Effective Field Theory for Two-Body Systems with Shallow S-Wave Resonances
EEffective Field Theory for Two-Body Systems with Shallow S -Wave Resonances J. Balal Habashi and S. Fleming
Department of Physics, University of Arizona, Tucson, AZ 85721, USA
S. Sen
Department of Physics & Astronomy, Iowa State University, Ames, IA 50011, USA
U. van Kolck
Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France andDepartment of Physics, University of Arizona, Tucson, AZ 85721, USA
Resonances are of particular importance to the scattering of composite particles in quantummechanics. We build an effective field theory for two-body scattering which includes a low-energy S -wave resonance. Our starting point is the most general Lagrangian with short-range interactions.We demonstrate that these interactions can be organized into various orders so as to generate asystematic expansion for an S matrix with two low-energy poles. The pole positions are restrictedby renormalization at leading order, where the common feature is a non-positive effective range.We carry out the expansion explicitly to next-to-leading order and illustrate how it systematicallyaccounts for the results of a toy model — a spherical well with a delta shell at its border. I. INTRODUCTION
In quantum mechanics, observables are encoded in the S matrix or, equivalently, in the T matrix. In the complexmomentum plane, poles of the S matrix lying on the positive imaginary axis with positive residue correspond to boundstates [1], while poles on the negative imaginary axis are negative-energy virtual states, which are not normalizable.Pairs of poles can also appear at complex momentum with equal, negative imaginary parts and equal but opposite realparts. These poles can be thought of as resonance states with a finite lifetime , which affect scattering significantlywhen they lie close to the real axis. Resonances are common in the nonrelativistic scattering of atoms and nuclei,sometimes appearing in the S wave (see, for example, Ref. [2]). Shallow resonances can lead to significant variationof phase shifts at low energies. Examples include a very low-energy resonance in proton-proton scattering [3] andthe Be ground state in the scattering of two alpha particles [4]. Here we develop a systematic treatment of shallow S -wave resonances, or more generally two shallow poles, based on effective field theory (EFT).EFTs exploit a separation of scales to generate a controlled expansion of observables in the small ratio(s) ofthese scales. The EFT framework was originally formulated in particle physics to allow a perturbative treatment oflow-energy processes involving strong-interacting particles [5], and immediately used to justify the successes of theperturbative Standard Model despite the wide range of possibilities for its underlying dynamics [6]. In the 1990s,as it was being applied to nuclear physics (for a review, see for example Refs. [7, 8]), the need arose for EFT tobe extended to shallow, nonrelativistic bound and virtual states, which cannot be treated in perturbation theory.Originally motivated by the two-nucleon problem, which exhibits a bound state in one S wave and a virtual statein another, an EFT description of shallow S -wave two-body states was achieved in the late 1990s [9–12]. This waspossible using either momentum-dependent contact interactions or a “dimeron” auxiliary field [13], which gives riseto energy-dependent interactions. Within a few years, shallow two-body resonances also yielded to an EFT approach,but only with a “dimeron” field [14–18]. While the corresponding energy-dependent interactions could be used inthree-body calculations [19–21], they are not easily incorporated in most ab initio methods that enable the solution ofthe Schr¨odinger (or equivalent) equation for more than three particles. Our goal here is to extend the EFT of shallow S -wave two-body resonances to momentum-dependent interactions, which should find wider use.To understand two-body resonant scattering at low energy in an EFT framework we need to identify the char-acteristic momentum scales inherent to the process . We consider the scattering of two particles at a momentum k ∼ M lo which is much smaller than the inverse range of the interaction 1 /R ∼ M hi . In the nonrelativistic regime,this problem is equivalent to the the scattering of a particle by a potential of finite range, but one in which the particlecannot resolve the details of the potential. It is sensible to make a “multipole” expansion of the potential, which We refer to a pole at complex (neither real nor purely imaginary) momentum as a “resonance”, regardless of whether it generates abump in a cross section. We use units such that (cid:126) = c = 1, so that mass, momentum, energy, inverse distance, and inverse time all have the same dimensions. a r X i v : . [ nu c l - t h ] J u l can be considered as a sum of Dirac delta functions with an increasing number of derivatives. Such a potential canbe formulated in terms of the most general Lagrangian density involving only the fields associated with the particlesunder study. The two-body potential is generated by interactions that involve four fields at the same spacetime pointand their derivatives. The coefficient of an operator with n derivatives — called Wilson coefficient in the particlephysics literature and low-energy constant (LEC) in hadronic and nuclear physics — is a real number 4 πC n /m thatencodes the details of the short-range dynamics. Here we consider for simplicity a single particle species of mass m and the most common case where the underlying dynamics is not only Lorentz invariant, but also symmetric underspatial parity and time reversal. We also neglect spin degrees of freedom, which bring no essential complications.In this case only interactions with an even number of derivatives appear. The more general case can be obtainedstraightforwardly following the same steps as we do below.We show in the following how scattering around a resonance can be described systematically in a series in powersof M lo /M hi , regardless of the details of the underlying interaction, as long as its range is short compared to themagnitude of the inverse momentum of the resonance. The two-body dynamics at low energies is obtained via theselective resummation of Feynman diagrams, or equivalently by solving the Schr¨odinger equation. In either case, as isusual in field theory, one has to impose a regularization procedure to cope with the singularity of the interactions. Theconceptually simplest regularization, which we employ below, imposes a momentum cutoff Λ. The choice of regulatoris arbitrary and thus makes for a particular model of the short-range dynamics. To ensure observables are insensitiveto this arbitrary choice and model independent, the theory has to be renormalized. The Λ dependence of each “bare”LEC that appears in the Lagrangian, 4 πC n (Λ) /m , is determined by imposing that a low-energy observable be Λindependent. Other low-energy observables then depend on Λ only through positive powers of 1 / Λ and approachfinite limits as Λ (cid:29) M hi . The expansion is organized in such a way that renormalization holds at each order, up toterms that have the same magnitude as higher-order terms when Λ > ∼ M hi . Thus, we can reproduce the effects of anypotential exhibiting a shallow resonance to arbitrary accuracy.The choice of low-energy observables used in the renormalization procedure is equally arbitrary, up to higher-orderterms. A particularly simple choice is offered by the effective-range expansion (ERE) [22]. From general considerationsit is possible to show [12] that an EFT for short-range interactions leads to a T matrix for S -wave scattering at on-shellmomentum k (cid:28) R − which can be expressed in the ERE form T ( k ) = − πm ( k cot δ ( k ) − ik ) − = − πm (cid:20) − a + r k − P (cid:16) r (cid:17) k + . . . − ik (cid:21) − , (1)where δ ( k ) is the phase shift, and a , r , P , . . . are known as, respectively, scattering length, effective range, shapeparameter , etc. The values of the ERE parameters can be extracted from the phase shifts, which in turn can beobtained from data with only mild theoretical input. As is done in much of the EFT literature [7, 8], we fix the LECsby forcing them to agree with the empirical values of these ERE parameters: C is related