Coupled-channel treatment of 7 Li(n,γ ) 8 Li in effective field theory
CCoupled-channel treatment of Li( n, γ ) Li in effective field theory
Renato Higa, ∗ Pradeepa Premarathna, † and Gautam Rupak ‡ Instituto de F´ısica, Universidade de S˜ao Paulo,R. do Mat˜ao Nr.1371, 05508-090, S˜ao Paulo, SP, Brazil Department of Physics & Astronomy and HPC Center for Computational Sciences,Mississippi State University, Mississippi State, MS 39762, USA
Abstract
The E1 contribution to the capture reaction Li( n, γ ) Li is calculated at low energies. We employa coupled-channel formalism to account for the Li (cid:63) excited core contribution. The halo effectivefield theory calculations show that the contribution of the Li (cid:63) degree of freedom is negligible atmomenta below 1 MeV and significant only beyond the 3 + resonance energy, though still compatiblewith a next-to-next-to-leading order correction. A power counting that accounts for the size of thiscorrection is proposed. We compare our formalism with a previous halo effective field theory[Zhang, Nollett, and Phillips, Phys. Rev. C , 024613 (2014)] that also treated the Li (cid:63) core asan explicit degree of freedom. Our formal expressions and analysis disagree with this earlier workin several aspects. Keywords: Coupled-channel, excited core, radiative capture, halo effective field theory ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ nu c l - t h ] S e p . INTRODUCTION Radiative capture reactions with light nuclei sharpen our knowledge about the primordialevolution of our universe, the fuel in the interior of stars, and explosive phenomena of astro-physical objects [1–3]. Among them features the long-studied Be( p, γ ) B reaction, crucialin determining the flux of solar neutrinos that oscillate into different lepton flavors on theirway to detection in Earth. The impact to neutrino oscillations and to the standard solarmodel of this reaction depends on the information of the respective cross section around theGamow energy ∼
20 keV. The Coulomb repulsion at such low energies makes it extremelydifficult, if not impossible, for direct measurements in laboratory, which are presently lim-ited to as low as ∼
100 keV. Therefore, theoretical guidance for reliable extrapolations ofexperimental data down to energies of astrophysical interest is an unavoidable necessity. Themirror-symmetry, backed by the accidental isospin symmetry of quantum chromodynamics(QCD) at low energies, is often invoked to constrain the strong nuclear part of the reaction.Besides the mirror connection, the Li( n, γ ) Li reaction may reveal its importance in someastrophysical scenarios, such as inhomogeneous Big-Bang nucleosynthesis or neutron-richexplosive environments [1, 4].Experimental studies on the Li( n, γ ) Li reaction date back to the 40’s [5] followed bya handful others between the 90’s and 2010’s [6–10]. Due to the absence of the Coulombrepulsion, many measurements were done in the sub-keV region with good precision, thusserving as testbed for theoretical low-energy extrapolations. Theoretical descriptions of thisreaction vary in degree of sophistication and accuracy — ab initio /microscopic models [11–14] are tied to the choice of two- and three-nucleon interactions and numerical precision ofthe method whereas two-body [15–22] and three-body [23, 24] cluster configuration withphenomenological potentials are much simpler and more flexible with adjustable parametersto data. Halo/cluster effective field theory (halo EFT) relies on the same cluster approach,but uses quantum field theory techniques and a small momentum ratio to provide a model-independent, systematic and improvable expansion with controlled theoretical uncertainties.Halo EFT ideas emerge from Refs. [25, 26] and were extended to several loosely-boundnuclear systems (see Refs. [27–29] and references therein).Halo EFT was first applied to Li( n, γ ) Li reaction in [30], which pointed out inconsisten-cies of potential-model extrapolations of fitted high-energy data to low energies, its origin,and ways to overcome it. The follow-up work [31] included E1 capture to the 1 + stateof Li and M1 capture from the initial 3 + resonant state, while the effects of the − Liexcited-core state was argued to be of higher-order. A subsequent work by Zhang, Nollett,and Phillips [32] combined halo EFT formalism with asymptotic normalization coefficients(ANCs) obtained from ab initio variational Monte Carlo (VMC) method. Zhang et al. con-sidered only E1 transitions, but included the − excited state of Li as an explicit degreeof freedom. Their overall results are qualitatively similar to ours within the leading-order(LO) theoretical uncertainties. They also find the contribution of the excited core of Licompatible with a next-to-leading order (NLO) correction, meeting the expectations of ourinitial assumptions [31].Nevertheless, Zhang et al. raised critical comments to our work, namely, (a) that theexcited state of Li must be formally included in a LO calculation given its excitationenergy of ∼ . ∼ Li, (b) thatthe couplings of the different final state spin channels were made equal, which influences therate of total initial spin S i = 2 contribution to the capture reaction, and (c) that capture2o the 1 + excited state of Li was not a prediction, but an input to constrain our EFTparameters. Point (c) was our choice and true in certain sense, though the best we cando within halo EFT itself: in [31] we fixed two remaining renormalization constants (afterfixing the usual ones from scattering lengths and binding energies) to just two input data,the thermal neutron capture cross section to the ground 2 + and first excited 1 + states of Li [6], and postdict capture data at other energies. Zhang et al. , on the other hand, fixtheir remaining renormalization constants from their ab initio model. Answer to points (a)and (b) was the motivation of the present work.We include the − excited state of Li core ( Li (cid:63) ) as an explicit degree of freedom ina coupled-channel formalism, along similar lines of Refs. [33, 34]. In our formulation, the Li (cid:63) excited core influences only the channel with total spin S = 1, which is regarded asa subleading contribution. Our capture calculations show that the contributions from theexcited core degree of freedom are negligible for momenta in the keV regime, and startsbeing noticeable only beyond the 3 + resonant state of Li, albeit compatible with a typicalnext-to-next-to-leading (NNLO) correction. A power-counting is proposed that naturallyincorporates the NLO contributions of the S = 1 channel and Li (cid:63) excited core, the latterwith kinematic imprints only at NNLO. We also compare our coupled-channel expressionsand the respective low-energy expansions with those from Zhang et al. and comment on thediscrepancies found.The paper is organized as follows. In Sec. II we present the framework of halo EFT for twoscenarios: without [30, 31] and with the Li (cid:63) excited core as an explicit degree of freedom, thelatter with the pertinent channels that couple. There we highlight the differences betweenour and Zhang et al. ’s formulations for the Li (cid:63) excited core in EFT. Section III containsthe relevant formulas for the Li( n, γ ) Li capture reaction in both theories with and without Li (cid:63) . A survey of possible families of EFT parameters compatible with given observables isdiscussed in Sec. IV, followed by the proposed power-counting in Sec. V. Numerical resultsand analyses are presented in Sec. VI with concluding remarks in Sec. VII. II. INTERACTION
In this section we construct two halo EFTs: one without the explicit Li (cid:63) degree offreedom that we refer to as just EFT and one with that we refer to as EFT (cid:63) . In the former,the low-energy degrees of freedom would be the spin-parity
12 + neutron, the − ground stateof Li, and the photon. In the latter we also have the − excited Li (cid:63) state as an additionaldegree of freedom. The 2 + ground and 1 + excited states of Li are represented as p -wavebound states of the neutron and the Li/ Li (cid:63) core with binding momenta γ ≈ .
