The Form of the Effective Interaction in Harmonic-Oscillator-Based Effective Theory
TThe Form of the Effective Interaction in Harmonic-Oscillator-Based Effective Theory
W. C. Haxton
Institute for Nuclear Theory and Department of Physics,University of Washington, Seattle, WA 98195, USA ∗ I explore the form of the effective interaction in harmonic-oscillator-based effective theory (HO-BET) in leading-order (LO) through next-to-next-to-next-to-leading order (N LO). As the includedspace in a HOBET (as in the shell model) is defined by the oscillator energy, both long-distance(low-momentum) and short-distance (high-momentum) degrees of freedom reside in the high-energyexcluded space. A HOBET effective interaction is developed in which a short-range contact-gradientexpansion, free of operator mixing and corresponding to a systematic expansion in nodal quantumnumbers, is combined with an exact summation of the relative kinetic energy. By this means thevery strong coupling of the included ( P ) and excluded ( Q ) spaces by the kinetic energy is removed.One finds a simple and rather surprising result, that the interplay of QT and QV is governed by asingle parameter κ , the ratio of an observable, the binding energy | E | , to a parameter in the effectivetheory, the oscillator energy ¯ hω . Once the functional dependence on κ is identified, the remainingorder-by-order subtraction of the short-range physics residing in Q becomes systematic and rapidlyconverging. Numerical calculations are used to demonstrate how well the resulting expansion re-produces the running of H eff from high scales to a typical shell-model scale of 8¯ hω . At N LOvarious global properties of H eff are reproduced to a typical accuracy of 0.01%, or about 1 keV, at8¯ hω . Channel-by-channel variations in convergence rates are similar to those found in effective fieldtheory approaches.The state-dependence of the effective interaction has been a troubling problem in nuclear physics,and is embodied in the energy dependence of H eff ( | E | ) in the Bloch-Horowitz formalism. It isshown that almost all of this state dependence is also extracted in the procedures followed here,isolated in the analytic dependence of H eff on κ . Thus there exists a simple, Hermitian H eff thatcan be use in spectral calculations.The existence of a systematic operator expansion for H eff , depending on a series of short-rangeconstants augmented by κ , will be important to future efforts to determine the HOBET interactiondirectly from experiment, rather than from an underlying NN potential. PACS numbers: 21.30.Fe,21.45.BcKeywords: Nucleon-nucleon interaction; Effective theory; N LO interactions
I. INTRODUCTION
In nuclear physics one often faces the problem of determining long-wavelength properties of nuclei, such as bindingenergies, radii, or responses to low-momentum probes. One approach would be to evaluate the relevant operatorsbetween exact nuclear wave functions obtained from solutions of the many-body Schroedinger equation. Becausethe NN potential is strong, characterized by anomalously large NN scattering lengths, and highly repulsive at veryshort distances, this task becomes exponentially more difficult as the nucleon number increases. Among availablequasi-exact methods, the variational and Green’s function Monte Carlo work of the Argonne group has perhaps setthe standard [1], yielding accurate results throughout most of the 1 p shell.Effective theory (ET) potentially offers an alternative, a method that limits the numerical difficulty of a calculationby restricting it to a finite Hilbert space (the P - or “included”-space), while correcting the bare Hamiltonian H (andother operators) for the effects of the Q - or “excluded”-space. Calculations using the effective Hamiltonian H eff within P reproduce the results using H within P + Q , over the domain of overlap. That is, the effects of Q on P -spacecalculations are absorbed into P ( H eff − H ) P .One interesting challenge for ET is the case of a P -space basis of harmonic oscillator (HO) Slater determinants.This is a special basis for nuclear physics because of center-of-mass separability: if all Slater determinants containingup to N oscillator quanta are retained, H eff will be translationally invariant (assuming H is). Such bases are alsoimportant because of powerful shell-model (SM) techniques that have been developed for iterative diagonalizationand for evaluating inclusive responses. The larger P can be made, the smaller the effects of H eff − H . If one could ∗ Electronic address: [email protected] a r X i v : . [ nu c l - t h ] D ec fully develop harmonic-oscillator based effective theory (HOBET), it would provide a prescription for eliminating theSM’s many uncontrolled approximations, while retaining the model’s formidable numerical apparatus.The long-term goal is a HOBET resembling standard effective field theories (EFTs) [2, 3]. That is, for a given choiceof P , the effective interaction would be a sum of a long-distance “bare” interaction whose form would be determinedby chiral symmetry, augmented by some general effective interaction that accounts for the excluded Q space. Thateffective interaction would be expanded systematically and in some natural way, with the parameters governing thestrength of successive terms determined by directly fitting to experiment. There would be no need to introduce orintegrate out any high-momentum NN potential, an unnecessary intermediate effective theory between QCD and theSM scale.One prerequisite for such an approach is the demonstration that a systematic expansion for the HOBET effectiveinteraction exists. This paper explores this issue, making use of numerically generated effective interaction matrixelements for the deuteron, obtained by solving the Bloch-Horowitz (BH) equation for the Argonne v potential, anexample of a potential with a relatively hard core ( ∼ < H eff is a Hermitian but energy-dependentHamiltonian satisfying H eff = H + H E − QH QHH eff | Ψ P (cid:105) = E | Ψ P (cid:105) | Ψ P (cid:105) = (1 − Q ) | Ψ (cid:105) . (1)Here H is the bare Hamiltonian and E and Ψ are the exact eigenvalue and wave function (that is, the solution ofthe Schroedinger equation in the full P + Q space). E is negative for a bound state. Because H eff depends on theunknown exact eigenvalue E , Eqs. (1) must be solved self-consistently, state by state, a task that in practice provesto be relatively straightforward. If this is done, the P -space eigenvalue will be the exact energy E and the P -spacewave function Ψ P will be the restriction of the exact wave function Ψ to P . This implies a nontrivial normalizationand nonorthogonality of the restricted ( P -space) wave functions. If P is enlarged, new components are added to theexisting ones, and for a sufficiently large P space, the norm approaches ones. This convergence is slow for potentialslike v , with many shells being required before norms near one are achieved [5, 6]. Observables calculated with therestricted wave functions and the appropriate effective operators are independent of the choice of P , of course. All ofthese properties follow from physics encoded in H eff .In HOBET P and thus H eff are functions of the oscillator parameter b and the number of included HO quantaΛ P . In this paper I study the behavior of matrix elements (cid:104) α | H eff | β (cid:105) generated for the Argonne v potential, asboth b and Λ P are varied. In particular, Λ P is allowed to run from very high values to the “shell-model” scale of 8¯ hω , in order to test whether the physics above a specified scale can be efficiently absorbed into the coefficients of somesystematic expansion, e.g., one analogous to the contact-gradient expansions employed in EFTs (which are generallyformulated in plane wave bases). There are reasons the HOBET effective interaction could prove more complicated: • An effective theory defined by a subset of HO Slater determinants is effectively an expansion around a typicalmomentum scale q ∼ /b . That is, the P -space omits both long-wavelength and short-wavelength degrees offreedom. The former are connected with the overbinding of the HO, while the latter are due to absence in P of thestrong, short-range NN interaction. As any systematic expansion of the effective interaction must simultaneouslyaddress both problems, the form of the effective interaction cannot be as simple as a contact-gradient expansion(which would be appropriate if the missing physics were only short-ranged). • The relative importance of the missing long-wavelength and short-wavelength excitations is governed by thebinding energy, | E | , with the former increasing as | E | →
0. These long-range interactions allow nuclear statesto de-localize, minimizing the kinetic energy. But nuclei are weakly bound – binding energies are very smallcompared to the natural scales set by the scalar and vector potentials in nuclei. One concludes that the effectiveinteraction must depend delicately on | E | . • An effective theory is generally considered successful if it can reproduce the lowest energy excitations in P . Butone asks for much more when one seeks to accurately represent the effective interaction, which governs all ofthe spectral properties within P . The HO appears to be an especially difficult case in which to attempt such arepresentation. The kinetic energy operator in the HO has strong off-diagonal components which raise or lowerthe nodal quantum number, and thus connect Slater determinants containing Λ P quanta with those containingΛ P ±