to the scattering length, C to the effective range, etc. A crucial aspect of EFT is “power counting” — the argument that justifies the expansion of the amplitude in powersof M lo /M hi or “orders”: leading order (LO), next-to-leading order (NLO), next-to-next-to-leading order (N LO), andso on. The simplest assumption is that of “naturalness”, where the size of observables is set solely by the largemomentum scale M hi : | a | ∼ /M hi , | r | ∼ /M hi , | P | ∼ etc. (For a review, see Ref. [23].) This is assured if therenormalized LECs scale as C n = O (1 /M n +1 hi ). In this case T ( k ) is purely perturbative, with LO consisting of C infirst order in perturbation theory, NLO of C in second order, N LO of C in third order and C in first order, and soon. (Starting at N LO contributions to higher waves are also present.) Poles characterized by momentum | k | (cid:28) M hi thus require a certain amount of fine tuning in the underlying theory such that at least one interaction is large enoughto demand a nonperturbative treatment.It is relatively easy to find examples of a fine tuning that produces a large scattering length, | a | ∼ /M lo (cid:29) /M hi .This situation corresponds to a single, shallow S -wave bound or virtual state near threshold. This is an intrinsicallyquantum-mechanical phenomenon: the particles in the bound state are on average at distances much larger than therange of the interaction, which in classical physics determines the size of orbits. For example, for a square well offixed range R , one can produce such states at values of the depth β /mR for which β is close to an odd multipleof π/
2. In this case, the LECs scale as C = O (1 /M lo ) and C n ≥ = O (1 /M lo M n − hi ) [9–12]. For k ∼ / | a | , theinteraction with no derivatives and coefficient C is as important as the unitarity term ik , and it needs to be treatednonperturbatively. All remaining terms in the ERE can still be treated as perturbations in a distorted-wave Bornexpansion. The resulting S -wave T matrix is an expansion of Eq. (1). It yields either a bound state or a virtual state, In much of the literature the coefficient of k is defined to be P r . With our choice P takes on more natural values. depending on the sign of the scattering length, at k (cid:39) i/a ∼ ± iM lo . This is exactly what happens in nucleon-nucleonscattering at low energies. Just as in the natural case, higher waves appear at higher orders.Neither of these two power-counting schemes can produce a shallow resonance. For this to occur we need, in general,to have two fine tunings, with | a | ∼ /M lo and | r | ∼ /M lo . We expect the physics of such a scaling to be producedby an effective Lagrangian where the two leading operators in the derivative expansion are of a size much largerthan what is expected from naturalness and hence have to be treated nonperturbatively. In contrast to other cases, C = O (1 /M lo ) and C = O (1 /M lo ), with other LECs smaller, being suppressed by powers of M hi . We will arguethat C n ≥ = O (1 /M n/ lo M n/ − hi ). These are the central ideas of our paper.Proper renormalization is the cornerstone of our approach. This is not the first time that the two leading contactinteractions are solved exactly. In fact our LO solution reproduces the amplitude of Beane, Cohen, and Phillips[24, 25]. These authors have shown that renormalization requires r < ∼ / Λ for Λ (cid:29) M lo , which is an example ofWigner’s bound on phase shifts [26]. However, in those early days of nuclear EFT, Beane et al. were concerned withbound states. The fact that two-nucleon data require r > and, when treated perturbatively, it can accommodateeither sign of r . Like Beane et al. , we take renormalizability and its constraint r ≤ S -wave resonance or, moregenerally, two low-energy poles.With this interpretation, we use the power counting outlined above to predict the positions of poles with controllederrors, which can be improved by adding higher-order operators perturbatively to the LO effective Lagrangian. Therenormalization requirement r ≤ T -matrix poles, the positions of whichare determined in the complex momentum plane by the relative sizes of a and r ≤
0. The two poles can lie • both below the real axis with non-vanishing, equal and opposite real parts — a resonance; • on top of each other on the negative imaginary axis — a double virtual pole; • both on the negative imaginary axis, but separated — two virtual states; • one on the negative imaginary axis and the other at the origin — a virtual state and a “zero-energy resonance”;or • one on the negative and the other on the positive imaginary axis — one virtual and one bound state.At NLO, the four-derivative contact interaction enters, which leads to the shape parameter | P | ∼ M lo /M hi . Wedemonstrate the systematic character of our approach by considering an example where the underlying interactiontakes a particularly simple form: a spherical well of range R with a delta-shell potential at its edge. By varyingthe strengths of the two components of the underlying potential at fixed R we can produce two poles in each of thearrangements mentioned above. Fixing the EFT parameters from the ERE parameters that appear at each order, weshow how the toy-model phase shifts and pole positions are approximated with increasing accuracy when we go fromLO to NLO. In the future we hope to include the Coulomb interaction as well, so as to be able to consider not onlya toy model, but also an S -wave resonance of phenomenological interest such as the Be ground state.The organization of the paper is as follows. In Sec. II we show the form of the relevant effective Lagrangian inthe derivative expansion, as well as the potential in LO and NLO. In Sec. III we derive the scattering amplitudefor the EFT at LO using the Schr¨odinger equation, followed by the nonperturbative renormalization of the bareparameters C (Λ) and C (Λ) of the two leading operators in the effective Lagrangian. We then find the scatteringamplitude for the EFT at NLO and perturbatively renormalize the amplitude using C (Λ), the bare parameter of thefour-derivative contact interaction. (An alternative derivation of the results of this section using field theory is offeredin App. A, while some details of the renormalization procedure are given in App. B.) In the following section, Sec.IV, we demonstrate that our EFT successfully reproduces shallow resonant states in the ERE. We also analyze thesensitivity of the positions of the poles for bound and virtual states to perturbations, and the error in the position ofthe poles is estimated. We compare in Sec. V the results from EFT with those of a toy model which among otherstates includes two shallow poles. We conclude in Sec. VI. For a recent proposal on how to iterate corrections in limited cutoff ranges while retaining the EFT expansion, see Ref. [27].
II. EFT FOR TWO-BODY SCATTERING
In this section we construct an EFT for scattering that exhibits a shallow S -wave resonance. The energy of scatteringparticles is restricted to be on the order of the resonance energy, assumed to correspond to momentum much smallerthan the inverse of the underlying interaction range. The scattering process is assumed to be elastic, meaning thereis a single open channel with particles taken to be in their ground states both before and after scattering. (Forscattering with more open channels, see Ref. [28].) For simplicity we limit ourselves to one particle species notaffected by the exclusion principle, but generalization is straightforward. As a consequence, we are concerned witha single “heavy” field [29] ψ , which encodes the annihilation of a particle, and contact operators in the Lagrangianthat are Hermitian, as there is no source or sink. Furthermore, our EFT is invariant under Lorentz (in the form ofreparameterization invariance [30]), parity and time-reversal transformations, and conserves particle number. Themost general Lagrangian for scattering without spin which respects these symmetries and constraints can be writtenin the form [31] L = ψ † (cid:32) i ∂∂t + −→∇ m + . . . (cid:33) ψ − πm (cid:26) C ( ψψ ) † ( ψψ ) − C (cid:104) ( ψψ ) † (cid:16) ψ ←→∇ ψ (cid:17) + H . c . (cid:105) + C (cid:20) ( ψψ ) † (cid:16) ψ ←→∇ ψ (cid:17) + H . c . + 2 (cid:16) ψ ←→∇ ψ (cid:17) † (cid:16) ψ ←→∇ ψ (cid:17)(cid:21) + . . . (cid:27) , (2)where ←→∇ ≡ −→∇ − ←−∇ . Here we display explicitly only terms contributing to the (on-shell) S -wave scattering oftwo particles: the term quadratic in the field ψ is the nonrelativistic kinetic term, while the others correspond tointeractions with LECs 4 πC n /m . The “ . . . ” represent operators with more derivatives and fields, and/or operatorsthat only contribute to two-body scattering off shell and in higher partial waves. For example, there is an independentoperator with four derivatives, which vanishes when the two-body system is on shell [32], which allows us to take thecoefficient of ( ψ ←→∇ ψ ) † ( ψ ←→∇ ψ ) to be − πC / m [31]. Similar choices can be made for higher-derivative operators.Interactions that vanish on shell in the two-body system cannot be separated from operators involving more fields,for example three-body forces of the type ( ψ † ψ ) . Since we have integrated out antiparticles, operators with six ormore fields do not contribute to the two-body system.For simplicity we work in the center-of-mass frame, where the scattering particles have relative momenta (cid:126)p and (cid:126)p (cid:48) before and after scattering, respectively. From the Feynman diagrams corresponding to the Lagrangian (2) we canobtain the nonrelativistic potential in momentum space [12, 24, 25], (cid:104) (cid:126)p (cid:48) | ˆ V | (cid:126)p (cid:105) = 4 πm (cid:20) C + C (cid:0) (cid:126)p (cid:48) + (cid:126)p (cid:1) + C (cid:0) (cid:126)p (cid:48) + (cid:126)p (cid:1) + . . . (cid:21) , (3)and, from that, the potential in coordinate space, (cid:104) (cid:126)r (cid:48) | ˆ V | (cid:126)r (cid:105) = 4 πC m δ ( (cid:126)r (cid:48) ) δ ( (cid:126)r ) − πC m (cid:2)(cid:0) ∇ (cid:48) δ ( (cid:126)r (cid:48) ) (cid:1) δ ( (cid:126)r ) + δ ( (cid:126)r (cid:48) ) (cid:0) ∇ δ ( (cid:126)r ) (cid:1)(cid:3) + πC m (cid:2)(cid:0) ∇ (cid:48) δ ( (cid:126)r (cid:48) ) (cid:1) δ ( (cid:126)r ) + 2 (cid:0) ∇ (cid:48) δ ( (cid:126)r (cid:48) ) (cid:1) (cid:0) ∇ δ ( (cid:126)r ) (cid:1) + δ ( (cid:126)r (cid:48) ) (cid:0) ∇ δ ( (cid:126)r ) (cid:1)(cid:3) + . . . . (4)Our first task is to order these interactions according to their effects on observables. In general the same operatorcontributes to various orders, so we decompose the LECs as C n = C (0) n + C (1) n + . . . , (5)where C ( N ) n is the part of C n that contributes at order N .Since we are interested in situations where there are two low-energy S -wave poles, the denominator of the LOscattering amplitude must be quadratic in momentum. In our EFT with momentum-dependent interactions we needto treat both C and C nonperturbatively ( i.e. at LO) to reproduce this non-analytic behavior. In momentum space,the LO potential is therefore (cid:104) (cid:126)p (cid:48) | ˆ V (0) | (cid:126)p (cid:105) = 4 πm (cid:34) C (0)0 + C (0)2 (cid:0) (cid:126)p (cid:48) + (cid:126)p (cid:1)(cid:35) . (6)In other words, C (0) n ≥ = 0. We seek an exact, nonperturbative solution of this potential under a momentum cut-offregulator Λ, with C (0)0 , (Λ) determined from the requirement that two observable quantities be reproduced.At higher orders, we expect small, perturbative corrections from higher-derivative operators. The fact that zero-and two-derivative operators are taken as LO might suggest that there is no expansion that suppresses the higher-derivative operators. However, as discussed in the introduction, we are facing a fine-tuned situation where a low-energyscale M lo enhances all operators, but not in the same way. Subleading interactions are suppressed by powers of thehigh-energy scale M hi , which are not entirely obvious at first. A powerful guide in this situation is the renormalizationgroup (RG). At any given order, observables not used in the determination of LECs have residual cutoff dependence:they depend on M lo and M hi in a form dictated by the explicit solution of the EFT interactions up to that order,and additionally on inverse powers of Λ. This residual cutoff dependence will be removed at higher orders by otherLECs, which will have bare components that scale as inverse powers of Λ. Naturalness assumes that the magnitudeof a renormalized LEC is determined by changes in the cutoff of relative O (1). This implies that for cutoffs thatdo not intrude in the region where we want the EFT to converge, Λ > ∼ M hi , we can determine the magnitude of therenormalized LEC, and hence its order, by the replacement Λ → M hi in the bare LEC.We will show below that the assumption of naturalness for subleading operators leads to a controlled expansion.In particular, the NLO momentum-space potential is (cid:104) (cid:126)p (cid:48) | ˆ V (1) | (cid:126)p (cid:105) = 4 πm (cid:34) C (1)0 + C (1)2 (cid:0) (cid:126)p (cid:48) + (cid:126)p (cid:1) + C (1)4 (cid:0) (cid:126)p (cid:48) + (cid:126)p (cid:1) (cid:35) , (7)which is to be treated in first-order perturbation theory. The appearance of quartic momentum operators at thisorder means that an additional observable is needed to fix C (1)4 (Λ). The latter induces changes in the observablesfitted at LO. To compensate, we include perturbative changes in C , so that C (1)0 , (Λ) produce opposite changes inthese observables, which then remain fixed at the values chosen at LO. For the remaining interactions, C (1) n ≥ = 0.An analogous procedure is followed at higher orders. The removal of regularization dependence — up to effects nolarger than those of the truncation in the potential — can be performed by a finite number of parameters at eachorder . That means that the theory is renormalizable in the modern sense, which generalizes the old-fashioned conceptof a finite set of interactions at all orders. III. SOLVING THE SCHR ¨ODINGER EQUATION
Equipped with the potential we now obtain the S -wave scattering amplitude in an expansion T ( k ) = T (0)0 ( k ) + T (1)0 ( k ) + . . . , (8)where successive terms are suppressed by an additional power of M lo /M hi . In order to renormalize the resultingamplitude, we demand that the C ( N ) n (Λ) appearing in the expansion1 T ( k ) = 1 T (0)0 ( k ) (cid:32) − T (1)0 ( k ) T (0)0 ( k ) + . . . (cid:33) (9)reproduce the ERE parameters a , r , P , etc. in the inverse of Eq. (1). Then other observables, such as the polepositions, can be predicted. Other renormalization conditions can be imposed instead. For example, one can fit thepole positions, if known, at LO and predict ERE parameters. As long as only low-energy input is used, differentrenormalization conditions differ only by higher-order effects.We can obtain the expansion of T ( k ) using Feynman diagrams (see App. A). In this case, we have to calculateloop diagrams which involve the Schr¨odinger propagator. After renormalization, when positive powers of Λ have beenremoved from loops, each loop effectively contributes O ( mk/ π ), while interaction vertices contribute 4 πC n k n /m .The LO potential has to be iterated to all orders if C (0) n = O (1 /M n +1 lo ), since then diagrams involving these verticesform a series in k/M lo which, once resummed, can give rise to a pole with | k | ∼ M lo . The resummation of Feynmandiagrams is equivalent to the exact solution of the Schr¨odinger equation. By definition, higher-order LECs haveadditional inverse powers of M hi , C ( N ) n = O (1 /M n − N +1 lo M Nhi ). Subleading interactions are dealt with in distorted-wave perturbation theory, which is equivalent to a finite number of insertions of subleading vertices in Feynmandiagrams that include all possible LO vertices. We find the Schr¨odinger equation easier to implement, especially atsubleading orders, and we present explicitly here the calculation of T (0 , ( k ). A. Leading order
