Universal physics of bound states of a few charged particles

We study few-body systems of charged particles subject to attractive short-range plus repulsive Coulomb interparticle forces. First, we focus on attractive potentials with a vanishing range. We discuss properties of the corresponding bound states, in particular, binding energies and sizes. Second, we investigate weakly-bound systems for attractive potentials with a finite range. We show that, in contrast to neutral particles, an accurate description of charged particles requires both the Coulomb modified scattering length and the effective range. Our findings are relevant for shallow bound states whose spatial extent is significantly larger than the range of the attractive potential. These states enjoy universality, i.e., their character is independent of the shape of the short-range potential. The only relevant parameters for two-body systems are the Sommerfeld parameter as well as the Coulomb modified scattering length and effective range; for three- and four-body systems, a three-body parameter must be added.

Introduction.-Shallow bound states of two neutral particles with zero angular momentum live in a classically forbidden region and retain almost no information about binding interactions [1]. As a consequence, any short-range attractive potential, V S , can be used to model these states as long as it fixes a few relevant parameters (e.g., the scattering length, effective range) to their physical values. A celebrated V S is a zero-range potential tuned to reproduce the scattering length [2][3][4]. It provides a powerful starting point for studying universal bound states (i.e., independent of the shape of V S ) in nuclear and atomic physics [1][2][3][4][5][6][7][8][9][10][11][12].
In this Letter we consider particles that interact via V S +V C , where V S is defined as for neutral particles and V C is a repulsive Coulomb potential. Potentials V S + V C are typical for cluster models of nuclei [5,13], e.g., in 17 F between 16 O and a proton [14,15]. Furthermore, they provide an effective description of interactions between charged quasi-particles, e.g., between dressed electrons in crystals [16][17][18][19][20][21]. We focus on V S of zero range, and explore such potentials as a possible starting point for understanding realistic charged systems.
The main finding of our study is that zero-range potentials for charged particles lead to a family of few-body bound states whose properties are fully determined by the Sommerfeld parameter, the Coulomb modified scattering length and the three-body parameter. We present and discuss these bound states. Note that, in contrast to neutral particles, the zerorange approximation to V S is not guaranteed to be useful for realistic shallow bound states: The Coulomb barrier makes the spatial extent of the wave function finite [22], forcing particles to explore the landscape of the short-range binding potential. Therefore, we must investigate finite-range corrections. For shallow two-body bound states, we prove that finite-range corrections to the energy can be accounted for by an effective range parameter. For weakly-bound three-and four-body systems, we study these corrections numerically using the Gaussian Expansion Method [23,24] and Stochastic Variational Method with Gaussians [24][25][26][27][28]. Our findings suggest that a single finite-range parameter suffices to describe these systems. FIG. 1. The probability density |u| 2 for two particles with charges Q1 and Q2 (Q1Q2 > 0) from Eq. (2) in the limit of vanishing binding energy. If the characterstic Coulomb length D is much larger than the range of the potential R, the wave function u is essentially characterized by its tail. Similarly to neutral particles the system becomes universal, i.e., independent of the shape of VS.
