aa r X i v : . [ nu c l - t h ] J a n Nuclear reactions in artificial traps
Peng Guo
1, 2, 3, ∗ and Bingwei Long † College of Physics, Sichuan University, Chengdu, Sichuan 610065, China Department of Physics and Engineering, California State University, Bakersfield, CA 93311, USA Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: January 12, 2021)Coupled-channel two-particle systems bound by a harmonic trap are discussed in the presentpaper. We derive the formula that relates the energy levels of such trapped systems to phase shiftsand inelasticity of coupled-channel reactions. The formula makes it possible to extract amplitudesof inelastic nuclear reactions from ab initio calculations of discrete levels of many-nucleon systemsin a harmonic trap.
PACS numbers:Keywords:
I. INTRODUCTION
Studies of nuclear structure have entered a stage wherebound states of many nucleons can be solved for withmicroscopic nuclear forces and with high precision [1–8]. For some of those many-body solvers, the compu-tation typically involves expansion of wave functions ona chosen basis, e.g., the harmonic-oscillator (HO) basis.This poses a challenge for immediate extension of basis-expansion methods to continuum-state problems like res-onances and reactions. The need for synergy of bound-state and continuum computational methods has beenrecognized by the nuclear-theory community [9–13].One of these ab initio methods for elastic scatteringof two clusters, normally stable nuclei, utilizes the en-ergy spectrum of all the constituent nucleons, trappedby an artificial field, to generate directly the phase shiftsat certain discrete energies decided by the spectrum ofthe trapped [14–17]. This method is mostly enabled by aset of so-called quantization conditions (QCs) that mapenergy levels to the phase shifts at momenta related tothose energy eigenvalues [18, 19]. The harmonic trap isan attractive choice for the artificial field because the HOwave functions are analytically known, so such mappingformulas can be presented in closed forms to a great ex-tent. Existing vast resource of software for the HO basisis also an important incentive [20]. After the pioneeringwork in Ref. [18] to connect the spectrum of the trappedtwo-body systems to their elastic scattering amplitude,referred to as the BERW formula in the present paper,more theoretical works to extend the formula and appli-cations to nucleon-nucleon and nucleon-cluster scatteringfollowed [10, 14–17, 19, 21, 22]. Here we turn our atten-tion to coupled-channel problems where at least one ofthe two participating particles changes its species afterthe reaction. This is relevant to nuclear reactions wherethe nuclei fuse and then break up to different nucleus ∗ Electronic address: [email protected] † Electronic address: [email protected] than before.Relating discrete energy spectrum of trapped compos-ite particles to their scattering amplitudes has been atopic of great interests in other fields of strong-interactionphysics. In these applications, the form of traps may dif-fer, such as periodic cubic boxes in lattice quantum chro-modynamics (QCD) [23] and spherical hard wall in someof the lattice implementations of chiral effective field the-ory [24–26]. Because the underlying theory to computethe energy spectrum is model-independent and the com-putation often consumes much computing resources, itis highly desired that the said connection is constructedwithout uncontrolled modeling of interactions betweenthe composite particles. This can be achieved when twoscales are well separated; one is the wideness of the trap b and the other the interaction range between the compos-ite particles R , b ≫ R . The separation makes it possibleto locate a domain inside the trap where the compositeparticles do not interact appreciably while the confiningforce is sufficiently weak so that the trapped particles’wave function resembles that of the scattering state ininfinite volume. One then matches the asymptotic formof the scattering wave function, which has undeterminedscattering parameters, say, phase shifts, to the trappedwave function. Doing so pins down the phase shift atthe particular energy associated with the trapped wavefunction. This yields the QC, a compact relation betweenthe discrete energy eigenvalues and the scattering param-eters, which often takes the following schematic form:det [cot δ ( E ) − g ( E ; b )] E = E k = 0 , (1)where E k is the k -th the energy eigenvalue and δ ( E ) refersto the diagonal matrix with phase shifts as the elements.The analytic matrix function g ( E ; b ) depends only on theconfiguration of the trap, such as the functional form ofthe confining force and the wideness b , but not on detailof the inter-particle interactions.Matching wave functions was the fundamental method-ology used by L¨uscher to derive his seminal QC formulafor periodic cubic boxes [23]. Once we are convinced thatthe QC is irrespective of the interactions, manipulationof wave functions, however, does not have to be the ubiq-uitous technique of deriving the QC. Actively researchedin the field of lattice QCD, other methods have been de-veloped to tackle the problem (see, e.g., Refs. [27–53]).We will follow the variational approach spelled out inRefs. [50, 52, 53]. In particular, contact interactions areused in describing dynamics in both harmonic trap andinfinite volume, and the QC is derived by eliminating thecontact coupling.The paper is organized as follows. To establish our no-tation, derivation of the BERW formula is discussed inSec. II. The extension to coupled-channel reactions is pre-sented in Sec. III, followed by discussions and summaryin Sec. IV. II. BERW FORMULA
Using the approach in Refs. [50, 52, 53], we re-derivethe BERW formula for elastic scattering of two spinlessparticles in this section. We consider cases where thekinematics is such that the reaction does not takes placesnear threshold. So the long-range Coulomb repulsion, ifany, can be neglected and one is left with only the stronginteraction, which is finite-ranged.