If the incoming free-particle state with energy E ≡ k /m is denoted by | (cid:126)k (cid:105) , the scattering amplitude from the LOpotential V (0) , Eq. (6), is (see, for example, Ref. [2]) T (0)0 ( k ) = (cid:104) ψ (0) − | ˆ V (0) | (cid:126)k (cid:105) = 4 πm (cid:34)(cid:32) C (0)0 + C (0)2 k (cid:33) ψ (0) − ∗ (0) − C (0)2 ψ (0) − (cid:48)(cid:48)∗ (0) (cid:35) = (cid:104) (cid:126)k | ˆ V (0) | ψ (0)+ (cid:105) = 4 πm (cid:34)(cid:32) C (0)0 + C (0)2 k (cid:33) ψ (0)+ (0) − C (0)2 ψ (0)+ (cid:48)(cid:48) (0) (cid:35) , (10)where | ψ (0)+ (cid:105) ( | ψ (0) − (cid:105) ) is the incoming (outgoing) scattering wavefunction and ψ (cid:48)(cid:48) ≡ ∇ ψ . The scattering wavefunctionsare combinations of homogeneous (free-field) and particular (potential term with free-field Green’s function) solutionsof the Schr¨odinger equation, ψ (0) ± ( (cid:126)r ) = e i(cid:126)k · (cid:126)r − π (cid:90) d q (2 π ) e i(cid:126)q · (cid:126)r q − k ∓ i(cid:15) (cid:34)(cid:32) C (0)0 + C (0)2 q (cid:33) ψ (0) ± (0) − C (0)2 ψ (0) ± (cid:48)(cid:48) (0) (cid:35) . (11)We define the integrals I ± n ( k ) = − π (cid:90) d q (2 π ) q n q − k ∓ i(cid:15) = − ∞ (cid:88) (cid:96) =0 L n − (cid:96) ) k (cid:96) ∓ ik n +1 , (12)where L (cid:96) = θ (cid:96) Λ (cid:96) , (13)with θ (cid:96) a regulator-dependent number — for example, θ (cid:96) = 2 / ( (cid:96)π ) for a sharp-cutoff regulator. These integrals allowus to write ψ (0) ± (0) = 1 − (cid:32) C (0)0 ψ (0) ± (0) − C (0)2 ψ (0) ± (cid:48)(cid:48) (0) (cid:33) I ± ( k ) − C (0)2 ψ (0) ± (0) I ± ( k ) , (14) ψ (0) ± (cid:48)(cid:48) (0) = − k + (cid:32) C (0)0 ψ (0) ± (0) − C (0)2 ψ (0) ± (cid:48)(cid:48) (0) (cid:33) I ± ( k ) − C (0)2 ψ (0) ± (0) I ± ( k ) . (15)Using Eqs. (12), (14) and (15) we solve for ψ (0) ± (0) and ψ (0) ± (cid:48)(cid:48) (0) in terms of C (0)0 , , L , ( k ) and I +0 ( k ). We have4 πm T (0)0 ( k ) = 1 v (0)0 + v (0)2 k − I +0 ( k ) , (16)where v (0)0 = C (0)0 − C (0)22 L / C (0)2 L / , (17) v (0)2 = C (0)2 C (0)2 L / C (0)2 L / , (18)a result obtained in Refs. [24, 25]. Equation (16) is the same T matrix we would have obtained with an energy-dependent potential v (0)0 + v (0)2 k . With our momentum-dependent interactions, however, v (0)0 , differ from C (0)0 , bycutoff-dependent factors that trace back to the increased singularity of the two-derivative term.As it stands, the T matrix (16) depends on the regulator through I +0 ( k ), L , , and C (0)0 , . In particular, the linearcutoff dependence of I +0 ( k ) needs to be eliminated. Our renormalization procedure involves expanding Eq. (16) inpowers of k/ Λ and equating it to the ERE in Eq. (1). The unitarity term ik stems from I +0 ( k ), and the first twopowers of k allow us to express the bare parameters C (0)0 (Λ) and C (0)2 (Λ) as functions of the L n and the observables a and r . Using Eq. (13) (details can be found in App. B), we obtain for r (cid:54) = 0 C (0)0 (Λ) = θ θ Λ (cid:20) ∓ ε + (cid:18) − θ θ θ (cid:19) ε ± (cid:18) − θ θ − θ − θ r θ a (cid:19) ε + O (cid:18) ε , r a ε (cid:19)(cid:21) , (19) C (0)2 (Λ) = − θ Λ (cid:20) ∓ ε ± (cid:18) − θ θ − θ − θ r θ a (cid:19) ε O (cid:18) ε , r a ε (cid:19)(cid:21) , (20)where we introduced ε = (cid:18) − θ θ r Λ (cid:19) / . (21)There are two solutions indicated by the ± signs accompanying the unusual inverse powers of ( − r Λ) / . For real C (0)0 and C (0)2 , we see from Eqs. (19) and (20) that we must have r <
0. This result is consistent with the Wigner bound[26], which puts a condition on the rate of change of the phase shift with respect to the energy for a finite-range,energy-independent potential. It translates [33, 34] into a constraint on the effective range r , r ≤ R (cid:18) − Ra + R a (cid:19) . (22)Interpreting the range R of the potential as the inverse cutoff in momentum space, R ∼ / Λ, the limit Λ → ∞ corresponds to a zero-range interaction ( R → C (0)0 and C (0)2 in terms of the scattering length, the effective range and thecutoff into the expression (16) and expand T (0)0 ( k ) about k/ Λ → T (0)0 ( k ) = − πm (cid:18) − a + r k − ik (cid:19) − (cid:34) (cid:18) − a + r k − ik (cid:19) − r θ Λ k + . . . (cid:35) (23)= − πm (cid:18) − a + r k − ik + r θ Λ k (cid:19) − + . . . . (24)Because the renormalized C (0)0 and C (0)2 involve only M lo , the ERE parameters scale as a ∼ r ∼ /M lo , as neededfor a shallow resonance. For k ∼ M lo , the first three terms in the denominator on the right-hand side of Eq. (24) are O ( M lo ), while the fourth term can be made arbitrarily small as Λ increases. The first three terms generate two poles,including a resonance, which we discuss in Sec. IV, and give rise to the LO phase shift k cot δ (0)0 ( k ) = − a + r k . (25)For Λ > ∼ M hi , the fourth term is no larger than O ( M lo /M hi ). Barring further fine tuning, this term is comparable toNLO interactions which will remove its residual cutoff dependence. In fact, it can be thought as an induced shapeparameter 1 / (4 θ r Λ). Taking Λ ∼ M hi in ∆ (cid:16) k cot δ (0)0 ( k ) (cid:17) = r θ Λ k (26)gives an estimate of the error in k cot δ (0)0 ( k ). In this case, the right-hand side of Eq. (26) is indeed O ( M lo /M hi )relative to Eq. (25).At this point we have shown, following Refs. [24, 25], that the EFT can produce the first two terms in the ERE atLO, as long as the effective range r <
0. In the next section we show how subleading contributions systematicallyimprove the LO result.
B. Subleading order
Adding perturbations to the LO implies that the corresponding changes in the scattering amplitude should be treatedin (distorted-wave) perturbation theory. The first correction to the T matrix, T (1)0 ( k ), is linear in the parameters ofthe NLO potential, the C (1)0 , , defined in Eq. (5). After renormalization, C (1)4 = O (1 /M lo M hi ) since its contributionmust be comparable to the LO error in Eq. (26). And, since they are of the same order, C (1)0 = O (1 /M hi ) and C (1)2 = O (1 /M lo M hi ).Explicitly (see Ref. [2] again), T (1)0 ( k ) = (cid:104) ψ (0) − | V (1) | ψ (0)+ (cid:105) = (cid:104) ψ (0)+ | V (1) | ψ (0) − (cid:105) = 4 πm (cid:34) C (1)0 ψ (0) − ∗ (0) ψ (0)+ (0) − C (1)2 ψ (0) − ∗ (0) ψ (0)+ (cid:48)(cid:48) (0) + C (1)4 (cid:18) ψ (0) − ∗ (0) ψ (0)+ (cid:48)(cid:48)(cid:48)(cid:48) (0) + ψ (0) − ∗(cid:48)(cid:48) (0) ψ (0)+ (cid:48)(cid:48) (0) (cid:19)(cid:35) , (27)where ψ (0)+ (cid:48)(cid:48)(cid:48)(cid:48) ≡ ∇ ψ (0)+ is given by ψ (0) ± (cid:48)(cid:48)(cid:48)(cid:48) (0) = k − (cid:32) C (0)0 ψ (0) ± (0) − C (0)2 ψ (0) ± (cid:48)(cid:48) (0) (cid:33) I ± ( k ) − C (0)2 ψ (0) ± (0) I ± ( k ) . (28)Once this is substituted in Eq. (27), T (1)0 ( k ) T (0)20 ( k ) = m π v (1)0 + v (1)2 k + v (1)4 k ( v (0)0 + v (0)2 k ) , (29)where v (1)0 = 1(1 + C (0)2 L / (cid:40) C (1)0 − C (1)2 C (0)0 L + C (0)2 L /
21 + C (0)2 L / C (1)4 / C (0)2 L / (cid:104) C (0)0 L (cid:16) C (0)0 L + C (0)2 L / (cid:17) − C (0)0 L + C (0)22 (cid:16) C (0)2 L / (cid:17) (cid:0) L − L L (cid:1) / − C (0)2 L / (cid:105)(cid:41) , (30) v (1)2 = 1(1 + C (0)2 L / (cid:34) C (1)2 − C (1)4 (cid:32) C (0)0 L + C (0)2 L + 2 C (0)0 L + C (0)2 L /
21 + C (0)2 L / (cid:33)(cid:35) , (31) v (1)4 = C (1)4 C (0)2 L / C (0)2 L / . (32)Again, Eq. (29) is the result we would obtain by treating an energy-dependent potential v (1)0 + v (1)2 k + v (1)4 k infirst-order perturbation theory. In our case, however, the additional singularity of two- and four-derivative interactionsleads to the relations (30), (31), and (32).As before, we renormalize the NLO amplitude by expanding the right-hand side of Eq. (9) in powers of k/ Λand matching it to the ERE, Eq. (1). The presence of C (1)4 = O (1 /M lo M hi ) now ensures that the quartic powerof momentum can be made cutoff-independent by demanding that it reproduce the empirical value of the shapeparameter P . The other two NLO parameters, C (1)0 , , can be chosen so that the terms independent of and quadraticin momentum remain unchanged. Again, with details given in App. B, we find C (1)0 (Λ) = θ θ (cid:0) θ θ − θ (cid:1) P r (cid:40) − (cid:20) − θ ( θ θ − θ ) θ ( θ θ − θ ) (cid:21) ε − (cid:20) θ θ − θ θ θ + θ θ ( θ θ − θ ) − θ θ − θ − θ r θ a (cid:21) ε + O (cid:18) ε , r a ε (cid:19)(cid:41) + θ θ Λ (cid:0) θ θ − θ (cid:1) [1 + O ( ε )] , (33) C (1)2 (Λ) = 3 θ θ θ P r Λ (cid:20) − (cid:18) − θ θ θ (cid:19) ε − (cid:18)