2. This means that P and Q are strongly coupled through low-energy excitations, a situation that isusually problematic for an effective theory.All of these problems involve the interplay, governed by | E | , of QT (delocalization) and QV (corrections for short-range repulsion). The explicit energy dependence of the BH equation proves to be a great advantage in resolving theproblems induced by this interplay, leading to a natural factorization of the long- and short-range contributions to theeffective interaction, and thereby to a successful systematic representation of the effective interaction. (Conversely,techniques such as Lee-Suzuki [7] will intermingle these effects in a complex way and obscure the underlying simplicityof the effective interaction.) The result is an energy-dependent contact-gradient expansion at N LO that reproducesthe entire effective interaction to an accuracy of about a few keV. The contact-gradient expansion is defined in a waythat is appropriate to the HO, eliminating operator mixing and producing a simple dependence on nodal quantumnumbers. The coefficients in the expansion play the role of generalized Talmi integrals.The long-range physics residing in Q can be isolated analytically and expressed in terms of a single parameter, κ = (cid:112) | E | / ¯ hω , remarkably the ratio of an observable ( | E | ) to a parameter one chooses in defining the ET. Thedependence of H eff on κ is determined by summing QT to all orders. The resulting H eff is defined by κ and by thecoefficients of the short-ranged expansion.This same parameter governs almost all of the state dependence that enters when one seeks to describe multiplestates. Thus it appears that there is a systematic, rapidly converging representation for H eff in HOBET thatcould be used to describe a set of nuclear states. The short-range parameters in that representation are effectivelystate-independent, as the state-dependence usually attacked with techniques like Lee-Suzuki is isolated in κ . II. LONG- AND SHORT-WAVELENGTH SEPARATIONS IN H eff In Refs. [5, 6] a study was done of the evolution of matrix elements (cid:104) α | H eff | β (cid:105) , for the deuteron and for He/ H,from the Λ P → ∞ limit, where H eff → H , down to Λ P characteristic of the shell model (SM), e.g., small P spaceswith 4, 6, or 8 ¯ hω excitations, relative to the naive 1 s -shell ground state. As noted above, this definition of P interms of the total quanta in HO Slater determinants maintains center-of-mass separability and thus leads to an H eff that is translationally invariant, just like H . Indeed, the HO basis is the only set of compact wave functions with thisattractive property.But this choice leads to a more complicated ET, as P excludes both short-distance and long-distance componentsof wave functions. This problem was first explored in connection with the nonperturbative behavior of H eff : theneed to simultaneously correct for the missing long- and short-distance behavior of Ψ P is the reason one cannot tune P to make H eff converge rapidly. For example, while it is possible to “pull” more of the missing short-range physicsinto P by choosing a small b , this adjustment produces a more compact state with very large Q -space corrections tothe kinetic energy. Conversely, one can tune b to large values to improve the description of the nuclear tail, but at thecost of missing even more of the short-range physics. At no value of b are both problems handled well: Fig. 1 showsthat a poor minimum is reached at some intermediate b , with a 10¯ hω “bare” calculation failing to bind the deuteron.The solution found to this hard-core correlation/extended state quandary is an a priori treatment of the overbindingof the harmonic oscillator. The BH equation is rewritten in a form that allows the relative kinetic energy operator tobe summed to all orders. (This form was introduced in the first of Refs. [6]; a detailed derivation can be found in theAppendix of the third of these references. The kinetic energy sum can be done analytically for calculations performedin a Jacobi basis.) This reordered BH equation has the form H eff = H + HQ E − QH QH = EE − T Q (cid:20) T − T QE T + V + V E − QH QV (cid:21) EE − QT (2)where the bare H is the sum of the relative kinetic energy and a two-body interaction H = 12 A (cid:88) i,j =1 ( T ij + V ij ) , with T ij = ( p i − p j ) AM . (3)This effective interaction is to be evaluated between a finite basis of Slater determinants | α (cid:105) ∈ P , which is equivalentto evaluating the Hamiltonian (cid:101) H eff ≡ T − T QE T + V + V E − QH QV (4)between the states | (cid:101) α (cid:105) ≡ EE − QT | α (cid:105) (5)By summing QT to all orders, the proper behavior at large r can be built in, which then allows b to be adjusted,without affecting the long-wavelength properties of the wave function. Fig. 1, from Ref. [6], shows that the resulting bare T -summed6,8,10 h4 h 10 h8 h6 h b (fm) -20246 U np e r t u r b e d D e u t e r on E g . s . ( M e V ) FIG. 1: (Color online) Deuteron ground-state convergence for “bare” calculations in small P -spaces, which omit all effects dueto the multiple scattering of V in Q . The three curves on the upper right were calculated from the standard BH equation,which identifies the bare interaction as P ( T + V ) P . These calculations fail to bind the deuteron, even with Λ P = 10, for allvalues of the HO size parameter b : the P -space estimate for V is poor if b is much above 1fm, while the estimate for T is poorif b is below that value. The lower four curves were evaluated for the bare interaction of the reordered BH given by Eq. (2),which incorporates the long-range effects of QT to all orders, building in the correct asymptotic form of the wave function. Thisallows one to reduce b to small values, pulling most of the effects of V into P , without distorting the long-distance behaviorof the wave function or, therefore, the estimate for T . Rather remarkably, this bare calculation reproduces the correct bindingenergy for P spaces as small as Λ P =6. That is, by the combination of the summation of QT to all orders and the adjustmentof b to an optimal value characteristic of the hard core radius of v , the effective interaction contribution can be driven to suchsmall values that it can be ignored. decoupling of the long- and short-wavelength physics can greatly improve convergence: a “bare” 6 ¯ hω calculation thatneglects all contributions of QV gives an excellent binding energy. This decoupling of QV and QT is also importantin finding a systematic expansion for H eff .This reorganization produces an H eff with three terms operating between HO Slater determinants, (cid:104) α | T EE − QT | β (cid:105) = (cid:104) α | EE − T Q T | β (cid:105) nonedge −→ (cid:104) α | T | β (cid:105)(cid:104) α | EE − T Q V EE − QT | β (cid:105) nonedge −→ (cid:104) α | V | β (cid:105)(cid:104) α | EE − T Q V E − QH QV EE − QT | β (cid:105) nonedge −→ (cid:104) α | V E − QH QV | β (cid:105) . (6)The ladder properties of QT make E/ ( E − QT ) the identity operator except when it acts on an | α (cid:105) with energy Λ P ¯ hω or (Λ P − hω . These are called the edge states. For nonedge states, the new grouping of terms in H eff reduces tothe expressions on the right-hand side of Eq. 6, the conventional components of H eff . Thus the summation over QT alters only a subset of the matrix elements of H eff , while leaving other states unaffected.Figure 2 shows the extended tail of the relative two-particle wave function that is induced by E/ ( E − QT ) acting r(fm) -101234 R , (r) FIG. 2: (Color online) A comparison of the radial wave functions for the HO state | nl (cid:105) (dashed) and for the extended state( E/E − QT ) | nl (cid:105) (solid), for ( n, l ) = (6 ,
0) in a Λ P = 10 deuteron calculation. The extended tail of the latter is apparent. Notethat the normalization of the extended state has been adjusted to match that of | nl (cid:105) at r =0, in order to show that the shapesdiffer only at large r . Thus a depletion of the extended state at small r is not apparent in this figure. on an edge HO state [6]. As will become apparent from later expressions, this tail has the proper exponential fall-off, ∼ e − κr κr (7)where κ = (cid:112) | E | / ¯ hω and r = | (cid:126)r − (cid:126)r | / √ b is the dimensionless Jacobi coordinate, not the Gaussian tail of the HO.At small r the wave function is basically unchanged (apart from normalization). III. THE HOBET EFFECTIVE INTERACTION
Contact-gradient expansions are used in approaches like EFT to correct for the exclusion of short-range (high-momentum) interactions. The most general scalar interaction is constructed, consistent with Hermiticity, parityconservation, and time-reversal invariance, as an expansion in the momentum. Such an interaction for the two-nucleon system, expanded to order N LO (or up to six gradients), is shown in Table I. (Later these operators will beslightly modified for HOBET.)The “data” for testing such an expansion for HOBET are deuteron matrix elements (cid:104) α | P ( H eff − H ) P | β (cid:105) evaluatedas in Refs. [5, 6] for v . I take an 8¯ hω P -space (Λ P = 8). The evolution of the matrix elements will be followed ascontributions from scattering in Q are integrated out progressively, starting with the highest energy contributions. Toaccomplish this, the contribution to H eff coming from excitations in Q up to a scale Λ > Λ P is defined as H eff (Λ),obtained by explicitly summing over all states in Q up to that scale: H eff (Λ) ≡ H + H E − Q Λ H Q Λ H Q Λ ≡ Λ (cid:88) α =Λ P +1 | α (cid:105)(cid:104) α | Q Λ P ≡ . (8)Thus H eff = H eff (Λ → ∞ ) and H eff (Λ P ) = H . The quantity∆(Λ) ≡ H eff − H eff (Λ) = H E − QH QH − H E − Q Λ H Q Λ H (9)represents the contributions to H eff involving excitations in Q above the scale Λ. For Λ >> Λ P , one expects ∆(Λ)to be small and well represented by a LO interaction. As Λ runs to values closer to Λ P , one would expect to find that TABLE I: Contact-gradient expansion for relative-coordinate two-particle matrix elements. Here → D M = ( → ∇ ⊗ → ∇ ) M , → D =[( σ (1) ⊗ σ (2)) ⊗ D ] , → F M = ( → ∇ ⊗ → D ) M , → F M = [( σ (1) ⊗ σ (2)) ⊗ F ] M , → G M = ( → D ⊗ → D ) M , → G M = [( σ (1) ⊗ σ (2)) ⊗ G ] M ,and the scalar product of tensor operators is defined as A J · B J = (cid:80) M = JM = − J ( − M A JM B J − M . Transitions LO NLO NNLO N LO S ↔ S a S LO δ ( r ) a S NLO ( ← ∇ δ ( r ) + δ ( r ) → ∇ ) a S , NNLO ← ∇ δ ( r ) → ∇ a S , N LO ( ← ∇ δ ( r ) → ∇ + ← ∇ δ ( r ) → ∇ )or S ↔ S a S , NNLO ( ← ∇ δ ( r ) + δ ( r ) → ∇ ) a S , N LO ( ← ∇ δ ( r ) + δ ( r ) → ∇ ) S ↔ D a SDNLO ( δ ( r ) → D + ← D δ ( r )) a SD, NNLO ( ← ∇ δ ( r ) → D + ← D δ ( r ) → ∇ ) a SD, N LO ( ← ∇ δ ( r ) → D + ← D δ ( r ) → ∇ ) a SD, NNLO ( δ ( r ) → ∇ → D + ← D ← ∇ δ ( r )) a SD, N LO ( ← ∇ δ ( r ) → ∇ → D + ← D ← ∇ δ ( r ) → ∇ ) a SD, N LO ( δ ( r ) → ∇ → D + ← D ← ∇ δ ( r )) D ↔ D a D NNLO ← D · δ ( r ) → D a D N LO ( ← D ← ∇ · δ ( r ) → D + ← D · δ ( r ) → ∇ → D )or D J ↔ D J D ↔ G a DGN LO ( ← D · δ ( r ) → G + ← G · δ ( r ) → D ) P ↔ P a P NLO ← ∇ · δ ( r ) → ∇ a P NNLO ( ← ∇ ← ∇ · δ ( r ) → ∇ + ← ∇ · δ ( r ) → ∇ → ∇ ) a P , N LO ← ∇ ← ∇ · δ ( r ) → ∇ → ∇ or P J ↔ P J a P , N LO ( ← ∇ ← ∇ · δ ( r ) → ∇ + ← ∇ · δ ( r ) → ∇ → ∇ ) P ↔ F a PFNNLO ( ← ∇ · δ ( r ) → F + ← F · δ ( r ) → ∇ ) a PF, N LO ( ← ∇ ← ∇ · δ ( r ) → F + ← F · δ ( r ) → ∇ → ∇ ) a PF, N LO ( ← ∇ · δ ( r ) → ∇ → F + ← F ← ∇ · δ ( r ) → ∇ ) F ↔ F a F N LO ← F · δ ( r ) → F or F J ↔ F J