A. Scattering dynamics
In infinite volume, the Lippmann-Schwinger (LS)equation for two-particle scattering has the following op-erator form,ˆ T ∞ ( E ) = − ˆ V + ˆ V ˆ G ∞ ( E ) ˆ T ∞ ( E ) , (2)where ˆ T ∞ is the T -matrix, E the energy, and ˆ V the short-range interaction between the clusters. The infinite-volume Green’s function is given in the operator formby ˆ G ∞ ( E ) = 1 E − ˆ H , (3)with ˆ H defined in terms of the reduced mass µ and theLaplacian: ˆ H = − ∇ µ . (4)The solution to the LS equation can be formally writtenas ˆ T ∞ ( E ) = − h ˆ V − − ˆ G ∞ ( E ) i − . (5)The on-shell elements of partial-wave projected T -matrix are usually parameterized in the center-of-mass(CM) frame by the phase shift δ L as T ( ∞ ) L ( q, q ) = (4 π ) µq δ L ( q ) − i , (6)where q = √ µE is relative momentum of particles. B. Two particles trapped by a HO potential
One may still define the T -matrix for the trapped two-particle system for any values of E , even though thereare no scattering states:ˆ T ω ( E ) = − ˆ V + ˆ V ˆ G ω ( E ) ˆ T ω ( E ) , (7)where the Green’s function of the HO potential with an-gular frequency ω is given byˆ G ω ( E ) = 1 E − ˆ H ω = X n | n ih n | E − ω ( n + ) . (8)Here ˆ H ω is the HO Hamiltonian,ˆ H ω = − ∇ µ + 12 µω r , (9)and | n i stands for eigenstates of ˆ H ω operator:ˆ H ω | n i = ω ( n + 32 ) | n i , n = 0 , , · · · . (10)The wideness of the harmonic trap can be characterizedby the so-called oscillator length: b ≡ (cid:18) µω (cid:19) − . (11)The bound state sustained by the harmonic trap canbe determined by poles of T ω ( E ), andˆ T ω ( E ) = − h ˆ V − − ˆ G ω ( E ) i − . (12)Therefore, any eigenvalue of the full Hamiltonian ( ˆ H ω +ˆ V ) E = E k satisfiesdet h ˆ V − − ˆ G ω ( E ) i E = E k = 0 . (13) C. Contact potentials and the BERW formula
Combining Eqs. (5) and (13) by eliminating ˆ V , onefinds symbolicallydet h ˆ T − ∞ ( E ) + ˆ G ω ( E ) − ˆ G ∞ ( E ) i E = E k = 0 . (14)Although the above equation is formally correct, it isdifficult to use in practical calculations because inverting T ∞ ( E ) directly is highly non-trivial.As long as R/b is small, as we argued, a model-independent QC must exist. So one can use whateverform of two-body interaction to deduce the QC. Wechoose a sum of contact interactions, with one term re-sponsible for each partial wave, to describe the cluster-cluster interaction: V ( q ′ , q ) = X L L + 14 π ( q ′ q ) L V L P L (ˆ q ′ · ˆ q ) , (15)where P L ( x ) is the Legendre polynomial and q ′ ( q ) is theoutgoing (incoming) momentum.In the CM frame for infinite volume, the separableform of the potential in Eq.(15) allows for a closed-formpartial-wave solution to Eq.