Fate of the neutron-deuteron virtual state as an Efimov level

The emergence of Efimov levels in a three-body system is investigated near the unitarity limit characterized by resonating two-body interaction. No direct evidence of Efimov levels is seen in the three-nucleon system since the triton is the only physical bound state. We provide a model-independent analysis of nucleon-deuteron scattering at low energy by formulating a consistent effective field theory. We show that virtual states evolve into shallow bound states, which emerge as excited triton levels as we drive the system towards unitarity. Even though we consider this specific system, our results for the emergence of the Efimov levels are universal.


I. INTRODUCTION
A distinctive feature of three-body physics is the Efimov effect [1][2][3][4]. In the presence of resonant two-body scattering, the three-body system supports a tower of bound states whose binding energies display geometric scaling. The number of three-body bound states was predicted to scale as ln(a 0 /r 0 ), where a 0 is the resonating two-body s-wave scattering length, and r 0 ≪ a 0 is the range of the interaction. Experimental verification through the detection of an excited state was provided three decades later in cold-atom experiments [5][6][7][8], where the ratio a 0 /r 0 could be increased by varying magnetic fields near a Feshbach resonance.
Recently, the first excited Efimov state was identified [9] for atomic 4 He, where a 0 /r 0 is sufficiently large even in the absence of external magnetic fields. In complex nuclei the ratio a 0 /r 0 is often unknown, but in some cases it might be large enough to accommodate an excited state: halo nuclei such as 11 Li, 22 C, or even as heavy as 62 Ca, where two neutrons are weakly bound to a tight nuclear core, provide opportunity to observe the Efimov effect [10][11][12].
Neutron-deuteron (nd) at low energy is another system that could provide evidence of Efimov physics. Lowenergy scattering is dominated by s waves with spins S = 3/2 (quartet) and S = 1/2 (doublet). In the quartet channel, where spins are aligned, the Pauli exclusion principle prevents a three-nucleon (3N) bound state. In the doublet channel, the strong nuclear force leads to an attractive interaction that supports a 3N bound state, the triton ( 3 H) with a binding energy B 3 ≃ 8.48 MeV. The next deeper Efimov state would appear only at ∼ 4 GeV, beyond the regime where a description in terms of nucleons makes sense. Here we search for a remnant of the next shallower Efimov state.
The low-energy phase shift in the doublet channel was analyzed by van Oers and Seagrave [13], who suggested a modified effective range expansion (ERE) to describe the data below the deuteron breakup momentum. The presence of a virtual state with binding energy ≃ 0.515 MeV was inferred [14]. Adhikari et al. [15] showed in a separable potential model that this virtual state is related to the Efimov spectrum. A similar connection for 20 C [16,17] and for atomic systems [18] was investigated within a zero-range model.
In contrast, we use the effective field theory (EFT) formalism to provide a model-independent analysis of the nd system at low energies. All interactions allowed by symmetries are constructed with the relevant low-energy degrees of freedom, without modeling the high-energy physics. A systematic scheme for calculations is formulated by expressing observables as an expansion in the small ratio p/Λ b , where p is a typical low momentum scale associated with the processes of interest and Λ b is a high momentum scale that marks the breakdown of the EFT.
In the so-called pionless EFT (/ πEFT) the relevant degrees of freedom are nonrelativistic nucleons (and other light particles such as photons, electrons and neutrinos) with Λ b ∼ m π , the pion mass, associated with the pion physics that is not included explicitly. / πEFT has been very successful in describing two-and three-nucleon systems -see for example Refs. [19][20][21][22][23][24][25][26] -when the typical momentum is taken to be p ∼ γ ∼ a −1 t , with γ ≃ 45.7 MeV the deuteron binding momentum and a t ≃ 5.4 fm the scattering length in the two-nucleon (2N) 3 S 1 channel.
