Energies and widths of Efimov states in the three-boson continuum
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Energies and widths of Efimov states in the three-boson continuum
A. Deltuva ∗ Institute of Theoretical Physics and Astronomy, Vilnius University, Saul˙etekio al. 3, LT-10257 Vilnius, Lithuania (Received April 21, 2020)Three-boson Efimov physics is well known in the bound-state regime, but far less in the three-particle continuum at negative two-particle scattering length where Efimov states evolve into reso-nances. They are studied solving rigorous three-particle scattering equations for transition operatorsin the momentum space. The dependence of the three-boson resonance energy and width on thetwo-boson scattering length is studied with several force models. The universal limit is determinednumerically considering highly excited states; simple parametrizations for the resonance energy andwidth in terms of the scattering length are established. Decreasing the attraction, the resonancesrise not much above the threshold but broaden rapidly and become physically unobservable, evolvinginto subthreshold resonances. Finite-range effects are studied and related to those in the bound-stateregime.
PACS numbers: 21.30.-x, 21.45.-v
I. INTRODUCTION
Fifty years ago V. Efimov studied theoreticallythe three-body system with large two-body scatteringlengths [1] and laid the foundations of the universalphysics, also called Efimov physics. Since then a largenumber of theoretical and experimental works with ap-plications to nuclear, cold atom, and molecular physicshas been performed, and the properties of universal few-body systems have been determined; see recent reviews[2, 3] for summary and further references. In partic-ular, the bound-state energy levels of three identicalbosons and their dependence on the two-boson scatter-ing length is among the best-known manifestations ofthe Efimov physics, even with semi-analytic parametriza-tions available [2, 4], the most refined version being pre-sented in Ref. [5]. However, the universal properties ofthe three-boson system were studied less extensively inthe regime of negative scattering length above the free-particle threshold where the trimers become unbound.Reference [4] considered only the limit of vanishing en-ergy, while Refs. [6–8] investigated the recombinationinto deeply bound dimers. It is known that in this regimethe trimers become resonant states [9]; their energies,widths, and their evolution with the scattering lengthwere determined by Bringas et al. [9], but not in astrictly universal regime where the scattering length issufficiently large such that the finite-range effects are neg-ligible. A systematic study of finite-range effects is alsostill missing. The aim of the present work is to resolvethis situation and to present a high-precision study ofuniversal three-boson resonant states in the three-particlecontinuum with no bound dimers, neither shallow nordeep.The study is based on exact Faddeev-type equations[10] for three-body transition operators in the version ∗ [email protected] proposed by Alt, Grassberger, and Sandhas (AGS) [11],that are solved in the momentum space. The methodhas been used for the search for three-neutron resonances[12]. An important advantage of the direct continuumapproach is its ability to estimate not only the reso-nance position but also its effect on scattering amplitudesthat lead to observables in collision processes. Additionalcomplications in the context of Efimov physics arise dueto very different sizes of the interaction range and thecharacteristic interparticle distances.Section II describes three-particle scattering equationsand some details of calculations whereas Sec. III containsresults. The summary is presented in Sec. IV. II. THEORY