(5) (see also Refs. [52, 53]): − µq L +1 (4 π ) [cot δ L ( q ) − i ]= 1 V L − L +1 Γ ( L + )(2 π ) G ( ∞ ) L ( r, r ′ ; q )( rr ′ ) L | r,r ′ → , (16)where G ( ∞ ) L ( r, r ′ ; q ) = − i µqj L ( qr < ) h (+) L ( qr > )is the free-particle Green’s function in the L -th partialwave.In the harmonic trap, the separable form also facili-tates a closed-form solution to Eq.(13):1 V L = 2 L +1 Γ ( L + )(2 π ) G ( ω ) L ( r, r ′ ; ǫ ( ω ) )( rr ′ ) L | r,r ′ → , (17)where G ( ω ) L is the partial-wave HO Green’s function andis given in Ref. [54] by G ( ω ) L ( r, r ′ ; ǫ ( ω ) ) = − ω ( rr ′ ) Γ( L + − ǫ ( ω ) ω )Γ( L + ) × M ǫ ( ω )2 ω , L + ( µωr < ) W ǫ ( ω )2 ω , L + ( µωr > ) . M and W are the Whittaker functions as defined inRef. [55]. ǫ ( ω ) represents the energy shift relative to theHO levels and is related to relative momentum q by q µ = ǫ ( ω ) + ω ( n + 32 ) = E . where ω ( n + ) is the CM energy in the harmonic trap,and the CM frame in infinite volume is assumed.By eliminating V L from Eqs. (16) and (17), and usingasymptotic forms of partial-wave Green’s functions G ( ω ) L and G ( ∞ ) L , one obtains the BERW formula:cot δ L ( q ) = g L ( q ) , (18)where g L ( q ) = ( − L +1 (cid:18) µωq (cid:19) L + Γ( + L − ǫ ( ω ) ω )Γ( − L − ǫ ( ω ) ω ) . (19) III. TWO-CHANNEL REACTIONS
For nuclear reactions where two nuclei fuse, break up,and rearrange, the BERW formula can not be immedi-ately applied. Extension of the derivation in Sec. II is needed to extract inelastic scattering amplitudes. We fo-cus on reactions that involve two channels and assumethat each channel is made up of two distinguishable par-ticles A + B and C + D . Labelling two channels by 1 and2, we can write, for instance, reaction 1 → A + B → C + D . (20)The kinetic energies in both channels are related by q µ + P M = q µ + P M + ∆ = E , (21)where P is the total momentum, q , relative momenta, M , the total mass of each channel,∆ ≡ M − M , and µ , the reduced mass. For simplicity, we will assumethe CM frame in what follows: P = 0. A. Coupled-channel energy levels in a harmonictrap
We start with the coupled Schr¨odinger equations fortwo interacting spinless particles bound by the HO po-tential: h ˆ H ω + V ( r ) i (cid:20) ψ (1 ,ω ) ( r ) ψ (2 ,ω ) ( r ) (cid:21) = " ǫ ( ω )1 ǫ ( ω )2 ψ (1 ,ω ) ( r ) ψ (2 ,ω ) ( r ) (cid:21) , (22)where ˆ H ω is given byˆ H ω = " − ∇ r µ + µ ω r − ∇ r µ + µ ω r . (23)Here r is the relative coordinate and the potential matrix V ( r ) is short-ranged and describes inter-cluster interac-tions and transition between the channels: V ( r ) = (cid:20) V (1 , ( r ) V (1 , ( r ) V (2 , ( r ) V (2 , ( r ) (cid:21) . (24) ǫ ( ω )1 and ǫ ( ω )2 are energy shifts in their respective channeland are related to the full energy E by ǫ ( ω )1 + ω ( n + 32 ) = ǫ ( ω )2 + ω ( n + 32 ) + ∆ = E .