23 + θ θ θ − θ θ − θ − θ r θ a (cid:19) ε + O (cid:18) ε , r a ε (cid:19)(cid:21) + 3 θ θ θ Λ [1 + O ( ε )] , (34) C (1)4 (Λ) = − θ θ P r Λ (cid:20) − ε − (cid:18) − θ θ − θ − θ r θ a (cid:19) ε + O (cid:18) ε , r a ε (cid:19)(cid:21) − θ θ Λ [1 + O ( ε )] . (35)The terms independent of P are those induced by the residual cutoff dependence in the quartic term of Eq. (24).The others are all linear in the NLO physical parameter P , whose sign is not constrained by renormalization at thisorder. The situation here is analogous to the one-pole case [9–12], where the perturbative treatment of C places norenormalization constraints on the sign of r .At this point we have tuned C , C and C in such a way that they reproduce the ERE in Eq. (1) exactly in thelimit Λ → ∞ . For a large but finite Λ, the NLO EFT reproduces Eq. (1) up to a correction that goes as the sixthpower of momentum, T (0+1)0 ( k ) = − πm (cid:20) − a + r k − P (cid:16) r (cid:17) k − ik − P r θ Λ k (cid:21) − + . . . . (36)Because C = O (1 /M lo M hi ), here the shape parameter P = O ( M lo /M hi ). The phase shift up to NLO is given by k cot δ (0+1)0 ( k ) = − a + r k − P (cid:16) r (cid:17) k , (37)with an error that can be estimated from the k term once we take Λ → M hi in∆ (cid:16) k cot δ (0+1)0 ( k ) (cid:17) = | P | r | θ | Λ k . (38)This error shows that at N LO a ( ψ † ψ )( ψ † ←→∇ ψ ) interaction with LEC 4 πC /m is needed, where after renormalization C = O (1 /M lo M hi ). At this order this interaction is solved in first-order distorted-wave perturbation theory, whileNLO interactions need to be accounted for in second order. The procedure continues at higher orders in an obviousway. IV. POLES AND RESIDUES
The distinctive feature of our LO is the existence of two S -wave S -matrix poles, allowing for the possibility of aresonance. Our S -wave S matrix can be written as S ( k ) = 1 − imk π T ( k ) = e iφ ( k ) ( k + k )( k + k )( k − k )( k − k ) , (39)where k , (cid:54) = 0 denote the pole positions and φ ( k ) is the nonresonant or “background” contribution to the phase shift.These quantities can be expanded order by order, and we now obtain them up to NLO. A. Leading order
The S matrix corresponding the LO T matrix (24), S (0)0 ( k ), is nothing but that corresponding to the effective-rangeapproximation to the ERE, that is, Eq. (39) with [35] k (0)1 , = 1 r (cid:18) i ± (cid:114) r a − (cid:19) (40)and φ (0) ( k ) = 0 . (41)While the effective range r < a can be positive or negative, giving riseto five qualitatively distinct cases:1. − | r | < a <
0: there are two resonance poles k (0)1 = k R − ik I , (42) k (0)2 = − k R − ik I , (43)0with k I = 1 | r | > , (44) k R = 1 | r | (cid:114) r a − > . (45)The two poles are thus forced to be in the lower half of the complex momentum plane by the requirement ofrenormalizability. This is in agreement with the general requirement on the S matrix [1, 36, 37] that leads tostates decaying with time. In the limit Λ → ∞ we obtain the well-known form S (0)0 ( E ) = E − E (0)0 − i Γ (0) ( E ) / E − E (0)0 + i Γ (0) ( E ) / , (46)where E (0)0 = k R + k I m = 2 ma r > , (47)Γ (0) ( E ) = 4 kk I m = 4 m | r | √ mE > . (48)The residues of iS (0)0 at the poles are complex,Res (cid:16) iS (0)0 (cid:17)(cid:12)(cid:12)(cid:12) ± k R − ik I = 2 k I (cid:18) ∓ i k I k R (cid:19) . (49)Our power counting describes the situation | a | ∼ | r | ∼ /M lo , where the resonance is shallow but broad in thesense that k I ∼ k R ∼ M lo (cid:28) M hi . We cannot exclude a situation where r (cid:29) a and the resonance is narrow,that is, k R (cid:29) k I with nearly real and positive residues [35]. However, our power counting is somewhat artificialin this case, since the unitarity term − ik in Eq. (24) is small compared to the inverse scattering length andthe effective range, except near the pole. A narrow resonance arises more naturally from a dimeron field whereresidual mass and kinetic terms are treated as LO, while loops are included perturbatively except in the vicinityof the resonance [15].2. a = − | r | : there is a double pole on the negative imaginary axis [38], k (0)1 = k (0)2 ≡ k (0)1 ≡ = − i | r | , (50)with positive residue Res (cid:16) iS (0)0 (cid:17)(cid:12)(cid:12)(cid:12) k ≡ = 4 ik ≡ > . (51)This represents a virtual state. A double pole on the upper plane is again excluded by the requirement ofrenormalizability, in agreement with other arguments [1, 36, 38].3. a < − | r | : there are two virtual states represented by poles on the negative imaginary axis, k (0)1 = − iκ − , (52) k (0)2 = − iκ + , (53)where κ ± = 1 | r | (cid:18) ± (cid:114) − r a (cid:19) > . (54)They have residues of opposite signs, Res (cid:16) iS (0)0 (cid:17)(cid:12)(cid:12)(cid:12) − iκ ± = ± κ ± κ + + κ − κ + − κ − , (55)the shallowest pole ( k ) with the negative sign.1 a , r − , − +, − − , + +, +Im k , Im k − , − +, − − , + +, +Res( iS (0)0 ) | k , Res( iS (0)0 ) | k − , + +, + − , − +, − pole 1, pole 2 V, V B, V V, R B, RTABLE I: Character of (simple) poles on the imaginary axis according to the signs of the scattering length a and effectiverange r for | Im k | > | Im k | . In the last row V stands for virtual state, B for bound state, and R for redundant pole. Onlythe first two columns are allowed by renormalization of the EFT. a >
0: the two poles are on opposite sides of the imaginary axis, k (0)1 = iκ − , (56) k (0)2 = − iκ + , (57)where κ ± = 1 | r | (cid:115) | r | a ± > . (58)Both residues are positive, Res (cid:16) iS (0)0 (cid:17)(cid:12)(cid:12)(cid:12) ∓ iκ ± = 2 κ ± κ + − κ − κ + + κ − > . (59)This indicates that the pole k on the positive imaginary axis is a bound state [1]. The constraint r < a > r or a <
0, see Table I. The interpretation of a redundant pole isunclear: it has a non-normalizable wavefunction [40, 42], but carries information about the asymptotic behaviorof continuum states [43]. Since at least in a nonrelativistic setting its position is determined by the range of thepotential [35, 44, 45], it is comforting that renormalization of our EFT prevents a shallow redundant pole.5. | a | − = 0: this is the boundary between the previous two cases. It is essentially the limit | a | → ∞ of thesecases, except that the S matrix has a single pole k (0)2 = − iκ + = − i | r | , (60)with positive residue Res (cid:16) iS (0)0 (cid:17)(cid:12)(cid:12)(cid:12) − iκ + = 2 κ + > . (61)The other, would-be S -matrix pole is only a pole of the T matrix which is sometimes called a zero-energyresonance. When | r | ∼ /M hi , the corresponding EFT [9–12] is scale invariant at LO with no other low-energy T -matrix pole. Here, the dimensionful parameter | r | ∼ /M lo explicitly breaks scale invariance at LO,generating a virtual pole.When we use effective-range parameters to fix the LECs, the pole positions have an uncertainty ∆ k (0) due to theneglect of higher orders. We can estimate the magnitude of the error from the residual cutoff dependence in Eq. (24)and then varying the cutoff from the theory’s breakdown scale to much larger values. We find | ∆ k (0)1 , | = r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (0)41 , θ (cid:113) r a − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , a (cid:54) = 2 r , (62)= 1 | r | (cid:115) | θ r | Λ , a = 2 r , (63)with Λ ∼ M hi . Evidently ∆ k (0) = 0 if we use the pole positions as input.2 B. Subleading order
If their positions are not used as input, the pole positions will move slightly at subleading orders and approach, ifthe theory is converging, their exact locations. As before, the procedure is systematic and we illustrate it only forNLO.When both poles are simple, that is, for a (cid:54) = 2 r , the pole positions can be written as k (0+1)1 , = k (0)1 , + k (1)1 , (64)in terms of the NLO shift k (1)1 , = P | r | ∓ (cid:113) r a − (cid:18) r a − r a + 1 (cid:19) + i (cid:18) − r a (cid:19) . (65)This is precisely what is needed to cancel the spurious double pole in T (1)0 (the “good-fit condition” of Ref. [46]).Now the error is reduced to the size of N LO interactions when Λ is above the breakdown scale of the theory, | ∆ k (1)1 , | = r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P k (0)61 , θ (cid:113) r a − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (66)In contrast, when a = 2 r the double nature of the pole leads to an expansion in half powers for its position, k (0+1)1 , = k (0)1 ≡ + k (1 / , + k (1)1 ≡ , (67)where k (1 / , = ± √ P | r | , (68) k (1)1 ≡ = i P | r | . (69)We will refer to the half-power correction as N / LO. While the smaller NLO correction is imaginary, the N / LOcorrection is real for P >
0, indicating that in the underlying theory the resonant poles have not coalesced. Conversely, P < / LO displacement. As an estimate for the error we take | ∆ k (1 / , | = | P | | r | , (70)which is √ P ∼ O (( M lo /M hi ) / ) smaller than k (1 / , in Eq. (68). This estimate could easily be off by a factor of O (1), but it accidentally exactly coincides with the magnitude of the NLO shift (69). The Λ-dependent error of thelatter scales as the 3/2 power of the expansion parameter, | ∆ k (1)1 , | = (cid:112) | P | r | θ | Λ . (71)In either case, up to higher-order terms the NLO S matrix S (0+1)0 ( k ) is given by Eq. (39) but with the poles attheir NLO positions k (0+1)1 , and φ (0+1) ( k ) = P r k ≡ − c k . (72)The nonresonant contribution to the phase shift is a subleading effect linear in k . This form of the S matrix forshort-range forces with two poles has been arrived at by causality-type arguments [36, 47], with c ≥ R of the force. For c ≥ P ≥
