Electric properties of the Beryllium-11 system in Halo EFT

# Electric properties of the Beryllium-11 system in Halo EFT

H.-W. Hammer Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany    D. R. Phillips Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany
###### Abstract

We compute E1 transitions and electric radii in the Beryllium-11 nucleus using an effective field theory that exploits the separation of scales in this halo system. We fix the leading-order parameters of the EFT from measured data on the 1/2 and 1/2 levels in Be and the B(E1) strength for the transition between them. We then obtain predictions for the B(E1) strength for Coulomb dissociation of the Be nucleus to the continuum. We also compute the charge radii of the 1/2 and 1/2 states. Agreement with experiment within the expected accuracy of a leading-order computation in this EFT is obtained. We also discuss how next-to-leading-order (NLO) corrections involving both s-wave and p-wave Be-neutron interactions affect our results, and display the NLO predictions for quantities which are free of additional short-distance operators at this order. Information on neutron-Be scattering in the relevant channels is inferred.

preprint: INT-PUB-11-007

## I Introduction

The first excitation of the Beryllium-10 nucleus is 3.4 MeV above the ground state, and that ground state has spin and parity quantum numbers . Meanwhile, the Beryllium-11 nucleus has a state whose neutron separation energy is 500 keV, and a state whose neutron separation energy is 180 keV AjK90 (). The shallowness of these two states of Be compared to the bound states of Be suggests that they have significant components in which a loosely-bound neutron orbits a Be core. In this “one-neutron halo” picture the is predominantly an s-wave bound state, while the is predominantly a relative p-wave between the neutron and the core. In this paper, we discuss efforts to use effective field theory (EFT) to systematically implement such a halo picture of the Be nucleus.

This halo viewpoint is reinforced by the fact that the scattering volume of n-Be scattering in the , channel has been determined to be TB04 ()

 a1=(457±67) fm3. (1)

The corresponding length scale of order 8 fm is large compared to the natural length-scale of core-neutron interactions, which is fm.

The datum (1), together with the information on the bound-state energies in the Be and Be systems, helps us to estimate the expansion parameter in our Halo EFT. This is the binding energy of the halo nucleus, as compared to the energy required to excite the core, i.e. . Converting this to an estimate of the different distance scales involved, we infer that a majority of the probability density of Be occupies a region outside the Be core: , which is consistent with the ratio implied by the numbers in the previous paragraph. This ratio of distance scales is the formal expansion parameter for the EFT, and since it is not particularly small, leading-order calculations are only a first step. We therefore present calculations up to next-to-leading order for several quantities, in order to confirm that the series is converging as expected.

In particular we apply this EFT to electromagnetic reactions in the Be system. The B(E1)() transition has recently been measured to be

 B(E1)(1/2+→1/2−)=0.105(12) e2 fm2 (2)

using intermediate-energy Coulomb excitation Su07 (). This is consistent with the older number

 B(E1)(1/2+→1/2−)=0.116(12) e2 fm2 (3)

from lifetime measurements Mi83 (). There are also two recent data sets on the Coulomb-induced breakup of the Be nucleus Pa03 (); Fu04 () (see also Ref. An94 ()). Both experiments extracted the excitation function B(E1) as a function of the energy of the outgoing neutron . For low neutron energies this excitation function is affected by the final-state interaction in the p-waves, and can be predicted in the halo picture TB04 (). Ref. Pa03 () also extracted a neutron radius for the ground state of Be from their data:

 ⟨r2⟩1/2=5.7(4) fm. (4)

This is consistent with the recent atomic-physics measurement of the Be charge radius No09 ():

 ⟨r2E⟩1/211Be=2.463(16) fm. (5)

All of these measurements can be addressed within the Halo EFT we will use here. In this theory the s- and p-wave states of the Beryllium-11 nucleus are generated by core-neutron contact interactions. The theory does not get the interior part of the nuclear wave function correct, but, by construction, it reproduces the correct asymptotics of the wave functions of these states:

 u0(r) = A0exp(−γ0r), u1(r) = A1exp(−γ1r)(1+1γ1r), (6)

for the keV and 180 keV states, respectively. The quantities and are determined by the neutron separation energies of the states in question. At leading order (LO) in the expansion the Asymptotic Normalization Coefficients (ANCs) and are fixed. (In the case of the p-wave this is related to the theorem discussed in Ref. Lee ().) However, at next-to-leading order and become, in essence, free parameters of the theory, and must themselves be extracted from data.

