Two particles interacting via a contact interaction on $S^2$

We consider two particles interacting via a contact interaction that are constrained to a sphere, or $S^2$. We determine their spectrum to arbitrary precision and for arbitrary angular momentum. We show how the non-inertial frame leads to non-trivial solutions for different angular momenta. Our results represent an extension of the finite-volume L\"uscher formulas but now to a non-trivial geometry. We apply our results to predict the spectrum of select two-nucleon halo nuclei and compare with experimental results.


I. INTRODUCTION
The eigenvalue solutions to two interacting particles is a standard topic introduced to beginning students of quantum mechanics. Typical first examples include two particles interacting via a contact interaction and the Coulombic solutions of oppositely charged particles. These examples serve as a stepping stone to more complicated quantum mechanical many-body systems whose solutions are usually not known.
Besides serving a great pedagogical introduction to many-body quantum mechanics, the twobody system itself plays an important role in multiple fields of physics. For example, when the particles are placed within a finite cubic volume their eigenvalue solutions satisfy Lüscher's quantization formula [1][2][3][4]. Lattice Quantum Chromodynamics (LQCD) calculations of composite two-body systems within a finite volume utilize this relation to extract infinite-volume interaction parameters between these particles [5,6]. Sometimes the finite-volume is dictated by the experimental setup as opposed to numerical convenience, as is the case with cold-ion traps. Here the confinement of the two particles can be satisfactorily approximated by an external harmonic oscillator well. Again, the energy solutions here [7,8] provide information on the interacting properties of the particles within this confinement, and in particular whether the two particles undergo a Feshbach resonance when their scattering length diverges [9]. As a final example, solutions exist for two interacting particles within a hard spherical wall [10], providing a means for "tuning" interaction parameters for many-nucleon simulations using nuclear lattice effective field theory [11,12].
The examples above refer to systems residing in three spatial dimensions. But in all these cases there are corresponding solutions in both one-and two-dimensions. A tacit assumption here is that the interaction between the two particles only depends on their relative coordinates.
When this is the case, and if the geometry allows it, one can readily separate the system into its relative (Jacobi) and center-of-mass (CM) coordinates. This provides a great simplification to the eigenvalue solutions since one can work solely within the inertial frame.
In this paper we consider two particles confined to a sphere of arbitrary radius R (i.e. confined to S 2 ) and that interact via a contact interaction. Though this interaction again only depends on the relative coordinates, the surface S 2 is a non-inertial frame that affords no general separation of relative and CM coordinates and as such there is no simplification to the eigenvalue solution. 1 Yet we show how solutions of arbitrary precision to this system can be found. Furthermore, because 1 The exception is the case with zero total angular momentum [13]. of the non-inertial frame, we find an infinite tower of solutions depending on the total angular momentum L. Our results are not completely academic-we consider two-nucleon halo nuclei and show how our results can be used to extract interaction parameters between the loosely bound nucleons. We view this system as another excellent pedagogical example of two quantum mechanical interacting particles, but this time within a non-inertial frame.
Our paper is organized as follows. In Sect. II we define our problem. We derive the quantization conditions for arbitrary total angular momentum L in terms of direct summations in Sect. II C.
When considering explicit values for L, we find analytic expressions for the summations, which we provide for L = 0 up to L = 2 in Sect. II D. We stress, however, that all solutions for L > 2 can be readily found using our method. We then apply our results to extract interaction parameters of two-nucleon halo nuclei in Sect. III. We recapitulate in Sect. IV. Detailed and lengthy derivations are reserved for the appendices.

II. PROBLEM SETUP
We consider two particles of equal mass m confined to the surface of ball of radius R, as shown in Fig. 1. The particles' positions are then solely dictated by their anglesr 1 andr 2 , which in turn can be expanded in a basis of spherical harmonics r|l, m l = Y lm l (r). The kinetic term of the Hamiltonian describing such particle movement is well known and is that of a rigid motor, 2 T |l, m l = l(l + 1) 2mR 2 |l, m l ≡ ε l |l, m l . (1)

