Radiative {}^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be} reaction in Halo Effective Field Theory

# Radiative 3He(α,γ)7Be reaction in Halo Effective Field Theory

Renato Higa    Gautam Rupak    Akshay Vaghani Instituto de Física, Universidade de São Paulo, R. do Matão 1371, 05508-090, São Paulo, SP, Brazil
Department of Physics Astronomy and HPC Center for Computational Sciences, Mississippi State University, Mississippi State, MS 39762, U.S.A.
###### Abstract

In this work we study the radiative capture of on within the halo effective field theory framework. At leading order the capture amplitude comprises the initial state -wave strong and Coulomb interactions summed to all orders, and depends on four parameters that can, in principle, be extracted from elastic - scattering alone. At next-to-leading order, - to -wave initial state radiation with non-perturbative Coulomb and two-body currents contribute, with two extra parameters from the latter that are fitted to capture data. We perform three different fits of our parameters to available scattering data and most recent capture data. Our astrophysical -factor, , is slightly above the average in the literature, though consistent within current error bars.

radiative capture, halo nuclei, effective field theory
###### pacs:
25.40.Lw, 25.20.-x
preprint: INT-PUB-16-055

## I Introduction

Low-energy reaction rates involving light nuclei became a recurrent subject nowadays, given their importance in many astrophysical processes. As astronomical observations aim at more accuracy, comparable improvements from experiments and theoretical estimates are called for these reactions. An example is the process that takes place in the interior of our Sun. The astrophysical -factor for this reaction within the Gamov window keV is the main source of uncertainty in the solar neutrino flux detected on Earth. For instance, the flux of neutrinos from the decay of and from the electron capture on are proportional to and , respectively Cyburt and Davids (2008); Bahcall and Ulrich (1988); Adelberger et al. (1998). The first weak decay provides energetic solar neutrinos that were detected by Super-K Fukuda et al. (2001) and SNO Aharmim et al. (2007) but depends also on the reaction rate. Electron capture on , on the other hand, provides a solar neutrino flux three orders of magnitude higher than the former process Bahcall and Serenelli (2005); Bahcall et al. (2005), with less energetic neutrinos possible to be measured by the BOREXINO experiment at Gran Sasso Arpesella et al. (2008), and depends exclusively on the reaction. In either case, the need for a better description of the latter at very low energies is of prime importance in order to improve potential constraints from solar neutrinos, like mass hierarchy, flavor mixing angles and CP violating phases. Besides neutrino physics, the chain reaction is the main source of production during big bang nucleosynthesis (BBN), with a Gamov window . The primordial abundance of calculated from BBN and WMAP cosmic baryon density measurements is a factor of 3 to 4 times larger than observations of metal-poor stars in our galaxy Cyburt and Davids (2008); Cyburt (2004); Fields (2011), which constitutes the so-called lithium problem. Many proposals to solve this puzzle, that involves alternative astronomical measurements and modeling, nuclear, and particle physics, can be strongly constrained with more reliable information on , since the abundance ratio  Cyburt and Davids (2008); Cyburt (2004).

Given its importance to the topics mentioned above, several measurements of the reaction were done in the past (see Adelberger et al. (1998) and references therein) and most recent years Cyburt and Davids (2008); Adelberger et al. (2011). As pointed out in Cyburt and Davids (2008), measurements done prior to the review Adelberger et al. (1998) fall into two discrepant groups—those based on induced activity, and those relying on prompt -ray detection. Due to improvements in detectors and background suppressions, this discrepancy is no longer present in the most recent measurements Adelberger et al. (2011). Nevertheless, error bars are still relatively large in the low-energy regime of astrophysical interest, due to the strong suppression of events by the Coulomb repulsion. The higher energy data where statistics are better shall therefore be theoretically extrapolated down to astrophysically relevant energies in as much less model-dependent way as possible.