Halo EFT is well-suited for this task. It is not intended to compete with ab initio calculations of this halo nucleus (see, e.g. Fo05 (); Fo09 ()) or of Be-n scattering QN08 (), or with microscopic descriptions of the Be E1 strength (see, e.g. Mi83 ()). Instead, Halo EFT is complementary to such approaches, since it takes , , , and as input, rather than seeking to predict them via a detailed description of the system. The EFT’s goal is to ensure that the long-distance properties of the halo are correctly taken care of, and then to elucidate the relationships between different observables in the Be-n system that result. Such correlations flow directly from the existence of the shallow and bound states in this system. In particular, below we will show how and are correlated with neutron-Be scattering observables, as well as with and the Coulomb dissociation data. This, in turn, demonstrates how—and with what accuracy—the ANCs can be inferred from these experimental quantities.

A preliminary version of these findings appeared in Ref. HP10 (). The presentation here corrects and expands upon that earlier work. A related study of radiative neutron capture on Lithium-7 was recently carried out in Rupak:2011nk (). The mechanism for the cancellation of divergences in the s-to-p transition in this reaction and in the Beryllium-11 system is the same.

## Ii Halo EFT for Beryllium-11

We use the “Halo EFT” developed in Refs. Be02 (); Bd03 () to calculate the properties of the Beryllium-11 nucleus. The degrees of freedom in our Halo EFT treatment are the Be core and the neutron. The EFT expansion in this case is an expansion in powers of . Here is, e.g. the excitation energy of states in Be, and so is of order a few MeV, and is the energy of the photon exciting the electromagnetic transition of interest.

### ii.1 Lagrangian: strong sector

In our calculation, the Be core and the neutron are represented by the bosonic field and a spinor field , respectively. We include the strong s-wave and p-wave interactions that lead to the shallow bound states in the Be system through the incorporation of additional fields, which encode physics of these states. Therefore the state is constructed as a spinor field, . In contrast, the field representing the state is a pseudo-spinor, i.e. it is parity odd. We denote it as . Its behavior under parity restricts the types of couplings which can appear in the Lagrangian. In particular, only combinations of nucleon and core fields with an odd number of derivatives can couple to . Moreover, when constructing these operators we must decompose them into their irreducible representations under the rotation group. The portion that couples to is then just the part. For example, if we construct where , this combination has both and parts. We project out the part by defining:

 [n(i\lx@stackrel↔∇)c]12,s=∑βj(12β1j∣∣∣(121)12s)nβ(i\lx@stackrel↔∇j)c, (7)

where is the Clebsch-Gordan coefficient to couple and to .

With the masses of the neutron and the core are denoted by and , respectively, and is the total mass of the n-Be system, we then have:

 L = c†(i∂t+∇22M)c+n†(i∂t+∇22m)n (8) −g0[c†n†sσs+σ†snsc]−g12(π†s[n(i\lx@stackrel↔∇)c]12,s+[c†(i\lx@stackrel↔∇)n†]12,sπs) −g12M−mMnc(π†s[i\lx@stackrel→∇(nc)]12,s−[i\lx@stackrel→∇(n†c†)]12,sπs)+…,

where we adopt the convention that repeated spin indices are summed. Note that the last line of Eq. (8) represents an additional p-wave interaction necessary to maintain Galilean invariance. It is required because the and fields have different masses. The dots represent higher-order interactions not considered here. One such interaction involves the part of the operator:

 L3/2 = −C(3/2)4[n(i\lx@stackrel↔∇)c]†32,β[n(i\lx@stackrel↔∇)c]32,β = −C(3/2)4∑αs1js2k(12s11j∣∣∣(121)32α)(12s21k∣∣∣(121)32α)(c†(i\lx@stackrel↔∇j)n†s1)(ns2(i\lx@stackrel↔∇k)c).