NLO, NNLO, N LO, .... contributions become successively more important. If one could formulate some expansionthat continues to accurately reproduce the various matrix elements of ∆(Λ) as Λ → Λ P , then a successful expansionfor the HOBET effective interaction ∆(Λ P ) = H eff − H would be in hand.Figure 3a is a plot of ∆(Λ) for the 15 S matrix elements in the chosen P-space. For typical matrix elements∆(Λ P ) = H eff − H ∼ -12 MeV – a great deal of the deuteron binding comes from the Q-space. Five of the matrixelements involve bra or ket edge states. The evolution of these contributions with Λ appears to be less regular thanis observed for nonedge-state matrix elements.One can test whether the results shown in Fig. 3a can be reproduced in a contact-gradient expansion. At each Λthe coefficients a S LO (Λ), a S NLO (Λ), etc., would be determined from the lowest-energy “data,” those matrix elements (cid:104) α | ∆(Λ) | β (cid:105) carrying the fewest HO quanta. Thus, in LO, a S LO (Λ) would be determined from the ( n (cid:48) , n ) = (1 , P -space matrix elements are then predicted, not fit; in NNLO four coefficientswould be determined from the (1,1), (1,2), (1,3), and (2,2) matrix elements, and eleven predicted. Figures 3b-dshow the residuals – the differences between the predicted and calculated matrix elements. For successive LO, NLO,and NNLO calculations, the scale at which residuals in ∆ are significant, say greater than 10 keV, is brought downsuccessively, e.g., from an initial ∼ hω , to ∼ hω (LO), to ∼ hω (NLO), and finally to ∼ hω (NNLO),except for matrix elements involving edge states. There the improvement is not significant, with noticeable deviationsremaining at ∼ hω even at NNLO. This irregularity indicates a flaw in the underlying physics of this approach– specifically the use of a short-range expansion for H eff when important contributions to H eff are coming fromlong-range interactions in Q . So this must be fixed. A. The contact-gradient expansion for HOBET
The gradient with respect to the dimensionless coordinate (cid:126)r ≡ ( (cid:126)r − (cid:126)r ) /b √ −→∇ . The coefficients a LO , a NLO , ... in Table I then carry the dimensions of MeV.The contact-gradient expansion defined in Table I is that commonly used in plane-wave bases, where one expandsaround (cid:126)k = 0 with −→∇ exp i(cid:126)k · (cid:126)r (cid:12)(cid:12)(cid:12) (cid:126)k =0 = 0 . (10) S m.e. contributions from Qa) -14-12-10-8-6-4-20 ( M e V ) S m.e.: LO b) -20246810 M . E . R e s i du a l s ( M e V ) S m.e.: NLO c)
20 40 60 80 100 120 140-2-101234 M . E . R e s i du a l s ( M e V ) S m.e.: NNLOd) -0.50.00.51.0 M . E . R e s i du a l s ( M e V ) S m.e.: N LO e) -0.50.00.51.0 M . E . R e s i du a l s ( M e V ) S RMS m.e. resultsf)
20 40 60 80 100 120 14002468101214 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOLOBare
FIG. 3: In a) the contributions to H eff − H from excitations in Q above Λ are plotted for a calculation with Λ P = 8 and b =1.7 fm. Each line describes the running of one of the 15 independent P -space matrix elements (cid:104) n (cid:48) l (cid:48) = 0 | H eff − H | nl = 0 (cid:105) ,1 ≤ n ≤ n (cid:48) ≤
5. Ten of the matrix elements are between nonedge states (solid), four connect the n (cid:48) = 5 edge state to the n =1,2,3,4 nonedge states (dashed), and one is the diagonal n (cid:48) = n = 5 edge-edge case (dot dashed). b)-e) show the residualsfor naive LO, NLO, NNLO, and N LO fits (see text). f) shows the RMS deviation for the set of P -space matrix elements. Theexpected systematic improvement with increasing order is apparent only for matrix elements between nonedge states. HOBET begins with a lowest-energy 1 s Gaussian wave packet with a characteristic momentum ∼ /b . An analogousdefinition of gradients such that −→∇ ψ s ( b ) = 0 (11)is obtained by redefining each operator appearing in Table I by O → ¯ O ≡ e r / Oe r / . (12)The gradients appearing in the operators of Table I then act on polynomials in r . This leads to two attractiveproperties. First is the removal of operator mixing. Once a S LO is fixed in LO to the ( n (cid:48) , n ) = (1 ,
1) matrix element,this quantity remains fixed in NLO, NNLO, etc. Higher-order terms make no contributions to this matrix element.Similarly, a NLO , once fixed to the (1 ,
2) matrix element, is unchanged in NNLO. That is, the NLO results contain theLO results, and so on. Second, this definition gives the HOBET effective interaction a simple dependence on nodalquantum numbers, −→∇ ∼ − n − −→∇ ∼ n − n − . (13)(The Appendix describes this expansion in some detail.) In each channel, this dependence agrees with the plane-wave result in lowest contributing order, but otherwise differs in terms of relative order 1 /n . This HO form of thecontact-gradient expansion is connected with standard Talmi integrals [8], generalized for nonlocal potentials, e.g., a LO ∼ (cid:90) ∞ (cid:90) ∞ e − r [ V ( r , r )] e − r r r dr dr a NLO ∼ (cid:90) ∞ (cid:90) ∞ e − r (cid:2) r V ( r , r ) (cid:3) e − r r r dr dr = (cid:90) ∞ (cid:90) ∞ e − r (cid:2) V ( r , r ) r (cid:3) e − r r r dr dr (14)and so on. B. Identifying terms with the contact-gradient expansion
The next question is the association of the operators in Table I with an appropriate set of terms in H eff − H , sothat the difficulties apparent in Fig. 3 are avoided. The reorganized BH equation of Eq. (2) H eff = EE − T Q (cid:20) T − T QE T + V + V E − QH QV (cid:21) EE − QT → EE − T Q (cid:20) T − T QE T + V + (cid:88) i = LO,NLO,... ¯ O i (cid:21) EE − QT (15)isolates V ( E − QH ) − QV , a term that is sandwiched between short-range operators that scatter to high-energystates: one anticipates this term can be successfully represented by a short-range expansion like the contact-gradientexpansion. This identification is made here and tested later in this paper.This reorganization only affects the edge-state matrix elements, clearly. As the process of fitting coefficients usesmatrix elements of low ( n (cid:48) , n ), none of which involves edge states, the coefficients are unchanged. But every matrixelement involving edge states now includes the effects of rescattering by QT to all orders. Thus a procedure forevaluating these matrix elements is needed. C. Matrix element evaluation
There are several alternatives for evaluating Eq. (15) for edge states. One of these exploits the tri-diagonal form of QT for the deuteron. If | nl (cid:105) is an edge state in P then EE − QT | n l (cid:105) = | n l (cid:105) + 1 E − QT QT | n l (cid:105) = | n l (cid:105) + (cid:112) n ( n + l + 1 /
2) 1 − κ − QT ¯ hω | n + 1 l (cid:105) (16)where E < κ = (cid:113) | E | ¯ hω depends on the ratio of the binding energy | E | to the HO energy scale. Note that the second vector on the right in Eq. (16) lies entirely in Q . Now2¯ hω QT | n + 1 l (cid:105) = (2 n + l + 3 / | n + 1 l (cid:105) + (cid:112) ( n + 1)( n + l + 3 / | n + 2 l (cid:105) hω QT | n + 2 l (cid:105) = (cid:112) ( n + 1)( n + l + 3 / | n + 1 l (cid:105) + (2 n + l + 7 / | n + 2 l (cid:105) + (cid:112) ( n + 2)( n + l + 5 / | n + 3 l (cid:105) hω QT | n + 3 l (cid:105) = (cid:112) ( n + 2)( n + l + 5 / | n + 2 l (cid:105) + (2 n + l + 11 / | n + 3 l (cid:105) + (cid:112) ( n + 3)( n + l + 7 / | n + 4 l (cid:105) hω QT | n + 4 l (cid:105) = ... (17)So the operator 2 QT / ¯ hω in the basis {| n + i l (cid:105) , i = 1 , , ... } has the form2¯ hω QT = α β β α β β α β · · · β α ... (18)where α i = α i ( n, l ) = 2 n + 2 i + l − / , β i = β i ( n, l ) = (cid:112) ( n + i )(( n + i + l + 1 / . (19)As is well known, if this representation of the operator 2 QT / ¯ hω is truncated after k steps, the 2 k − { α i , β i } determine the 2 k − | n + 1 l (cid:105) , (cid:104) n + 1 l | (cid:18) QT ¯ hω (cid:19) i | n + 1 l (cid:105) , i = 1 , ...., k − | n + 1 l (cid:105) of such a tri-diagonal matrix, allowing us to write (cid:112) n ( n + l + 1 /
2) 1 − κ − QT ¯ hω | n + 1 l (cid:105) = (cid:101) g ( − κ ; n, l ) | n + 1 l (cid:105) + (cid:101) g ( − κ ; n, l ) | n + 2 l (cid:105) + (cid:101) g ( − κ ; n, l ) | n + 3 l (cid:105) + · · · (21)The coefficients { (cid:101) g i } can be obtained from an auxiliary set of continued fractions { g (cid:48) i } that are determined by downwardrecursion g (cid:48) k ( − κ ; n, l ) ≡ − κ − α k ( n, l ) g (cid:48) i − ( − κ ; n, l ) = 1 − κ − α i − ( n, l ) − β i − ( n, l ) g (cid:48) i ( − κ ; n, l ) , i = k, .... (cid:101) g ( − κ ; n, l ) = (cid:112) n ( n + l + 1 / g (cid:48) ( − κ ; n, l ) (cid:101) g i ( − κ ; n, l ) = (cid:101) g i − ( − κ ; n, l ) β i − g (cid:48) i ( − κ ; n, l ) , i = 2 , ..., k (23)Defining (cid:101) g ( − κ ; n, l ) ≡ EE − QT | n l > = k →∞ (cid:88) i =0 (cid:101) g i ( − κ ; n, l ) | n + i l (cid:105) , edge state= | n l >, otherwise (24)where it is understood that k is made large enough so that the moments expansion for the Green’s function is accuratethroughout the region in coordinate space where E/ ( E − QT ) | n l (cid:105) is needed. Note that the first line of Eq. (24) canbe viewed as the general result if one defines g i ( − κ ; n, l ) ≡ , i = 1 , ...k, if | n l (cid:105) is not an edge state . (25)0(For A ≥ A -1)-dimensional HO, with the role of the spherical harmonics replaced bythe corresponding hyperspherical harmonics.) Eq. (24) can now be used to evaluate the various terms in Eq. (15). Matrix elements for the contact-gradient operators:
The matrix elements have the general form (cid:104) n (cid:48) l (cid:48) | EE − T Q ¯ O EE − QT | nl (cid:105) = (cid:88) i,j =0 (cid:101) g j ( − κ ; n (cid:48) , l (cid:48) ) (cid:101) g i ( − κ ; n, l ) (cid:104) n (cid:48) + j l | ¯ O | n + i l (cid:105) (26)where ¯ O is formed from gradients acting on the bra and ket, evaluated at (cid:126)r =0. The general matrix element (anypartial wave) is worked out in the Appendix. For example, one needs for S -wave channels the relation( (cid:126) ∇ ) p e r / R nl =0 ( r ) Y (Ω r ) (cid:12)(cid:12) (cid:126)r → = ( − p ( n − n − − p )! 1 π (cid:20) Γ( n + 1 / n − (cid:21) / (27)from which it follows (cid:104) n (cid:48) ( l (cid:48) = 0 S = 1) J = 1 | EE − T Q (cid:20) (cid:88) i = LO,...,N LO ¯ O S ,i (cid:21) EE − QT | n ( l = 0 S = 1) J = 1 (cid:105) =2 π (cid:88) i,j =0 (cid:101) g j ( − κ ; n (cid:48) , l (cid:48) = 0) (cid:101) g i ( − κ ; n, l = 0) (cid:20) Γ n (cid:48) + j + 1 / n + i + 1 / n (cid:48) + j − n + i − (cid:21) (cid:20) a S LO − n (cid:48) + j −
1) + ( n + i − a S NLO +16 (cid:110) ( n (cid:48) + j − n + i − a S , NNLO + (( n (cid:48) + j − n (cid:48) + j −