The Be nucleus has a predominant He- cluster structure. Its ground state binding energy,  MeV, is considerably smaller than the proton separation energy in He ( MeV) and the energy of the first excited state of the particle ( MeV). The distinct two-cluster configuration of Be, with tight constituents and the low-energy regime one is interested in, make this reaction very suitable for a halo effective field theory (halo EFT) approach. Halo EFT was first formulated in Refs. Bertulani et al. (2002); Bedaque et al. (2003) in their study of the shallow -wave neutron-alpha resonance and applied to other systems, such as the -wave alpha-alpha resonance Higa et al. (2008); Gelman (2009), three-body halo nuclei Canham and Hammer (2008, 2010), coupled-channel proton- scattering Lensky and Birse (2011), electromagnetic transitions Hammer and Phillips (2011) and capture reactions Rupak and Higa (2011); Fernando et al. (2012); Rupak et al. (2012); Zhang et al. (2014a, b); Ryberg et al. (2014); Zhang et al. (2015). In this work, we apply the same ideas to the radiative reaction, following a two-cluster approach of point-like objects at leading order (LO) approximation. Corrections due to the structure of each cluster and higher order electromagnetic interactions are taken into account in perturbation theory. In halo EFT, therefore, a systematic and model-independent expansion of observables is achieved through the use of an expansion parameter —formed by the ratio of a soft momentum scale , associated with the shallowness of the binding of the clusters, and a hard momentum scale , related to the tightness of the cores. Moreover, the formalism guarantees unambiguous inclusion of electromagnetic interactions that preserve the required symmetry constraints, such as gauge invariance. In this study we find that the LO process, given by the sum of initial strong interactions to all orders with intermediate one-body current radiation, provides nearly all of the radiative capture contribution. The LO term depends on four low-energy constants that can be determined, in principle, from the He- elastic scattering alone. A better partial wave analysis of this elastic process with reduced uncertainties would therefore put stronger constraints on the radiative capture. Next-to-leading order (NLO) corrections come from the initial state radiation and electromagnetic two-body currents, this latter bringing two extra low-energy parameters that are fit to capture data. Our -factor values are consistent with other determinations within error bars, though with a slightly higher mean value.

The paper is organized as follows. In Sec. II we briefly comment on the energy scales, degrees of freedom, and channels relevant to the dominant E1 transition, as well as the construction of the corresponding interaction lagrangian. Sec. III presents the main elements necessary to deal with Coulomb interactions between the He and nuclei. Elastic scattering for both physical initial state and bound final state interactions are obtained in the halo EFT framework in Sec. IV. There we also relate the EFT couplings to the effective range parameters and set the power counting. Sec. V collects the relevant expressions of the EFT capture amplitude and cross section, whose numerical results are shown and discussed in Sec. VI. Our concluding remarks are presented in Sec. VII.

## Ii Interaction

The halo EFT we construct treats the Be nucleus as a bound state of point-like nuclear clusters He and . The ground state has a binding energy MeV, and the first excited state has a binding energy MeV. The next excited state of Be is about 3 MeV above the He- threshold tun (). In halo EFT the ground and first excited states are included respectively as and in the spectroscopic notation . The states beyond the first excited states are not included in the low-energy theory. Similarly, only the ground states of He and are relevant at astrophysical energies.

Early works on radiative capture  indicate it is dominated by E1 transition from initial -wave state at low energies. Thus we consider the following Lagrangian for the calculation,

 L= ψ†[i∂0+∇22mψ]ψ+ϕ†[i∂0+∇22mϕ]ϕ +χ(ζ)[j]†[Δ(ζ)+i∂0+∇22M]χ(ζ)[j]+h(ζ)[χ(ζ)[j]†ψP(ζ)[j]ϕ+h.c.], (1)

where the spin-1/2 fermion field represents the He nucleus field with mass MeV, and the scalar field represents the spinless field, with mass MeV. is the total mass. We use natural units with . Note, in the intermediate steps, we display more significant digits than appropriate given theory and fitting errors, in the final result. The projectors and the auxiliary fields carry vector and spinor indices to specify the relevant spin-angular momentum channels for the incoming state , final ground state , and final excited state , described below.