As we shall discuss in the next section, this interaction is assumed to be natural in our power counting, in contrast to the interactions mediated by and fields, which are enhanced.

### ii.2 s-wave 10Be-neutron interactions

In order to treat the shallow s-wave state in the Be-neutron system we adopt the counting that has been successfully developed to treat shallow s-wave states in the nucleon-nucleon system vK99 (); Ka98A (); Ka98B (); Ge98 (); Bi99 (). In leading order, the field is static and its bare propagator is simply . Dividing out the leading term, the one-loop correction to the bare propagator is times an bubble which has a typical size of order  111In a suitable regularization scheme, e.g. power-law divergence subtraction Ka98A (); Ka98B (), this is true for both the real and imaginary parts of the loops.. Since , this correction is of order one. Consequently, the loops must be resummed when computing the full propagator.

This can be achieved through the Dyson equation shown in Fig. 1, which leads to:

 Dσ(p)=1Δ0+η0[p0−p2/(2Mnc)+iϵ]−Σσ(p), (10)

with the one-loop self-energy for the field.

This one-loop self-energy is calculated as:

 Σσ(p)=−g20mR2π[i√2mR(p0−p22Mnc)+μ], (11)

when computed in power-law divergence subtraction (PDS) with a scale  Ka98A (); Ka98B (). Here we have introduced the reduced mass of the neutron-core system:

 mR=mMm+M, (12)

and the limit in the end is understood.

Substituting Eq. (11) into Eq. (10), we can set the parameters and by computing the s-wave neutron-core scattering amplitude in the theory defined by Eq. (8) (see Fig. 2):

 t0(E)=g20Dσ(E,0), (13)

in the two-body center-of-mass frame with . This is then matched to the effective-range expansion in this channel:

 t0(E)=2πmR11/a0−12r0k2+ik, (14)

producing

 Dσ(p)=2πγ0m2Rg2011−r0γ01p0−p22Mnc+B0+Rσ(p), (15)

where is regular at the pole . In Eq. (15), the position of the pole is determined by the binding energy , and is the positive root of the equation:

 1a0+12r0γ20−γ0=0. (16)

The wave-function renormalization for the dressed propagator can be read off as the residue of the pole in Eq. (15):

 Zσ=2πγ0m2Rg20(1−γ0r0)−1. (17)

This result is valid to NLO in the expansion in . It yields a wave function (6) with

 A0=√2γ01−γ0r0. (18)

### ii.3 p-wave 10Be-neutron interactions

We proceed similarly for the p-wave state. The propagator, , for this state obeys the Dyson equation depicted in Fig. 3. Rather than computing the self-energy of the field directly it is easier to compute the self-energy for a field and then couple the neutron spin and the relative momentum in the appropriate way to project out the piece of the result. Hence we now consider the one-loop self-energy, for such a p-wave field. We first observe:

 Σπ(p)ij,αβ=δijδαβΣ(p). (19)

The scalar function:

 Σ(p)=−mRg216π2mR(p0−p22Mnc)[i√2mR(p0−p22Mnc+iϵ)+μ], (20)

where the PDS scheme has been employed and momentum traces have been performed in three dimensions. From this we can construct a self-energy for transitions from the -field state to the -field state :

 Σπs′s(p)=∑βj(12β1j∣∣∣(121)12s)(12β1j∣∣∣(121)12s′)Σ(p) (21)

since is independent of and we can use completeness of the Clebsch-Gordon coefficients to show that is diagonal in and , i.e. . It follows that takes the form:

 Dπ(p)=1Δ1+η1[p0−p2/(2Mnc)]−Σ(p). (22)

We note that since the self-energy loop is cubically divergent both parameters, and are mandatory for renormalization at LO Be02 (). This time we are interested in the p-wave core-neutron scattering amplitude in the center-of mass frame:

 t1(p′,p;E) = g21p′⋅pDπ(E,0) (23) = 6πmRp′⋅p1/a1−12r1k2+ik3,

with for on-shell scattering. Consequently we obtain:

 Dπ(p)=−6πm2Rg211r1+3γ11p0−p2/(2Mnc)+B1 + regular. (24)

Here is the solution of

 1a1+12r1γ21+γ31=0, (25)

where is the scattering volume, and the p-wave “effective range”, which, in fact, has dimensions of 1/length. Both parameters are required to leading order in the Halo EFT. The wave-function renormalization for the dressed propagator can be read off as the residue of the pole in Eq. (24):

 Zπ=−6πm2Rg211r1+3γ1. (26)

The propagator (23) has three poles corresponding to the zeroes of Eq. (25). Using the NLO parameter values for the Be-n system obtained in Sec. II.6 below, we find two bound-state poles corresponding to typical momenta and . The first is that which we identified with the Be excited state in the previous paragraph. The second is a spurious bound state, which is outside the domain of validity of Halo EFT, and is not physically realized in the Be system. The third solution of Eq. (25) represents a virtual state with a typical momentum .

The power counting for the propagator that we adopt here is that of Ref. Bd03 (). We take . The propagator then has a pole at , which occurs “kinematically” when . then must obviously be of order for such a “kinematic” pole to occur. We note that for the unitarity piece of the propagator has a size . Thus, away from the pole, the dominant contribution to the propagator now comes from the bare part (after appropriate renormalization) and so:

 t1(p′,p;E)=6πmRp′⋅p−12r1(k2+γ21), (27)

where we have also used Eq. (25) to re-express in terms of and then dropped the term relative to the (larger) piece.

The result (27) can be easily re-expressed as:

 t1(p′,p;E)=−6πm2Rr1p′⋅pE+B1. (28)

The amplitude (28) has only a pole at —a pole that corresponds to the 1/2 state of Be. (This pole actually occurs on both sheets of the complex -plane, since (27) exhibits poles in both the lower and upper half of the complex -plane.) In contrast, the spurious deep pole from Eq. (23) has disappeared from the expressions (27) and (28). Our power counting therefore reproduces the spectrum of the Be system. The p-state wave-function (6) is then obtained, with

 A1=√2γ21−r1. (29)

The p-wave phase shifts in this theory are given by:

 k3cotδ1=γ31+12r1(k2+γ21), (30)

if no expansions are made. However, for we can, once again, drop to leading order in a power counting in , with the result that

 cotδ1=r12(1k+γ21k3)+O(RcoreRhalo). (31)

Since we have large, which implies that is approximately zero. Indeed

 δ1=2r1k3k2+γ21+O(RcoreRhalo), (32)

for all . Small phase shifts imply small unitarity corrections, which is why the imaginary part of can be treated perturbatively in this regime.

The only exception to this occurs if we consider . In that case we are close to the pole and the two terms in Eq. (27) cancel, or come close to doing so. It then becomes necessary to resum the pieces and which were dropped in order to obtain Eq. (28PP03 (); Bd03 (). In particular, if (i.e. the pole is at positive energy) then these corrections shift the pole off the real axis and mean that it represents a resonance. The propagator (23) thus describes a p-wave resonance if the pole is at positive energy and the width of the resonance at energy is of order  Be02 (); Bd03 (); Sak (). This is thus a narrow resonance if . The case of p-wave resonances will not be discussed further here, since we will restrict ourselves to , as is relevant for Be. This is the case of a shallow p-wave bound state.