2) + ( n + i − n + i − a S , NNLO (cid:111) − (cid:110) ( n (cid:48) + j − n + i − (cid:0) ( n (cid:48) + j −
2) + ( n + i − (cid:1) a S , N LO + (( n (cid:48) + j − n (cid:48) + j − n (cid:48) + j −
3) + ( n + i − n + i − n + i − a S , N LO (cid:111)(cid:21) . (28)In the case of nonedge states, (cid:101) g i ≡ (cid:101) g ≡
1. Thus it is apparent that the net consequence ofthe rearrangement of the BH equation and the identification of the contact-gradient expansion with V ( E − QH ) − QV ,is effectively a renormalization of the coefficients of that expansion for the edge HO states. That renormalization isgoverned by κ = 2 | E | / ¯ hω , e.g., a LO ( n (cid:48) , l (cid:48) , n, l ) → a (cid:48) LO ( E ; n (cid:48) , l (cid:48) , n, l ) = a LO ( n (cid:48) , l (cid:48) , n, l ) (cid:88) i,j =0 (cid:101) g j ( − κ ; n (cid:48) , l (cid:48) ) (cid:101) g i ( − κ ; n, l ) × (cid:20) Γ( n (cid:48) + j + 1 / n + i + 1 / n (cid:48) + 1 / n + 1 / (cid:21) / (cid:20) ( n (cid:48) − n − n (cid:48) + j − n + i − (cid:21) / . (29)This renormalization is large, typically a reduction in strength by a factor of 2-4, for | E | =2.224 MeV, and alsoremains substantial for more deeply bound systems, as will be illustrated later. (The binding energy for this purposeis defined relative to the lowest particle breakup channel, the first extended state.) The effects encoded into | (cid:101) α (cid:105) bysumming QT to all orders are nontrivial: they depend on a nonperturbative strong interaction parameter | E | as wellas QT , and they alter effective matrix elements of the strong potential. For a given choice of Λ P , the renormalizationdepends on a single parameter, 2 | E | / ¯ hω , not on | E | or b separately. In the plane-wave limit b → ∞ , this parameteris driven to ∞ , so that a (cid:48) LO → a LO . No renormalization is required in this limit. The dependence on | E | is discussedin more detail later, including its connection to the state-dependence inherent in effective theory. Matrix elements of the relative kinetic energy:
The relative kinetic energy operator couples P and Q via strong matrixelements that grow as n . As Ref. [6] discusses, this coupling causes difficulties with perturbative expansions in H even in the case of P spaces that contain almost all of the wave function (e.g., Λ P ∼ r that is nonperturbative, involving matrix elements of T that exceed Λ P ¯ hω/ (cid:104) α | T + T E − QT QT | β (cid:105) = (cid:104) (cid:101) α | T − T QE T | (cid:101) β (cid:105) = (cid:104) α | T | (cid:101) β (cid:105) = (cid:104) (cid:101) α | T | β (cid:105) (30)where the last two terms show that the transformation to states | (cid:101) α (cid:105) = E/ ( E − QT ) | α (cid:105) reduces the calculation of therescattering to that of a bare matrix element. It follows from this expression (cid:104) n (cid:48) l | T + T E − QT QT | n l (cid:105) = (cid:104) n (cid:48) l | T | n l (cid:105) + ¯ hω δ n (cid:48) n (cid:112) n ( n + l + 1 / (cid:101) g ( − κ ; n, l ) . (31)1Thus, rescattering via QT alters the diagonal matrix element of the effective interaction for edge states, as determinedby (cid:101) g ( − κ ; n, l ). Matrix elements of the bare potential:
The P -space matrix element of V becomes (cid:104) (cid:101) α | V | (cid:101) β (cid:105) which, as is illustrated inFig. 2, involves an integral over a wave function that, apart from normalization, differs from the HO only in the tail,where the potential is weak. It can be evaluated by generating the wave functions | (cid:101) α (cid:105) and | (cid:101) β (cid:105) as HO expansions, (cid:88) j =0 (cid:101) g j ( − κ ; n (cid:48) , l (cid:48) ) (cid:104) n (cid:48) + j l (cid:48) | V (cid:34)(cid:88) i =0 | n + i l (cid:105) (cid:101) g i ( − κ ; n, l ) (cid:35) = (cid:88) i,j =0 (cid:101) g j ( − κ ; n (cid:48) , l (cid:48) ) (cid:101) g i ( − κ ; n, l ) (cid:104) n (cid:48) + j l (cid:48) | V | n + i l (cid:105) . (32)though the alternative Green’s function expression, discussed below, is simpler. Use of the free Green’s function:
An alternative to an expansion in an HO basis is generation of | (cid:101) α (cid:105) with the free(modified Helmholtz) Green’s function. For any P-space state | n l (cid:105) ,( E − QT ) | (cid:101) α (cid:105) = E | α (cid:105) ⇒ ( E − T ) | (cid:101) α (cid:105) = E | α (cid:105) − P T | (cid:101) α (cid:105) (33)That is, both E − QT and E − T project | (cid:101) α (cid:105) back into the P -space. The free Green’s function equation can be written( E − T ) | (cid:101) α (cid:105) = P (cid:20) E − T EE − QT (cid:21) P | α (cid:105) = (cid:20) P E − T P (cid:21) − | α (cid:105) . (34)Either of the driving terms on the right-hand side is easy to manipulate. The second expression requires inversion ofa P-space matrix, one most easily calculated in momentum space, as the HO is its own Fourier transform and as theresulting momentum-space integrals can be done in closed form. This form was used in the three-body calculationsof Ref. [10].Here I will use the first expression above, rewriting the right-hand-side driving term in terms of | α nlm l (cid:105) , P (cid:20) E − T EE − QT (cid:21) P | α (cid:105) = ¯ hω (cid:20)(cid:18) − κ − (2 n + l − / − (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 / (cid:19) | n l m l > − (cid:112) ( n − n + l − / | n − l m l (cid:105) − (cid:112) n ( n + l + 1 / P | n + 1 l m l (cid:105) (cid:21) ≡ ¯ hω | α nlm l (cid:105) , (35)where the driving term has been kept general, valid for either edge or nonedge states: the latter can be a helpfulnumerical check, verifying that a HO wave function is obtained, for such cases, from the expression below. For anedge state, (cid:101) g is nonzero and P | n + 1 l (cid:105) ≡
0; for a nonedge state, (cid:101) g = 0 and P =1. Labeling the corresponding edgestate as | (cid:101) α nlm l (cid:105) , (cid:104) (cid:126)r | (cid:101) α nlm l (cid:105) = (cid:90) d (cid:126)r (cid:48) π | (cid:126)r − (cid:126)r (cid:48) | e − κ | (cid:126)r − (cid:126)r (cid:48) | (cid:104) (cid:126)r (cid:48) | α nlm l > = − Y lm (Ω r ) (cid:20) √ r I l +1 / ( κr ) (cid:90) ∞ r d (cid:126)r (cid:48) ( r (cid:48) ) / K l +1 / ( κr (cid:48) ) (cid:104) (cid:126)r (cid:48) | α nlm l > + 1 √ r K l +1 / ( κr ) (cid:90) r d (cid:126)r (cid:48) ( r (cid:48) ) / I l +1 / ( κr (cid:48) ) (cid:104) (cid:126)r (cid:48) | α nlm l > (36)where I and K denote the standard modified Bessel functions. By expressing the HO radial wave functions in termsof the underlying Laguerre polynomials and integrating the polynomials term by term, alternative expressions areobtained for the various quantities previously expressed as expansions in the (cid:101) g i . This is detailed in the Appendix.One finds, for example, (cid:104) (cid:126)r = 0 | (cid:101) α nlm l (cid:105) = δ l, δ m l , (cid:114) ( n − n + 1 / π n (cid:88) k =0 ( − k k !( n − k )!Γ( k + 3 / × (cid:20) ( n − k )( κ + 3 n − / − k + (cid:101) g ( − κ ; n, (cid:112) n ( n + 1 / P [ n + 1 , l = 0] n ( n + 1 / (cid:21) × (cid:20) −√ κ Γ( k + 3 / F [ k + 3 /
2; 3 / κ /
2] + k ! F [ k + 1; 1 / κ / (cid:21) (37)2where F is Kummer’s confluent hypergeometric function, and where P [ n + 1 , l = 0] = 1 if | n + 1 l (cid:105) is in P , and0 otherwise. Similar expressions can be derived to handle all of the operators ¯ O appearing in the contact-gradientexpansion (see the Appendix). D. Numerical Tests
In this subsection channel-by-channel N LO results are presented for H eff based on Eqs. (12) and (15), whichisolate a short-range operator that plausibly can be accurately and systematically expanded via contact-gradientoperators.For Λ P = 8, the fitting procedure determines all N LO coefficients from nonedge matrix elements, leaving all edgematrix elements and a substantial set of nonedge matrix elements unconstrained. Thus one can use these matrixelements to test whether the expansion systematically accounts for the “data,” the set of numerically generated v matrix elements of H eff . One test is the running of the results as a function of Λ: a systematic progression throughLO, NLO, etc., operators should be observed as Λ is lowered to the SM scale. A second test is the “Lepage plot”[11], which displays residual errors in matrix elements: if the improvement is systematic, these residual errors shouldreflect the nodal-quantum-number dependence of the operators that would correct these results, in next order.Eq. (15) includes “bare” terms – the matrix elements (cid:104) α | T | (cid:101) β (cid:105) and (cid:104) (cid:101) α | V | (cid:101) β (cid:105) – and a term involving repeated scatteringby H in Q , but sandwiched between the short-range operator QV . To test the dependence on Λ, the rescatteringterm is decomposed in the manner of Eq. (9),∆ QT (Λ) = EE − T Q (cid:20) V E − QH QV − V E − Q Λ H Q Λ V (cid:21) EE − QT , to isolate the contribution of scattering above the scale Λ . ∆ QT (Λ) is evaluated numerically for v at each requiredΛ . The long-wavelength summation is always done to all orders – the running with Λ thus reflects the behavior of theshort-range piece, V ( E − QH ) − QV . The full P -space effective interaction is obtained as Λ → Λ P .As outlined before, coefficients are fitting to the longest wavelength information. For example, in S channels, a LO is fixed to the ( n (cid:48) , n ) = (1,1) matrix element; the absence of operator mixing then guarantees this coefficient remainsfixed, as higher order terms are evaluated. The single a NLO coefficient is fixed to (2,1) (or equivalently (1,2)) ; a NNLO and a NNLO are determined from (2,2) and (3,1); and finally a N LO and a N LO are fixed to (3,2) and (4,1). So atN LO there are a total of 6 parameters. This procedure is repeated for a series of Λ ranging from 140 to Λ P =8.The results in each order, and the improvement order by order, are thus obtained as a function of Λ. P contains 15independent matrix elements in the S − S channel, nine of which play no role in the fitting: these test whetherthe improvement is systematic.Figures 4 and 5 show the results for S − S and S − S . Panel a) shows the evolution of the matrix elements (cid:104) α | ∆ QT (Λ) | β (cid:105) for each of the 15 independent matrix elements. Matrix elements involving only nonedge states, a singleedge state, or two edge states are denoted by solid, dashed, and dash-dotted lines, respectively. Progressively morebinding is recovered as Λ → Λ P . In the S − S case, the contribution at Λ P is ∼ ∼ ∼ (cid:104) n = 5 l = 0 | ∆ QT (Λ P ) | n = 5 l = 0 (cid:105) double-edge matrix element.Panels b)-e) show the residuals – the difference between the matrix elements of ∆ QT (Λ P ) and those of the contact-gradient potential of Eq. (15) – from LO through N LO. The trajectories correspond to the unconstrained matrixelements (14 in LO, 9 in N LO): the fitted matrix elements produce the horizontal line at 0. Unlike the naiveapproach in Fig. 3, the improvement is now systematic in all matrix elements. In the S − S channel, a LOtreatment effectively removes all contributions in Q above Λ ∼