78 MeV and γ ≈ .
56 MeV, respectively [35]. We identify the relative center-of-mass (c.m.) momentum p and the binding energies with a low momentum scale Q ∼ γ ∼ γ (cid:38) p . The breakdownscale Λ ∼
150 MeV is associated with pion physics, He- H break up of the core, etc [30, 31].The calculation is organized as an expansion in the small ratio Q/ Λ ∼ /
3. We only considerthe non-resonant E1 capture. The M1 contribution from the 3 + resonance initial state hasbeen considered in Ref. [31], and would be revisited in a future publication [36].A radiative capture calculation requires a description of the initial scattering states, thefinal bound states, and the electroweak transition operators. The dominant contributionto capture at low momentum is from initial s -wave states to the final p -wave bound statethrough the E1 transition. The electroweak operators relevant to our calculation are one-3ody currents produced through minimal substitution by gauging incoming momenta q → q + eZ c A where Z c = 3 is the charge of the Li/ Li (cid:63) core. Thus we start with a frameworkfor the strong interaction that describes the initial scattering states, the final bound states,and the E1 operators.The strong interaction in the first halo EFT is given by the Lagrangian L = N † (cid:20) i∂ + ∇ m n (cid:21) N + C † (cid:20) i∂ + ∇ m c (cid:21) C + χ ( ζ )[ j ] † (cid:20) ∆ ( ζ ) + h ( ζ ) (cid:18) i∂ + ∇ M (cid:19)(cid:21) χ ( ζ )[ j ] + (cid:114) πµ (cid:104) χ ( ζ )[ j ] † N T P ( ζ )[ j ] C + h . c . (cid:105) , (1)where N represents the
12 + neutron with mass m n = 939 . C represents the − Li corewith mass m c = 6535 . M = m n + m c is the total mass, and µ = m n m c /M the reduced mass. We use natural units (cid:126) = 1 = c . The dimer fields χ ( ζ )[ j ] are auxiliarythat are introduced for convenience. Repeated indices are summed over. The neutron and Li core interact with each other through the exchange of the χ ( ζ )[ j ] fields. These auxiliaryfields can be integrated out of the theory to generate neutron-core contact interactions. Theunknown couplings h ( ζ ) are included in the dimer propagator [39], for convenience, insteadof an equivalent formulation where they appear in the dimer-particle interaction [30, 31]. P ( ζ )[ j ] are the projectors for the relevant s - and p -wave channels in the spectroscopic notation S +1 L J : S , S , P , P , P , P [30, 31] indicated by the superscript ζ . We providetheir explicit form in Appendix A. The subscript [ j ] is a single index or double indices asappropriate for J = 1 and J = 2 states, respectively. For example, with ζ = P one shouldread χ ( ζ )[ j ] = χ ( P ) ij and P ( ζ )[ j ] = P ( P ) ij . The couplings ∆ ( ζ ) , h ( ζ ) can in principle be relatedto elastic scattering phase shifts if available. The E1 contribution to Li( n, γ ) Li has beencalculated using this theory in Refs [30, 31]. We will present the results in Section III.The second halo EFT with excited Li (cid:63) core (EFT (cid:63) ) can be described with the Lagrangian L (cid:63) = N † (cid:20) i∂ + ∇ m n (cid:21) N + C † (cid:20) i∂ + ∇ m c (cid:21) C + C † (cid:63) (cid:20) i∂ − E (cid:63) + ∇ m c (cid:21) C (cid:63) + χ ( ζ )[ j ] † (cid:20) Π ( ζζ (cid:48) ) + t ( ζζ (cid:48) ) (cid:18) i∂ + ∇ M (cid:19)(cid:21) χ ( ζ (cid:48) )[ j ] + (cid:114) πµ (cid:104) χ ( ζ )[ j ] † N T P ( ζ )[ j ] C + χ ( ζ (cid:48) )[ j ] † N T P ( ζ (cid:48) )[ j ] C (cid:63) + h . c . (cid:105) . (2)Here the C (cid:63) field represents the excited Li (cid:63) core with excitation energy E (cid:63) = 0 . S (cid:63) , P (cid:63) , and P (cid:63) involving the C (cid:63) field. We also allow for the possibility of mixing between channelswhich is induced by the off-diagonal terms in the dimer field χ ( ζ )[ j ] (inverse) propagators. Thecouplings Π ( ζζ (cid:48) ) and t ( ζζ (cid:48) ) can again be related to scattering phase shifts if available. Thegeneric index ζ is used to represent all the channels, and one has to appropriately considerinteractions only in the relevant channels as discussed below. We associate a low-momentumscale γ ∆ = √ µE (cid:63) ≈ . ∼ Q with the excited core [32].We consider mixing in three pairs of channels: S - S (cid:63) , P - P (cid:63) and P - P (cid:63) . Howeverthe formalism can be extended to other scattering channels as well. One could do the same to4nclude mixing between all S = 1 and S = 2 p -wave channels. However, given that capturein the S = 2 channel is 4 times that in the S = 1 channel [41–43], any possible mixingbetween the different spin channels would be subleading, and beyond the order consideredin this work. A. S - S (cid:63) Coupled-Channel
A coupled-channel calculation involving s -wave states was presented in Ref. [33]. SeeRef. [34] as well for a coupled-channel calculation with and without Coulomb interaction for s -wave bound states. Here we present a slightly different derivation using the dimer fieldsinstead of nucleon-core contact interactions. Some of the renormalization conditions are alittle different but the final results expressed in terms of scattering parameters are the same.The coupled-channel s -wave scattering amplitude is a 2 × i A ( ab ) ( p ) = − πµ i D ( ab ) ( E, , (3)where E = p / (2 µ ) is the c.m. energy and the superscripts are the row-column indices ofthe amplitude matrix. We identify the S state as channel 1, and the S (cid:63) state as channel2. The dimer propagator is given by the integral equation D ( E,
0) = D ( E,
0) + D ( E, E, D ( E, , (4)which is conveniently calculated from its inverse D − = D − − Σ , (5)where D − is the inverse free dimer propagator and Σ is the self-energy. We have the freeinverse dimer propagator directly from Eq. (2):[ D ( E, − = (cid:18) Π (11) Π (12) Π (12) Π (22) (cid:19) , (6)where we only kept the couplings Π ( ij ) in a low-momentum expansion. In a single-channelcalculation this would correspond to keeping only the scattering length contribution. Theself-energy is − Σ( E,
0) = − πµ (cid:18) J ( (cid:112) − p − i + ) 00 J ( (cid:112) − p + γ − i + ) (cid:19) ,J ( x ) = − µ (cid:18) λ (cid:19) − D (cid:90) d D − q (2 π ) D − q + x = − µ π ( λ − x ) , (7)where λ is the renormalization group (RG) scale. We use dimensional regularization in theso-called power divergence subtraction (PDS) scheme [44] that removes all divergences inspace-time dimensions D ≤