The integral representation of Eq.(22) is the coupledLS equation: (cid:20) ψ (1 ,ω ) ( r ) ψ (2 ,ω ) ( r ) (cid:21) = Z d r ′ G ( ω ) ( r , r ′ ; E ) V ( r ′ ) (cid:20) ψ (1 ,ω ) ( r ′ ) ψ (2 ,ω ) ( r ′ ) (cid:21) , (25)where G ( ω ) denotes the coupled-channel version of theHO Green’s function: G ( ω ) ( r , r ′ ; E ) = " G (1 ,ω ) ( r , r ′ ; ǫ ( ω )1 ) 00 G (2 ,ω ) ( r , r ′ ; ǫ ( ω )2 ) , (26)and G ( α,ω ) satisfies (cid:20) ǫ ( ω ) α + ∇ r µ α − µ α ω r (cid:21) G ( α,ω ) ( r , r ′ ; ǫ ( ω ) α ) = δ ( r − r ′ ) . (27)The partial-wave projection of the HO Green’s functionis similar to the uncoupled-channel case: G ( α,ω ) ( r , r ′ ; ǫ ( ω ) α ) = X L L + 14 π G ( α,ω ) L ( r, r ′ ; ǫ ( ω ) α ) P L ( ˆr · ˆr ′ ) , (28)where G ( α,ω ) L ( r, r ′ ; ǫ ( ω ) α ) = − ω ( rr ′ ) Γ( L + − ǫ ( ω ) α ω )Γ( L + ) × M ǫ ( ω ) α ω , L + ( µ α ωr < ) W ǫ ( ω ) α ω , L + ( µ α ωr > ) . (29)We are going to take one more step to write the integralequation (25) in momentum space: T ( ω ) ( q ′ ) = Z d q (2 π ) d k (2 π ) e V ( q ′ − k ) e G ( ω ) ( k , q ; E ) T ( ω ) ( q ) , (30)where T ( ω ) ( q ′ ) matrix is defined by T ( ω ) ( q ′ ) = − Z d r e − i q ′ · r V ( r ) (cid:20) ψ (1 ,ω ) ( r ) ψ (2 ,ω ) ( r ) (cid:21) . (31) e V and e G ( ω ) are the Fourier transform of V and G ( ω ) , re-spectively. The partial wave expansion of Eq.(30) yields T ( ω ) L ( q ′ ) = Z ∞ q dq (2 π ) k dk (2 π ) e V L ( q ′ , k ) e G ( ω ) L ( k, q ; E ) T ( ω ) L ( q ) . (32)With the assumption of separable potential, e V L ( q ′ , q ) = ( q ′ q ) L V L , (33)where V L is a constant 2 × T ( ω ) L ( q ′ ) matrix must be of the form T ( ω ) L ( q ) = q L t ( ω ) L , where t ( ω ) L matrix does not depend on q . Therefore, thequantization condition is given bydet (cid:20) V − L − Z ∞ q dq (2 π ) k dk (2 π ) ( kq ) L e G ( ω ) L ( k, q ; E ) (cid:21) = 0 . (34)Using the limiting form of the spherical Bessel func-tions q L = 2 L +1 Γ( L + ) √ π j L ( qr ) r L | r → , (35)one can show that Z ∞ q dq (2 π ) k dk (2 π ) ( kq ) L e G ( ω ) L ( k, q ; E )= 2 L +1 Γ ( L + )(2 π ) G ( ω ) L ( r, r ′ ; E )( rr ′ ) L | r,r ′ → . (36) Therefore, we can rewrite Eq.(34) asdet " V − L − L +1 Γ ( L + )(2 π ) G ( ω ) L ( r, r ′ ; E )( rr ′ ) L | r,r ′ → = 0 , (37)which is just the generalization of Eq.(17) in the case ofcoupled-channel reactions. B. Coupled-channel reaction amplitude
In infinite volume, the coupled-channel reaction is de-scribed by the two-channel LS equation: T ( ∞ ) ( q ′ , q ) = − e V ( q ′ − q )+ Z d k (2 π ) e V ( q ′ − k ) e G ( ∞ ) ( k ; E ) T ( ∞ ) ( k , q ) , (38)where T ( ∞ ) is a 2 × T ( ∞ ) = (cid:20) T ((1 , , ∞ ) T ((1 , , ∞ ) T ((2 , , ∞ ) T ((2 , , ∞ ) (cid:21) , (39)and the free Green’s function is also in matrix form, e G ( ∞ ) ( k ; E ) = q µ − k µ q µ − k µ , (40)with q µ = q µ + ∆ = E .