The -wave interaction can be written using a spin-1/2 auxiliary field as:

 χα,s†[Δ(s)+i∂0+∇22M]χα,s+h(s)[χα,s†Pαβ,sψβϕ+h.c.], (2)

where the spinor indices , on the fields and are contracted using the diagonal -wave projector . The spinor index . The two -wave couplings and can be fitted to scattering length and effective range for elastic scattering of He and . We discuss this in more detail when we consider the relevant power counting.

For the final state, we want to project the vector index , 2, 3 for the -wave and the spinor index for the He spin into total angular momentum and pieces. This can be done for a generic auxiliary field as follows:

 χαi=13(σiσj)αβχβj+[δijδαβ−13(σiσj)αβ]χβj, (3)

where the two pieces are the irreducible forms representing the and channels respectively. The Pauli matrices ’s act on the spinor indices. The two -wave interactions can then be written as

 χα,ζi†[Δ(ζ)+i∂0+∇22M]χα,ζi+√3h(ζ)[χα,ζi†Pαβ,ζijψβ(\lx@stackrel→∇mϕ−\lx@stackrel←∇mψ)kϕ+h.c.], (4)

where the -wave projectors are

 Pαβ,ζij =13(σiσj)αβfor ζ=2P1/2, (5) and  Pαβ,ζij =δijδαβ−13(σiσj)αβfor ζ=2P3/2.

The two couplings , in each of the two -wave channels can be determined from the corresponding binding momentum and effective range. For bound states, both of these couplings contribute at leading order Bertulani et al. (2002); Bedaque et al. (2003). This remains true even in the presence of long-range Coulomb interaction as we have here.

The capture calculation proceeds through E1 transition. At LO, we couple the external photon through minimal substitution, that corresponds to gauging the momentum of the charged particle, , where is the charge number. for both the He and nuclei. We include the long-range Coulomb interaction between the He and nuclei to all orders in perturbation by summing the Coulomb ladder as described below. At NLO, two-body currents that are not related to elastic scattering operators by gauge invariance contribute. Two-body currents that contribute to E1 transition between -wave, and -wave ground and excited states can be written using the auxiliary fields as

 eL(ζ)E1√3h(s)h(ζ)χα,ζi†Pαβ,ζijχβ,sEj. (6)

We include factors of , in the definition of the coupling for convenience. In the absence of Coulomb interaction this reduces to a factor of that has been suggested earlier Beane and Savage (2001).

For scattering of two charged particles He and at low energy, the relevant quantity that provides the strength of Coulomb photon exchanges is the Sommerfeld parameter , where and are the charge number of He and , respectively, is the electromagnetic fine structure constant, is the reduced mass, and is the relative center-of-mass (c.m.) momentum. The inverse of the Bohr radius of the system defines the momentum scale and the Sommerfeld parameter is written as the ratio . Each photon exchange is proportional to . In the high-energy side where photon exchanges can be treated in perturbation theory. However, in the low momentum region that we are interested in, , multiple photon exchanges contribute at least at the same order, forcing the summation of Coulomb ladder diagrams, Fig. 1.

The Coulomb scattering amplitude satisfies the following useful relations

 |χ(±)p⟩ =|p⟩+ˆG(±)0ˆTC|p⟩, (7) ˆG(±)C =ˆG(±)0+ˆG(±)0ˆTCˆG(±)0,

where is a plane wave state in the two-particle center-of-mass (c.m.) system and () is the incoming (outgoing) Coulomb scattering state. The free and Coulomb Green’s functions as operators are respectively written as

 ˆG(±)0=(E−ˆH0±iϵ)−1andˆG(±)C=(E−ˆH0−ˆVC±iϵ)−1. (8)