If the two-particle potential v is tuned to yield a nega-tive two-boson scattering length a and no bound dimers,then the only possible process in the three-boson contin-uum is the 3 → U = (1 + P ) t (1 + P ) + (1 + P ) tG U G t (1 + P ) (1)contains two-body and three-body parts represented bythe first and second terms, respectively; only the lattercorresponds to a genuine three-particle process. All op-erators in Eq. (1) refer to the relative three-particle mo-tion. They are the combination of particle permutationoperators P = P P + P P , the free resolvent G = ( E + i − H ) − (2)at the available energy E with the kinetic energy opera-tor H , and the two- and three-particle transition oper-ators t and U , respectively. They are related to the two-particle potential v via integral equations by Lippmann-Schwinger, t = v + vG t (3)and AGS U = P G − + P t G U. (4)As is well known, a true bound state of the few-bodysystem (stable with respect to decay into several clusters)corresponds to the pole of the respective transition oper-ator at a real negative energy E = − E b in the physicalenergy sheet, E b being the binding energy. In the sys-tem of three spinless bosons, bound Efimov states havezero total angular momentum J = 0 and positive parityΠ. When this bound state, via variation of the inter-action, crosses the free-particle threshold and becomes aresonance, the pole of the transition operators (both U and U ) moves to E = E R − i Γ / E R being the resonance positionand Γ its width. As long as E R > U J Π = ∞ X n r = − ˜ U ( n r ) J Π ( E − E R + i Γ / n r , (5)the n r = − n r ≥ U and T = tG U G t are calculated as functions of the energy E , within nu-merical accuracy all exhibiting the same resonant be-havior, much like that shown in Ref. [12] for the three-neutron system with enhanced interaction. The positionand width of the resonance is determined fitting thoseenergy-dependent matrix elements to Eq. (5). Differencesbetween the resonance parameters obtained from transi-tion matrix elements with different initial and final teststates, as well as variations due to the different maximalpower n r in the expansion (5), serve as an estimationof the error for this procedure. Typically, for small Γthe pole is close to the physical scattering region, theresonant behavior is pronounced very clearly, and, con-sequently, the error is extremely small. In contrast, verybroad resonances become hardly distinguishable from thebackground, resulting also slower convergence of the ex-pansion (5) where n r up to 5 or 6 was employed in thepresent work. This leads to decreasing accuracy, and renders a reliable extraction of the resonance parametersimpossible when Γ is too large. On the other hand, thefact that resonant behavior cannot be seen in transitionmatrix elements indicates that the resonance cannot bedistinguished from background and becomes physicallyunobservable. III. RESULTS
The present work aims to study the evolution of three-boson Efimov resonances, i.e., to establish universal re-lations between their energies and widths and the two-boson scattering length, similar to what is well knownfor bound states [2, 4, 5]. The quantity connecting thetrimer bound- and resonant-state regimes is the nega-tive two-boson scattering length a − n where the n -th Efi-mov trimer crosses the free-particle threshold. There isa universal relation between a − n and the binding momen-tum k un of the n -th Efimov trimer in the unitary limit.Reference [4], where also the elastic three-boson scatter-ing in the zero energy limit was considered, predicted k un a − n = − . k un a − n = − . k un a − n = − . n = 5. Tomake the connection with the latter work, the two-bosoninteraction model is taken from Ref. [15]. It has only the S -wave component h p f | v s | p i i =(1 + βp f / Λ ) e − p f / Λ × πm (cid:26) a − (cid:20) β (cid:18) β (cid:19)(cid:21) Λ √ π (cid:27) − × e − p i / Λ (1 + βp i / Λ ) (6)where p f ( p i ) is the final (initial) momentum, m is theboson mass, and Λ is the momentum cutoff parameterthat controls the range of the force; the ~ = 1 conventionis used. The parameter β enables variations in the bal-ance of low- and high-momentum components as will beneeded later on; for the following results it is, however,chosen as β = 0 unless explicitly stated otherwise. Forthe extraction of universal relations highly excited stateshave to be considered with vanishing finite-range correc-tions. This can be characterized by the ratio of the two-boson S -wave effective range r s and the scattering length a . At the trimer and free-particle threshold intersectionpoints these ratios r s / | a − n | are 0 . , . , . × − ,and 3 . × − for n = 0, 1, 2, and 3, respectively. Thetrimer ground state n = 0 is therefore expected to ex-hibit significant finite-range corrections. However, finite-range effects should become very small for n ≥