4. The scattering amplitude has to be independent of λ whichcan be accomplished with the renormalized couplingsΠ ( ij ) = 1 a ij − λδ ij , (8)5here we introduced the scattering lengths a ij following Ref. [33]. Then we get[ D ( E, − = a + ip a a a − (cid:112) − p + γ − i + , (9)and in particular the s -wave amplitude A (11) ( p ) = 2 πµ − a + (cid:112) − p + γ − i + ( − a − ip )( − a + (cid:112) − p + γ − i + ) − a , (10)which reduces to the single-channel amplitude when the mixing coupling ∆ = 1 /a is setto zero. The off-diagonal amplitude mixing channels 1 and 2 is A (12) ( p ) = 2 πµ /a ( − a − ip )( − a + (cid:112) − p + γ − i + ) − a . (11)These expressions agree with Eq. (2.19) from Ref. [33].At momenta p (cid:28) γ ∆ , the scattering amplitudes should be analytic around p = 0, and inparticular A (11) should be given by the effective range expansion (ERE). So we need − a − a − / ( − a + (cid:113) − p + γ − i + ) ≈ − a + a a − − a γ ∆ − a a − γ ∆ (1 − a γ ∆ ) p + . . . = − a (1)0 + 12 r (1)0 p + . . . , (12)where a (1)0 and r (1)0 are the scattering length and effective range in the S channel, respec-tively. Matching the EFT expression to the ERE one obtains a = a (1)0 − a γ ∆ a ( a (1)0 a − − γ ∆ ) ,a − = − r (1)0 γ ∆ (1 − a γ ∆ ) a . (13)We can fix a from a (1)0 and a from r (1)0 , leaving a as an undetermined parameter whichcould in principle be obtained from the low-momentum measurement of A (12) [45]. Oneshould note that in the halo EFT (cid:63) with Li (cid:63) , an effective range r (1)0 is generated dynamicallythough we started with a momentum-independent interaction because there is a momentumassociated with the excitation energy of the core. Ref. [33] considered the situation wherethe scattering lengths a ij ∼ /Q are fine-tuned. Then it produces a fine-tuned negativeeffective range | r (1)0 | ∼ /Q as well for γ ∆ ∼ Q . In the Li system a (1)0 = 0 . / Λ. Thus we assume that all the scattering lengths a ij ∼ / Λ areof natural size as well which still produces a fine-tuned negative effective range | r (1)0 | ∼ /Q .6hen we can expand in the Q/ Λ ratio and write A (11) ( p ) = 2 πµ − a (1)0 + r (1)0 γ ∆ 1 − a γ ∆ a − r (1)0 γ ∆ (1 − a γ ∆ ) a (1 − a √ − p + γ − i + ) − ip ≈ − πµ a (1)0 (cid:20) − ia (1)0 p + a (1)0 r (1)0 γ − a (1)0 r (1)0 γ ∆ (cid:113) − p + γ − i + + . . . (cid:21) , A (12) ( p ) = 2 πµ a (1)0 (cid:113) − r (1)0 γ ∆ (1 /a − γ ∆ ) × (cid:20) a − (cid:113) − p + γ − i + − a (1)0 r (1)0 γ ∆ ( 1 a − γ ∆ )( γ ∆ − (cid:113) − p + γ − i + )+ ia (1)0 p ( 1 a − (cid:113) − p + γ − i + ) (cid:21) − ≈ πµ a (1)0 a (cid:115) − r (1)0 γ ∆ a (cid:20) − ia (1)0 p + (cid:18) γ ∆ − (cid:113) − p + γ − i + (cid:19) × (cid:16) a (1)0 r (1)0 γ ∆ − a (cid:17) + . . . (cid:105) . (14)The Q/ Λ expansion in Eq. (14) implies that the non-analyticity from the open channelinvolving the excited core Li (cid:63) is a subleading effect. B. P - P (cid:63) Coupled-Channel
The coupled-channel calculation for p -wave states is very similar to the s -wave states insubsection II A. An important difference is that for p -waves we need effective momentumcorrections at LO [25, 26]. Now we write the free inverse dimer propagator as[ D ( E, − = (cid:18) Π (11) + Et (11) Π (12) + Et (12) Π (12) + Et (12) Π (22) + Et (22) (cid:19) , (15)where we identify P as channel 1 and P (cid:63) as channel 2. The p -wave self-energy term isgiven by − Σ( E,
0) = − πµ (cid:18) J ( (cid:112) − p − i + ) 00 J ( (cid:112) − p + γ − i + ) (cid:19) ,J ( x ) = − µD − (cid:18) λ (cid:19) − D (cid:90) d D − q (2 π ) D − q q + x = − µ π (cid:18) x − x λ + π λ (cid:19) . (16)The RG conditions (no sum over repeated indices intended) µ Π ( ij ) = 1 a ij − π λ δ ij + 32 γ λδ i δ ij ,µ t ( ij ) = − µ ( r ij + 3 λδ ij ) , (17)make the inverse propagator (and the scattering amplitude) λ -independent:[ D ( E, − = 1 µ a − r p + ip a − r p a − r p a − r p + ( − p + γ − i + ) / , (18)7here a ij are the p -wave scattering volumes, and r ij the p -wave effective momenta. The p -wave coupled-channel amplitude is given by A ( p ) = − πµ p µ D ( E, . (19)The wave function renormalization constant that enters the capture calculation is[ Z ] − = ddE [ D ( E, − (cid:12)(cid:12)(cid:12) E = − B = − µ (cid:18) r + 3 γ r r r + 3 (cid:112) γ + γ (cid:19) , πµ Z = − π √ γ + γ + r (3 γ + r ) (cid:16) √ γ + γ + r (cid:17) − r − r (3 γ + r ) (cid:16) √ γ + γ + r (cid:17) − r − r (3 γ + r ) (cid:16) √ γ + γ + r (cid:17) − r γ + r (3 γ + r ) (cid:16) √ γ + γ + r (cid:17) − r , (20)with binding momentum γ = √ µB . The wave function renormalization constant reducesto the single-channel result [30, 31] when r vanishes. In our calculation we will not assume r to be small. We will simply fit the constants Z , Z to the ANCs without attemptingto interpret what it implies in terms of the scattering parameters because the two P , P (cid:63) ANCs are not sufficient to determine the three effective momenta r ij . The coupled-channel P - P (cid:63) calculation is very similar.The calculation by Zhang et al. [32] differs from our result for the Li (cid:63) contribution inseveral manners. In the s -wave initial states, Ref. [32] uses a very specific interaction wherethe scattering amplitude A (12) in the inelastic channel S - S (cid:63) is assumed to be determinedonce the amplitudes A (11) and A (22) are given. In particular, they use a single dimer field thatcouples to the S and S (cid:63) states with two different couplings. By construction, their theoryuses two couplings to describe the coupled-channel calculation. When the auxiliary dimerfield is integrated out of the field theory, it will generate neutron-core contact interactionswhere the S - S (cid:63) coupling is simply the geometric mean of the couplings in the S and S (cid:63) channels. Accordingly, their corresponding scattering length a is not a free parameter butgiven by a , and a . The coupled-channel calculations presented here and in Refs. [33, 34],in contrast, require three renormalized couplings at LO that depend on three independentscattering lengths a , a , and a . A priori there is no expectation that a is a functionof a and a .In the final p -wave 2 + ground state calculation, the interaction used by Zhang et al. [32]produces mixing between all p -wave channels: P , P and P (cid:63) , see Eq. (14) in Ref. [32].The interaction they use is such that the amplitudes in the inelastic channels are specifiedonce the elastic channels are known. The wave function renormalization constant is writtenin terms of a single p -wave effective momentum. However, we note that the renormalizationcondition imposed in deriving Eq. (15) of Ref. [32] for the wave function renormalizationconstant ignores the contribution from the mixing between P - P that is present in theirformulation. This would affect their result. In contrast, we consider only a coupled-channelcalculation in P - P (cid:63) for the 2 + ground state of Li. The coupled-channel p -wave amplitudesdepend on three scattering volumes a ij and three effective momenta r ij . The associated wavefunction renormalization constant depends on the three effective momenta r ij . A coupled-channel calculation mixing three channels in general would involve 6 effective momenta forthe wave function renormalization, and not one effective momentum as suggested [32]. Inthe EFT without explicit Li (cid:63) core, the P and P channels do not mix and we require twoindependent effective momenta [30, 31]. There are some other minor technical differences in8ur calculations such that we subtract both the linear and cubic divergences for the p -wavesin PDS [30, 31] instead of just the linear divergences [32]. III. CAPTURE CALCULATION