60; NLO lowers this scale to ∼
40, and NNLO is ∼
20. The magnitude of N LO residuals at Λ P is typically ∼ < S − S channel, where the convergence (in terms of the size of the residuals) issomewhat faster. The N LO RMS deviation among the unconstrained matrix elements at Λ P is ∼ LO are given in Figs. 6 through 11: S − D (leading order contribution NLO); D − D , D − D , D − D , and D − D (NNLO); and D − G (N LO).Table II gives the resulting fitted couplings at N LO for all contributing channels, along with numerical results forthe root-mean-square Q -space contributions to ∆ QT (Λ P ) and the root-mean-square residuals (the deviation betweenthe contact-gradient prediction and ∆ QT (Λ P ) for the remaining unconstrained effective interactions matrix elements).The quality of the agreement found in the S and S channels is generally typical – residuals at the few kilovoltlevel – though there are some exceptions and some general patterns that emerge. One of these is the tendency of the3 S m.e. contributions from Qa) -14-12-10-8-6-4-20 Q T ( M e V ) S m.e.: LO b) -20246810 M . E . R e s i du a l s ( M e V ) S m.e.: NLO c)
20 40 60 80 100 120 140-2-101234 M . E . R e s i du a l s ( M e V ) S m.e.: NNLOd) -0.20.00.20.4 M . E . R e s i du a l s ( M e V ) S m.e.: N LO e) -0.0020.00.0020.0040.0060.0080.01 M . E . R e s i du a l s ( M e V ) S RMS m.e. resultsf)
20 40 60 80 100 120 1400246810 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOLOBare
20 40 600.00.010.02 R e s i d . ( M e V ) FIG. 4: As in Fig. 3, but for the QT -summed reordering of H eff . The contributions to the effective interaction from excitationsin Q above Λ, denoted ∆ QT (Λ) in the text, are plotted. Each line gives the running of a P -space matrix element. b)-e) show theresiduals for LO, NLO, NNLO, and N LO fits (see text). f) shows the RMS deviation for the set of P -space matrix elements.The improvement with increasing order is systematic and rapid: at N LO the RMS deviation for unconstrained matrix elementsas Λ → Λ P is about 3 keV. That is, the entire effective interaction is reproduced to a few parts in 10 . S m.e. contributions from Qa) -12-10-8-6-4-20 Q T ( M e V ) S m.e.: LO b) -1.0-0.50.00.51.01.52.0 M . E . R e s i du a l s ( M e V ) S m.e.: NLO c)
20 40 60 80 100 120 140-0.5-0.4-0.3-0.2-0.10.00.10.2 M . E . R e s i du a l s ( M e V ) S m.e.: NNLOd) -0.010.00.010.02 M . E . R e s i du a l s ( M e V ) S m.e.: N LO e) -0.0020.00.0020.0040.0060.0080.01 M . E . R e s i du a l s ( M e V ) S RMS m.e. resultsf)
20 40 60 80 100 120 14002468 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOLOBare
20 40 600.00.0020.0040.0060.008 R e s i d . ( M e V ) FIG. 5: As in Fig. 4, but for the S channel. The N LO results are seen to reproduce the entire effective interaction to theaccuracy of about a keV, or one part in 10 . S - D m.e. contributions from Qa) -0.2-0.10.00.10.20.30.4 Q T ( M e V ) S - D m.e.: NLOb) -0.4-0.3-0.2-0.10.00.1 M . E . R e s i du a l s ( M e V ) FIG. 6: As in Figs. 4 and 5, but for the S − D channel.As with the cases described before, the N LO results remainaccurate at the few keV level, as the integration is brought downto the shell-model scale, Λ → Λ P . S - D m.e.: NNLOc) -0.050.00.050.10.15 M . E . R e s i du a l s ( M e V ) S - D m.e.: N LOd) -0.0020.00.0020.0040.0060.0080.01 M . E . R e s i du a l s ( M e V ) S - D RMS m.e. resultse)
20 40 60 80 100 120 1400.00.10.2 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOBare
20 40 600.00.0050.010.015 R e s i d . ( M e V ) D m.e. contributions from Qa) -0.1-0.08-0.06-0.04-0.020.0 Q T ( M e V ) D m.e.: NNLOb)
20 40 60 80 100 120 140-0.010.00.010.020.03 M . E . R e s i du a l ( M e V ) D m.e.: N LOc) -0.004-0.0020.00.002 M . E . R e s i du a l ( M e V ) D RMS m.e. resultsd)
20 40 60 80 100 120 1400.00.020.040.06 R M S M . E . R e s i du a l s ( M e V ) N LONNLOBare
20 40 600.00.0050.01 R e s i d . ( M e V ) FIG. 7: As in Fig. 4, but for the D channel. triplet channels with spin and angular momentum aligned ( S − S , P − P , D − D , and F − F
4) to exhibitlarger residuals than the remaining
S, P, D and F channels, respectively. The D − D , which has contributions atNNLO and N LO, stands out as the most difficult channel, with a residual of 122 keV, one to two orders of magnitudegreater than the typical scale of N LO residuals.Figures 12 through 18 show the convergence for the various channels involving odd-parity states and contributingthrough N LO: P − P , P J − P J , P − F , F − F , and F J − F J . While the spin-aligned channels showslightly large residuals, overall the RMS errors at N LO are at the one-to-few keV level. Thus a simple and essentiallyexact representation for the effective interaction exists.
Expansion parameters, naturalness:
The approach followed here differs from EFT, where the formalism is based onan explicit expansion parameter, the ratio of the momentum to a momentum cutoff. The input into the presentcalculation is a set of numerical matrix elements of an iterated, nonrelativistic potential operating in Q . Potentialslike v are also effectively regulated at small r by some assumed form, e.g., a Gaussian, matched smoothly to theregion in r that is constrained by scattering data. Thus there are no singular potentials iterating in Q .Intuitively it is clear that the convergence apparent in Table II is connected with the range of hard-core interactions(once edge states are transformed by summing T ). A handwaving argument can be made by assuming rescattering7 D m.e. contributions from Qa) -0.2-0.15-0.1-0.050.0 Q T ( M e V ) D m.e.: NNLOb)
20 40 60 80 100 120 140-0.010.00.010.020.030.040.05 M . E . R e s i du a l ( M e V ) D m.e.: N LOc) -0.002-0.0010.00.0010.002 M . E . R e s i du a l ( M e V ) D RMS m.e. resultsd)
20 40 60 80 100 120 1400.00.050.10.150.2 R M S M . E . R e s i du a l s ( M e V ) N LONNLOBare
20 40 600.00.0020.0040.006 R e s i d . ( M e V ) FIG. 8: As in Fig. 4, but for the D channel. in Q effectively generates a potential of the form V e − r /a , where r = | (cid:126)r − (cid:126)r | . This ansatz is local, so there is some arbitrariness in mapping it onto contact-gradient expansioncoefficients, which correspond to the most general nonlocal potential. But a sensible prescription is to equate termswith equivalent powers of r , in the bra and ket, when taking HO matrix of this potential. Then one finds, for S-wavechannels a ( m (cid:48) , m ) ≡ a S, m (cid:48) mN m (cid:48) + m LO = 14 m (cid:48) + m m (cid:48) ! m ! (2 m (cid:48) + 2 m + 1)!!(2 m (cid:48) + 1)!!(2 m + 1)!! V (cid:20) πa a + 2 b (cid:21) / (cid:20) a a + 2 b (cid:21) m (cid:48) + m (38)where b is the oscillator parameter. (The notation is such that, e.g., a ( m (cid:48) = 3 , m = 0) = a S, N LO .) The last term isthus the expansion parameter: if the range of the hard-core physics residing in Q is small compared to the naturalnuclear size scale b , then each additional order in the expansion should be suppressed by ∼ ( a/b ) .One can use this crude ansatz to assess whether the convergence shown in Table II is natural, or within expectations.The LO and NLO S − S results effectively determine V and a ; thus the strengths of four NNLO and N LO8 D m.e. contributions from Qa) -0.5-0.4-0.3-0.2-0.10.0 Q T ( M e V ) D m.e.: NNLOb)
20 40 60 80 100 120 140-0.10.00.10.20.30.40.5 M . E . R e s i du a l ( M e V ) FIG. 9: As in Fig. 7 and 8, but for the D channel. This“stretched” configuration generates much larger residuals thanthe other l = 2 channels. Consequently a calculation to N LOwould be needed to reduce typical matrix element errors to ∼
10 keV, in the limit Λ → Λ P . D m.e.: N LOc) -0.2-0.15-0.1-0.050.00.05 M . E . R e s i du a l ( M e V ) D m.e.: N LOc) M . E . R e s i du a l ( M e V ) D RMS m.e. resultsd)
20 40 60 80 100 120 1400.00.10.20.3 R M S M . E . R e s i du a l s ( M e V ) N LON LONNLOBare
20 40 600.00.010.02 R e s i d . ( M e V ) D m.e. contributions from Qa) -0.05-0.04-0.03-0.02-0.010.0 Q T ( M e V ) D m.e.: NNLOb)
20 40 60 80 100 120 140-0.0050.00.0050.010.0150.02 M . E . R e s i du a l ( M e V ) D m.e.: N LOc) -0.002-0.0010.00.001 M . E . R e s i du a l ( M e V ) D RMS m.e. resultsd)
20 40 60 80 100 120 1400.00.010.020.03 R M S M . E . R e s i du a l s ( M e V ) N LONNLOBare
20 40 600.00.0020.0040.006 R e s i d . ( M e V ) FIG. 10: As in Fig. 4, but for the D channel. potentials can be predicted relative to that of a LO and a NLO . The predicted hierarchy a LO : a NLO : a NNLO : a NNLO : a N LO : a N LO of1 : 6 . × − : 6 . × − : 2 . × − : 3 . × − : 4 . × − matches the relative strengths of the couplings in the table quite well,1 : 6 . × − : 6 . × − : 3 . × − : 2 . × − : 5 . × − , including qualitatively reproducing the ratios of the two NNLO and two N LO coefficients. The parameters derivedfrom a LO and a NLO are a ∼ .
39 fm and V ∼ − . S channel the bare Argonne v potential at small r can be approximated by a Gaussian with a ∼ .
33 fm and V ∼ . V s are correct – the P -space lacks theappropriate short-range repulsion and thus samples the iterated bare potential at small r , a contribution that thenmust be subtracted off when H eff is evaluated.]A similar exercise in the S channel yields the predicted hierarchy1 : 2 . × − : 8 . × − : 2 . × − : 13 . × − : 1 . × − D - G m.e. contributions from Qa) -0.0050.00.0050.010.0150.02 Q T ( M e V ) D - G m.e.: N LOb)
20 40 60 80 100 120 140-0.02-0.015-0.01-0.0050.00.005 M . E . R e s i du a l ( M e V ) D - G m.e.: N LOc) -0.0020.00.0020.0040.006 M . E . R e s i du a l ( M e V ) D - G RMS m.e. resultsd)
20 40 60 80 100 120 1400.00.0050.010.015 R M S M . E . R e s i du a l s ( M e V ) N LON LOBare
20 40 600.00.0010.002 R e s i d . ( M e V ) FIG. 11: As in Fig. 4, but for the D − G channel. The N LO contribution is also shown. which compares with the coupling ratios calculated from Table II1 : 2 . × − : 8 . × − : 1 . × − : 9 . × − : 2 . × − . The convergence is very regular but slower: in this case the effective Gaussian parameter needed to describe thesetrend is a ∼ .
75 fm. The overall strength, V ∼ − .