EFT enables us to study nd scattering in the limit a t → ∞, where the 2N scattering amplitude is only bounded by unitarity. We first fit the parameters of / πEFT to reproduce the relevant 2N and 3N experimental data. / πEFT is then treated as the underlying theory that is used to generate artificial "data" at increasingly large values of a t . We develop a new low-energy EFT, which we refer to as halo EFT, to provide a modelindependent analysis of the 3N phase shift at low momentum generated from / πEFT. This halo EFT, which treats the deuteron as an "elementary" particle and is thus applicable only below the deuteron breakup, is modeled after other halo EFTs where different clusters of nucleons are treated as relevant degrees of freedom [27,28]. Here, the deuteron is the core and the neutron forms the "halo" around it, consisting of shallow virtual and bound states. The halo EFT provides a theoretical basis for the modified ERE obtained empirically by van Oers and Seagrave, and allows us to track the virtual state at unphysical scattering lengths. As we drive / πEFT towards the unitarity limit, the binding energy of the virtual state decreases till it becomes the first excited bound state of the triton, thus demonstrating its Efimov character. Higher Efimov states appear in the same way if the scattering lengths are increased further. Although we focus on the 3N system, our framework could be applied to study the emergence of Efimov levels in other systems as well.

II. PIONLESS EFT
The doublet nd scattering amplitude was first calculated at leading order (LO) in / πEFT in Ref. [21]. It receives contributions from the LO 2N interactions, which consist of a single non-derivative contact operator in each 2N s wave ( 3 S 1 and 1 S 0 ) with strength determined in terms of the respective scattering length. In addition, there is a contribution from a 3N non-derivative contact interaction, which is needed to render the amplitude well-defined. Next-to-leading-order (NLO) corrections introduce a two-derivative interaction in each 2N s wave, which can be constrained by the corresponding effective range. No new 3N interaction contributes at this order [21,23]. A momentum-dependent 3N interaction enters at NNLO [24]. To this order, the EFT expansion has been shown to be convergent [25], and to reproduce both experimental data (when available) and results from sophisticated phenomenological potentials.
The unitarity limit corresponds not only to arbitrarily large 2N scattering lengths but also to vanishing 2N effective ranges and higher ERE parameters. This removes higher-order corrections in the 2N interaction, so that a LO calculation is sufficient to explore the connection to the Efimov spectrum. The LO nd T -matrix T t (p) is ob-tained from two coupled integral equations [21]: In the 3 S 1 channel we use the deuteron binding momentum to set γ t = g t γ, and in the 1 S 0 channel, where no bound state exists, we use the scattering length a s ≃ −23.714 fm to fix γ s = g s /a s , which is consistent with the LO power counting [19,29]. The physical point corresponds to g t = g s = 1. The 3N amplitudes A t,s = m N T t,s /(3π) do not converge when the regulator is removed, λ → ∞, unless we fix the 3N parameter h 0 (λ) to guarantee that one 3N observable is made regulator independent. For any given cutoff λ, we tune h 0 such that we reproduce the doublet scattering length a 3 = 0.65 fm [30] for g t = g s = 1. This determines the independent parameter Λ ⋆ appearing in the log-periodic 3N force [21]. The doublet phase shift is obtained from p cot δ = ip + 1/A t (p, p). In Fig. 1, we show the phase shift calculated from the three-body integral equations in Eq. (1) at the physical point and a few results as we approach the unitarity limit. The only physical parameters that enter the microscopic calculation using / πEFT are g t , g s , and Λ ⋆ . We approach unitarity taking g s = 0 and making the deuteron arbitrarily shallow, g t → 0, while keeping Λ ⋆ fixed.
For short-ranged interactions, p cot δ is an analytic function of p 2 . However, p cot δ rises rapidly at low momenta -see panel (a) of Fig. 1 -and for g t = g s = 1 a simple Taylor series expansion around p = 0 gives a poor description even at relatively small momenta p ∼ 10 MeV. Instead, the modified ERE [13] p cot δ = −1/a + rp 2 /2 + sp 4 /4 + · · · 1 + p 2 /p 2 works remarkably well up to about the deuteron breakup momentum 2γ/ √ 3 ≃ 53 MeV. While bound and virtual states correspond to poles of the T -matrix T t (p) at imaginary momenta p = iκ j , there is also a pole in p cot δ at p 2 = −p 2 0 that corresponds to a zero of T t (p). When we fit the modified ERE to the LO phase shift in panel (a) of Fig. 1, we get the fit parameters a ≈ 0.65 fm = a 3 , r ≈ −141 fm, s ≈ 62 fm 3 , and p 0 ≈ 16.1 MeV, which translate into a shallow virtual state at κ 1 ≈ −26.8 MeV with a binding energy ≈ 0.574 MeV. Naively one might expect all the scattering parameters to scale with some power of the range of nuclear interaction ∼ m −1 π ≃ 1.4 fm. The fitted parameters p −1 0 , a −1 and r are unusually large compared to the naive expectation. The standard ERE holds only for p ≪ p 0 , with large inverse scattering length a −1 , effective range r + 2(ap 2 0 ) −1 , etc.