The E1 capture reaction Li( n, γ ) Li is given by the diagrams in Fig. 1. Capture in the S = 2 channel dominates and in our formulation it doesn’t mix with the S = 1 channel. Theexcited core Li (cid:63) contributes only in the S = 1 channel. Thus the S = 2 channel calculationin both EFT and EFT (cid:63) are very similar. ( a ) ( b )( c ) ( d ) FIG. 1. Double solid line represents a Li or Li (cid:63) core as appropriate, single solid line a neutron,double dashed line a dressed dimer propagator, wavy line a photon, and ⊗ represents the finalbound state. We start with the calculation in the S = 2 channel in EFT without Li (cid:63) first. The squaredamplitude for the capture from the initial S state to the 2 + ground state Li is [30, 31] (cid:12)(cid:12)(cid:12) M ( P )E1 (cid:12)(cid:12)(cid:12) = (2 j + 1) (cid:18) Z c m n M (cid:19) παM µ πµ Z ( P ) × (cid:12)(cid:12)(cid:12)(cid:12) − p p + γ − A ( a (2)0 , p )[ B ( p, γ ) + J ( − ip )] (cid:12)(cid:12)(cid:12)(cid:12) , (21)where j = 2. The initial state strong interaction in the s -wave is given by the amplitude A ( a (2)0 , p ) = 2 πµ − a (2)0 − ip , (22)with the scattering length a (2)0 = − . /Q . The loop contributionfrom diagram ( b ) of Fig. 1 is contained in the function B ( p, γ ) = 4 µD − (cid:18) λ (cid:19) − D (cid:90) d D − q (2 π ) D − q ( q − p − i + )( q + γ )= µ (cid:18) λ π + 13 π ip − γ p + γ (cid:19) . (23)9ig. 1 ( d ) is proportional to J ( − ip ). The combination B ( p, γ ) + J ( − ip ) is RG scaleindependent. The wave function renormalization constant in the P channel has the simpleform 2 πµ Z ( P ) = − πr ( P )1 + 3 γ , (24)where r ( P )1 is the p -wave effective momentum in the P channel. In this EFT withoutexplicit Li (cid:63) , the capture from initial S state to the 2 + ground state is given by a similarexpression as above in Eq. (21) with the replacements a (2)0 → a (1)0 for the scattering lengthin the S channel, and r ( P )1 → r ( P )1 for the effective momentum in the P channel.It is straightforward to extend this EFT calculation for the capture to the 1 + excitedstate—one replaces j = 1 in Eq. (21), γ → γ for the 1 + state binding momentum, anduses the appropriate p -wave effective momenta in the P and P final states.Next we discuss the capture process in EFT (cid:63) that contains explicit Li (cid:63) core contributions.The capture in the spin S = 2 remains exactly the same as Eq. (21). The additionalcontributions in EFT (cid:63) come from Figs. 1 ( b ) and ( d ) in the spin S = 1 where the initialstate s -wave interactions have to be described in terms of coupled-channel amplitudes A (11) and A (12) derived earlier in Eq. (14). The contribution from A (12) also entails modifying themomentum of the core in the loops with a photon attached in Fig. 1 ( b ) and ( d ). A directcalculation of capture to the 2 + ground state in spin S = 1 channel is then given by (cid:12)(cid:12)(cid:12) M ( P )E1 ,(cid:63) (cid:12)(cid:12)(cid:12) = (2 j + 1) (cid:18) Z c m n M (cid:19) παM µ πµ Z ( P ) × (cid:12)(cid:12)(cid:12)(cid:12) − p p + γ − A (11) ( p ) [ B ( p, γ ) + J ( − ip )] −A (12) ( p ) (cid:20) B (cid:18) i (cid:113) − p + γ − i + , (cid:113) γ + γ (cid:19) + J (cid:18)(cid:113) − p + γ − i + (cid:19)(cid:21) √Z P (cid:63) √Z ( P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (25)The extension to the capture to the 1 + state with j = 1 is straightforward with γ → γ andthe appropriate modification of the wave function renormalization constants Z ( P ) → Z ( P ) , Z ( P (cid:63) ) → Z ( P (cid:63) ) for the P and P (cid:63) states, respectively.The total c.m. cross section for the capture to the 2 + state is given by σ (2 + )E1 ( p ) = 116 πM k p (cid:20) | a | (cid:12)(cid:12)(cid:12) M ( P )E1 (cid:12)(cid:12)(cid:12) + (1 − | a | ) (cid:12)(cid:12)(cid:12) M ( P )E1 (cid:12)(cid:12)(cid:12) (cid:21) , (26)in the EFT without excited Li (cid:63) core and by σ (2 + )E1 ,(cid:63) ( p ) = 116 πM k p (cid:20) | a | (cid:12)(cid:12)(cid:12) M ( P )E1 (cid:12)(cid:12)(cid:12) + (1 − | a | ) (cid:12)(cid:12)(cid:12) M ( P )E1 ,(cid:63) (cid:12)(cid:12)(cid:12) (cid:21) , (27)in the EFT (cid:63) with Li (cid:63) core as explicit degree of freedom. We use the normalized boundstate | + , m (cid:105) = a | S = 2 , L = 1 , J = 2 , m (cid:105) + (cid:112) − | a | | S = 1 , L = 1 , J = 2 , m (cid:105) , (28)10n both EFT and EFT (cid:63) . The choice a = 1 / √
2, makes | + (cid:105) a p / valance neutron statethat we use in this work [8, 24]. The mixing with the | P (cid:63) (cid:105) state in the coupled-channelcalculation is included through the interaction. One views the relation in Eq. (28) above asan interpolating field that only needs to have the correct quantum numbers of the groundstate. The details of the ground state are built in through the interaction in the theory, suchas by the factors of Z ( P (cid:63) ) / Z ( P ) in Eq. (25). The capture cross section to the 1 + state of Li is similar with the appropriate change of parameters as detailed earlier. We take | + , m (cid:105) = b | S = 2 , L = 1 , J = 1 , m (cid:105) − (cid:112) − | b | | S = 1 , L = 1 , J = 1 , m (cid:105) , (29)with b = (cid:112) /
6, making it a p / valance neutron configuration [8, 24]. The choice b = 1 / √ IV. A SURVEY - (cid:1)(cid:2)(cid:3) - (cid:4)(cid:2)(cid:5) - (cid:4)(cid:2)(cid:3) - (cid:6)(cid:2)(cid:5) - (cid:6)(cid:2)(cid:3) - (cid:1)(cid:2)(cid:3) - (cid:4)(cid:2)(cid:5) - (cid:4)(cid:2)(cid:3) - (cid:6)(cid:2)(cid:5) - (cid:6)(cid:2)(cid:3) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) ( (cid:7)(cid:8) - (cid:1) ) (cid:1) (cid:1) ( (cid:1) (cid:1) (cid:2) ( (cid:1) (cid:2) - (cid:1) ) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:5)(cid:2)(cid:1) (cid:5)(cid:2)(cid:6)(cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:7)(cid:7) ⨯ FIG. 2. EFT without Li (cid:63) . Dashed curves: Contour plot of capture cross section to the 2 + stateat thermal energy normalized to data [6]. Solid curves: Contour plot of branching ratio of capturecross section to the 2 + state in the spin S = 2 channel at thermal energy. The × marks theparameter value r ( P )1 = − .
47 fm − = r ( P )1 used in Ref. [31]. The boxed numbers indicate thevalues on the corresponding contour lines. The E1 cross section in Eq. (26) was calculated earlier in Refs. [30, 31]. It depends on fourscattering parameters a (2)0 , a (1)0 , r ( P )1 and r ( P )1 . The two effective momenta are not knownexperimentally, and the capture to the 2 + 8 Li ground state is sensitive to only a combinationof these. In the following we define the thermal ratio as the theory calculation of the capturecross section to the 2 + 8
Li ground state at thermal energy, divided by the correspondingexperimental value [6]. Fig. 2 shows that a single parameter family of p -wave effectivemomenta can reproduce a given thermal ratio (dashed curves). The solid curves showhow a single parameter family of p -wave effective momenta can produce a given branchingratio of the capture to the S = 2 channel at thermal energy. In Refs. [30, 31] a common11ffective momentum r (2 + )1 ∼ Λ was used for both r ( P )1 and r ( P )1 for convenience whichreproduced the thermal capture rate. This also gave a branching ratio that is consistentwith the experimental lower bound of W + ≥ .
