strength together with the virtual photon number allowed us to calculate the differential cross section for E1 breakup of 31 Ne. In order to take into account a realistic limited energy resolution in experiment, we have folded our results with a Gaussian energy resolution of width 0.1 MeV. Comparing these results to future data will help us to further determine unknown parameters which in turn enables us to improve our Halo EFT for 31 Ne. Finally, an application of our formalism to 37 Mg [12], which is also a candidate for a deformed P -wave halo nucleus appears promising. with A c denoting the mass number of the core, while a, b and c are functions given by where the definition of the mass ratio y is given in Eq. (25). The dipole moment of a J = 1/2 dimer also depends on an unknown counterterm L 1/2 M (µ) and is given by In contrast, the "renormalized magnetic radius" µ D r 2 M1 is independent of L 1/2 M (µ) and the result for a J = 1/2 state reads For the derivation of the wave function, we use where G i i is the fully interacting Green's function, B 1 is the binding energy, and |ψ i denotes the corresponding P -wave bound state. Furthermore, we use where G 0 and T i i are the free Green's function and the P -wave T-matrix, respectively. Since the free Green's function gives no contribution to the pole, we find from Eq. (D1) and Eq. (D2) We consider where we have used with D 1 (E) denoting the full dimer propagator for the P -wave bound state. Making use of the expansion of the full dimer propagator around the bound state energy we obtain Comparing this result to Eq. (D3) yields the wave function in momentum space Finally, coupling the orbital angular momentum with quantum numbers (1i) with the spin of the neutron with quantum numbers ( 1 2 α) to the total angular momentum ( 3 2 β) leads to