42, differs substantially from that found for the S channel,though the underlying v potentials for S − S and S − S scattering are quite similar (see Fig. 19).The S − S behavior is similar to that found in the other spin-aligned channels, such as D − D and P − P ,where the scattering in Q includes contributions from the tensor force. The tensor force contributes to the LO s-wavecoupling through intermediate D -states in Q , e.g., (cid:104) n (cid:48) l (cid:48) = 0 | V SD Q | n (cid:48)(cid:48) l = 2 (cid:105) (cid:104) E (cid:105) (cid:104) n (cid:48)(cid:48) l = 2 | QV SD | nl = 0 (cid:105) , as the product of two tensor operators has an s-wave piece. The radial dependence of V SD for v , shown in Fig. 19,is significantly more extended than in central-force S − S and S − S cases. This has the consequences that(1) the mean excitation energy (cid:104) E (cid:105) for S − D will be lower (enhancing the importance of the tensor force) and(2) the P -space (cid:104) S | H eff | S (cid:105) matrix element will reflect the extended range.1 P m.e. contributions from Qa) -1.2-1.0-0.8-0.6-0.4-0.20.00.2 Q T ( M e V ) P m.e.: NLO b)
20 40 60 80 100 120 140-0.050.00.050.10.150.2 M . E . R e s i du a l ( M e V ) FIG. 12: As in Fig. 4, but for the P channel. P m.e.: NNLOc) -0.009-0.006-0.0030.00.003 M . E . R e s i du a l ( M e V ) P m.e.: N LO d) -0.0010.00.001 M . E . R e s i du a l ( M e V ) P RMS m.e. resultse)
20 40 60 80 100 120 1400.00.20.40.6 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOBare
20 40 600.00.00020.0004 R e s i d . ( M e V ) P m.e. contributions from Qa) -2.5-2.0-1.5-1.0-0.50.00.5 Q T ( M e V ) P m.e.: NLO b)
20 40 60 80 100 120 140-0.10.00.10.20.30.40.5 M . E . R e s i du a l ( M e V ) FIG. 13: As in Fig. 4, but for the P channel. P m.e.: NNLOc) -0.05-0.04-0.03-0.02-0.010.00.01 M . E . R e s i du a l ( M e V ) P m.e.: N LO d) -0.0010.00.0010.0020.0030.004 M . E . R e s i du a l ( M e V ) P RMS m.e. resultse)
20 40 60 80 100 120 1400.00.20.40.60.81.01.21.4 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOBare
20 40 600.00.0010.002 R e s i d . ( M e V ) P m.e. contributions from Qa) -3.0-2.5-2.0-1.5-1.0-0.50.00.5 Q T ( M e V ) P m.e.: NLO b)
20 40 60 80 100 120 140-0.10.00.10.20.30.4 M . E . R e s i du a l ( M e V ) FIG. 14: As in Fig. 4, but for the P channel. P m.e.: NNLOc) -0.02-0.015-0.01-0.0050.00.005 M . E . R e s i du a l ( M e V ) P m.e.: N LO d) -0.0010.00.001 M . E . R e s i du a l ( M e V ) P RMS m.e. resultse)
20 40 60 80 100 120 1400.00.40.81.21.6 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOBare
20 40 600.00.0002 R e s i d . ( M e V ) P m.e. contributions from Qa) -0.5-0.4-0.3-0.2-0.10.00.1 Q T ( M e V ) P m.e.: NLO b)
20 40 60 80 100 120 140-0.10.00.10.20.3 M . E . R e s i du a l ( M e V ) FIG. 15: As in Fig. 4, but for the P channel. P m.e.: NNLOc) -0.06-0.04-0.020.00.02 M . E . R e s i du a l ( M e V ) P m.e.: N LO d) -0.0020.00.0020.0040.0060.008 M . E . R e s i du a l ( M e V ) P RMS m.e. resultse)
20 40 60 80 100 120 1400.00.10.20.3 R M S M . E . R e s i du a l s ( M e V ) N LONNLONLOBare
20 40 600.00.0020.0040.006 R e s i d . ( M e V ) TABLE II: The effective interaction for LO through N LO, with Λ P = 8 and b =1.7 fm. † Channel Couplings (MeV) (cid:104)
M.E. (cid:105)
RMS (MeV) (cid:104)
Resid. (cid:105)
RMS (keV) a SLO a SNLO a S, NNLO a S, NNLO a S, N LO a S, N LO S − S -32.851 -2.081E-1 -2.111E-3 -1.276E-3 -7.045E-6 -1.8891E-6 7.94 0.53 S − S -62.517 -1.399 -5.509E-2 -1.160E-2 -5.789E-4 -1.444E-4 11.97 2.71 a SDNLO a SD, NNLO a SD, NNLO a SD, N LO a SD, N LO a SD, N LO S − D a DNNLO a DN LO D − D -6.062E-3 -1.189E-4 0.027 1.21 D − D -1.034E-2 -1.532E-4 0.051 2.27 D − D -3.048E-2 -5.238E-4 0.141 1.20 D − D -9.632E-2 -4.355E-3 0.303 122 ‡ a SDN LO D − G ‡ a PNLO a PNNLO a P, N LO a P, N LO P − P -8.594E-1 -7.112E-3 -6.822E-5 1.004E-5 0.694 0.11 P − P -1.641 -1.833E-2 -2.920E-4 -1.952E-4 1.283 2.26 P − P -1.892 -1.588E-2 -1.561E-4 -6.737E-6 1.526 0.08 P − P -4.513E-1 -1.257E-2 -5.803E-4 -1.421E-4 0.285 5.61 a PFNNLO a PF, N LO a PF, N LO P − F -4.983E-3 1.729E-5 -5.166E-5 0.034 1.43 a FN LO F − F -3.135E-4 0.007 1.03 F − F -8.537E-4 0.020 2.34 F − F -2.647E-4 0.006 0.61 F − F -5.169E-4 0.008 6.23 † The appropriate LO, NLO, and NNLO interactions are obtained by truncating the table at the desired order. ‡ An N LO calculation in the D − D channel yields a D , N LO =-2.510E-4 MeV and a D , N LO = -7.550E-5 MeV, and reduces (cid:104) Resid . (cid:105) RMS to 22.3 keV; and in the D − G channel yields a DG, N LO = -2.141E-5 MeV and a DG, N LO = 1.180E-5 MeV andreduces (cid:104) Resid . (cid:105) RMS to 3.26 keV.
Once this point is appreciated – that the effective expansion parameter are naturally channel-dependent because ofeffects like the tensor force – the results shown in Table II are very pleasing: • In each channel the deduced couplings a LO , a NLO , a
NNLO , a N LO , ... evolve in a very orderly, or natural,fashion: one can reliably predict the size of the next omitted term. The convergence appears related to aneffective range characterizing scattering in Q . • The convergence varies from channel to channel, but this variation reflects underlying physics, such as role ofthe tensor force, governing the channel’s range. One does not find, nor perhaps should one expect to find, somesingle parameter p/ Λ to characterize convergence independent of channel. • The convergence is very satisfactory in all channels: the measure used in Table II, (cid:104)
Resid. (cid:105)
RMS , is an exceedingly conservative one, as discussed below. But even by by this standard, in only one channel ( D − D ) do the RMSresidual discrepancies among unconstrained matrix elements exceed ∼
10 keV. Given the arguments above, itis perfectly sensible to work to order NNLO in rapidly-converging channels like S − S and N LO in slowlyconverging channels like D − D . As noted in the table, at N LO the residual in the D − D channel isreduced to 22 keV. Convergence and the “Lepage” plot:
The procedure often followed in an effective theory is to use information aboutthe low-lying excitations to parameterize an effective Hamiltonian, which is then used to predict properties of otherstates near the ground state. In contrast, the goal here has been to characterize the entire effective interaction to highaccuracy. As described below, the residual errors in the procedure are typically dominated by matrix elements with6 P - F m.e. contributions from Qa) -0.08-0.06-0.04-0.020.00.02 Q T ( M e V ) P - F m.e.: NNLOb)
20 40 60 80 100 120 140-0.0030.00.0030.0060.0090.012 M . E . R e s i du a l s ( M e V ) P - F m.e.: N LOc) -0.0010.00.0010.0020.0030.004 M . E . R e s i du a l s ( M e V ) P - F RMS m.e. resultsd)
20 40 60 80 100 120 1400.00.010.020.030.04 R M S M . E . R e s i du a l s ( M e V ) N LONNLOBare
20 40 600.00.0010.002 R e s i d . ( M e V ) FIG. 16: As in Fig. 4, but for the P − F channel. the largest n and n (cid:48) , corresponding to minor components in the deuteron ground state, for example. The differencein the deuteron binding energy using exact matrix elements of H eff versus using the N LO expansion is quite small( ∼
40 eV).Order-by-order improvement should be governed by nodal quantum numbers. For example, in LO in the S channel the omitted NLO term would be − a S NLO π ( n (cid:48) + n − (cid:20) Γ[ n (cid:48) + 1 / n + 1 / n (cid:48) − n − (cid:21) / n (cid:48) ,n large −→ − a S NLO π [(4 n (cid:48) − n − / ( n (cid:48) + n −
2) (39)Thus the fractional error associated with the omission of the NLO terms relative to LO should to be linear in thesum of the nodal quantum numbers, if the expansion is capturing the correct physics. That is, the expected absolute(e.g., in keV) error for ( n (cid:48) , n )=(5,5) would be about 16 times that for (1,2).In higher orders this distinction between large and small n grows. At LO+NLO, the expected fractional errors inmatrix elements from omitted NNLO terms would be quadratic in n and n (cid:48) : the explicit functional dependence is nolonger simple as there are two NNLO operators, and one would not know a priori the relevant quadratic combinationof n (cid:48) and n governing the error. At NNLO the fractional error would be a cubic polynomial in n and n (cid:48) .While beyond LO the expected fractional errors have a dependence on both n and n (cid:48) , it is still helpful to display7 F m.e. contributions from Qa)
20 40 60 80 100 120 140-0.010.0 Q T ( M e V ) F m.e.: N LO b)
20 40 60 80 100 120 140-0.0010.00.0010.002 M . E . R e s i du a l ( M e V ) FIG. 17: The lowest contributing order to the 1 F channel is N LO. ∆ QT (Λ) and the N LO residuals for the five unconstrainedmatrix elements are shown. results as a 2D “Lepage plot” using n + n (cid:48) – proportional to the average (cid:104) p (cid:105) of bra and ket – as the variable. Such aplot makes clear whether improved fits in an effective theory are systematic – that is, due to a correct description ofthe underlying physics, not just additional parameters. The use of a single parameter, n + n (cid:48) , of course maps multiplematrix elements onto the same x coordinate, when the ET indicates this is a bit too simple beyond LO. Nevertheless,the right panel in Fig. 19 still shows rather nicely that the nuclear effective interactions problem is a very well behavedeffective theory. In LO the residual errors do map onto the single parameter n + n (cid:48) to very good accuracy, and theresidual error is linear. The steepening of the convergence with order is consistent with the expected progression fromlinear to quadratic to cubic behavior in nodal quantum numbers. By NNLO, errors in unconstrained matrix elementsfor small n + n (cid:48) are tiny, compared to those with high n + n (cid:48) . That is, the expansion converges most rapidly formatrix elements between long-wavelength states, as it should. However, improvement is substantial and systematiceverywhere, including at the largest n + n (cid:48) . IV. PROPERTIES AND ENERGY DEPENDENCE OF THE EFFECTIVE INTERACTION
The results of the previous section demonstrate the existence of a simple systematic operator expansion for theHOBET effective interaction. Its behavior order-by-order and in the Lepage plot indicates that the short-wavelengthphysics is being efficiently captured in the associated operator coefficients.The error measure used in the N LO fit is dominated by the absolute errors in matrix elements involving thehighest nodal quantum numbers: these matrix elements are large even though they may not play a major role indetermining low-lying eigenvalues. (It might have been better to use the fractional error in matrix elements, a measurethat would be roughly independent of n (cid:48) and n .) Other possible measures of error are the ground state energy; thefirst energy-moment of the effective interaction matrix (analogous to the mean eigenvalue in the SM); the fluctuationbetween neighboring eigenvalues of that matrix (analogous to the level spacing in the SM); and the overlap of theeigenfunctions of that matrix with the exact eigenfunctions (analogous to wave function overlaps in the SM). TheN LO interaction in the coupled S − D channel produces a ground state energy accurate to ∼