III. HALO EFT AND MODIFIED ERE
We now formulate the halo EFT that describes nd scattering below the deuteron breakup momentum 2γ t / √ 3, and provides a justification for Eq. (2). In this theory the deuteron is treated as a fundamental particle, and so the breakup momentum sets the breakdown scale Λ b ∼ γ t . A shallow s-wave pole by itself can be accounted for in the standard ERE by a large scattering length such as in the 2N 1 S 0 and 3 S 1 channels, requiring a fine-tuned interaction to be treated nonpertubatively at LO [19,29]. A shallow amplitude zero by itself requires another finetuning that leads to a perturbative amplitude with a large effective range and a small scattering length [29]. Here we identify two momentum scales associated with the presence of both zero and virtual pole, respectively |p 0 | ∼ Q and |κ 1 | ∼ ℵ. The parameters a, r and p 0 are fine-tuned, and we require three fine-tuned couplings to reproduce the desired modified ERE. The halo EFT is conveniently written using two auxiliary fields as Here, n is the spin-doublet neutron field with mass m N , d is the spin-triplet deuteron field with mass m d ≈ 2m N , and ψ (j) with j = 1, 2 are two auxiliary spin-1/2 fields with total mass M = m N + m d ≈ 3m N and residual masses ∆ j . σ are Pauli matrices that act on the (suppressed) spinor indices of n and ψ (j) . We chose to fix the couplings of both auxiliary fields to neutron and deuteron in terms of the reduced mass µ = m N m d /M ≈ 2m N /3, transferring their strength to parameters c j [31]. In the power counting discussed below, c 2 ≪ c 1 and the corresponding interaction appears at subleading orders together with the interactions lumped into the "· · · ". Integrating out ψ (2) one recovers the Lagrangian from Ref. [32], but the scaling of parameters is different here. Our approach inspired a reformulation of chiral EFT in the 2N 1 S 0 channel where not only the shallow virtual state but also the amplitude zero is taken into account [33]. A straightforward calculation of the nd scattering amplitude T t (p) at LO in the halo EFT, where the contribution from the loops is resummed, gives The loop contribution, was, for simplicity, evaluated in dimensional regularization using minimal subtraction. The final result is independent of the regularization method. We obtain the modified ERE (2) from Eq. (4) with the shape-like parameter s appearing at higher order.

IV. ANALYSIS
To develop a consistent power counting for the halo EFT, we start with the analytic structure of the T -matrix T t (p). The T -matrix poles here are the roots of with Using the parameters a, r, p 0 fitted earlier for g t = g s = 1, the three roots iκ 1 ≈ −27i MeV, iκ 2 ≈ 35i MeV, and iκ 3 ≈ 83i MeV are imaginary. As we move towards unitarity (g s = 0, g t → 0), we refit the / πEFT results with Eq. (7), as shown in Fig. 1. The roots remain imaginary. The third root is always deeper than the breakdown scale Λ b ∼ γ t of the halo EFT, and is, therefore, not relevant.
Near the poles we can write the S-matrix S t (p) = exp[i2δ(p)] ≈ j R j /(p − iκ j ) + f (p), where R j are the residues at the poles, and f (p) some finite piece. We provide an interpretation of the shallower roots based on the residues R 1,2 . The wavefunction normalizations for the two shallowest poles are , and their evolution towards unitarity is shown in Fig.  2. For a bound state, the normalization must be nonnegative. At the physical point g t = g s = 1, the shallowest root iκ 1 with κ 1 < 0 is a pole on the second Riemann energy sheet with a negative normalization iR 1 < 0, and thus describes the virtual state that has been identified in the past [14]. The second root iκ 2 is a pole on the first energy sheet since κ 2 > 0. However, its normalization is also negative, iR 2 < 0 -it is a "redundant pole" [34,35]  and does not describe a physical state. As we take the limit g t → 0 at g s = 0, the second root iκ 2 remains a redundant pole and the first root iκ 1 evolves from a virtual to a real bound state. In devising the power counting for the halo EFT, the three physical scales identified earlier, |p 0 | ∼ Q, |κ 1 | ∼ ℵ and Λ b ∼ γ t , have to be taken into account. The scale Q starts as the smallest, gets smaller in size and then grows. When |p 0 | grows beyond the breakdown scale, |p 0 | > ∼ Λ b , it stops being relevant in halo EFT. We find it convenient to separate the evolution into three intervals of g t to account for different relative sizes of Q. Moreover, as the modified ERE is most easily expressed in terms of the scattering parameters a, r, p 0 and s, we use these to write the power counting for the renormalized halo EFT couplings c 1 , c 2 , ∆ 1 and ∆ 2 .