86 [43], once the theory errors from the EFTin Ref. [31] are also taken into consideration. We mention that earlier works have estimatedthe lower bound as 0.80 [42] and 0.75 [41], respectively. Applying the branching ratio lowerbound W + ≥ .
86 to the experimental constraint on thermal capture cross section wouldrestrict r ( P )1 ∼ − . − but leave r ( P )1 (cid:46) − − completely unbound from below.The result from Ref. [31], without initial state d -wave contribution, is plotted as EFT Ain Fig. 3, where data from Refs.[6, 7, 9] are also shown. As pointed out in Ref. [32], onecould use the known ANCs C ,ζ to fit the effective momenta using the relation: C ,ζ = γ π πµ Z ( ζ ) , (30)where γ is the appropriate binding momentum in the channel ζ . We use the following ANCvalues: C , P = 0 . − , C , P =0 . − , C , P = 0 . − , C , P =0 . − from neutron transfer reaction [47] and C , P (cid:63) = 0 . − C , P (cid:63) =0 . − from an ab initio calculation [32].The experimental ANCs from Ref. [47] give r ( P )1 ∼ − .
37 fm − , r ( P )1 ∼ − . − whichare consistent with the expectation from the analysis of Fig. 2. The corresponding result forthe capture cross section is plotted as EFT ANC in Fig. 3. Though the curves EFT A andEFT ANC differ, this difference is within the expected theoretical error. We provide a morerobust theory error estimate later in Section V when we develop the EFT power countingrelevant to E1 transition at low momenta. Table I lists some of the fit parameters, and thethermal and branching ratios.In Fig. 3 we also plot the capture calculation to the 2 + state in EFT (cid:63) with the explicit Li (cid:63) degree of freedom. It is labeled as EFT (cid:63) ANC. In the spin S = 2 channel this cross sectiondepends on a (2)0 and r ( P )1 . In the S = 1 channel we need the parameters a (1)0 , a , a , andoverall normalization constants Z ( P ) , Z ( P (cid:63) ) . The unknowns r ( P )1 , Z ( P ) are constrainedfrom the experimental ANCs [47], and Z ( P (cid:63) ) is constrained from the calculated ANC [32].We take a = − / ± .
4) MeV − and get a from Eq. (13) with r (1)0 = − ± .
4) fm.We saw earlier in Eq. (13) that r (1)0 is negative and fine-tuned to be large. A negative a appear to give a better description of data but this is not very crucial for the analysis.Using a = 1 / ± . − changes the cross section by a few percent (well withina NNLO contribution we discuss later). In this plot we did not approximate the s -wavescattering amplitudes A (11) , A (12) in Eq. (14) by the Q/ Λ expansion. One notices that thecurves EFT ANC and EFT (cid:63)
ANC are compatible with each other within the expected errorsregarding their momentum dependence. In particular, the opening of the inelastic thresholdat p = γ ∆ = 28 . Li (cid:63) degree of freedom gives the largest contributions is where the M1 capturedue to the 3 + resonance is already predominant [31]. Given that p R < γ ∆ , the inclusionof the 3 + degree of freedom precedes that of the Li (cid:63) in the formulation of the low-energyeffective theory for the Li( n, γ ) Li reaction if we attempt to describe data. Nevertheless,one can analyze the non-resonant capture theoretically. We interpret the plot EFT (cid:63)
ANCas indicative of the fact that the Li (cid:63) contribution to the initial s -wave scattering at theseenergies is a subleading effect. The contribution of the Li (cid:63) to the bound state wave functioncannot be separated completely with the two ANCs for P and P (cid:63) states to constrain the12hree relevant p -wave effective momenta r , r , and r in Eq. (20). In Table I, we list the r ( P )1 value, and the thermal and branching ratios for the EFT (cid:63) ANC fit. (cid:1)(cid:2)(cid:3)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:3) (cid:4)(cid:5)(cid:14)(cid:15)(cid:9)(cid:16)(cid:9)(cid:17) (cid:4)(cid:18)(cid:19) (cid:20)(cid:21)(cid:22) (cid:23)(cid:20)(cid:21)(cid:22) (cid:23)(cid:15)(cid:24)(cid:20)(cid:21)(cid:22) * (cid:23)(cid:15)(cid:24) (cid:1)(cid:2)(cid:3)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:1)(cid:2)(cid:5)(cid:4)(cid:1)(cid:2)(cid:6)(cid:1)(cid:1)(cid:2)(cid:6)(cid:4)(cid:1)(cid:2)(cid:7)(cid:1) (cid:1) (cid:1) σ (cid:1) (cid:2) (cid:3) (cid:2) μ (cid:1) ) (cid:3)(cid:1) - (cid:1) (cid:3)(cid:1) - (cid:2) (cid:3)(cid:1) (cid:3) (cid:3)(cid:1) (cid:2) (cid:1) ( (cid:1)(cid:2)(cid:3) ) (cid:1)(cid:2)(cid:3) (cid:4)(cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:1)(cid:2)(cid:3) * (cid:4)(cid:5)(cid:6) (cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:3)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:1)(cid:2)(cid:6)(cid:6)(cid:1)(cid:2)(cid:6)(cid:7)(cid:1)(cid:2)(cid:6)(cid:4) (cid:1) (cid:1) σ (cid:1) (cid:2) (cid:3) (cid:2) μ (cid:1) ) (cid:8) (cid:3)(cid:1) (cid:3)(cid:8) (cid:6)(cid:1) (cid:6)(cid:8) (cid:9)(cid:1) (cid:9)(cid:8) (cid:7)(cid:1) (cid:1) ( (cid:1)(cid:2)(cid:3) ) FIG. 3. Capture cross section to the 2 + state σ g . s . scaled by the neutron c.m. velocity v n vs c.m.relative momentum p . The (black) dot-dashed curve is the EFT A result, the (red) dashed curveis the EFT ANC result, and the (blue) solid curve is the EFT (cid:63) ANC result, respectively. The gridlines are at the 3 + resonance momentum p R = 19 . Li (cid:63) inelasticity γ ∆ = 28 . V. EFT AND EFT (cid:63)
POWER COUNTING
We start with the capture to the dominant 2 + state. Capture in the spin S = 2 channelis about a factor of 4 larger than in the spin S = 1 channel [48]. Thus we count capturein the S = 1 channel to be NLO (a subleading effect). In the dominant spin channel, the s -wave scattering length a (2)0 ∼ /Q is large. At low momentum the contribution from theinitial state interaction scales as a (2)0 ( B + J ). The loop integral combination scales as13 ABLE I. Li( n, γ ) Li capture to the 2 + state. We estimate the parameters as described in thetext. Thermal ratio is the EFT cross section normalized to Lynn data [6]. Branching ratio is thecapture in the S = 2 spin channel compared to the total cross section at thermal momentum. Weshow results to 3 significant figures. Theory r ( P )1 (fm − ) Z ( P ) Z ( P ) Thermal ratio Branching ratioEFT A − . . . . − . . . . . (cid:63) ANC − . . . . . (cid:63) Lynn LO − . . (cid:63) Lynn-ANC NLO − . . . . (cid:63) ANC LO − . . . (cid:63) ANC NLO − . . . . . B + J ∼ Q . Thus the contributions from the diagrams with and those without initialstate strong interactions in Fig. 1 are of the same size O (1) in the Q/ Λ expansion. Thisconstitutes the LO contribution in both the EFT (cid:63) with Li (cid:63) and the EFT without.The cross section in the S = 2 channel is proportional to the wave function renormal-ization constant Z ( P ) that depends on the effective momentum r ( P )1 ∼ Λ and bindingmomentum γ ∼ Q . Discounting factors of 3, we can expand Z ( P ) in powers of Q/ Λ. Herewe use the so-called zed-parameterization introduced in Ref. [49] for s -wave bound statesthat can also be applied to p -wave bound states [31]. A consequence of this parameterizationis that the wave function renormalization constant is exact at NLO with no other higher-order corrections but there is a substantial change in the parameters from LO to NLO,see r ( P )1 values at LO and NLO in Table I. At NLO, there is an s -wave effective range r (2)0 ∼ / Λ correction. The capture data at low energy is not sensitive to this parameter.Fig. 4 shows the sensitivity to r (2)0 in the range − S = 1 channel starts at NLO, and the power counting for theEFT (cid:63) with Li (cid:63) and EFT without has to be discussed separately though they are the sameup to NLO. We start with the theory without the excited Li (cid:63) core. In this theory, thecontributions from Figs. 1 ( b ), ( d ) from the initial state interactions scale as a (1)0 ( B + J ). Given the smaller a (1)0 ∼ / Λ in this channel, these initial state interactions are Q/ Λsuppressed compared to the contributions from Figs. 1 ( a ), ( c ), thus they constitute aNNLO contribution. The wave function renormalization constant Z ( P ) has the same formas Eq. (24) for the S = 2 channel. Here we again use the zed-parameterization.In EFT (cid:63) with explicit Li (cid:63) core contribution there are two differences in the S = 1channel from the previous discussion due to the mixing in the initial S - S (cid:63) scattering stateand final P - P (cid:63) bound state. The scattering state contribution scales as A (11) ( B + J ), A (12) ( B + J ). In the S = 1 channel, a (1)0 ∼ / Λ and assuming all the scattering lengthparameters a ∼ a ∼ a ∼ / Λ to be natural as well, we see from Eq. (14) that theinitial state interaction in this channel also scales as Q/ Λ compared to the contributionsFig. 