40 eV; a spectral firstmoment accurate to 1.81 keV; an RMS average deviation in the level spacing of 3.52 keV; and wave function overlapsthat are unity to better than four significant digits. As rescattering in Q contributes ∼ -10 MeV to eigenvalues, theaccuracy of the N LO representation of the effective interaction is, by these spectral measures, on the order of 0.01%.As the best excited-state techniques in nuclear physics currently yield error bars of about 100 keV for the lightestnontrivial nuclei, this representation of the two-body effective interaction is effectively exact [1, 12].The approach requires one to sum QT to all orders, producing a result that depends explicitly on | E | – whichin this context should be measured relative to the first breakup channel. While the associated effects increase withdecreasing | E | , it will be shown later that the renormalization is substantial even for well-bound nuclear states. The8 F m.e. contributions from Qa) -0.03-0.02-0.010.00.01 Q T ( M e V ) F m.e. contributions from Qa) -0.010.0 Q T ( M e V ) F m.e. contributions from Qa)
20 40 60 80 100 120 140-0.010.0 Q T ( M e V ) F m.e.: N LO b) -0.0010.00.0010.0020.0030.004 M . E . R e s i du a l ( M e V ) F m.e.: N LO b) -0.0010.00.001 M . E . R e s i du a l ( M e V ) F m.e.: N LO b)
20 40 60 80 100 120 140-0.0020.00.0020.0040.0060.0080.01 M . E . R e s i du a l ( M e V ) FIG. 18: As in Fig. 17, but for the F J − F J channels. As has been noted in other cases, the stretched F case has the largestresidual. Av18 potential S S S - D r (f) -500050010001500200025003000 V (r)( M e V ) S - S n+n’ -325 -225 -125 | E / E | LO NLO NNLO
FIG. 19: (Color online) The left panel shows the radial dependence of the Argonne v potential in the S − S , S − S ,and S − D (tensor) channels. The last is clearly more extended. The right panel is a “Lepage plot” displaying fractionalerrors as a function of the order of the calculation, on log scales. The steepening of the slope with order is the sign of a wellbehaved, converging effective theory. deuteron is definitely not an extreme case. The effects are also sensitive to the choice of P , through b , which controlsthe mean momentum within P – a small b reduces the missing hard-core physics, but exacerbates the problems atlong wavelengths, and conversely. Figure 1 suggests factor-of-two changes in the Q -space contribution to the deuteronbinding energy can result from ∼
20% changes in b . At the outset, the dependence on | E | and b seems like a difficultyfor nuclear physics, as modest changes in these parameters alter predictions.One of the marvelous properties of the HO is that the QT sum can be done. The two effects discussed above turnout to be governed by a single parameter, κ . The associated effects are nonperturbative in both QT and QV . In thecase of QT an explicit sum to all orders is done. The effects are also implicitly nonperturbative in QV , because ofthe dependence on | E | . This is why the BH approach is so powerful: because | E | is determined self-consistently, it issimple to incorporate this physics directly into the iterative process (which has been shown to converge very rapidlyin the HOBET test cases A=2 and 3). When this is done, one finds that κ affects results in three ways: • the rescattering of QT to all orders, T ( E − QT ) − QT , is absorbed into a new “bare” matrix element (cid:104) α | T | (cid:101) β ( κ ) (cid:105) ; • the new “bare” matrix element (cid:104) (cid:101) α ( κ ) | V | (cid:101) β ( κ ) (cid:105) captures the effects of QT in all orders on the contributionfirst-order in V ; and • the matrix elements of the short-range operators ¯ O , which contain all the multiple scattering of QV , are similarlymodified, (cid:104) (cid:101) α ( κ ) | ¯ O | (cid:101) β ( κ ) (cid:105) .So far the discussion has focused on the problem of a single bound state of fixed binding energy | E | , the deuteronground state. No discussion has occurred of expectations for problems in which multiple bound states, each with adifferent H eff ( | E | ), might arise. But 1) the dependence of H eff ( | E | ) on κ arises already in the single-state case,which was not a priori obvious; and 2) state dependence (energy dependence in the case of BH) must arise in the caseof multiple states, as this is the source of the required nonorthogonality of states when restricted to P , a requirementfor a proper effective theory. So a question clearly arises about the connection between the explicit κ dependencefound for fixed | E | , and the additional energy dependence that might occur for a spectrum of states.Because other techniques, like Lee-Suzuki, have been used to address problem 2), it is appropriate to first stress therelationship between κ and the strong interaction parameters provided in Table II. The choice Λ P =8 is helpful, as itshows there is no relation. Every short-range coefficient arising through order N LO was determined from nonedgematrix elements: the fitting procedure matches the coefficients to the set of matrix elements with n (cid:48) + n ≤
5, andthere are no edge states satisfying this constraint. Nothing in the treatment of the strong interaction “knows” aboutedge states. This then makes clear how efficiently κ captures the remaining missing physics. Without κ one would0have, in the contact-gradient expansion to N LO, a total of 78 poorly reproduced edge-state matrix elements, 10 ofwhich would be S -state matrix elements with errors typically of several MeV. With κ – a parameter nature (and thechoice of b ) determines – all of the 78 matrix elements are properly reproduced, consistent with the general ∼ keVaccuracy of the N LO description of H eff .Suppose someone were to prefer an H eff free of any dependence on | E | , again in the context of an isolated stateof energy | E | . Could this be done? Yes, but at the cost of a cumbersome theory that obscures the remarkably simplephysics behind the proper description of the edge state matrix elements. Suppose one wanted merely to fix the five S − S edge state matrix elements, those where n (cid:48) = 5 couples to n =1, 2, 3,4, and 5. One could introduce operatorscorresponding to the coefficients a S, N LO , a S, N LO , a S, N LO , a S, N LO , a S, N LO to correct these matrix elements. It is clear all five couplings would be needed – that’s the price one would pay formocking up long-range physics (a long series of high-order Talmi integrals) with a set of short-range operators of thissort.This would be a rather poorly motivated exercise: • The problems in these matrix elements have nothing to do with high-order generalized Talmi integrals of thestrong potential, as was demonstrated in the previous section. • This approach does not “heal” the effective theory: the poor running of matrix elements would remain. Therewould be no systematic improvement, for all matrix elements, as a function of Λ, as one progresses from LO, toNLO, etc. The five parameters introduced above would remove the numerical discrepancies at Λ P , but not fixthe running as a function of Λ, even for just the edge-state matrix elements. • This approach amounts to parameter fitting, in contrast to the systematic improvement demonstrated in theLepage plot. The parameter a S, N LO introduced to fix the n = 1 to n = 5 matrix element will not properlycorrect the n = 2 to n = 5 matrix element, as the underlying physics has nothing to do with the r r -weightedTalmi integral of any potential. • If Λ P is increased, the number of such edge-state matrix elements that will need to be corrected by the fictitiouspotential increases. This contrasts with the approach where | E | is explicitly referenced: there the number ofshort-range coefficients needed to characterize Q will decrease (that is, the LO, NLO, ... expansion becomes morerapidly convergent), while κ remains the single parameter governing the renormalization of those coefficients foredge-state matrix elements.While these reasons are probably sufficient to discard any such notion of building a κ -independent H eff , considernow the consequences of changing b – which after all is an arbitrary choice. The short-range coefficients in Table IIwill change: there is an underlying dependence on QV ( (cid:126)r /b ). This governs natural variations in the coefficients –one could estimate those variations based on some picture of the range of multiple scattering in Q , as was done in the“naturalness” discussion. But there would be additional changes in the ratios of edge to nonedge matrix elements,reflecting the changes in κ . This would induce in any κ -independent potential unnatural evolution in b . That is, thefake potential would look fake, as b is changed.The arguments above apply equally well to the case of the state-dependence associated with techniques like Lee-Suzuki. To an accuracy of about 95%, the κ -dependence isolated in H eff is also the state-dependence that oneencounters when | E | is changed. This is a lovely result: the natural κ -dependence that is already present in the caseof a short-range expansion of H eff for a fixed state, also gives us “for free” the BH state-dependence. The resultis not at all surprising, physically: changes in | E | will alter the balance between QT and QV , and that is preciselythe physics that was disentangled by introducing κ . Mathematically, it is also not surprising: changing b at fixed | E | alters κ , just as changes in | E | for fixed b would. Thus all of the QT effects identified above, in considering a singlestate, must also arise when one considers spectral properties.This argument depends on showing that other, implicit energy dependence in H eff is small compared in the explicitdependence captured in κ . Such implicit dependence can reside in only one place, the fitted short-range coefficients. A. Energy Dependence
The usual procedure for solving the BH equation, H eff = H + HQ E − QH QH, E yields an H eff ( E ) and thus an eigenvalue E (cid:48) , which then can be used in a new calculation of the interaction H eff ( E (cid:48) ).This procedure is iterated until the eigenvalue coresponds to the energy used in calculating H eff . In practice, theconvergence is achieved quite rapidly, typically after about five cycles.As the BH procedure produces a Hermitian H eff , this energy dependence is essential in building into the formalismthe correct relationship between the P -space and full-space wave functions, that the former are the restrictions of thelatter (and thus cannot form an orthonormal set). This relationship allows the wave function to evolve smoothly tothe exact result, in form and in normalization, as Λ P → ∞ .Generally this energy dependence remains implicit because the BH equation is solved numerically: one obtainsdistinct sets of matrix elements (cid:104) α | H eff ( E i ) | β (cid:105) for each state i , but the functional dependence on E i is not immediatelyapparent. But that is not the case in the present treatment, where an analytic representation for the effectiveinteraction has been obtained.While significant energy-dependent effects governed by κ have been isolated, additional sources remain in the caseof a spectrum of bound states. The identified energy-dependent terms are • (cid:104) α | T + EE − QT QT | β (cid:105) = (cid:104) α | T | (cid:101) β ( κ ) (cid:105) ; • (cid:104) α | EE − T Q V EE − QT | β (cid:105) = (cid:104) (cid:101) α ( κ ) | V | (cid:101) β ( κ ) (cid:105) ; and • (cid:104) α | EE − T Q ¯ O EE − QT | β (cid:105) = (cid:104) (cid:101) α ( κ ) | ¯ O | (cid:101) β ( κ ) (cid:105) The implicit energy dependence not yet isolated resides in the coefficients of the contact-gradient expansion, • (cid:104) α | V EE − QH QV | β (cid:105) = (cid:104) α | ¯ O ( E ) | β (cid:105) .To isolate this dependence, one must repeat the program that was executed for the deuteron ground state at a varietyof energies, treating H eff ( | E | ) as a function of | E | . The resulting variations in the extracted coefficients will thendetermine the size of the implicit energy dependence. Of course, all of the explicit energy dependence is treated asbefore, using the appropriate κ. The simplest of the explicit terms is the “bare” kinetic energy (cid:104) n (cid:48) l | T | (cid:101) n (cid:101) l ( κ ) (cid:105) ≡ (cid:104) n (cid:48) l | T + T E − QT QT | nl (cid:105) = (cid:104) n (cid:48) l | T | nl (cid:105) + ¯ hω δ n (cid:48) n (cid:112) n ( n + l + 1 / (cid:101) g ( − κ ; n, l ) . where effects only arise in the double-edge-state case. Two limits define the range of variation. As | E | → ∞ , (cid:101) g → n + l − / hω/