Two-Body System.-First, we consider two particles whose relative motion is described by the radial Schrödinger equation where µ is the reduced mass, E 2 = − 2 κ 2 2µ , κ > 0 is the two-body energy, V C = k Q1Q2 r is the Coulomb potential energy (Q 1 , Q 2 are the particle charges, k is Coulomb's constant), and V S is a binding potential of range R. We consider only zero angular momenta since our focus is on the bound states for R → 0 (later referred to as the zero-range or universal limit). Moreover, we are free to choose any shape of V S , which is irrelevant as long as the limit R → 0 is well-defined for neutral particles interacting via V S (cf. [3,29]). For simplicity, we assume that V C + V S is a square well for r ≤ R, i.e., V S (r) = − 2 g 2µR 2 − V C (r), and V C otherwise. The dimensionless parameter g > 0 sets the interaction strength. The arXiv:1904.00913v1 [nucl-th] 1 Apr 2019 wave function u for this potential reads where A is a normalization constant is the Sommerfeld parameter, and W is the Whittaker W -function [30]. The values of κ that lead to a continuous derivative of u at r = R define allowed bound states. In the limit R → 0, κ is a root of the equation where γ is Euler's constant, and ψ is the digamma function [30]. Note that for neutral particles (η = 0) Eq. (2) depends only on κR, hence, the result of taking the limit R → 0 with fixed κ is identical to that with κ → 0 and fixed R. In other words, for weakly-bound states of neutral particles one may always rely on the zero-range limit. For charged particles with κ → 0, by contrast, this limit is not necessarily accurate. Using Eq. (3) we derive the identity which is model-independent, as it connects one observable (the binding energy) to another (the scattering length). The Coulomb modified scattering length, a C , is defined as [29] is the length scale associated with the Coulomb potential. Previously, Eq. (4) was derived to research the possibility of proton-proton bound states [31,32]. It specifies the poles of the scattering amplitude defined as in Refs. [33,34]. In this Letter, we consider Eq. (4) in the context of bound states in zero-range potentials. The left-handside of Eq. (4) is a monotonic negative function. Therefore, if a C > 0 a zero-range potential can support at most one bound state, and if a C < 0 there can be no bound states. Let us investigate Eq. (4) in the two limits: D a C (weak Coulomb potential) and a C D (strong Coulomb potential). In the former limit, we obtain This equation features a logarithmic correction to the standard expression κ = 1/a C for neutral particles [1]. In the opposite limit, we have which describes shallow bound states. The fact that κ 2 ∼ 1/a C in the limit a C → ∞ will be of utmost importance for determining finite-range corrections to the universal limit.
To calculate other observables, we note that for R → 0 the particles move almost exclusively in the classically forbidden  region; see Fig. 1. Indeed, the probability to find particles with r > R is approaching unity: To derive the limiting value, we notice that the integral dr is finite in the limit R → 0, whereas R 0 u 2 dr < u 2 (R)R. Therefore, observables for zero-range interactions are described with the wave function W −η,1/2 (2κr), defined by η and κ. As an example, we use the root-mean-square (rms) radius, r 2 ≡ u 2 r 2 dr, -a standard observable in few-body physics -given by where the subscript 0 refers to the zero-range limit. The righthand-side of Eq (8) is a monotonically increasing function of η. The maximum value is attained at 1/η = 0 where r 2 0 /D = 0.507 . . . ; in this limit the size of the bound state is fully determined by D. The boundedness of the rms radius is relevant for charged halo nuclei [22]; it also supports the predicted discontinuous behavior of the mean distance between two polarons across the unbound-polarons to bipolaron transition [17,35]. For η → 0 the rms radius is determined from the standard relation 2 r 2 0 κ 2 → 1 [5]. Finite-range corrections.-The rms radius in Eq. (8) does not diverge for κ → 0, indicating that the inclusion of finiterange corrections is unavoidable for charged systems. These corrections must be small if r 2 R, which, according to Eq. (8), is satisfied for weakly-bound systems if D R (cf. Fig. 1). For comparison, D 57.6 fm for two free protons and D 0.1 nm for free electrons. Therefore, if two protons (or a proton with a light nucleus) formed a shallow bound state it would be universal [36], since natural values of R in this case are around 1 fm. Two dressed electrons in solids could represent another universal system since the effective mass, the strength of the Coulomb potential (hence D), as well as R depend on the material. In contrast, a shallow bound state of two free atomic ions (D 57.6 fm and R ∼ 0.1 nm) cannot be universal.
To estimate finite-range corrections to the energy, we notice that the right-hand-side of Eq. (4) is the first term of the Coulomb-modified effective-range expansion. To account for the next term, one must use −1/(a C κ) − r eff κ/2 instead of −1/(a C κ), where r eff is the effective range [38,39].