### ii.4 Rescaled fields and naive dimensional analysis

At this point it is useful to rescale the piece of the Lagrangian which encodes the enhanced s- and p-wave interactions. We rewrite in terms of fields with non-canonical dimensions, which absorb factors of , , , and  BS01 (). We define:

 ~σs=σsg0mR;~πs=πsg1mR. (33)

The scaling of the absorbed factors and can be obtained by recalling the matching between (13) and (14) for s-waves, and in Eq. (23) for p-waves. At leading order, this yields:

 g20m2R≃−2πη0r0,g21m2R≃−6πη1r1. (34)

In our counting, we have , which then determines how and scale with . Note also that, since and for n-Be interactions, we have , in this system.

We analyze because this operator is dimension 5, and has no powers of the mass in it anymore LM97 (). The expression for the product of the total mass and the pertinent piece of the Lagrangian then becomes, in terms of these fields:

 MncL = 1g20m2R~σ†s[η0(iMnc∂t+∇2)+MncΔ0]~σs−MncmR[c†n†s~σs+~σ†snsc] (35) +1g21m2R~π†s[η1(iMnc∂t+∇2)+MncΔ1]~πs −Mnc2mR(~π†s[n(i\lx@stackrel↔∇)c]12,s+[c†(i\lx@stackrel↔∇)n†]12,s~πs),

where the pieces of that restore Galilean invariance have been suppressed. In terms of these new fields all the coefficients—even those in the “enhanced” interactions which generate shallow bound states (and resonances) are natural. The shallowness of these states is now encoded in the fact that the fields associated with them have non-canonical dimensions: , , and non-canonical wave-function normalization—even at tree level.

### ii.5 Lagrangian: electromagnetic sector

Photons are then included in the Lagrangian via minimal substitution:

 ∂μ→Dμ=∂μ+ie^QAμ. (36)

The charge operator takes different values, depending on whether it is acting on a field or an field. for the neutron, and below we denote the eigenvalue of the operator for the field as . in the case of interest here, where the core is Beryllium-10.

Here our focus is on electric properties (and form factors), and the dominant pieces of the electric response can be derived by looking at how the Lagrangian (8) is affected by the substitution (36). But, at higher orders in the computation of these properties, operators involving the electric field and the fields , , , and which are gauge invariant by themselves contribute to observables. Possible one- and two-derivative operators with one power of the photon field are:

 LEM = −L(σ)C0σ†l(∇2A0−∂0(∇⋅A))σl−L(1/2)E1∑ll′jσlπ†l′(12l12l′∣∣∣1j)(∇jA0−∂0Aj) (37) −L(π)C0π†l(∇2A0−∂0(∇⋅A))πl −L(3/2)E1∑ll′jσl[n(i\lx@stackrel↔∇)c]†3/2l′(12l32l′∣∣∣1j)(∇jA0−∂0Aj).

Note that if magnetic properties are to be considered we would also include operators involving and the neutron, core, and bound-state fields.

The electric interactions in Eq. (37) are gauge invariant by themselves, and so we must determine the order at which they occur. To do this we rewrite the Lagrangian (37) in terms of the rescaled fields (33). In terms of these fields we assume scaling by naive dimensional analysis with respect to the scale of the operators that appear in . We then obtain the following scaling of the coefficients written above:

 L(σ)C0 ∼ R3corel(σ)C0g20m2R, (38) L(1/2)E1 ∼ Rcorel(1/2)E1g0g1m2R, (39) L(π)C0 ∼ Rcorel(π)C0g21m2R, (40) L(3/2)E1 ∼ R4corel(3/2)E1g0mR, (41)

where the parameters are all of order one.