0, although this constraint does not follow from renormalization of ourEFT to NLO.As we have seen, the renormalization condition on the EFT allows only the standard cases of a (decaying) resonance,a bound state, and virtual states. The pole positions can be determined from the effective-range parameters withincreasing precision as the order increases. In order to show explicitly that the accuracy also improves, we consideran explicit example of underlying theory next.3
V. TOY MODEL
The inclusion of all interactions allowed by symmetries ensures that any underlying dynamics producing the samelow-energy pole structure can be accommodated in the EFT. Information about the dynamics at short-distance scalesis encoded in the LECs. We now consider a simple model for the short-distance physics, in order to illustrate how theEFT captures the long-distance dynamics associated to the existence of two shallow poles.As a toy model we take a potential consisting of an attractive spherical well of range R and depth β /mR with arepulsive delta shell with strength α/mR at its edge: V ( r ) = αmR δ ( r − R ) − β mR θ ( R − r ) , (73)with α > β >
0. This model was used for resonances in Ref. [17]. The Schr¨odinger equation is easily solved inthe S -wave in the standard fashion inside ( r < R ) and outside ( r > R ), with the wavefunctions and their derivativesmatched at the range R . One findscot δ ( k ) = − κR cot( κR ) cot( kR ) + α cot( kR ) + kRκR cot( κR ) + α − kR cot( kR ) , (74)where κ = (cid:112) k + β /R . Equivalently, the phase shift is given by δ ( k ) = − kR + arctan (cid:18) kRκR cot( κR ) + α (cid:19) . (75) A. Scattering length, effective range, and pole positions
We are interested in the dynamics for k (cid:28) R − ≡ M hi . Equations (74) and (75) reduce to the well-knownexpressions for the attractive spherical well [1] when α = 0. In this case there is a single low-energy pole, either abound or a virtual state, which is captured by the EFT where LO consists of only C and NLO of C [9–12]. For α > α and β yields the various cases (bound, virtual, resonant) consideredabove.Expanding the inverse of the T matrix in kR (cid:28) a R = 1 − α + β cot β , (76) r R = 2( α + β cot β ) ( α + β cot β − − β − (cot β ) /β − α + β cot β − (77)= 1 − (cid:18) Ra (cid:19) − Rβ a − αβ (cid:34) α + 1 − α Ra + ( α − (cid:18) Ra (cid:19) (cid:35) . (78)The next term in the expansion of T − ( k ) gives an expression for the shape parameter P as a function of theparameters α and β . We plot a and r as functions of β for various values of α in Fig. 1. For most values of thepotential parameters, a ≈ R and r ≈ R as expected from dimensional analysis. Only some specific regions haveunnaturally large magnitudes. In the regions where | a | (cid:29) R , a can be positive or negative, but it is still verylikely that r ≈ R . This is the situation previously investigated in EFT where only C is enhanced. Only in narrowparameter ranges where β cot β ≈ − α (cid:28) − | a | (cid:29) R and | r | (cid:29) R . For a pure spherical well( α = 0), r /R is given by the first three terms in Eq. (78) and it is easy to see that | a | (cid:29) R leads to r ≈ R . Theadditional parameter α > | r | (cid:29) R . But in this case r <
0, just as obtained from renormalization of theEFT considered here, where also C is enhanced.For given values of a and r , we can solve Eqs. (76) and (77) for α and β , and then use the exact expression for k cot δ in Eq. (74) to find the poles of T . In the regions where | a | /R (cid:29) | r | /R (cid:29) α = 4 . β ≈
7. In the plot we begin with β = 6 .
69, which gives us two resonance poles,and increase β , which corresponds to increasing the attraction of the well. The two poles collide on the negativeimaginary axis creating a double pole, and then one pole moves down as a virtual state while the other moves up untilit eventually becomes a bound state. We continue following the increase in binding till β = 7 .
06. The pole evolutioncovers the situations we considered in the previous section. Note that a similar pole evolution exists for the spherical4 α = α = α = α = β - - / R α = α = α = α = β - - - - - r / R FIG. 1: Scattering length a (left panel) and effective range r (right panel) in units of the potential range R as functions ofthe square of the strength β of the spherical well, for various values of the strength α of the delta shell. ●●●●●●●●●● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ - - ( Rk )- - - ( Rk ) β ≈ β ≈ β ≈ β ≈ FIG. 2: Pole positions in the kR complex plane for a given strength of the delta-shell potential, α = 4 . β of the spherical well increases from 6.69 to 7.06, the two resonance poles (blue circles) approach the imaginary axis and thenturn into two virtual poles: one that remains a virtual state (green diamonds), another that eventually becomes a bound state(red squares). well alone ( α = 0) [48], but in the latter case the coalescence on the imaginary axis happens at k = k = − i/R ,which is outside the EFT range. The possibility of this evolution for a general potential well surrounded by a barrierwas studied in Ref. [38]. We now turn to a quantitative comparison with the EFT. B. Comparison with the EFT
We now compare the predictions of the EFT at LO and NLO, using a sharp-cutoff regulator, with the toy model.We choose values for a and r , from which we extract β and α , and calculate P . We fit the EFT to these EREparameters and compare the resulting phase shifts and pole positions. Since the phase shift is very sensitive to changesin momentum k and there are periodic discontinuities in its derivatives, it is better to work with k cot δ instead of δ itself. For the toy model we evaluate k cot δ from Eq. (74). For the EFT we use Eq. (25) at LO and Eq. (37) atNLO, with their corresponding errors in Eqs. (26) and (38). We also compare the position of the poles determinednumerically in the toy model with the EFT predictions and their errors given in Sec. IV.For illustration, we keep the effective range fixed at r /R = − a /R so as to reproduce the various casesdiscussed in the previous section. We start with a /R = − > r /R . The toy-model parameters for this choice are β = 7 . α = 7 . P = 0 . LO - EFTNLO - EFTToy Model ( kR ) - - - - - ( δ ) ● ●●■ ■■◆◆◆ ● LO - EFT ■ NLO - EFT ◆ Toy Model - - ( kR )- - - ( kR ) FIG. 3: Comparison between EFT and toy model for kR cot δ as a function of ( kR ) (left panel) and resonance pole positionson the complex kR plane (right panel), when α = 7 . β = 7 . k R k R LO EFT (0 . ± . − (0 . ± . i − (0 . ± . − (0 . ± . i NLO EFT (0 . ± . − (0 . ± . i − (0 . ± . − (0 . ± . i Toy model 0 . − . i − . − . i TABLE II: Position of resonance poles k , in units of R − , when α = 7 . β = 7 . shifts at LO and NLO compared to the toy-model values. As expected, the LO EFT agrees with the toy model withinan error that increases with energy. The NLO perturbation improves the agreement for the central value and theerror is decreased compared to LO. For this choice of parameters the EFT has two resonance poles, just as the toymodel. The pole positions are shown on the right panel of Fig. 3 and given in Table II. Again the EFT error barsdecrease with order and central values approach the exact result. This example demonstrates the power of EFT toapproximate in a systematic and controlled way the T matrix for resonant states.We now increase the attraction so that a /R = −