The sign in the definitions correspond to retarded and advanced Green’s functions. Taking expectation values between final and initial momentum states, one can derive several useful relations,

 G(±)0(E;p′,p) ≡⟨p′|ˆG(±)0|p⟩=(2π)3δ(p′−p)E−p′2/(2μ)±iϵ, (9) TC(E;p′,p)E−p′2/(2μ)±iϵ ≡⟨p′|ˆG0ˆTC|p⟩=χ(±)p(p′)−(2π)3δ(p′−p), G(±)C(E;p′,p) ≡⟨p′|ˆG(±)C|p⟩=G(±)0(E;p′,p)+TC(E;p′,p)[E−p′2/(2μ)±iϵ][E−p2/(2μ)±iϵ],

with . The diagrammatic relation between and from Fig. 1 can be expressed as

 tC(E;p′,p) =(2π)3δ(p′−p)(E−p′22μ±iϵ)+TC(E;p′,p), (10)

and helps write in terms of and .

The Coulomb wave function and the retarded Green’s function are known in closed form in coordinate space Abramowitz and Stegun (1965); Thompson (),

 ~χ(±)p(r) ≡e−ηpπ2Γ(1±iηp)1F1(∓iηp,1;±ipr−ip⋅r)eip⋅r, (11) =∞∑l=0(2l+1)ileiσlPl(^r⋅^r′)Fl(ηp,pr)pr, G(+)C(E;r′,r) ≡∞∑l=0(2l+1)Pl(^r⋅^r′)G(l)C(E,r′,r), G(l)C(E,r′,r) =−iμp2πFl(ηp,r′p)r′pFl(ηp,rp)rp,

where

 Fl(ηp,ρ) =Cl(ηp)ρl+1e−iρM(l+1−iηp,2l+2,+2iρ), (12) H(+)l(ηp,ρ) =(−i)leπηp/2√Γ(l+1+iηp)Γ(l+1−iηp)W−iηp,l+1/2(−i2ρ), Cl(ηp) =2le−πηp/2|Γ(l+1+iηp)|Γ(2l+2),

with conventionally defined Kummer and the Whittaker functions. is the regular Coulomb wave function, the irregular wave function is given by , and is the Coulomb phase shift. We define the Coulomb Green’s function for a bound state with binding energy as

 G(l)C(−B,r′,r)=−iμγ2πFl(ηiγ,iγr′)iγr′H(+)l(ηiγ,iγr)iγr, (13)

where is the binding momentum. The coordinate space definitions assume , In the limit for charged neutral particles (), we recover the expected result

 G(0)C(−B,r′→0,r)∼−μ2πre−γr. (14)

## Iv Elastic scattering

The elastic scattering amplitude in the presence of both short-range strong and long-range Coulomb interaction is traditionally written as

 T(E;p′,p)=TC(E;p′,p)+TSC(E;p′,p), (15)

where the purely Coulomb contribution can be written as

 TC(E;p′,p) =∞∑l=0(2l+1)T(l)C(E;p)Pl(^p⋅^p′)=−2πμ∞∑l=0(2l+1)e2iσl−12ipPl(^p⋅^p′), (16)

using the incoming (outgoing) c.m. momentum ().

The on-shell Coulomb subtracted amplitude can also be expanded in partial waves as

 T(l)SC =−2πμe2iσlpcotδl−ip, (17)

where the full phase shift is simply . The Coulomb subtracted phase shift is usually expressed in terms of a modified effective range expansion

 [Γ(2l+2)2lΓ(l+1)]2[Cl(ηp)]2p2l+1(cotδl−i) =−1al+12rlp2−2kCp2lΓ(l+1)2|Γ(l+1+iηp)|2|Γ(1+iηp)|2H(ηp), (18) H(η) =ψ(iη)+12iη−ln(iη),

with the digamma function.