Below we show that the leading effects in the E1() matrix element have parametric dependence . Including the proper wave-function renormalization factors, the operator yields an effect , and so occurs already at NLO in that quantity. Similarly the leading effects in the charge-radius-squared of the state in Be are . The operator proportional to above produces effects in this charge radius of order , and so affects the prediction for the p-wave radius at next-to-leading order. Thus if we desire NLO accuracy for quantities involving the shallow p-wave bound state there are two parameters in the Halo EFT description of the electric structure of the Beryllium-11 which cannot be fixed from Be-neutron scattering information alone.

### ii.6 Fixing parameters

Using the values keV, keV from Ref. AjK90 (), we infer fm, and fm, which are both of the expected size . From the power counting discussed in Sec. II.3, we have at leading order:

 r1=−2γ21a1. (42)

It follows that if we adopt the value extracted in Ref. TB04 () from experimental data, Eq. (1), we have fm. This number should, however, be taken as indicative rather than definitive, since in the end we will fit to data at both LO and NLO, and deduce corresponding values for from our results. But already we see that the order of magnitude estimate implied by our counting is borne out by the numbers.

At NLO, Eq. (42) is corrected to:

 r1=−2γ21a1−2γ1, (43)

which, if we again use Eq. (1) to get an idea of the effect, alters to fm. Such a 30% correction is in line with the anticipated expansion parameter of Halo EFT in the Be system.

In the s-waves the situation is more straightforward: there we count , and . In consequence we can set at LO, and obtain from Eq. (16)

 γ0=1a0. (44)

At leading order, in s-waves we have:

 Zσ=2πγ0m2Rg20 (45)

and all other pertinent results can be obtained by taking the limit of the formulae in Sec. II.2.

Thus, the parameters in Halo EFT for Beryllium-11 bound states at LO are , (or equivalently ), and . At NLO these are to be supplemented by , and the electromagnetic contact interactions for the E1 transition and -state radius.

## Iii Results for bound-state observables

Using Eq. (8) plus minimal substitution (36), we obtain a Lagrangian that describes interactions amongst the core, the neutron, the ground and excited states of the Be nucleus, and photons. In this section we use this Lagrangian to compute the form factor of the and states and the E1 transition from the to the state.

### iii.1 s-wave form factor

The s-wave form factor is computed by calculating the contribution to the irreducible vertex for interactions shown in Fig. 4. This is the only diagram it is necessary to consider at leading order. After the application of wave-function renormalization, the irreducible vertex for the photon coupling to the state is equal to , where is the three-momentum of the virtual photon. (Such an interpretation is valid provided the computation is carried out in the Breit frame, where the four-momentum of the virtual photon .) A straightforward calculation yields:

 G(σ)E(|q|)=2γ0f|q|arctan(f|q|2γ0), (46)

with . Note that , as it should. For the deuteron, we have , and this reduces to the LO result of Ref. Ch99 ().

The electric radius of the s-wave state can be extracted according to:

 G(σ)E(|q|)≡1−16⟨r2E⟩(σ)q2+…, (47)

and an expansion of Eq. (46) in powers of then yields

 ⟨r2E⟩(σ)=f22γ20. (48)

Eq. (48) gives the electric radius of the Be ground state relative to the electric radius of Be. In order to compare with the experimental radius, we therefore have to add our result and the charge radius of Be in quadrature:

 ⟨r2E⟩11Be=⟨r2E⟩10Be+f22γ20. (49)

This relation can be derived by writing the charge distribution of Be as a convolution of the charge distribution of Be with that of the Be-n halo system. Using the convolution theorem for the Fourier transform, one finds that the total rms radius squared can be written as the sum of the squared radii for Be and for the Be-n halo system. The latter effect can be calculated in the Halo EFT. Inserting the value fm obtained in the previous section and using the experimental result for the Be charge radius No09 (), we find fm from this leading-order HEFT computation. This is 2–3% smaller than the atomic physics measurement which yields fm No09 (). In fact, comparing our result for ( fm) with the experimental result for this quantity ( fm), the agreement looks poor. But, this difference is actually consistent with the nominal size of NLO effects when Halo EFT is applied to this system.