A. The contact interaction
We assume that the particles interact via a contact interaction only, which in this geometry is given in coordinate space bŷ Here C 0 (Λ) is a coefficient that is tuned to reproduce a particular observable of the two-particle system and the variable Λ represents a momentum cutoff scale. The procedure for tuning this coefficient is non-trivial but has been done previously in [14,15], and we only mention some salient features of this procedure relevant to our analysis in Sect. III. For a more thorough description of this tuning we recommend the reader consult the aforementioned references.
The relevant physical observable is the s-wave scattering lengthã, which in two dimensions is dimensionless, despite its name [16]. To a certain degree, the magnitude and sign of this parameter dictates how strongly the particles repulsively or attractively interact with one other. We can define a 'reduced scattering length' a that is dimensionful by introducing an arbitrary length scale. We set this length scale to be the radius R of our sphere. The relation between the physical scattering lengthã (dimensionless) and reduced scattering length a (dimension of length) is given by [16] Note that this definition implies that a ≥ 0. The tuning of C 0 (Λ) then follows the procedures described in [14,15]. Assuming a hard-cutoff regulator in momentum space, the coefficient is We note that though the interaction Eq. (2) is both cutoff and scheme dependent by virtue of the coefficient in Eq. (4), observables are not. We ultimately take the limit Λ → ∞ in all our subsequent calculations. 2 We seth = c = 1 in all our expressions.
Our task then is to solve the eigenvalue equation 3 where our eigenstates are states with good total angular momentum L and M since our interaction preserves total angular momentum. To do this, we first recast Eq. (5) into integral form, We then project the eigenstate onto where l 1 , m 1 ; l 2 , m 2 |LM is a Clebsch-Gordan coefficient. This gives On the RHS above we have inserted the closure relation1 = ∑ l 1 l 2 |(l 1 l 2 )LM (l 1 l 2 )LM| and used the fact that |l 1 m 1 and |l 2 m 2 are eigenstates ofT 1 andT 2 , respectively, with eigenenergies given in Eq. (1).

C. Quantization condition for general L
To continue further we require the explicit form of the matrix element (l 1 l 2 )LM V 12 (l 1 l 2 )LM .
As this derivation is quite tedious, we leave it for the appendices (App. A) and only provide the end result here: 3 The eigenvalue E includes both rotational and vibrational energies.
is a Wigner 3-j symbol [17] and we definel i ≡ 2l i + 1 for brevity. The condition that l i (l i + 1) ≤ (ΛR) 2 and l i (l i + 1) ≤ (ΛR) 2 for i = 1, 2 comes from the momentum hard cutoff condition of our interaction. We now plug this expression into Eq. (7), giving The equality above holds for all l i such that l i (l i + 1) ≤ (ΛR) 2 The equality now holds non-trivially if the term in square brackets vanishes. Using the exact form of C 0 (Λ) from Eq. (4) and equating the term in square brackets to zero gives the desired quantization condition for arbitrary total angular momentum L: where x ≡ 2mER 2 . The summation above is over all l i such that The eigenvalues E, or equivalently x, of Eq. (5) are those that satisfy the equality in Eq. (12).
This represents Lüscher's formula on S 2 for each rotational band L under the assumption of a pure contact interaction.

D. Closed expressions for select L
When we concentrate on specific values of L and take the limit Λ → ∞ we can further simplify Eq. (12) and obtain closed expressions. We do this explicitly for L = 0 and 1, and provide the closed expression for L = 2 without derivation. In principle it is possible to obtain closed expressions for concrete values of L > 2, but the derivation becomes much more tedious and onerous.
To start, note that the sums over l 1 and l 2 in Eq. (12) are restricted by the triangle inequalities of the Wigner 3-j symbol For L = 0 this implies that l 1 = l 2 ≡ l. The 3-j symbol simplifies to (−1) l √ 2l+1 and Eq. (12) becomes If we identify the cutoff with some maximum angular momentum λ via λ (λ + 1) ≡ (ΛR) 2 , then our expression above can be written as The sum can be explicitly expressed in terms of the digamma function ψ(x) = d dx log(Γ(x)), In the limit λ → ∞ the first two terms on the RHS above exactly cancel the logarithm term in Eq. (15) What remains gives us our closed-form expression, The triangle inequality in this case requires that, given l 1 ≡ l, the sum over l 2 is restricted to the values |l − 1|, l, and l + 1. However, the 3-j symbol vanishes for l 1 = l 2 = l (when L = 1), and so Eq. (12) becomes the sum over two expressions only, The first term on the RHS above comes from the l = 0 contribution. After simplifying the 3-j symbols the sums can be performed and analytically expressed in terms of digamma functions.
The Λ → ∞ limit can be subsequently taken, giving The fact that there exists a non-trivial quantization condition for L = 1, despite the interaction being a pure contact interaction, comes from the fact that our general expression in Eq. (12) is derived using single-particle coordinates as opposed to relative coordinates.