1 ( a ), ( c ) without initial state interaction. Thus in this theory also the scaling forthe s -wave interaction is similar to the theory without explicit Li (cid:63) degree of freedom upto NLO. There is one difference, however, from before. Now we have two wave function14enormalization constants Z ( P ) , Z ( P (cid:63) ) that are shown in Eq. (20). Given that the ANCsfor P and P (cid:63) [32, 50] are of similar size, we do not attempt any perturbative expansionin the mixing parameter r .To summarize, the LO contribution to the capture to the 2 + ground state is from the spin S = 2 channel. At this order the cross section depends only on the S scattering length a (2)0 and the P effective momentum r ( P )1 in both EFT and EFT (cid:63) . The NLO correctionscome from the S effective range r (2)0 and from the capture in the spin S = 1 channelwithout initial interaction in either of the two theories, EFT or EFT (cid:63) . Thus the momentumdependence in EFT and EFT (cid:63) are indistinguishable at NLO. Up to this order, the differencebetween the two theories lies in the interpretation of the wave function renormalizationconstant Z ( P ) —in the EFT without Li (cid:63) one directly relates Z ( P ) with the p -wave effectivemomentum r ( P )1 . In EFT (cid:63) , Z ( P ) (and Z ( P (cid:63) ) ) is a function of three effective momenta r , r , r .The proposed power counting for the capture to the 1 + state of Li in this work is verysimilar.
VI. RESULTS AND ANALYSIS
In Fig. 4, the EFT/EFT (cid:63)
Lynn LO curve was generated by constraining the wavefunction renormalization constant Z ( P ) from the thermal capture data [6]. We use thezed-parameterization. The fitted effective momentum r ( P )1 value is in Table I. We took a (2)0 = − . Z ( P ) was constrainedby the ratio of the ANCs C , P /C , P = 0 . − [47], and then Z ( P ) was fitted tothermal capture data. This is the curve labeled EFT/EFT (cid:63) Lynn-ANC NLO in Fig. 4. Asdiscussed earlier, see Fig. 2, it is important to independently constrain the wave functionrenormalization constants in the S = 2 and S = 1 channels since the capture data andthe branching ratio are not sufficient. We see a large change in the r ( P )1 value at NLOas expected in the zed-parameterization, see Table I. Z ( P ) can be expressed in terms of asingle effective momentum in EFT but in EFT (cid:63) it depends on three effective momenta. Asmentioned earlier, we do not attempt to write Z ( P ) in terms of these effective momenta.The LO and NLO curves and the error bands, from errors in the input parameters only,overlap. This is primarily a consequence of constraining both the LO and NLO results tothe same thermal capture data. As alternative fitting procedure one determines the un-known couplings from just the ANCs, like the last two rows of Table I. The errors in Fig. 4were propagated in quadrature from the errors in the input parameters.In the bottom panel of Fig. 4, we show only the error associated with varying the s -wave effective range r (2)0 ∼ / Λ in the S channel in the range − r (2)0 has anoticeable impact on the capture cross section only at larger momenta where precision datais lacking, and also happens to be in a momentum range where the 3 + resonance contributionis significant. The resonance contribution can be added in EFT as discussed earlier [31]. Inthis work we only include the non-resonant contribution.The plot in Fig. 5 includes capture to both the 2 + ground- and 1 + excited-state of Liat NLO. The 2 + capture cross section is the same as in Fig. 4. The wave function renor-malization constant Z ( P ) in the 1 + capture is constrained from the ratio C , P /C , P =0 . − [47], and Z ( P ) to the thermal capture data [6]. We include the expected 10%15 (cid:2)(cid:3)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:3) (cid:4)(cid:5)(cid:14)(cid:15)(cid:9)(cid:16)(cid:9)(cid:17) (cid:4)(cid:18)(cid:19) (cid:1)(cid:2)(cid:3) / (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:6) (cid:4)(cid:7)(cid:1)(cid:2)(cid:3) / (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:6) - (cid:8)(cid:9)(cid:10) (cid:9)(cid:4)(cid:7) (cid:1)(cid:2)(cid:3)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:1)(cid:2)(cid:5)(cid:4)(cid:1)(cid:2)(cid:6)(cid:1)(cid:1)(cid:2)(cid:6)(cid:4)(cid:1)(cid:2)(cid:7)(cid:1) (cid:1) (cid:1) σ (cid:1) (cid:2) (cid:3) (cid:2) μ (cid:1) ) (cid:3)(cid:1) - (cid:1) (cid:3)(cid:1) - (cid:2) (cid:3)(cid:1) (cid:3) (cid:3)(cid:1) (cid:2) (cid:1) ( (cid:1)(cid:2)(cid:3) ) (cid:1)(cid:2)(cid:3) / (cid:1)(cid:2)(cid:3) * (cid:4)(cid:5)(cid:6) (cid:1) (cid:1) ( (cid:2) ) (cid:1)(cid:2)(cid:3)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:1)(cid:2)(cid:5)(cid:4)(cid:1)(cid:2)(cid:6)(cid:1)(cid:1)(cid:2)(cid:6)(cid:4)(cid:1)(cid:2)(cid:7)(cid:1) (cid:1) (cid:1) σ (cid:1) (cid:2) (cid:3) (cid:2) μ (cid:1) ) (cid:3)(cid:1) (cid:1) (cid:1) ( (cid:1)(cid:2)(cid:3) ) FIG. 4. Capture cross section to the 2 + state. The LO (red) dashed and NLO (blue) solid curves,and error bands (red dashed and blue dashed curves) overlap. Both of these were constrained bythermal capture data [6]. We varied r (2)0 in the range − r (2)0 . The grid lines are at the 3 + resonance momentum p R = 19 . Li (cid:63) inelasticity γ ∆ = 28 . EFT/EFT (cid:63) error in Fig. 5 from NNLO corrections. We show some parameters for the cap-ture to 1 + in Table II. As in the 2 + capture, we express the wave function renormalizationonly in the spin S = 2 channel Z ( P ) in terms of an effective momentum r ( P )1 . We see thatthe extrapolation of the EFT/EFT (cid:63) curves to low momentum gives an accurate postdictionof the sub-thermal datum [8].The higher momentum data points in Fig. 5 from Refs. [8, 9] are associated with the M1transition from the initial 3 + resonance state. As mentioned before, this contribution can beincluded in EFT as presented in Ref. [31] where the resonance is described as a P state ofthe valance neutron and the ground state of Li. The excited state Li (cid:63) does not contributeto the initial P scattering state. To keep the discussion more focused we do not includethe M1 contribution. 