Initially, for 1 g t 0.55, we have Q ≪ ℵ ≪ Λ b . The sizes of |κ 1 | and |κ 2 | are similar and we also identify |κ 2 | ∼ ℵ to avoid introducing another scale. As we make g t smaller, we find that κ 3 gets deeper while κ 1 gets shallower, so we take κ 3 ∼ A ∼ Λ 2 b /ℵ. These roots arise if the halo EFT couplings are large, we see that p 0 comes out shallow as assumed, while 1/a ∼ R ∼ ℵΛ 2 b /Q 2 ≫ A and r ∼ Λ 2 b /(ℵQ 2 ) are large. For p ≪ ℵ, Eq. (7) is dominated by the a term containing the amplitude zero, and for p ≪ Q the effective range is ∼ 2(ap 2 0 ) −1 . In contrast, for p > ∼ ℵ the r term and the unitarity term (−ip) become comparable to the 1/a contribution and generate the T -matrix poles. If we take c 2 /(2µ) ∼ Q 2 /Λ 3 b ≪ c 1 /(2µ), then the shape-like parameter s = −(c 1 c 2 /µ 2 )/(∆ 1 + ∆ 2 ) ∼ 1/Λ 3 b , which is consistent with its fit value at g t = 1. Its contribution for p ∼ ℵ is suppressed by a factor of sp 2 /r ∼ Q 2 ℵ 3 /Λ 5 b ≪ 1 compared to the r and a contributions. Other modified ERE parameters appear at even higher orders. The halo EFT with the power counting we propose leads to a model-independent derivation of the modified ERE, and describes the data accurately. As we make g t smaller, p 2 0 gets smaller and changes sign such that p cot δ develops a pole at real momentum around g t = 0.9, analogous to the Ramsauer-Townsend effect [36,37]. This is shown in panel (b) of Fig. 1. The halo EFT (and the modified ERE) still gives a good description of the phase shift through the pole in p cot δ even though the EFT couplings are fitted at momentum below the pole, as indicated in the figure. As g t gets smaller, |p 2 0 | gets larger and we look at the second interval below.
For 0.55 g t 0.35, |p 0 | continues to grow, approaching and exceeding Λ b . The first root is a progressively shallower virtual state with κ 1 < 0 and iR 1 < 0, while the second root stays a redundant pole with κ 2 > 0 and iR 2 < 0. Making Q → Λ b in the relations of the first interval leads to a ∼ r ∼ 1/ℵ. Numerically, this works well. It can be accomplished with ∆ 1 ∼ ℵ, ∆ 2 ∼ Λ 2 b /ℵ ≫ ∆ 1 , and c 1 /(2µ) ∼ 1/ℵ. From Eq. (4), one sees that the second auxiliary field contribution is suppressed by ℵ 2 /Λ 2 at small momenta p ∼ ℵ, and the modified ERE increasingly looks similar to the traditional ERE written as a Taylor series around p = 0. The situation is depicted in panel (c) in Fig. 1. With c 2 /(2µ) ∼ 1/Λ b ≪ c 1 /(2µ), the shape-like parameter contribution continues to be suppressed by sp 2 /r ∼ ℵ 3 /Λ 3 b ≪ 1. In the third interval, 0.35 g t 0.1, |p 0 | ∼ Q becomes very large. The fits to / πEFT scattering phase shift are not sensitive to p 0 which decouples from the theory. The S-matrix now has only two poles constrained by κ 1 +κ 2 = 2/r and κ 1 κ 2 = 2/(ar), and two residues The first root continues to get smaller, and at around g t ≃ 0.3 it vanishes. Then it moves to the first Riemann energy sheet with a positive normalization iR 1 > 0, signaling the emergence of a bound nd state. The second root remains a redundant pole, and eventually moves slightly beyond the breakdown scale Λ b . Near the emergence of the shallow bound state, the phase shift is characterized by a large scattering length and a small effective range, as seen in panel (d) of Fig. 1. Qualitatively, the phase shift goes from something similar to the 1 S 0 2N system with a shallow virtual state to the 3 S 1 2N channel with a shallow bound state. The scattering length scales as |a| ∼ 1/ℵ whereas the effective range r remains fixed at some other small momentum scale set by the second root, κ 2 ∼ ℵ ′ ∼ 1/r ≫ ℵ. We did not explore how ℵ ′ scales with variation of the nd input parameter a 3 (through Λ ⋆ ) in / πEFT. The halo EFT couplings scale as ∆ 1 ∼ ℵ, c 1 /(2µ) ∼ 1/ℵ ′ , and ∆ 2 → ∞. The second auxiliary field is integrated out of the low-momentum theory, and we recover the traditional ERE. The shape-parameter contribution can be included in the single auxiliary-field formulation as a higher-order operator. With a scaling s ∼ 1/Λ 3 b , the shape parameter is suppressed by asp 4 ∼ ℵ 3 /Λ 3 b compared to the LO scattering-length contribution, whereas the effectiverange correction is suppressed by arp 2 ∼ ℵ/ℵ ′ .