L = 2
The steps used for the L = 0, 1 cases can be analogously applied to L = 2 (and higher). Clearly the sum over l 2 for a given l 1 becomes more involved as L becomes larger, and as such, the expressions become more complicated and cumbersome to express. Therefore we do not show these steps here but instead provide the expression for L = 2 without derivation: E. Limits and zeros of the quantization relations for L = 0, 1 cases The structure of these quantization equations for energies x ∈ {−9, 40} is displayed in fig. 2.
These will be utilized to find solutions to the Schrödinger equation for two particles on a sphere in the following chapter. One may notice the divergent parts of each graph, which corresponds to the case of no interaction.
As already mentioned earlier, in two dimensions the scattering length a ≥ 0 [18]. In the limit a R, the solutions to Eq. (17) approach the non-interacting energies from below. In the limit a R we have, in addition to the deeply bound solution x → −∞ (i.e. the so-called "dimer solution"), solutions that also approach the non-interacting energies, but now from above. We can expand the solutions x about the non-interacting energies by considering the limit | log(a/R)| 1.
For the n th solution, where n ∈ Z ≥0 , we find for the L = 0 case This expression is valid for both limits a → ∞ and a → 0 (keeping R fixed). The bound dimer solution valid as a → 0 scales as Given that x = 2mER 2 , this corresponds to the standard dimer binding energy E = − 1 ma 2 . Similarly, for L = 1 we have Another interesting limit is to consider the case when a/R = 1, corresponding to the |ã| → ∞ limit. 4 Solutions to Eq. (12) in this case occur when the curves in Fig. 2 intersect the x-axis, corresponding to zeros of the quantization equations. We provide these zeros to machine precision for L = 0, 1, and 2 in Table I  is while for L = 1 it is where ψ (1) (z) ≡ d dz ψ(z). In Fig. 3 we plot these limiting expressions and compare them to the exact solution for the L = 0 case. F. Comparison with S 1 × S 1 topology and 2-D harmonic oscillator As mentioned earlier, the quantization condition for two particles interacting in a confined space has been determined in other 2-D systems. Here we take the opportunity to compare our L = 0 result Eq. (17) with its analog in the S 1 × S 1 geometry and the harmonic oscillator.
Busch et al. [7] have derived the case for the 2-D harmonic oscillator with frequency ω, where x = E/ω with E the eigenenergy and b = 1/ √ 2mω is the oscillator parameter.
For a 2-D square lattice of side L with periodic boundary conditions (i.e. the torus or S 1 × S 1 topology), a thorough derivation is provided in [14], giving Here S 2 (x) is the two dimensional zeta function, n = (n i , n j ) ∈ Z 2 , and x = mEL 2 /(4π 2 ). The

III. APPLICATION: PREDICTING ENERGY LEVELS OF TWO-NUCLEON HALO NUCLEI
Halo nuclei consist of a tightly bound core of nucleons surrounded by small group of loosely bound, or halo, nucleons. The resulting nuclei appear much larger than the radius of the original tightly bound core. A classic example is the 11 Li halo nucleus originally found by I. Tanihata et al. [19]. This nucleus can be decomposed into a three-body system, 9 Li + 2n, where the 9 Li constitutes the tightly bound core and the two neutrons the halo nucleons that are considered to be loosely bound and interacting. Another example is 6 He, which can also be decomposed into a tightly bound core, 4 He, plus two halo neutrons, again loosely bound and interacting. Both of these systems are only stable 5 as a three-body constellation, and therefore are considered borromean [20].
A simple, albeit crude, approximation to these systems is to assume that the two nucleons are constrained to interact on a sphere with halo radius R and that the core is located at the center of this sphere. The confinement of the halo nucleons is assumed to be due to some non-trivial interaction 5 Stable in this context applies only to the strong interaction. with the core, which we approximate as infinitely massive and therefore non-dynamical. 6 When the core has its own angular quantum numbers, we may couple the angular momentum of the halo nucleons with that of its core, but aside from that, the core has no other influence on the halo nucleons. If we further assume that the interaction between the nucleons is contact in nature, then our formalism of the previous section directly describes this situation. 7 Under this approximation radial excitations are not possible and therefore there are only vibrational excitations for each rotational band.
Nucleons are of course fermions with spin and isospin equal to 1/2. To incorporate our results from the previous section, we must take the nucleons' spins, isospins, and their Pauli-exclusion into account. For the two nucleons to 'feel' the s-wave interaction, we must couple their spins and isospins to total spin and isospin S = 0, T = 1 (e.g. 'spin-singlet' two-neutron system) or S = 1, T = 0 (i.e. 'spin-triplet' deuteron system), respectively. We then couple their total spin S and angular momentum L to make total angular momentum J NN . An anti-symmetric two-nucleon wavefunction requires and this in turn restricts the allowed angular momentum L of the two nucleons. The total angular momentum J NN of the halo nucleons is then coupled with the angular momentum of the core to obtain the total angular momentum of the halo system J. Finally, the parity of the two-nucleon system is and is multiplied with the parity of the core to obtain the overall parity π of the halo system.
Before we can use our formalism to predict energy levels, however, we have to tune the parameters (i.e.ã or equivalently a/R) of our theory. We now describe in detail how we use the low-energy spectrum of the 6 He and 11 Li halo nuclei to determine these parameters. In particular, these systems will allow us to determine the spin-singlet scattering lengthã 0 . We also consider the 6 Li system which will allow us to determine the spin-triplet scattering lengthã 1 . 6 Such an approximation has been used to describe doubly-excited atomic electrons interacting via a contact interaction [21,22] and via a modified coulomb interaction [23], for example. 7 Naturally there exist more sophisticated models and calculations of these systems, see e.g. Refs. [24][25][26] and references within.