16 = 2 r /R , when the toy-model parameters are β = 7 . α = 7 . P = 0 . k cot δ ( k ) is smaller at k = 0, but the similar values of toy-modelparameters lead to low-energy phase shifts that are not very different from the previous case. The pole positionsare given on the right panel of Fig. 4 and in Table III. The increased attraction brings the toy-model poles closerto the imaginary axis. The EFT has a double pole at LO with error bars that encompass the two resonance poles.The half-power correction corrects for the horizontal splitting, and the full NLO correction moves the poles slightlyupwards. The convergence pattern is clear, although the NLO error bars are underestimated by a factor of about 2.Had we estimated them by simply multiplying Eq. (69) with √ P , similarly to what we have done at N / LO, theNLO error would be about four times larger.Further increase of the attraction moves the two toy-model poles onto the negative imaginary axis, that is, itcreates two virtual states. Taking a /R = − < r /R , the values for the potential parameters are β = 7 . α = 7 . P = 0 . k R k R LO EFT − (0 . ± . i − (0 . ± . i N / LO EFT (0 . ± . − (0 . ± . i − (0 . ± . − (0 . ± . i NLO EFT (0 . ± . − (0 . ± . i − (0 . ± . − (0 . ± . i Toy model 0 . − . i − . − . i TABLE III: Position of resonance poles k , in units of R − , when α = 7 . β = 7 . / LO, andNLO is compared with the toy model. LO - EFTNLO - EFTToy Model ( kR ) - - - - - - ( δ ) ●●●■ ■■◆ ◆◆▲▲▲ ● LO - EFT ■ N / LO - EFT ◆ NLO - EFT ▲ Toy Model - - ( kR )- - ( kR ) FIG. 4: Comparison between EFT and toy model for kR cot δ as a function of ( kR ) (left panel) and pole positions on thecomplex kR plane (right panel), when α = 7 . β = 7 . / LO gives the (blue)point with intermediate-size error bar. Other notation as in Fig. 3. LO - EFTNLO - EFTToy Model ( kR ) - - - - - - ( δ ) ●●●■■■◆◆◆ ● LO - EFT ■ NLO - EFT ◆ Toy Model - - ( kR )- - - ( kR ) FIG. 5: Comparison between EFT and toy model for kR cot δ as a function of ( kR ) (left panel) and virtual-state pole positionson the complex kR plane (right panel), when α = 7 . β = 7 . left panel of Fig. 5 to be, again, very similar to the previous cases. The pole positions are nevertheless very different,as shown on the right panel and in Table IV. As expected, the EFT describes the shallower pole very well, but hasmuch larger errors for the deeper state.Finally, as an example of a /R >
0, we take a /R = 40, which translates into β = 7 . α = 8 . P = 0 . a > k R k R LO EFT − (0 . ± . i − (0 . ± . i NLO EFT − (0 . ± . i − (0 . ± . i Toy model − . i − . i TABLE IV: Position of virtual poles k , in units of R − , when α = 7 . β = 7 . LO - EFTNLO - EFTToy Model ( kR ) - - - ( δ ) ●●●■■■◆◆◆ ● LO - EFT ■ NLO - EFT ◆ Toy Model - - ( kR )- - - - ( kR ) FIG. 6: Comparison between EFT and toy model for kR cot δ as a function of ( kR ) (left panel), and virtual- and bound-statepole positions on the complex kR plane (right panel), when α = 8 . β = 7 . k R k R LO EFT (2 . ± . × − i − (0 . ± . i NLO EFT (2 . ± . × − i − (0 . ± . i Toy model 2 . × − i − . i TABLE V: Position of bound and virtual poles k , in units of R − , when α = 8 . β = 7 . VI. CONCLUSION
We have constructed an effective field theory that describes the scattering of two nonrelativistic particles whentwo shallow S -wave poles are present. Resonant nonrelativistic scattering has been considered before in EFT [14–18],but only with energy-dependent interactions. The characteristic feature of our formulation is the sole reliance onmomentum-dependent interactions, which are easier to employ in more-particle systems. The effective Lagrangian forany low-energy two-body scattering process involving short-range forces can be written as an expansion in contactoperators with an increasing number of spatial derivatives. The challenge, which we met above, is to order theseinteractions with a power counting appropriate to produce resonant poles in the scattering amplitude.It is well known that various observables — like the scattering length and the effective range in the effective-rangeexpansion — reflect the presence of resonant states. This feature, which we have illustrated with a toy model againstwhich we compare our EFT, motivates the power-counting scheme we use in this paper. We find that the low-energyscattering amplitude yields resonant poles only when the two leading operators in the derivative expansion are treatednonperturbatively. At higher orders, operators with an increasing number of derivatives contribute perturbatively.Our EFT successfully produces a controlled expansion about the resonant states and predicts the positions of poleswith an error that can be systematically reduced by adding higher-dimensional operators. The same is true moregenerally for other situations involving two shallow S -wave poles, such as two virtual states, or one virtual state andone bound state.As a bona fide EFT, ours obeys approximate renormalization-group invariance. It is remarkable that renormalizationat leading order forces the effective range to be negative [24, 25]. This is in agreement with Wigner’s bound [26] andallows no more than one pole in the upper half of the complex momentum plane. Thus, renormalization automaticallyincorporates the causality constraint that a resonance represents decaying, not growing, states [1, 36]. It also doesnot allow for a redundant pole on the positive imaginary axis [39–41] nor for a double bound-state pole, which isexcluded by other arguments [1, 36, 38]. In short, the resulting S matrix obeys the conditions expected on generalgrounds [1]. In contrast, the renormalization of the EFT with “dimer” auxiliary fields [14–18] allows for the moregeneral situation, believed to be unphysical, where two (or more) shallow poles appear in the upper half-plane.The situation does not change at higher orders, since the corrections are perturbative. The need to remove theresidual cutoff dependence gives clues about the orders corrections come at. We saw explicitly how the four-derivativeoperator enters at next-to-leading order, gives rise to a known form for the S matrix [36], and improves on the leading-order description systematically. There is no obvious obstacle to continuing this process beyond next-to-leading order.Power counting and renormalization here are significantly different than those [9–12] for a single shallow S -wavepole. The need to treat the two-derivative contact interaction perturbatively in the latter case has been known for a8 +++ ... T=T = FIG. 7: The two-body T matrix as a sum of Feynman diagrams. Particle propagation (A1) is represented by a solid line, whilethe dark oval stands for the potential (3). long time [9–12], but now we understand this need from the fact that the situation described by a nonperturbativetreatment of the two-derivative contact interaction is different in a physically meaningful way — it corresponds todifferent regions of the parameter space of an underlying potential, or to a different type of potential altogether.The EFT developed in this paper is directly applicable only to resonant S -wave scattering of two spin-zero particlesthat interact via a short-range force. Although at the two-body level it is equivalent to a particular ordering ofthe effective-range expansion, it is a Hamiltonian framework that allows for the investigation of the effects of thistwo-body physics in processes involving more than two particles. This is analogous to the single-pole theory, wherefor example the three-body system can be dealt with [49–53]. The next obvious step is to include spin and (in thenuclear case) isospin quantum numbers, as well as higher waves. Furthermore, we aim to study resonant scatteringof electrically charged particles by including the long-range Coulomb interactions in our EFT. We anticipate thata nonperturbative treatment of the Coulomb interaction will be necessary to describe resonant scattering of alphaparticles at low energies [16]. The EFT developed in this paper is a step in this direction. Acknowledgments
UvK is grateful to R. Higa for useful discussions. This research was supported in part by the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics, under award number DE-FG02-04ER41338.