At NLO a careful treatment of current conservation, which includes an operator associated with gauging the term in Eq. (8), still yields , but also produces an increased charge radius, as long as , cf. Ref. BS01 (); Ph02 ():

 ⟨r2E⟩11Be=⟨r2E⟩10Be+f22(1−r0γ0)γ20. (50)

Therefore NLO corrections improve the agreement with experiment. The precise size of the increase is fixed once the s-wave effective range is determined, as we shall do in Sec. V below.

We can also obtain from our leading-order calculation a number for the mean-square of the relative core-neutron co-ordinate, , i.e.:

 ⟨r2⟩=12γ20. (51)

To convert this to a neutron radius, we must insert the conversion factor . When this is done we find a LO neutron radius for the Be ground state of:

 ⟨r2n⟩1/2=4.3 fm. (52)

The neutron radius can be measured with a probe that couples only to the neutrons. To a very good approximation, the weak gauge boson constitutes such a probe. Thus one could, in principle, measure the rms neutron radius using parity-violating electron scattering—c.f. the PREX experiment for the case of Lead-208 PREX (). However, a measurement of parity-violating electron scattering from Be is certainly beyond present-day experimental capabilities.

### iii.2 p-wave form factor

In this subsection, we calculate the charge form factor of the excited state in Be. NLO corrections might be expected to be smaller there since its binding energy, and so its typical momentum, is lower. However, as we shall see, a counterterm enters already at NLO in this observable.

The p-wave form factor is computed by calculating the contribution to the irreducible vertex for interactions shown in Fig. 5. There are two diagrams at LO. The first diagram is analogous to that for the s-wave state while the second diagram represents a direct coupling of the photon to the field. The latter contribution is leading order for the p-wave state since the effective range is leading order for this state.

As with the self-energy of the -field, it is easier to compute the irreducible bubble for an photon coupling to the p-wave field . In this way, we find that, after the application of wave-function renormalization, the irreducible vertex for the photon coupling to the state in the Breit frame can be written as:

 ⟨π′s(p′)|J0|πs(p)⟩ = −ieQc∑αβij(12α1i∣∣∣(121)12s)(12β1j∣∣∣(121)12s) (53) δαβ[G(π)E(|q|)δij+12M2ncG(π)Q(|q|)(qiqj−q2δij3)],

where is the three-momentum of the virtual photon. Here we have expressed the form factor in terms of the charge and quadrupole form factors of a vector field. Choosing , and exploiting the fact that the neutron spin is unaffected by the charge operators that can occur up to the order we consider here, a brief calculation shows

 ⟨π′s(p′)|J0|πs(p)⟩=−ieQcδs′sG(π)E(|q|). (54)

The quadrupole form factor is thus unobservable in the state. It could be observed in a state. The lowest state in Be is 2.69 MeV above the ground state. However, this state is a n-Be scattering resonance that corresponds to typical momenta of order and thus is outside the range of applicability of the Halo EFT. Since the situation could be different in other one-neutron halo nuclei, we quote the result for the quadrupole form factor for completeness:

 G(π)Q(|q|)=2M2ncr1+3γ134|q|3f(2|q|fγ1+(|q|2f2−4γ21)arctan(f|q|2γ1)). (55)

In the case of Be, only the charge form factor is observable. A straightforward calculation yields:

 G(π)E(|q|)=1r1+3γ1[r1+1|q|f(2|q|fγ1+(|q|2f2+2γ21)arctan(f|q|2γ1))], (56)

where again . For a strict LO result should be replaced by in these expressions.

We have , as required by charge conservation. The electric radius of the p-wave state relative to the Be ground state can be extracted according to Eq. (47), and we obtain the electric radius for the case from an expansion of Eq. (56) in powers of :

 ⟨r2E⟩(π)=−5f22γ1(3γ1+r1). (57)

At LO in the situation of interest here