The amplitude is given in EFT by the set of diagrams in Fig. 2. The -wave amplitude for c.m. incoming momentum and outgoing momentum with and can be written as

 −iT(l=0)SC =(2μ)2[∫d3k(2π)3tc(E;p′,k)k2−p2−iϵ][−i(h(s))2D(s)(E,0)][∫d3l(2π)3tc(E;l,p)l2−p2−iϵ] (19) =−i[h(s)]2D(s)(E,0)~χ(−)∗p′(0)~χ(+)p(0)=−i[h(s)]2D(s)(E,0)C20(ηp)ei2σ0,

where the dressed dimer propagator is given by

 D(s)(p0,p=0) =1Δ(s)+p0−[h(s)]2J0(√2μp0), (20) J0(√2μE) =J0(p)=GC(E;r=0,r′=0)=−2μ∫d3q(2π)31[q2−p2−iϵ]2πηq[e2πηq−1].

Comparing the modified ERE in Eq. (18) to the -wave EFT expression, we can determine the two couplings in terms of and ,

 Δ(s) (21) 2 =−2πμ2r0,

where and is the renormalization scale within the power-divergence subtraction (PDS) scheme. The elastic scattering amplitude requires non-perturbative treatment of the Coulomb photons at low energies. However, it is not clear if the short-range interaction contained in the parameters , , etc., should be included in perturbation or not. Fitting the EFT expression to both elastic and capture data later, we find that both the scattering length and the effective range contribute at leading order. We propose a power counting where is fine tuned and is of natural size, which is very similar to the -wave - scattering Higa et al. (2008). Then the -wave scattering amplitude gets LO contributions from , for small momenta , and the contribution from a natural sized shape parameter , is suppressed by factors of . This is the largest source of theory error in the calculation.

The -wave amplitude is written as

 −iT(1)SC=−i[h(ζ)]2D(ζ)(E,0)[∇a~χ(−)∗p′(0)][∇a~χ(+)p(0)], (22)

with the -wave dressed dimer propagator written as

 D(ζ)(E,0)= 1Δ(ζ)+E−3[h(ζ)]2J1(p), (23) J1(p)= −2μn∫dnk(2π)n1k2−p2−iδ[∇a~χ(+)∗k(0)][∇a~χ(+)k(0)].

is given by a divergent integral that we regulate through dimensional regularization in space dimensions. Using the relations

 [∇a~χ(−)∗p(0)][∇a~χ(+)k(0)]= e−πηpΓ(2+iηp)2p′⋅p, (24) [∇a~ψ(+)k(0)]= (k2+β2)2πηke2πηk−1,

the PDS prescription for the integrals, and Eq. (18), one gets

 Δ(ζ) =h(ζ)22πμa1−h(ζ)22πμ{2k3C[1D−4−ln(λ√π2kC)−1+32CE] +k2C(3λ2−2kC3)+8π2k3Cζ′(−2)+π2λk2C2−3πλ2kC2+πλ32}, (25) 1[h(ζ)]2 =−r12π−1π{2kC[1D−4−ln(λ√π2kC)−1+32CE]+(3λ2−2kC3)}.

The dressed -wave dimer propagator defines the wave-function renormalization constant via

 1Z(ζ)=∂∂p0[D(ζ)(p0;\boldmathp)−1]∣∣p0=p2/(2μ)−B(ζ), (26)

where is the -wave binding energy. Straightforward calculation leads to

 −2πh(ζ)2Z(ζ) =r(ζ)1−4kCH(−ikCγ)−2k2Cγ3(k2C−γ2)[ψ′(kCγ)−γ22k2C−γkC]. (27)

We assign the c.m. momenta to the particle, and to the outgoing photon in the final state. From energy-momentum conservation in the EFT power counting. Thus in a typical loop-calculation a combination such as is approximated as where and is the loop energy-momentum. This approximation corresponds to zero-recoil of the final bound state Be. We count and neglect recoil effects in this calculation up to NLO.