The effective-range correction enters at leading order in the expansion of the energy for weakly bound states κ 2 6/(a C D)/(1 − 3r eff /D), because the leading contribution to κ 2 in Eq. (6) is proportional to 1/a C . Higher order corrections (e.g., due to the shape parameter) are not as important in the limit 1/a C → 0 because they are convoluted with k n where n > 2. Note that the factor (1 − 3r eff /D) in the denominator of κ 2 for 1/a C → 0 implies that for potentials with weaklybound states r eff < D/3 must hold, in agreement with the causality constraints of [40]. Our result can be used to define the leading order of an effective field theory for shallow bound states of charged particles where r eff contributes at leading order, while higher effective range parameters can be included perturbatively. The fact that range corrections are enhanced in systems with strong Coulomb interactions was already observed for 17 F [15] and 7 Be [41], and attributed to an additional fine tuning. The importance of finite-range effects in systems with strong Coulomb interactions was also observed in [42]. Here, we show that this enhancement is generic for strong Coulomb.
To illustrate finite-range effects, we use the Gaussian potential V G S = g G e −r 2 /(2R 2 G ) , where R G defines the range of the potential, and g G is used to fix the Coulomb modified scattering length for a fixed value of R G . For the sake of discussion, we use parameters of two α-particles 2 /µ = 20.73 MeV×fm 2 and kQ 1 Q 2 = 5.76 MeV×fm [D 3.6 fm]. We employ the Gaussian Expansion Method (GEM) [23] to calculate κ. The result is plotted in Fig. 2a) as a function of [a C (1 − 3r eff /D)] −1 (for consistency, we take r eff for V G S with κ = 0), which is the only relevant parameter for 1/a C → ∞; see Eq. (9). Figure 2 shows that even though the universal prediction does not describe finite values of r eff /D (see Fig. 2b)), it is still useful: Finite-range corrections for shallow bound states are captured by rescaling a C with 1 − 3r eff /D. [See Fig. 2a)] Expectation values of other observables ( O ≡ u 2 Odr) also acquire finite-range corrections when r eff = 0. It is par- ticularly easy to calculate these corrections for an observable O, which in the limit R → 0 satisfies R 0 Ou 2 dr ∼ R 1+δ , where δ > 0, e.g., the rms radius. After straightforward but tedious calculations we derive (in the leading order in r eff ) where O 0 is the universal prediction for the same κ and η.
As anticipated, the universal value O 0 is accurate if R D (typically r eff R). The left-hand-side depends weakly on energy (η) and for η > 2 it can be accurately written as 1 + r eff 2D (6 + 1.1 η 2 ). Note also that the correction in Eq. (10) is independent of O. Therefore, it can, in principle, be used to relate different measurements.
Three-Body System.-Now we consider a three-body system of charged particles with mass m i , charge Q i , and coordinates r i , i = 1, 2, 3, interacting via Coulomb and shortrange pair interactions as in Eq. (1). For Q i ≡ 0, this system features two hallmarks of few-body physics: the Efimov effect [43] and the Thomas collapse [44]. Both are connected in the hyperspherical formalism [45] to a super-attractive ρ −2potential in the hyperradius, ρ 2 = (r 2 12 + r 2 13 + r 2 23 )/3, with r ij = |r i − r j |. The Thomas collapse occurs due to the divergence of 1/ρ 2 at the origin [29], whereas the infinite tower of Efimov states is supported by the scale invariance of this potential. The 1/ρ 2 form of the potential strongly suggests that the Thomas effect is weakly modified by the Coulomb potential, but the shallow Efimov states must disappear.
Efimov Effect and Thomas Collapse.-To illustrate the fate of the Efimov effect in the presence of the Coulomb interaction, we consider two identical heavy charged particles, the third particle is neutral and light, i.e., m 1 = m 2 , m 3 /m 2 1, Q 1 = Q 2 and Q 3 = 0. This system is conveniently studied within the Born-Oppenheimer approximation: First, the Schrödinger equation with fixed heavy particles is solved for the light particle, which gives the energy (|r 1 − r 2 |). Then, the energy spectrum is found from the two-body equation where R ≡ r 1 − r 2 , and Φ is the wave function that describes the relative motion of the two heavy particles. For simplicity, we assume that V S is an infinite barrier for |R| < R and zero otherwise. The function (R) can be found analytically for a separable s-wave interaction between heavy and light particles [46]. For an infinite heavy-light scattering length it reads: (R → ∞) − 2 A m1R 2 where A > 1/4 generates infinitely many bound states if V C = 0. For charged particles the Schrödinger equation, cannot support infinitely many-bound states. It can support at most N bound states, where N can be estimated using the Bargmann inequality [47] where D is the Coulomb length for two heavy particles. It is clear that only if D/R 1 there can be many bound states. For example, for the parameters as in 9 Be described as an α + α + n system [48,49] we have N 1.5, where, for simplicity, we used natural values: A = 1.25 and R = 1 fm. It can be shown that the Coulomb potential dominates also long-range behavior of the lowest adiabatic potential in the hyperspherical formalism (cf. [50]), which leaves no room for the Efimov effect with charged particles. However, the lowlying Efimov states survive if the Coulomb interaction is sufficiently weak [51]. These low-lying states in the zero-range limit are of our interest.