A. Helium-6
Here we have two neutrons surrounding a 4 He core. The two neutrons are thus in the S = 0, T = 1 channel. The three lowest allowed angular momentum bands are L = 0, L = 1, and L = 2, with L = 1 being odd in parity and the others even. As the 4 He core has J π C C = 0 + angular momentum, the total angular momenta of the halo nucleus for these bands are simply J π = J π NN = 0 + , 1 − , and 2 + . Within our approximation the interaction of two neutrons on a 2d surface is described solely by the parameterã 0 , where we add the subscript 0 to denote that this parameter is for the spin-singlet S = 0 system. This parameter is independent of the halo nucleus. As it is also dimensionless, a single empirical (dimensionful) energy is not sufficient to constrain this parameter and therefore a second energy is required. We use the experimental J π = 0 + and 2 + energies of the 6 He halo nucleus [27,28], measured relative to the 4 He + n + n threshold, to constrain the dimensionful parameters a 0 and R of our theory, which we stress are halo nucleus dependent. We then obtainã 0 by the relation Eq. (3). We findã 0 = −5.58 (6) .
The experimental energies used to obtain this value, as well as the resulting a 0 , R, and predicted energy levels of our model for the J π = 0 + , 1 − and 2 + rotational bands, are given in Fig. 5. We take the mass of the neutron as m = 939.565 MeV.
To obtain the errors of the fit parameters quoted in Fig. 5, we first assume that the experimental errors for the J = 0 and J = 2 energies are uncorrelated and follow a normal distribution with width dictated by their respective errors. We then sample these energies from their distributions, each time performing our fit to obtain a 0 , R, andã 0 , and we tally these results. The mean of these tallies is our quoted values of these terms in Fig. 5, and the standard deviation their errors. Our sample size is 10,000.
The determined value ofã 0 then fixes log (a 0 /R) through the relation Eq. (3), which we show as the red line in the L = 0, 1 and 2 plots in Fig. 2. The intercept of this red line with the solid black curves in these plots gives us our energy solutions. Our fitting procedure is guaranteed to reproduce the lowest 0 + and 2 + experimental energies and their errors, as these were used to obtain our fit parameters. The higher intercepts then provide our predicted energy levels shown in As already mentioned above, the applicability of our model is quite limited due to its extreme simplicity, and this is quite obvious when looking at its predicted J π = 1 − energies. Our model 0.3656 (2) 1.828 (9) 4.1127 (19) 7.2097(33) −0.1244 (1) 1.0822 (5) 2.9643 (14) 5.6576 (26) 0.3128 (2) 0.9956 (4) 2.0424 (9) 2.5754 (11) 4.3524 (20) 4.9721 ( predicts as its lowest state a negative energy solution, although experimentally no such state exists. Furthermore, there exist positive energy solutions that are predicted in other rotational bands that have no obvious experimental counterparts. It is also interesting to compare our estimate of the halo radius R = 6.258 (15) fm which is nearly a factor of two larger than the experimental result of R exp = 3.08(10) fm [29]. Again, this disagreement is not surprising given the level of crudeness of our model.