2. In-deed, the product k un a − n is − . , − . , − . − . n = 0, 1, 2, and 3, respectively, in the n = 3 case being very close to the universal value of − . n = 3 should be sufficientalso for the determination of universal Efimov resonanceproperties with a good accuracy.To emphasize the universal character of the results,they will be given as dimensionless quantities. Thus, theenergy E Rn and width Γ n of the n -th Efimov state will beshown in the dimensionless forms ε n = E Rn m ( a − n ) / ~ and γ n = Γ n m ( a − n ) / ~ . In other words, ε n ( γ n ) is theenergy (width) of the n -th Efimov resonance in units of ~ / [ m ( a − n ) ]. For the two-boson scattering length themost meaningful reference point in the present contextis a − n for each n . To keep consistency with the stan-dard representation of the Efimov physics in terms ofthe inverse scattering length, the results will be shownas functions of the dimensionless quantity | a − n | /a thattakes values below (above) − ε n of four low-est three-boson states with n = 0 , ,
2, and 3 are shownin Fig. 1, partially including also the bound state re-gion where the binding energy was obtained by solvingthe standard bound-state Faddeev equation [15]. Thebound-state and continuum calculation results connectwell at | a − n | /a = −
1. The corresponding dimensionlessresonance widths γ n are shown in Fig. 2. In both casesthe convergence with increasing n is evident, suggestingthat, n = 2 and n = 3 results, being very close to eachother, accurately represent the universal limit. In con-trast, the ground state ( n = 0) results show significantdeviations from the universal limit due to finite-rangecorrections, as can be expected given the correspondingratio r / | a − n | = 0 . | a − n | /a , and, as a consequence,the theoretical error bars increase too. For clarity, thetheoretical error bars are given only for n = 3 results,they are roughly the same for all considered n values.Beyond | a − n | /a = − | a − n | /a > − κ rn − iκ in , related to energy andwidth as ε n − iγ n / κ rn − iκ in ) , i.e., ε n = ( κ rn ) − ( κ in ) , (7a) γ n = 4 κ rn κ in . (7b)Note that in the bound-state region κ rn = k n | a − n | sim-ply relates to the standard binding momentum k n while κ in = 0. The dependence of κ rn and κ in on the inversescattering length is presented in panels (a)-(b) of Fig. 3.A remarkable feature is that κ in depends on | a − n | /a al-most linearly. Furthermore, ( κ rn ) displayed in the panel(c) of Fig. 3 also exhibits a nearly linear dependence on -0.20.00.20.4 -2.0 -1.5 -1.0 ε n |a n- |/a n=0n=1n=2n=3 FIG. 1. (Color online) Dimensionless real energy part forground ( n = 0) and excited ( n = 1, 2, and 3) three-bosonstates as a function of the inverse two-boson scattering length. γ n |a n- |/a n=0n=1n=2n=3 FIG. 2. (Color online) Dimensionless width for ground ( n =0) and excited ( n = 1, 2, and 3) three-boson states as functionof the inverse two-boson scattering length. | a − n | /a in the resonance region | a − n | /a < −
1, while devi-ations from the linearity are more visible in the bound-state region. Thus, introducing for brevity the variable x a = − (1 + | a − n | /a ), in the resonance region ( x a >
0) thebehavior of the dimensionless momentum componentsroughly is κ in ∼ x a and ( κ rn ) ∼ x a . Based on theseobservations and Eq. (7), the lowest-order (LO) approx-imate expressions for energy and width are postulated,i.e., ε n ≈ c ε x a − c ε x a , (8a) γ n ≈ c γ x / a . (8b)The state label n on the right-hand side is suppressed forbrevity. Assuming strictly linear dependence of κ in and( κ rn ) on x a , only two out of three coefficients in Eqs. (8)would be independent. However, as there are deviations, (a) κ n r (b) κ n i n=0n=1n=2n=30.00.5-2.0 -1.5 -1.0 (c) ( κ n r ) |a n- |/a FIG. 3. (Color online) Components of complex dimensionlessresonance momentum for ground ( n = 0) and excited ( n = 1,2, and 3) three-boson states as functions of the inverse two-boson scattering length. the most evident one being for ( κ r ) , the quality of theapproximation is improved by considering the three coef-ficients as independent. The higher order (HO) approx-imation is obtained taking into account small deviationsfrom linearity in κ in and ( κ rn ) at small x a , i.e., near thethreshold. Proposed phenomenological relations( κ rn ) ≈ c r x a + c r x / a , (9a) κ in ≈ x a ( c i − c i e − c i x / a ) , (9b)are used in Eq. (7) for HO approximations of ε n and γ n .The coefficients in Eqs. (8) and (9) are obtained by fittingthe results presented in Figs. 1 - 3. In order to checkthe predictive power of the LO and HO approximations,only the data points in the regime − . < | a − n | /a < | a − n | /a < − .