The subsequent evolution of the new bound state is shown in Fig. 3, the "Efimov plot" calculated directly in / πEFT. The physical triton at g t = g s = 1 is seen below the diagonal line on the fourth quadrant that marks the breakup threshold. For this bound state, there is no significant difference between g s = 0 and g s = 1 as the scaling violation due to nonzero γ s = 1/a s ≃ −8 MeV is a small effect compared to the binding momentum of the triton ∼ 100 MeV [26]. As we move towards unitarity with g s = 0, g t → 0, the new, shallow Efimov state appears around g t ≃ 0.3. It occurs exactly at the place indicated earlier by the halo EFT based on the analytic structure of the S-matrix. At the unitarity point g t = 0, the ratio of binding momenta between the triton and the first-excited state gives the geometric factor 22.7 predicted by Efimov [1][2][3][4].
Efimov physics displays a limit-cycle behavior [38,39], and shallower bound states (not shown in Fig. 3) also appear. For example, we have found that, as we evolve towards unitarity, around g t ≃ 0.05 a new shallow virtual state is present and the phase shift goes through the same qualitative behavior as for the first excited state. Again a modified ERE with a new set of scattering parameters describes this virtual state, which becomes shallower, and finally emerges as the second-excited state of the triton. The same process repeats ad infinitum at progressively smaller g t intervals.

V. CONCLUSIONS
We studied the relation between virtual state and bound Efimov level in a three-body system. We chose nd scattering in the spin-doublet channel as it had been shown to support a virtual state and a bound triton. The geometric scaling between bound states had not been observed in this system because deeper Efimov levels are beyond the range of applicability of any reasonable nuclear theory. We have shown that a new shallow Efimov state emerges from the virtual state as we drive the system towards unitarity. The shallow state displays the geometric scaling predicted by Efimov at unitarity, and we find evidence that the accumulation of shallow Efimov levels involve pulling in shallow virtual states from the second Riemann energy sheet to the first sheet. Though we consider a specific nuclear system, our results are universal to any three-body system with resonating zero-ranged twobody interactions, at least one of them supporting a twobody bound state. In addition to atomic systems near a Feshbach resonance, recent lattice QCD calculations, even at unphysical quark masses, provide another interesting scenario where in the presence of a strong magnetic field the 2N interaction is driven towards unitarity [40].
Recently it has been argued [41] that nuclear ground states beyond the deuteron are characterized by a momentum scale intermediate between the pion mass and the inverse 2N scattering lengths. In this case, nuclei are accessible through / πEFT with an additional expansion around the unitarity limit of infinite 2N scattering lengths. The existence of a shallow virtual nd state that becomes the triton excited state supports this picture.
A low-energy halo EFT was formulated for a modelindependent description of the transition of the shallow virtual to the shallow bound state. We studied the analyticity of the S-matrix on the complex energy plane in order to interpret the various poles that correspond to bound, virtual or redundant states. The halo EFT formulated here could be useful in the study of low-energy pd scattering, for example for the model-independent extraction of doublet ERE parameters from / πEFT [42]. In halo EFT, the Coulomb interaction is simpler as pd is effectively a two-body system. The halo EFT could also be useful in the low-energy description of the reactions d(n, γ) 3 H and d(p, γ) 3 He, which are relevant in big-bang nucleosynthesis.