The partial-wave expansion of Eq.(38) produces T ( ∞ ) L ( q ′ , q ) = − e V L ( q ′ , q )+ Z ∞ k dk (2 π ) e V L ( q ′ , k ) e G ( ∞ ) ( k ; E ) T ( ∞ ) L ( k, q ) . (41)The separable form of the potential (33) enables semi-closed form solutions to the above integration equation: T ( ∞ ) L ( q ′ , q ) = ( q ′ q ) L t L ( E ) , (42)where t − L ( E ) = − V − L + Z ∞ k dk (2 π ) k L e G ( ∞ ) ( k ; E ) . (43)When it is on-shell, the matrix t L is usually parameter-ized by phase shifts, δ (1 , L , and inelasticity η L , t L ( E )(4 π ) = i (cid:18) η L e iδ (1) L − (cid:19) µ q L +11 √ − η L e i ( δ (1) L + δ (2) L ) µ q ( q q ) L √ − η L e i ( δ (1) L + δ (2) L ) µ q ( q q ) L i (cid:18) η L e iδ (2) L − (cid:19) µ q L +12 . (44)Using the following identity, Z ∞ k dk (2 π ) k L e G ( ∞ ) ( k ; E )= 2 L +1 Γ ( L + )(2 π ) G ( ∞ ) L ( r, r ′ ; E )( rr ′ ) L | r,r ′ → , (45)where G ( ∞ ) L ( r, r ′ ; E ) = " G (1 , ∞ ) L ( r, r ′ ; q ) 00 G (2 , ∞ ) L ( r, r ′ ; q ) ,G ( α, ∞ ) L ( r, r ′ ; q α ) = − i µ α q α j L ( q α r < ) h (+) L ( q α r > ) , (46)we find the coupled-channel version of Eq.(16): t − L ( E ) = − V − L + 2 L +1 Γ ( L + )(2 π ) G ( ∞ ) L ( r, r ′ ; E )( rr ′ ) L | r,r ′ → . (47) C. BERW formula for coupled-channel reactions
Using Eq.(47) in Eq.(37) and , one findsdet (cid:20) L +2 Γ ( L + ) π G ( ω ) L ( r, r ′ ; E ) − G ( ∞ ) L ( r, r ′ ; E )( rr ′ ) L | r,r ′ → + (4 π ) t − L ( E ) (cid:21) = 0 . (48)The cancellation of ultraviolet divergences manifested bythe Green’s functions is made clear if we use the limitingforms of both Green’s functions:2 L +2 Γ( L + ) π G ( α,ω ) L ( r, r ′ ; ǫ ( ω ) α )( rr ′ ) L | r,r ′ → = − ( µ α ω ) L + ω L +2 ( − L +1 Γ( L + − ǫ ( ω ) α ω )Γ( − L − ǫ ( ω ) α ω ) − L +1 Γ( L + )Γ( L + ) π µ α r L +1 | r → , (49)and2 L +2 Γ( L + ) π G ( α, ∞ ) L ( r, r ′ ; q α )( rr ′ ) L | r,r ′ → = − i µ α q L +1 α − L +1 Γ( L + )Γ( L + ) π µ α r L +1 | r → . (50)Equation (48) is thus reduced to the more compact form: η L (cid:16) g (1) L g (2) L (cid:17) cos (cid:16) δ (1) L − δ (2) L (cid:17) + (cid:16) − g (1) L g (2) L (cid:17) cos (cid:16) δ (1) L + δ (2) L (cid:17) − η L (cid:16) g (1) L − g (2) L (cid:17) sin (cid:16) δ (1) L − δ (2) L (cid:17) − (cid:16) g (1) L + g (2) L (cid:17) sin (cid:16) δ (1) L + δ (2) L (cid:17) = 0 , (51) where g ( α ) L = ( − L +1 (cid:18) µ α ωq α (cid:19) L + Γ( L + − ǫ ( ω ) α ω )Γ( − L − ǫ ( ω ) α ω ) . (52)The coupled-channel QC in a harmonic trap expressedby Eq.(51) resembles that in a periodic cube [see Eq.