67. For n = 3, which is expected to bea very accurate approximation of the universal limit, the values of fit parameters are c ε = 0 . ± . , (10a) c ε = 0 . ± . , (10b) c γ = 0 . ± . , (10c)and c r = 0 . ± . , (11a) c r = 0 . ± . , (11b) c i = 0 . ± . , (11c) c i = 0 . ± . , (11d) c i = 4 . ± . n = 3. Both LO andHO approximations seem to fit the original data very wellup to | a − n | /a > − .
67. However, a closer inspection ofdeviations ∆ ε n = ε n − ε n (XO) and ∆ γ n = γ n − γ n (XO),amplified by a factor of 10 and shown by thin curves nearthe zero line, demonstrates considerably better accuracyof the HO approximation, especially for the two pointsat the left that were excluded from the fit. Nevertheless,the LO still fits those data within their error bars.For curiosity, in Fig. 4 the LO and HO results are ex-trapolated to the | a − n | /a < − γ n (XO) increases rapidlywhile ε n (XO) reaches its maximum and then decreaseswith decreasing | a − n | /a , becoming a completely unobserv-able very broad subthreshold resonance. The trajectoryof the resonance in the complex plane ε n − iγ n / | a | de-creases.As mentioned, an earlier study of Efimov resonanceswas performed by Bringas et al. [9]. They solved bound-state equation with contour deformation into the complexplane. Instead of the inverse two-boson scattering length1 /a they used the square root of the virtual dimer energy, p | B | . Since in the universal regime B = − / ( ma ),these two representations differ by finite-range correc-tions only, allowing for an easy qualitative comparison.Bringas et al. studied n = 0, 1, and 2 Efimov states,the evolution of their energies and widths is qualitativelyfully consistent with present results. A perfect agree-ment between n = 1 and 2 results in Ref. [9] was notachieved, indicating that the universal regime was notyet reached with a good accuracy. This can be quanti-fied by ratios of virtual dimer energies B ( n )2 = B | a = a − n when the n -th trimer crosses the threshold. In detail, (a) ∆ ε n * 10 ε n (b) ∆ γ n * 10 γ n |a n- |/a LOHO -3-2-10 0.0 0.5 (c) - γ n / ε n FIG. 4. (Color online) Dimensionless energy (a) and width (b)for the excited ( n = 3) three-boson state as functions of theinverse two-boson scattering length. The LO (dashed-dottedcurves) and HO (solid curves) approximations are comparedwith the results of the direct calculation, represented by sym-bols with error bars. Thin curves around the zero line repre-sent the corresponding differences, amplified by a factor of 10.The panel (c) shows the resonance trajectory in the complexenergy plane. q B (1)2 /B (2)2 = 21 . q B (1)2 /B (2)2 = 21 . q B (2)2 /B (3)2 = 22 .
63 in this work, the universal limitbeing 22 . | a − n | /a > − .