In this section we present some of the Feynman diagrams that contribute to the capture process, see Figs. 3,  4,  5. In the next section when we present our analysis, we elaborate more on the power counting and discuss how these set of diagrams constitutes the EFT contribution up to NLO. We consider E1 transitions from the initial state to both final bound states, ground and excited . The external photon is minimally coupled to the charged clusters at LO.

The first set of diagrams only include Coulomb interactions for the incoming charged particles He and . We find

 (a1)+(a2) =−eiσ0A(p)μΓ(ζ)aa, (28) (a3) =−~χ(+)p(r=0)Γ(ζ)aa=−C0(ηp)eiσ0Γ(ζ)aa, A(p) =2γμ3Γ(2+kC/γ)∫∞0dr rW−kC/γ,3/2(2γr)∂r[F0(kC/p)pr],

where the index refers to the final state -wave bound states. The corresponding binding momentum is given by and for the ground and the first excited states, respectively. The Whittaker function is associated with the final -wave bound state, and the -wave Coulomb wave function is associated with the initial incoming scattering state.

The projection onto the -wave states is given by

 Γ(ζ)ab=(eZϕmϕ−eZψmψ)(h(ζ)√3√Z(ζ)√2mϕ)ϵ∗aUα,ζi(−→k)Pαβ,ζibUβ,ψ(−→p), (29)

where is the photon polarization vector, is given by Eq. (27), is the spinor field for the -wave final state with mass , and is the spinor field for the incoming He nucleus. The spinor fields satisfy the completeness relations

 Uα,ζi(p)[Uβ,ζj(p)]∗ =2MPαβ,ζij, (30) Uα,ψ(p)[Uβ,ψ(p)]∗ =2mψδαβ,

where , are vector indices, and , are spin indices.

The second set of diagrams from Fig. 4 involve initial state short-range interaction that is constrained by the -wave phase shift through the ERE. We find

 (b1)+(b2) =2πμC0(ηp)eiσ0−1a0+r0p2/2−2kCH(ηp)Bab(p)μΓab, (31) Bab(p) =3∫d3r∂G(+)C(E;0,r)∂rarbr⎡⎣G(1)C(−B;r,r′)r′⎤⎦∣∣r′=0, =−μγ6πrΓ(2+kC/γ)W−kC/γ,3/2(2γr), G(+)C(E;0,r) =−μ2πrΓ(1−ikC/p)WikC/p,1/2(−i2pr).

The integral is divergent, which is rendered finite when combined with the contribution from the third diagram,

 (b3) =2πμC0(ηp)eiσ0−1a0+r0p2/2−2kCH(ηp)J0(p)Γaa. (32)

We regulate the divergences using PDS, which is most conveniently done in this calculation in momentum space. The divergences come from zero and single Coulomb photon exchanges. Thus we analytically calculate the divergent pieces perturbatively up to and calculate the rest (more than a single Coulomb photon) that is not divergent numerically:

 Bab(p) ≡B(0)ab(p)+αeB(1)ab(p)+ΔBab(p), (33) B(0)ab ≡B(0)δab=μ2[λ2π+ip3π−γ23πip+γp2+γ2]δab, αeB(1)ab ≡αeB(1)δab=−kCμ22π[1ϵ+lnπλ24k2C−γE−23+ln4π]δab+kCC(p)δab, C(p) =μ26π2(p2+γ2)∫10dx∫10dy1√x(1−x)√1−y ×(xp2ln[π4k2C(−yp2+(1−y)γ2/x−iδ)] +p2ln[π4k2C(−yp2−(1−y)p2/x−iδ)] +xγ2ln[π4k2C(yγ2+(1−y)γ2/x−iδ)] +γ2ln[π4k2C(yγ2−(1−y)p2/x−iδ)]).

The double integral can be reduced further to a single integral that we evaluate numerically. The finite piece is evaluated numerically where we use spherical symmetry to write in the integral. Consequently,