We study the Thomas collapse with charged particles numerically [24]. We observe that the ground state energy behaves similarly to that for neutral particles, i.e., E ∼ −1/R 2 , in the vicinity of the zero-range limit (R → 0). Therefore, to study three-body states in the universal limit, we introduce a three-body force [52] to fix the ground state energy to a finite value. As we show below, this three-body force also allows us to study a four-body problem without any additional parameters. This is similar to neutral systems [53][54][55], Universal three-and four-body states.-We use the Gaussian Expansion Method [23] to study few-body states in the zero-range limit. We obtain the energies by perfoming a sequence of calculations with small values of R G and extrapolate to R G → 0, which gives the value at R G = 0 and the error bars. Figure 3 reports on the energies of two, three and four charged bosons. As before, the mass and charge of a boson are those of an alpha particle. The energies are fully determined by D, a C and an additional three-body parameter. The latter can be characterized either by a − C which determines the three-body binding threshold or by the three-body energy at 1/a C = 0, κ 3 (see Fig. 3), but a − C κ 3 is not constant for charged particles. Note that the energy of the three-body state at 1/a C = 0, and, hence, κ 3 , is fixed with the three-body force. For neutral particles another value of κ 3 would simply rescale the energies due to the discrete scale invariance. For charged particles the discrete scale invariance is broken (cf. Eq. (12)). Thus, we also should investigate the effect of the three-body force; see the inset of Fig. 3. We use two different three-body forces whose a − C differ by a factor of 10, and then rescale the x-axis and y-axis using a − C and κ 3 , correspondingly. We see that the effect of the three-body parameter leads to merely a rescaling of the axes for the considered cases. Therefore, we refrain from showing energies for other values of a − C . Finite-range corrections (three-body system).-We study finite-range corrections numerically; see Fig. 4. Even small values of R G immediately lead to significant corrections to the energy; see the inset of Fig. 4. Similarly to two particles, these corrections can be accounted for by rescaling a C . To demonstrate this, we define a − C (R G ) that determines the three-body binding threshold for a given value of R G . We use a − C (R G ) to rescale the x-axis. The rescaled curves coincide, which suggests that the universal limit is a good starting point for studying three-body charged systems.
Outlook.-More detailed studies are required to see in which systems the features discussed here can be observed. To this end, charged quasi-particles and nuclei close to the proton drip-line [14,15,17,20,22,[56][57][58][59][60] must be investigated. The concept of universality for neutral particles has also been explored in low and mixed spatial dimensions [61-63, 65, 66, 77], and with higher angular momenta [67][68][69]. It will be interesting to study the effect of the Coulomb potential on those universal states, especially in connection with lowdimensional bipolarons [70][71][72] and p-wave halo nuclei such as 8 B [73,74]. Finally, it would be interesting to formulate an effective field theory for shallow bound states of charged particles based on our findings and calculate corrections from higher effective range parameters perturbatively.
We thank Aksel Jensen, Dmitri Fedorov, Karsten Riisager and Wael Elkamhawy for useful discussions about proton halos, and Daniel Phillips for pointing us to [42]. This work has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project numbers VO 2437/1-1 and 279384907 -SFB 1245 and by the Bundesministerium für Bildung und Forschung (BMBF) through contract 05P18RDFN1. The three-body energy times R 2 G as a function RG for charged particles (see lower dots); the line shows the corresponding linear fit. For comparison, we also plot results for the corresponding neutral system (upper dots).