3, the quantitative differences beingof the order of 10%. This possibly indicates that deepdimers do not affect significantly the energy and dissoci-ation width of the Efimov resonance.Another important and interesting problem is thequantification of finite-range effects. Since the evolutionof the resonance can be reasonably reproduced by a fewparameters of the LO approximation (8), they are cho-sen for this study. Ji et al. showed that finite-rangecorrections to various Efimov features in the bound-stateregime can be expressed by terms proportional to r s and nr s [16], and related this result to an earlier work byKievsky et al. [17]. The connection between the two ap- proaches was updated recently by Gattobigio et al. [5]through the running Efimov parameter. It appears thatfinite-range effects for the resonance parameters c εj and c γ determined in the present work are compatible withthe pattern of Ref. [16]. For n = 0, 1, 2, and 3 theapproximate expressions c ε ≈ . − . r s /a − n − . nr s /a − n , (12a) c ε ≈ . − . r s /a − n − . nr s /a − n , (12b) c γ ≈ . − . r s /a − n − . nr s /a − n , (12c)reproduce the results of direct calculations with good ac-curacy, the deviations being below 0.05%, 1%, and 0.3%,respectively. However, the above relations involve non-universal coefficients and cannot predict finite-range cor-rections for other force models.To study finite-range effects, four additional calcula-tions with different force models are performed for low( n = 0 or 1) resonances. One of them uses the scaledMalfliet-Tjon (MT) I-III S potential [18], that hasan attractive long-range part and a strongly repulsiveshort-range part. However, the results turn out to bequite close to the ones based on the potential (6), i.e., r s / | a − n | = 0 .
369 and k un | a − n | = 2 . n = 0. There-fore other models are desired to have different r s / | a − n | and k un | a − n | values, spanning the gap between the previ-ously shown n = 0 and 1 results of the potential (6) with β = 0. This goal can be achieved by the variation of theparameter β in the potential (6) thereby enhancing thehigh-momentum components and enabling the control of r s / | a − n | and k un | a − n | values. With n = 1 and β = − , − − r s / | a − n | = 0 . k un | a − n | = 1 . c εj and c γ resultsfor these eight sets do not exhibit a clear dependenceon r s / | a − n | and/or nr s / | a − n | , but show linear correlationwith the corresponding k un | a − n | values as demonstrated inFig. 5. Most of the points are very close to the linesparameterized by c ε ≈ . . k un | a − n | − . , (13a) c ε ≈ . . k un | a − n | − . , (13b) c γ ≈ . . k un | a − n | − . , (13c)except for the two n = 0 points possessing largest r s / | a − n | and k un | a − n | values, for which the finite-range effects areexpected to be sizable, thereby leading to larger devi-ations. Nevertheless, Eqs. (13) may be useful for theestimation of the resonance position and width from thefeatures of the corresponding bound state. IV. SUMMARY
The three-boson continuum study was carried out inthe regime of negative two-boson scattering length with c ε j k nu |a n- | c γ c ε c ε FIG. 5. (Color online) Resonance energy and widthparametrization coefficients c εj and c γ for various levels andforce models as functions of the product k un | a − n | . In termsof the potential (6) parameter β and level n the points fromleft to right correspond to ( β, n ) being ( − , , , , − , − , , k un | a − n | = 1 . no shallow or deep dimers, where the three-particle Efi-mov states are not bound but resonant. Exact scatter-ing equations for three-particle transition operators were solved in the momentum-space framework using severalinteraction models. From the energy dependence of thevarious on-shell and off-shell transition matrix elementsthe resonance properties were determined, in particular,their universal limit was accurately calculated consider-ing highly excited Efimov states such that the finite-rangeeffects become negligible. Universal relations between theresonance energy and width and the two-particle scatter-ing length a were established, and simple parametriza-tions were proposed, based on a nearly linear depen-dence of the complex resonant momentum componentson the inverse scattering length. It was found that de-creasing | a | , i.e., decreasing the attraction, the resonancesrise not much above the threshold, the real part of en-ergy reaching roughly ~ / [2 m ( a − n ) ] or ~ / (8 ma ) around a ≈ a − n /