Appendix A: T matrix from Feynman diagrams Here we obtain the T matrix in field theory by summation of Feynman diagrams. The potential (3) is representedby four-legged vertices with an increasing number of powers of momenta. Since antiparticles are integrated out, thetwo-body amplitude is just a string of these vertices connected by two single-particle propagators, S F ( q ) = iq − (cid:126)q / m + i(cid:15) + . . . , (A1)where q ( (cid:126)q ) is the fourth (three-dimensional) component of the particle’s 4-vector and “ . . . ” represent relativisticcorrections. We will neglect the latter here, but their inclusion poses no additional conceptual problems [12]. Thesum of diagrams is shown in Fig. 7.We can write the S -wave T matrix in a compact way by defining in ( | p (cid:105) ) and out ( | p (cid:48) (cid:105) ) vectors and a vertex matrix( C ) through | p (cid:105) ≡ p p ... , | p (cid:48) (cid:105) ≡ p (cid:48) p (cid:48) ... , C ≡ πm C C / C / · · · C / C / C / · · · C / C / C / · · · ... ... ... . . . . (A2)(See also Ref. [27].) The tree diagram in Fig. 7 is then simply T ( p (cid:48) , p ) = V ( p (cid:48) , p ) = (cid:104) p (cid:48) | C | p (cid:105) . (A3)9The loop diagrams involve a 4-momentum integration. Since the vertices depend only on the 3-momenta, we canevaluate the q integrals in the center-of-mass frame, (cid:90) d q (2 π ) (cid:126)q n iq − (cid:126)q / m + i(cid:15) ik / m − q − (cid:126)q / m + i(cid:15) = − i m π I +2 n , (A4)where I +2 n is defined in Eq. (12). If we define a matrix of integrals, I ≡ − m (cid:90) d q (2 π ) | q (cid:105)(cid:104) q | k − q + i(cid:15) = − m π I +0 I +2 I +4 · · · I +2 I +4 I +6 · · · I +4 I +6 I +8 · · · ... ... ... . . . , (A5)the one-loop diagram takes the form T ( p (cid:48) , p ) = − m (cid:90) d q (2 π ) V ( p (cid:48) , q ) V ( q, p ) k − q + i(cid:15) = (cid:104) p (cid:48) | C I C | p (cid:105) . (A6)With a regulator on nucleon momenta, the multiple-loop diagrams separate and the sum of diagrams is T ( p (cid:48) , p ) = ∞ (cid:88) i =0 T i ( p (cid:48) , p ) = (cid:104) p (cid:48) | ( C + C I C + C I C I C + . . . ) | p (cid:105) = (cid:104) p (cid:48) | C (1 − I C ) − | p (cid:105) ≡ (cid:104) p (cid:48) | T | p (cid:105) . (A7)Since predictive power requires a finite number of parameters at each order, we need to truncate the amplitude(A7) at different orders in order to renormalize it. The LO T matrix results from taking C = C (0)0 , C = C (0)2 , and C n ≥ = 0 in the matrix C . We arrive at4 πmT (0)0 ( p (cid:48) , p ) = (4 C (0)0 − C (0)22 I ) I + (2 + C (0)2 I ) C (0)0 − C (0)22 I + C (0)2 (2 + C (0)2 I )( p (cid:48) + p ) − C (0)22 I p (cid:48) p , (A8)which gives Eq. (16) on-shell, that is, when p = p (cid:48) = k . To obtain the NLO T matrix, we take instead C = C (0)0 + C (1)0 , C = C (0)2 + C (1)2 , C = C (1)4 , and C n ≥ = 0, expanding in the subleading pieces. Retaining only termslinear in C (1)0 , , results on-shell in Eq. (29). The procedure can be continued in a straightforward way at higher orders.Note that the same results can be obtained from Feynman diagrams in a form that is somewhat closer to theprocedure of the main text. We rewrite the sum of diagrams (A7) as a Lippmann-Schwinger equation, (cid:104) p (cid:48) | T | p (cid:105) = (cid:104) p (cid:48) | C | p (cid:105) + (cid:104) p (cid:48) | C I T | p (cid:105) . (A9)This form is also shown in Fig. 7. The Lippmann-Schwinger equation can be solved at LO [24] with an ansatz motivated by the momentum structure of Eq. (A6), T (0)0 = τ (0)0 τ (0)2 · · · τ (0)2 τ (0)4 · · · · · · ... ... ... . . . . (A10)Inserting this form on both sides of Eq. (A9) and matching powers of momenta, one finds three algebraic equationsfor τ (0)0 , , . Solving these equations we again obtain Eq. (A8). At NLO, we expand both C and T in Eq. (A9) to linearorder. The NLO correction T (1)0 appears both directly on the left-hand side and inside an integral on the right-handside. To solve the resulting equation we make an ansatz analogous to (A10) but now including the sixth power ofmomenta. Appendix B: Renormalization procedure
In this appendix we give some of the details of our renormalization procedure, at both LO and NLO. In either casewe expand the amplitude calculated within the EFT in a power series in k/ Λ (cid:28) πmT ( k ) = ik + ∞ (cid:88) n =0 g n k n , (B1)0where the coefficients g n = g (0)2 n + g (1)2 n + . . . (B2)depend on the bare LECs C n (Λ) and the cutoff Λ. Since g n encodes short-range physics, only integer powers of theenergy appear in the expansion (B1). The only non-analytic behavior is represented by the unitarity term ik , whichstems from the Schr¨odinger propagation. The C n (Λ) are fixed by matching Eq. (B1) with the ERE in Eq. (1).At LO, the amplitude is given by Eq. (16). The first four non-zero g (0)2 n are: g (0)0 = L − C (0)22 L − C (0)0 (cid:16) C (0)2 L + 2 (cid:17) , (B3) g (0)2 = L − − C (0)2 C (0)2 L + 4( C (0)22 L − C (0)0 ) (cid:16) C (0)2 L + 2 (cid:17) , (B4) g (0)4 = L − − C (0)22 ( C (0)2 L + 4) ( C (0)22 L − C (0)0 ) (cid:16) C (0)2 L + 2 (cid:17) , (B5) g (0)6 = L − − C (0)32 ( C (0)2 L + 4) ( C (0)22 L − C (0)0 ) (cid:16) C (0)2 L + 2 (cid:17) . (B6)The expressions for g (0)0 , agree with those in Refs. [24, 25]. We demand that g (0)0 , reproduce given values of thescattering length a and effective range r , g (0)0 = 1 a , (B7) g (0)2 = − r . (B8)This is a set of two equations from which the two running values of C (0)0 and C (0)2 can be obtained in terms of the L n and the observables a and r as C (0)0 = − L L (cid:34) ∓ √ a L − (cid:112) − a ( r + 2 L − ) L + 2( a L − (cid:35) + 2 a ( a L − a ( r + 2 L − ) L − a L − , (B9) C (0)2 = − L (cid:34) ∓ √ a L − (cid:112) − a ( r + 2 L − ) L + 2( a L − (cid:35) . (B10)Equations (19) and (20) follow upon expanding the expressions above for a Λ (cid:29) r Λ (cid:29)
1. In addition, g (0)4 gives the residual dependence shown in Eq. (23).1At NLO, the amplitude is given by Eq. (29). Once we expand in k/ Λ the shifts in the first four g n are given by g (1)0 = − C (0)22 L − C (0)0 ) (cid:26) C (1)0 (cid:16) C (0)2 L + 2 (cid:17) − C (1)2 (cid:16) C (0)2 L + 2 (cid:17) (cid:16) C (0)2 L + 2 C (0)0 L (cid:17) + C (1)4 (cid:104) C (0)22 (cid:16) C (0)2 L + 4 (cid:17) (cid:0) L − L L (cid:1) − C (0)2 L + 4 C (0)0 L (cid:16) C (0)2 L − (cid:17) + 8 C (0)20 L (cid:105)(cid:41) , (B11) g (1)2 = 8( C (0)2 L + 2)( C (0)22 L − C (0)0 ) (cid:40) C (1)0 C (0)2 (cid:16) C (0)2 L + 2 (cid:17) (cid:16) C (0)2 L + 4 (cid:17) + C (1)2 (cid:104) C (0)2 (cid:16) C (0)2 L + 4 (cid:17) (cid:16) C (0)2 L + 2 C (0)0 L (cid:17) − C (0)22 L + 4 C (0)0 (cid:105) + C (1)4 (cid:20) C (0)32 (cid:16) C (0)2 L + 4 (cid:17) (cid:0) L − L L (cid:1) + 2 C (0)22 L (cid:16) C (0)2 L + 4 (cid:17) − C (0)0 C (0)22 L L − C (0)20 L (cid:16) C (0)2 L + 3 (cid:17)(cid:21)(cid:41) , (B12) g (1)4 = − C (0)2 L + 2)( C (0)2 L + 4)( C (0)22 L − C (0)0 ) (cid:40) C (1)0 C (0)22 (cid:16) C (0)2 L + 2 (cid:17) (cid:16) C (0)2 L + 4 (cid:17) − C (1)2 C (0)2 (cid:104) C (0)2 (cid:16) C (0)2 L + 4 (cid:17) (cid:16) C (0)2 L + 2 C (0)0 L (cid:17) − (cid:16) C (0)22 L − C (0)0 (cid:17)(cid:105) + C (1)4 (cid:20) C (0)42 (cid:16) C (0)2 L + 6 (cid:17) (cid:0) L − L L (cid:1) − C (0)32 L + 4 C (0)32 L (cid:16) C (0)0 L − C (0)2 L (cid:17) +32 C (0)20 (cid:16) C (0)2 L + 1 (cid:17) + 24 C (0)20 C (0)2 L (cid:16) C (0)2 L + 2 (cid:17)(cid:21)(cid:41) , (B13) g (1)6 = − C (0)2 L + 2)( C (0)2 L + 4) ( C (0)22 L − C (0)0 ) C (0)2 (cid:40) C (1)0 C (0)22 (cid:16) C (0)2 L + 2 (cid:17) (cid:16) C (0)2 L + 4 (cid:17) − C (1)2 C (0)2 (cid:104) C (0)2 (cid:16) C (0)2 L + 4 (cid:17) (cid:16) C (0)2 L + 2 C (0)2 L (cid:17) − (cid:16) C (0)22 L − C (0)0 (cid:17)(cid:105) + C (1)4 (cid:20) C (0)42 (cid:16) C (0)2 L + 6 (cid:17) (cid:0) L − L L (cid:1) + 8 C (0)32 L − C (0)32 L (cid:16) C (0)0 L − C (0)2 L (cid:17) +16 C (0)20 (cid:16) C (0)2 L + 1 (cid:17) + 4 C (0)20 C (0)2 L (cid:16) C (0)2 L + 3 (cid:17)(cid:21)(cid:41) . (B14)Now we demand that the shape parameter P be reproduced, without changes in the scattering length and effectiveranges; that is, we impose g (1)0 = 0 , (B15) g (1)2 = 0 , (B16) g (1)4 = P (cid:16) r (cid:17) − g (0)4 . (B17)2Solving for the three unknowns C (1)0 , , which appear linearly, C (1)0 = − (cid:18) P r − g (0)4 (cid:19) ( C (0)22 L − C (0)0 ) C (0)2 L + 2) ( C (0)2 L + 4) (cid:20) C (0)22 (cid:16) C (0)2 L + 4 (cid:17) (cid:0) L + L L (cid:1) + 4 C (0)2 L +6 C (0)0 C (0)2 L L (cid:16) C (0)2 L + 4 (cid:17) + 4 C (0)20 L (cid:16) C (0)2 L + 4 (cid:17) + 8 C (0)0 L (cid:21) , (B18) C (1)2 = − (cid:18) P r − g (0)4 (cid:19) ( C (0)22 L − C (0)0 ) C (0)2 L + 2)( C (0)2 L + 4) (cid:104) C (0)2 L (cid:16) C (0)2 L + 4 (cid:17) + C (0)0 L (cid:16) C (0)2 L + 6 (cid:17)(cid:105) , (B19) C (1)4 = − (cid:18) P r − g (0)4 (cid:19) ( C (0)22 L − C (0)0 ) C (0)2 L + 4) , (B20)where g (0)0 , C (0)0 and C (0)2 are given in Eqs. (B5), (B9) and (B10). 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