THOMAS COLLAPSE
If a zero-range two-body potential is set to reproduce a binding energy of two neutral particles, then the ground state of the corresponding three-body system is infinitely deep. This phenomenon is called the Thomas collapse [44]. A way to deal with this peculiarity in models with zero-range potentials is to introduce a three-body parameter. Here we show numerically that calculations with three charged particles, even in spite of a repulsive Coulomb potential, also require a three-body force. To this end, we compute three-body energies for two-body Gaussian potentials that have different values R G but lead to the same value of the two-body binding energy. For the sake of discussion, we use masses and charges of α-particles, and assume that the binding energy is 1 MeV. The results are presented in Fig. 5. The three-body bound state becomes unphysically deep for R G → 0 (the right panel of the figure suggests that E 3 ∼ 1/R 2 G for R G → 0), which shows the necessity of a three-body force.

NUMERICAL METHODS
We employed the Gaussian expansion method [23] for numerical calculations in the main text. It is a variational method in which the wave function is expanded as a sum of Gaussians. In this section we briefly illustrate the method for a three-body system; see [51,75] for a more detailed presentation. A variational wave function is written as where x = r 1 − r 2 and y = r 3 − (r 1 + r 2 )/2 are the Jacobi coordinates, S is a symmetrization operator [the main text considers only spinless bosons], and M is the basis size. The parameters α i and β i are chosen in a form of a geometric progression, i.e., α i = α 1 A i−1 and β i = β 1 B i−1 , where A and B are input parameters. Once the parameters A, B and M are given, the coefficients a i are found by minimizing the expectation value of the Hamiltonian, E Ψ = Ψ|H|Ψ / Ψ|Ψ . We vary the parameters A, B and M to find the minimal value of E Ψ , which is used in the main text as E. To benchmark our numerical calculations, we used known results for charged and neutral systems [25,27,76]. We also employed the stochastic variational method with correlated Gaussians (SVM) [26,28] to cross-check some of the energies presented in the main text. In our implementation, the SVM assumes the variational wave function in the form where the parameters α S i , β S i and γ S i [α S i β S i − γ S i 2 /4 > 0] are found by a stochastic search (cf. [26,28]); a S i are chosen to minimize the expectation value of the Hamiltonian. For weakly-bound states one might restrain these parameters to better reproduce the tails of the wave function (cf. [77]). In our explorations, we saw that this option does lead to a much faster convergence for charged systems, because the corresponding tails decay faster with distances than those of neutral systems.
Finally, we briefly explain why our results do not gain any systematic errors due to the choice of the basis. To this end, we show that an eigenstate with L = 0 (L for total angular momentum) of the Hamiltonian can be accurately approximated by the variational wave function (15). This is not a trivial observation, since Ψ SV M depends only on the three variables: x, y, and θ xy -the angle between x and y [78]. In general, an eigenstate might also depend on other combinations of angles that determine x and y (e.g., θ x ,θ y , φ x , φ y in spherical coordinates). A suitable angular basis for our discussion is Y l1m1 (θ y , φ y )Y l2m2 (θ x , φ x ), where Y is a real spherical harmonic. Any suitable variational function can be written as where f is the function that determines the expansion. We note that the Hamiltonian, H 3 , does not mix the subset {P l (x · y)} with the rest of the basis, P l is the Legendre polynomial [P l (x · y) = 4π 2l+1 l m=−l Y lm (θ y , φ y )Y lm (θ x , φ x )]. Indeed, since two-body potentials V depend only on x, y and θ xy , we derive where F l is an irrelevant for our discussion function that depends on x and y. Therefore, there are eigenstates of H 3 that can be written as φ = f i l (x, y)P l (x · y).
It is expected that a square integrable function φ can be well represented by a suitable Ψ SV M [26]. This becomes intuitively clear after writing Gaussian functions in the form of the corresponding Maclaurin series. One can confirm that Ψ SV M indeed describes the ground state by checking numerically that it does not change sign.