B. Lithium-11
The 9 Li core has angular quantum numbers 3/2 − , and for the 11 Li halo system there is only the measured J = 3/2 − ground state energy E 0 = −0.369 MeV [30,31] that has definitive quantum numbers assigned. However, given that we determinedã 0 in the previous section (which in our approximation is independent of halo nucleus), we have sufficient information to determine a 0 and R for this system. In this case we tally fit results obtained from uncorrelated samplings of E 0 and a 0 to arrive at a 0 and R, and then subsequently predict the higher energy levels. Our results are given in the right panel of Fig. 5. When coupling the angular momentum J NN of the halo nucleons with that of the 3/2 − core, our model predicts multiplets of energies in the L = 1 and L = 2 cases.
We label these multiplets in our figure.
As in the 6 He case, our model predicts another negative energy solution near threshold coming from the L = 1 case, which is not seen experimentally. Our model again predicts many positive energy solutions that have no obvious experimental counterparts. We compare our estimate of the halo radius R = 10.154 (24) with its experimental value, R exp = 6.5(3) fm [29]. Again, this level of disagreement is not surprising given the simplicity of our model.

C. Lithium-6
Though not technically considered a halo nucleus, the small separation energy for 4 He + n + p breakup (small compared to the binding energy of its 4 He core) suggests that the nucleus is extended in size. We therefore assume that the 4 He acts as the core and the 'halo' nucleons for this system consist of a neutron and proton. This system supports both S = 0, T = 1 and S = 1, T = 0 channels, therefore we expect the spectrum to be much richer than in the previous two examples.
We assume isospin charge symmetry, meaning that the (dimensionless) scattering length in the spin-singlet S = 0 channel is the same as that determined in the 6 He case given in Eq. (31).
To determine the spin-triplet S = 1 scattering length, we again use the two lowest experimental energies [28] of this system, measured relative to the 4 He + n + p breakup threshold. Here the two lowest energies have the quantum numbers 1 + and 3 + states. We find where the subscript 1 denotes the S = 1 spin-triplet system. Note the sign change compared to the spin-singlet case in Eq. (31). 8 As before, the determined value ofã 1 , along with the experimental energies, fixes log (a 1 /R) through the relation Eq. (3). We show this result as the blue line in the L = 0 and 2 plots in Fig. 2. The intersection of this blue line with the black curves gives us our predicted energy levels. The experimental energies, our determined a 0 , a 1 and R parameters for this system, as well as our predicted energy levels are shown in Fig. 6.
It is interesting to note that relative to the 4 He + n + p breakup threshold, the 6 Li system has three positive parity negative energy states. The two lowest energies, both in the T = 0 channel, are exactly captured in our model, since we use these energies to fit our parametersã 1 and the combination a 1 and R. But our model also predicts a third positive parity negative energy corresponding to the T = 1 channel. This is due to the spin-singlet scattering lengthã 0 and the energy of this state coincides with the intersection of the red line with the lowest black curve of L = 0 in Fig. 2. The correct ordering of these levels is captured in our model, however the experimental value of this state is much closer to the 4 He + n + p threshold, while our prediction is significantly lower in energy. Lastly, our model predicts a near threshold negative energy in the negative parity J π (T ) = 1 − (1) band due to the coupling with L = 1, which is not observed experimentally.

IV. CONCLUSION
In this paper we derived the quantization condition for two-particles constrained to a sphere, or S 2 , and under the assumption that they interact via a contact interaction. We show how the energy levels of the system are related to the reduced scattering length a and radius of the sphere R. As the constraint on S 2 represents a non-inertial frame, the system is not amenable to a separation of CM and relative coordinates. As such, there is an infinite tower of solutions for each total angular momentum L, each of which is different and not related by any constant offset from each other.
We then used these results to predict the higher-lying spectrum. Our results for the halo radius disagreed by up to a factor of two from experiment, but given the level of crudeness of our approximation this was not a surprising result.
Finally, our formalism, and its application to two-nucleon halo nuclei, provides another excellent pedagogical example of two quantum mechanical interacting particles, but this time within a non-inertial frame.
Note that the sum over L, M in the 3 j-symbols in Eq. (A2) does not allow a factorisation of terms between l 1 , m 1 , l 2 , m 2 and λ 1 , µ 1 , λ 2 , µ 2 . To condense our expression a little we will use the abbre- where we used the fact that for non-vanishing 3 j coefficient, l 1 + l 2 + L = must be even, which implies that the factor (−1) l 1 −l 2 −L = 1.
Inserting this result in Eq. (A1) leaves us with the final form of the matrix element,