16 (cid:2)(cid:3)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:3) ( (cid:14)(cid:10)(cid:9)(cid:8)(cid:15)(cid:16) ) (cid:4)(cid:5)(cid:17)(cid:18)(cid:15)(cid:19)(cid:8) (cid:4)(cid:5)(cid:20)(cid:21)(cid:9)(cid:22)(cid:9)(cid:19) (cid:4)(cid:23)(cid:24) (cid:25)(cid:26)(cid:27) / (cid:25)(cid:26)(cid:27) (cid:1)(cid:2)(cid:3)(cid:3) - (cid:28)(cid:21)(cid:29) (cid:21)(cid:1)(cid:30) (cid:6) + (cid:25)(cid:26)(cid:27) / (cid:25)(cid:26)(cid:27) * (cid:1)(cid:2)(cid:3)(cid:3) - (cid:28)(cid:21)(cid:29) (cid:21)(cid:1)(cid:30) (cid:27)(cid:13)(cid:31)(cid:9)(cid:8) (cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6) (cid:1) (cid:1) σ μ (cid:1) ) (cid:3)(cid:1) - (cid:1) (cid:3)(cid:1) - (cid:2) (cid:3)(cid:1) (cid:3) (cid:3)(cid:1) (cid:2) (cid:1) ( (cid:1)(cid:2)(cid:3) ) FIG. 5. Total capture cross section to the 2 + and 1 + state at NLO. Both of these were constrainedby the thermal capture data [6]. We varied r (2)0 in the range − + state.The error bands in the capture to the 1 + state are not shown but included in the total capturecross section. The grid lines are at the 3 + resonance momentum p R = 19 . Li (cid:63) inelasticity γ ∆ = 28 . + ground state which was scaled inthat work by a factor of 1/0.894 to represent the total capture rate.TABLE II. Li( n, γ ) Li capture to the 1 + state. We estimate the parameters as described in thetext. Thermal ratio is the EFT cross section normalized to Lynn data [6]. Branching ratio iscapture in the S = 2 spin channel compared to the total cross section at thermal momentum. Wecorrected the p -wave effective momenta values from Ref. [31] in EFT A below. We show results to3 significant figures. Theory r ( P )1 (fm − ) Z ( P )1 Z ( P )1 Thermal ratio Branching ratioEFT A − . . . . (cid:63) Lynn LO − . . (cid:63) Lynn-ANC NLO − . . . . (cid:63) ANC LO − . . . (cid:63) ANC NLO − . . . . . Finally in Fig. 6, we consider some recent Coulomb dissociation data that was used topredict the capture cross section. We extracted the experimental results digitally from Fig.10 of Ref. [51]. We assumed the lowest horizontal-axis tick mark on the log-log plot tobe at 10 keV (instead of 1 keV) which gives the expected results for the known Nagai [9]data. We expect our digital extraction to introduce errors of about a 1%. The dashed (red)curve is the EFT A result from Fig. 3 with the capture from initial d -wave states includedthat was published earlier [30, 31]. This is not a complete NNLO calculation in the EFT17ithout Li (cid:63) core as it lacks a two-body contribution and S = 2 channel s -wave effectiverange correction. However, one can see that the expected NNLO corrections to the NLOEFT/EFT (cid:63) solid (blue) curve from Fig. 5 moves the theory result in better agreement withthe data. The differences between the dashed and solid curve are about 4% at E ≈ (cid:1)(cid:2)(cid:3)(cid:4) (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:9)(cid:3) (cid:5)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17) (cid:5)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22) (cid:23) (cid:24)(cid:3)(cid:25)(cid:26) (cid:27) - (cid:24)(cid:9)(cid:28)(cid:2)(cid:20)(cid:21)(cid:22) / (cid:20)(cid:21)(cid:22) * (cid:29)(cid:30)(cid:31)(cid:31) - (cid:23)(cid:8) (cid:8)(cid:29)! (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) σ μ (cid:1) ) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:3) (cid:1) ( (cid:1)(cid:2)(cid:3) ) FIG. 6. Capture cross section via Coulomb dissociation. The Izs´ak data was extracted fromRef. [51] as explained in the text. The dashed (red) curve is the EFT A result from Fig. 3 togetherwith capture from d -wave initial state [30, 31]. Solid (blue) curve is the NLO EFT/EFT (cid:63) resultfrom Fig. 5. VII. CONCLUSIONS
We considered the E1 contribution to the Li( n, γ ) Li capture reaction at low energies.A coupled-channel calculation for the contribution from the excited Li (cid:63) core was presented.This theory was compared to an earlier calculation which did not include the excited coreas an explicit degree of freedom [30, 31]. We developed a power counting for both theEFTs—one with explicit excited Li (cid:63) core and one without—and show that the two theoriesare equivalent in their momentum-dependence up to NLO. This confirms the expectationthat the Li (cid:63) contribution which participates only in the sub-dominant spin S = 1 channelwith a branching ratio of ∼ . S = 1 channel in terms of p -wave effective momenta in the two theories is different. Inthe EFT without Li (cid:63) , we can relate the cross section to an overall normalization in termsof the P effective momentum. No such simple interpretation is possible in the other EFT (cid:63) with the excited Li (cid:63) core. We estimate the theoretical errors in our calculation to be about10% from Q / Λ NNLO corrections.The survey presented in Sec. IV relaxed the simplification r ( P )1 = r ( P )1 ∼ Λ, for the 2 +8 Li ground state, done in previous works [30, 31]. In the EFT without Li (cid:63) , Fig. 2 showshow r ( P )1 and r ( P )1 are correlated to reproduce a given value for the thermal capture to the18round 2 + state of Li and the branching ratio of the spin channel S = 2. The differencesfrom the simplified assumption is compatible with a higher-order correction. Alternatively r ( P )1 and r ( P )1 can be fixed via the corresponding ANCs C , P and C , P calculated inRef. [50] or taken from experiment [47], with similar numerical results. In the EFT (cid:63) withexplicit Li (cid:63) the wave function renormalization constants Z ( P ) , Z ( P ) , and Z ( P (cid:63) ) can befitted to the corresponding ANCs, though they are not enough to pin down the extra EFTcouplings that appear with the excited core as an explicit degree of freedom—while r ( P )1 isuniquely determined from Z ( P ) , this is not true for the three S = 1 effective momenta r ij in terms of Z ( P ) and Z ( P (cid:63) ) . Similar analysis for the p -wave channels corresponding to the1 + 8 Li excited state holds.At NNLO, there are several contributions to the capture to the 2 + ground state of Li.Two-body current contribution to capture from s -wave scales as k a (2)0 L E1 [52]. This is a Q / Λ NNLO contribution for a natural-sized coupling L E1 ∼
1, given the scalings for thescattering length a (2)0 ∼ /Q and the photon energy k = ( p + γ ) / (2 µ ) ∼ Q / Λ . Thecontributions from the S = 1 channel start with initial state s -wave strong interactions. Inthe EFT without Li (cid:63) core, the s -wave interactions can be parameterized by the scatteringlength a (1)0 . In the EFT (cid:63) with Li (cid:63) core, the initial state s -wave interactions involve a coupled-channel calculation parameterized by the three scattering lengths a , a , a . Capture from d -wave that scales as p / ( p + γ ) was included in Refs. [30, 31]. However, it is kinematicallysuppressed at low momentum, contributing around 10% at p (cid:38)
40 MeV. So this contributioncan also be included as NNLO in the dominant spin S = 2 channel. In the same spin channelthe initial state S receives a NNLO correction from two insertions of the leading derivativecoupling proportional to [ a (2)0 r (2)0 p / while the shape parameter P (2)0 enters at N LO. TheNNLO contributions for the capture to the 1 + excited state of Li are similar.The EFT formalism and the theory expressions for the cross section in this work, withthe excited Li (cid:63) core contributions, are different from those in Ref. [32]. We also differin the interpretation of the wave function renormalization constants in terms of the p -wave scattering parameters. However, given that the excited Li (cid:63) core contributions tothe momentum dependence of the cross section are a NNLO effect, it is expected that thenumerical results of Ref. [32] are similar to those obtained here which in turn do not differsignificantly (differences (cid:46) .