(25)in [33]]. In the harmonic trap, g ( α ) L functions play thesame role of L¨uscher zeta functions in the periodic cube.Given one energy eigenvalue E , we can not determineall of δ (1 , L and η L simultaneously. A possible way out isto gather several pairs of ( E, ω ) to build more constraintson δ (1 , L and η L , provided that energy eigenvalues E ’s aresomewhat close to each other and the ratio E/ω is dis-tinct from one another. Though this approach is model-independent, the requirement may only be met by smallportion of the calculated levels. Another useful approachwidely used in the lattice QCD community is to model δ (1 , L and η L with a few free parameters, which can befitted to more energy levels by using QC as defined inEq.(51), (see, e.g., discussions in Refs. [33, 34]). IV. DISCUSSIONS AND SUMMARY
The harmonic trap may be replaced with artificialfields of different configuration or boundary condition.For instance, hard-sphere boundary condition is used insome Monte Carlo simulations [24–26]. We can repeat allthe discussions in previous sections by substituting thefollowing external potential in place of the HO potential: V h.s. ( r ) = ( , r < b , ∞ , r > b , (53)where b is the radius of the sphere. The Green’s functionmust satisfy the boundary condition: G ( h.s. ) L ( r, r ′ ; q )= − µqj L ( qr < ) j L ( qr > ) (cid:20) n L ( qb ) j L ( qb ) − n L ( qr > ) j L ( qr > ) (cid:21) . (54)For instance, in the case of the single-channel elastic scat-tering, Eqs. (16) and (17) are replaced with2 L +2 Γ ( L + ) π G ( h.s. ) L ( r, r ′ ; q ) − G ( ∞ ) L ( r, r ′ ; q )( rr ′ ) L | r,r ′ → = − µq L +1 [cot δ L ( q ) − i ] . (55)Using the asymptotic form of G ( ∞ ) L (50) and12 µ L +2 Γ ( L + ) π G ( h.s. ) L ( r, r ′ ; q )( rr ′ ) L | r,r ′ → = − q L +1 n L ( qb ) j L ( qb ) − L +1 Γ( L + )Γ( L + ) π r L +1 | r → , (56)the quantization condition of the single-channel case forthe spherical hard wall is thus given by a form analogousto the BERW formula (or L¨uscher’s for that matter):cot δ L ( q ) = n L ( qb ) j L ( qb ) . (57)The hard-sphere geometry is encoded in n L ( qb ) /j L ( qb ), which plays the same role as g L ( q )to the harmonic trap. Similarly, the coupled-channelQC can be worked out for the spherical hard wall byreplacing g ( α ) L in Eq.(52) as follows g ( α ) L → n L ( q α b ) j L ( q α b ) . (58)In summary, we have derived the quantization condi-tion for coupled-channel two-particle systems trapped bythe harmonic potential. The formula by itself relates thecoupled-channel reaction amplitudes in infinite volume,parametrized by two phase shifts and one inelasticity, toenergy eigenvalues of the corresponding trapped system.Although only the quantization condition for the case of distinguishable spinless particles is shown, with the tech-nical detail spelled out in the paper, the extension to in-clude slight complications due to spin, identical particlesand so on should be straightforward. Coulomb repulsionis important, however, if reactions take place near thethreshold of charged particles. Quantization conditionsmay be derived for that circumstance in the harmonictrap, following the methodology developed in Ref. [53]. Acknowledgments
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