5% for p (cid:46)
40 MeV) from earlier calculations (excluding d -wavecontributions) in Refs. [30, 31].In this work, we only included the non-resonant capture. Future work would include theM1 contribution from the 3 + resonance to Li( n, γ ) Li reaction [36]. We would also applyour formalism to the related capture reaction Be( p, γ ) B. ACKNOWLEDGMENTS
We thank C. Bertulani and A. Horv´ath for discussing their work on Coulomb dissoci-ation with us. This work was supported in part by U.S. NSF grants PHY-1615092 andPHY-1913620 (PP, GR) and Brazilian agency FAPESP thematic projects 2017/05660-0 and2019/07767-1, and INCT-FNA Proc. No. 464898/2014-5 (RH). The cross section figures forthis article have been created using SciDraw [53].19 ppendix A: Projectors
The following are from Ref. [31] that we include for reference. For each partial wave weconstruct the corresponding projection operators from the relative core-nucleon velocity, thespin-1/2 Pauli matrices σ i ’s, and the following spin-1/2 to spin-3/2 transition matrices S = 1 √ (cid:18) −√ − √ (cid:19) ,S = − i √ (cid:18) √ √ (cid:19) ,S = 2 √ (cid:18) (cid:19) , (A1)which satisfy S i S † j = 23 δ ij − i (cid:15) ijk σ k , S † i S j = 34 δ ij − (cid:8) J (3 / i , J (3 / j (cid:9) + i (cid:15) ijk J (3 / k , (A2)where J (3 / i ’s are the generators of the spin-3/2. We construct the Clebsch-Gordan coeffi-cient matrices F i = − i √ σ S i , Q ij = − i √ σ (cid:0) σ i S i + σ j S i (cid:1) , (A3)for projections onto spin channels S = 1 and S = 2, respectively. Then in coordinate spacethe relevant projectors that appear in the Lagrangians involving the Li ground state in Eqs.(1), (2) are [30, 31] P ( S ) i = F j , P ( S ) ij = Q ij ,P ( P ) i = (cid:114) F x (cid:32) → ∇ m c − ← ∇ m n (cid:33) y (cid:15) ixy , P ( P ) ij = √ F x (cid:32) → ∇ m c − ← ∇ m n (cid:33) y R xyij ,P ( P ) i = (cid:114) Q ix (cid:32) → ∇ m c − ← ∇ m n (cid:33) x , P ( P ) ij = 1 √ Q xy (cid:32) → ∇ m c − ← ∇ m n (cid:33) z T xyzij . (A4)The tensors R ijxy = 12 (cid:18) δ ix δ jy + δ iy δ jx − δ ij δ xy (cid:19) ,T xyzij = 12 (cid:16) (cid:15) xzi δ yj + (cid:15) xzj δ yi + (cid:15) yzi δ xj + (cid:15) yzj δ xi (cid:17) , (A5)ensures total angular momentum j = 2 is picked.The new projectors to describe the interactions in Eq. (2) with the excited Li (cid:63) core are P ( S (cid:63) ) i = 1 √ σ i ,P ( P (cid:63) ) i = √ σ x (cid:32) → ∇ m c − ← ∇ m n (cid:33) y (cid:15) ixy , P ( P (cid:63) ) ij = (cid:114) σ x (cid:32) → ∇ m c − ← ∇ m n (cid:33) y R xyij . (A6)20or the external states we introduce the photon vector ( ε ( γ ) i ), excited state Li 1 + spin-1( ε j ), and ground state Li 2 + spin-2 ( ε ij ) polarizations, obeying the following polarizationsums [54, 55], (cid:88) pol . ε ( γ ) i ε ( γ ) ∗ j = δ ij − k i k j k , (cid:88) pol . ave . ε i ε ∗ j = δ ij , (cid:88) pol . ave . ε ij ε ∗ lm = R ijlm . (A7) [1] M. Wiescher, F. K¨appeler, and K. Langanke, Annu. Rev. Astron. Astrophys. , 165 (2012).[2] C. R. Brune and B. Davids, Ann. Rev. Nucl. Part. Sci. , 87 (2015).[3] C. Bertulani and A. Gade, Phys. Rept. , 195 (2010), arXiv:0909.5693 [nucl-th].[4] L. H. Kawano, W. A. Fowler, R. W. Kavanagh, and R. A. Malaney, Astrophys. J. , 1(1991).[5] D. J. Hughes, D. Hall, C. Eggler, and E. Goldfarb, Phys. Rev. , 646 (1947).[6] J. E. Lynn, E. T. Jurney, and S. Raman, Phys. Rev. C , 764 (1991).[7] J. C. Blackmon et al. , Phys. Rev. C , 383 (1996).[8] M. Heil, F. K¨appler, M. Wiescher, and A. Mengoni, Astrophys. J. , 1002 (1998).[9] Y. Nagai et al. , Phys. Rev. C , 055803 (2005).[10] R. Izs´ak et al. , Phys. Rev. C , 065808 (2013), arXiv:1312.3498 [nucl-ex].[11] P. Navratil, R. Roth, and S. Quaglioni, Phys. Rev. C , 034609 (2010), arXiv:1007.0525[nucl-th].[12] K. Bennaceur, F. Nowacki, J. Okolowicz, and M. Ploszajczak, Nucl. Phys. A , 289 (1999),arXiv:nucl-th/9901060.[13] K. Fossez, N. Michel, M. P(cid:32)l oszajczak, Y. Jaganathen, and R. Id Betan, Phys. Rev. C ,034609 (2015), arXiv:1502.01631 [nucl-th].[14] P. Descouvemont and D. Baye, Nucl. Phys. A , 341 (1994).[15] T. Tombrello, Nucl. Phys. , 459 (1965).[16] A. Aurdal, Nuclear Physics A , 385 (1970).[17] H. Esbensen, Phys. Rev. C , 047603 (2004).[18] F. C. Barker, Nucl. Phys. A768 , 241 (2006).[19] B. Davids and S. Typel, Phys. Rev. C , 045802 (2003), arXiv:nucl-th/0304054.[20] S. Typel and G. Baur, Nucl. Phys. A759 , 247 (2005), arXiv:nucl-th/0411069.[21] C. Bertulani, Z. Phys. A , 293 (1996).[22] J. T. Huang, C. A. Bertulani, and V. Guimaraes, At. Data Nuc. Data Tables , 824 (2010),arXiv:0810.3867 [nucl-th].[23] N. Shul’gina, B. Danilin, V. Efros, J. Bang, J. Vaagen, and M. Zhukov, Nucl. Phys. A ,197 (1996).[24] L. Grigorenko, B. Danilin, V. Efros, N. Shul’gina, and M. Zhukov, Phys. Rev. C , 044312(1999).[25] C. Bertulani, H. Hammer, and U. van Kolck, Nucl. Phys. A , 37 (2002), arXiv:nucl-th/0205063.[26] P. Bedaque, H. Hammer, and U. van Kolck, Phys. Lett. B , 159 (2003), arXiv:nucl-th/0304007.[27] G. Rupak, Int. J. Mod. Phys. E , 1641004 (2016).
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