2. Similarly one can show (cid:101) g ( − κ ; n, l ) → n as the binding energy | E | approaches zero. Thus for small binding, the matrix element approaches ( n + l − / hω/ n ¯ hω/
2, about 35 MeV for the parameters used in this paper. The behavior betweenthese limits can be calculated. The results over 20 MeV in binding are shown in the upper left panel of Fig. 20 for S , P , and D states. One finds that even deeply bound (E=-20 MeV) states have very significant corrections due to QT : the scattering in Q reduces the edge-state kinetic energy matrix elements by (2-3) ¯ hω/
2, which serves to lowerthe energy of the bound state. The kinetic energy decreases monitonically as | E | → κ -dependent term, the “bare” potential energy (cid:104) (cid:101) α ( κ ) | V | (cid:101) β ( κ ) (cid:105) , is displayed over the same range in thelower left panel of Fig. 20 for the five S − S edge-state matrix elements. These matrix elements are again quitesensitive to | E | , varying by 2-3 MeV over the 20 MeV range displayed in the figure.The third κ -dependence is the renormalization of the contact-gradient coefficients for edge states, (cid:104) n (cid:48) l (cid:48) | EE − T Q ¯ O EE − QT | n l (cid:105) = (cid:88) i,j =0 (cid:101) g j ( − κ ; n (cid:48) , l (cid:48) ) (cid:101) g i ( − κ ; n, l ) (cid:104) n (cid:48) + j l (cid:48) | ¯ O | n + i l (cid:105) (40)Here ¯ O is fixed, while the explicit energy dependence carried by the (cid:101) g i (i.e., the effects of the interplay between QT and QV ) is evaluated. The upper left panel in Fig. 20 gives the result for the diagonal edge-state matrix element, | n (cid:48) l (cid:48) (cid:105) = | n l (cid:105) = | (cid:105) . As has been seen in other cases, the reduction due to the QT − QV interplay is substantialthroughout the illustrated 20 MeV range. Thus the large effects observed for the deuteron, a relatively weakly boundstate, are in fact generic. But weakly bound states are more strongly affected, with the differences between thecorrections for the double-edge states changing by a factor of nearly two between | E | =20 MeV and | E | ∼ -20 -15 -10 -5 0 Energy (MeV) < n l | T + T / ( E - Q T ) Q T | n l > |4 2>|5 0>|4 1>| 5 0 >, | 4 2 >| 4 1 > S - S -20 -15 -10 -5 0 Energy (MeV) < n ’ l = | E / E - T QV E / E - Q T | n l = > ( M e V ) n’=5 - n=5n’=5 - n=4n’=5 - n=3n’=5 - n=2n’=5 - n=1 -20 -15 -10 -5 0 Energy (MeV) < | E / E - T QO E / E - Q T | > / < | O | > LONLONNLO22NNLO40N3LO42N3LO60
FIG. 20: Contributions to H eff with explicit energy depen-dence, for P defined by Λ P = 8 and b = 1 . (cid:104) α | T | (cid:101) α (cid:105) for the edge states | α (cid:105) = | n = 5 l = 0 (cid:105) , | n = 4 l = 1 (cid:105) , and | n = 4 l = 2 (cid:105) . The dots indicate the limiting values for verylarge and very small binding energies. The kinetic energy plot-ted is dimensionless, given in terms of ¯ hω/
2. The lower left panelgives the matrix elements of the bare potential V between S edge states, as a function of E . The upper right panel shows theevolution of the quantities a (cid:48) LO ( E ; n (cid:48) , l (cid:48) , n, l ) /a LO ( n (cid:48) , l (cid:48) , n, l ), a (cid:48) NLO ( E ; n (cid:48) , l (cid:48) , n, l ) /a NLO ( n (cid:48) , l (cid:48) , n, l ), etc., through N LO forthe diagonal matrix element with | n = 5 l = 0 (cid:105) . The generalsoftening of such matrix elements is apparent, for small bindingenergy – repeated scattering by T through high-energy oscilla-tor states in Q spreads the wave function and thus reduces theeffects of the strong potential at short range. This effect is car-ried by the edge states, because their renormalization is affectedby the missing long-range physics. See the text for further dis-cussion. In doing these calculations, some care is needed in going to the limit of very small binding energies. One can showfor edge states (cid:101) g i ( − κ ; n, l ) small κ −→ ( − i (cid:20) Γ( n + l + 1 / n − i )!Γ( n + l + 1 / i )( n − (cid:21) / (41)If one uses this in Eq. (40) with κ ≡
0, one finds that (cid:80) i =0 (cid:101) g i (0; n, l ) (cid:104) (cid:126)r = 0 | n + i l (cid:105)(cid:104) (cid:126)r = 0 | n l (cid:105) oscillates (for an edge state) between 0 and 1, with every increment in i . However, a nonzero κ acts as a convergencefactor. If it is quite small, but not zero, the ratio then goes smoothly to 1/2. Consequently, as Fig. 20 shows, a (cid:48) LO /a LO → κ .The effects illustrated in Fig. 20 – the three effects explicitly governed by κ – are associated with the couplingbetween P and Q generated by T . Because this operator connects states with ∆ n = ±
1, there is no large energy scale3
TABLE III: Spectral property variations in H eff ( E ) over 10 MeVTerm Parameter 1st Moment Shift (MeV) RMS Level Variation (MeV) Wave Function Overlaps (cid:104) α | T | (cid:101) β (cid:105) κ (cid:104) (cid:101) α | V | (cid:101) β (cid:105) κ (cid:104) (cid:101) α | ¯ O | (cid:101) β (cid:105) κ -0.239 0.957 99.51-99.99% (cid:104) α | ¯ O ( E ) | β (cid:105) implicit 0.135 0.107 99.95-100% associated with excitations. As the effects are encoded into a subset of the matrix elements, the overall scale of the κ dependence on spectral properties is, at this point, still not obvious.This leaves us with one remaining term that, qualitatively, seems quite different, V E − QH QV ↔ { a LO ( | E | ) , a NLO ( | E | ) , a NNLO ( | E ) | , a N LO ( | E | ) , ... } . (42)Here the energy dependence is implicit, encoded in the parameters fitted to the lowest energy matrix elements of H eff . The underlying potentials are dominated by strong, short-ranged potentials, much larger than nuclear bindingenergies. Thus the implicit ratio governing this energy dependence – | E | vs. the strength of the hard-core potential –is a small parameter. For this reason one anticipates that the resulting energy dependence might be gentler than inthe cases just explored.After repeating the fitting procedure over a range of energies, one obtains the results shown in Fig. 21. Becausethe energy variation is quite small, results are provided only for the channels that contribute in low order, S , S , P and P J . The variation is very modest and regular, varying inversely with | E | and well fit by the assumption(motivated by the form of V ( E − QH ) − QV ) a ( E ) = a (10 M eV )1 + α | E | . The variation is typically at the level of a few percent, over 20 MeV. The progression in the slopes within each channel,order by order, correspond to expectation: the lowest order terms, which account for the hardest part of the scatteringin Q , have the weakest dependence on | E | . Comparisons between channels also reflect expectations. In the earlierdiscussion of naturalness, the rapid convergence in the S channel, order by order, was consistent with very shortrange interactions in Q . Accordingly, a S LO varies by just 0.72% over a 10 MeV interval, and a S NLO by 1.10%. Thischannel contrasts with the S channel, where convergence in the contact-gradient expansion is slower, consistentwith somewhat longer range interactions in Q . For the S case one finds 2.64% variations in a LO ( S ) and 5.17%variations in a NLO ( S ) per 10 MeV interval.Are such variations of any numerical significance, compared to the explicit variations isolated in κ ? That is, if onewere to determine a HOBET interaction directly from bound-state properties of light nuclei, would the neglect ofthis implicit energy dependence lead to significant errors in binding energies? One can envision doing such a fit overbound-state data spanning ∼
10 MeV, finding the couplings as a function of | E | , so that the error induced by usingaverage energy-independent couplings a LO ( | ¯ E | ) can be assessed. These errors would reflect variations in the matrixelements to which these couplings are fit, following the procedures previously described. Such a study showed thatonly two channels exhibited drifts ∆ in excess of 15 keV over 10 MeV, a S LO : ∆ ∼ ±
21 keV a S LO : ∆ ∼ ±
148 keV a S NLO : ∆ ∼ ±
32 keVOne concludes that the S channel is, by a large factor, the dominant source of implicit energy dependence in theHOBET interaction.This allows one to do a more quantitative calculation that focuses on the most difficult channel ( S ) and comparesthe relative sizes of the κ -dependent and implicit energy dependences, as reflected in changes in the matrix H eff ( | E | ).Thus this matrix is constructed at | E | = 10 MeV and at | E | ∼ D ), and changes inglobal quantities of that matrix over 10 MeV are examined: shifts in the first moment (the average eigenvalue) , theRMS shifts of levels relative to the first moment (related to the stability of level splittings), and eigenvalue overlaps.The four energy-dependent effects discussed here are separately turned on and off. Thus this exercise should providea good test of the relative importance of these effects. The results are shown in Table III.Despite the selection of the worst channel, S , the implicit energy dependence is small, intrinsically and in compar-ison with the implicit energy dependence embedded in κ . The implicit dependence in the first moment – a quantity4 -33.0-32.8-32.6-32.4-32.2 S C oup li ng s ( M e V ) a LO (E)158 a NLO (E) -64-62-60-58 S C oup li ng s ( M e V ) a LO (E)1175 a NNLO,22 (E) 46 a
NLO (E) -20 -15 -10 -5 0
Energy (MeV) -0.86-0.84-0.82 P C oup li ng s ( M e V ) a NLO (E)122 a
NNLO (E) -1.64-1.62-1.6 P C oup li ng s ( M e V ) a NLO (E)90 a
NNLO (E) -1.92-1.88-1.84-1.8 P C oup li ng s ( M e V ) a NLO (E)120 a
NNLO (E) -20 -15 -10 -5 0
Energy (MeV) -0.46-0.44-0.42 P C oup li ng s ( M e V ) a NLO (E)36.5 a
NNLO (E)
FIG. 21: The calculated energy dependence of derived coefficients for the contact-gradient expansion are indicated by themarkers, for the various S − S and P − P channels. Over a 10 MeV interval typical of bound-state nuclear spectra, variationsare typically at the few percent level. The continuous lines represent simple linear fits, a (10 MeV)/ a ( E ) = 1 + α | E | , to theresults. The fit is generally excellent. important to absolute binding energies – is 5% that of the explicit dependence in (cid:104) α | T | (cid:101) β (cid:105) . The RMS shifts in levelsrelative the first moment are at the ∼ ∼
100 keV for the implicit term.Eigenfunction overlaps show almost no dependence on the implicit term, exceeding 99.95% in all cases: variations10-100 times larger arise from the analytical terms in κ .Thus a simple representation of the HOBET effective interaction exists: • The requirements for a state of fixed | E | are a series of short-range coefficients and a single parameter κ thatgoverns long-range corrections residing in Q , including certain terms that couple QV and QT . By variousmeasures explored here, an N LO expansion is accurate to about a few keV • The κ dependence found for a state of definite energy | E | also captures almost all of the energy dependenceresulting from varying | E | , the state-dependence in BH. Even in the most troublesome channel, calculationsshow that ∼
95% of the energy dependence associated with changes in | E | is explicit. It appears that neglectof the implicit energy dependence would induce errors of ∼ <
100 keV, for a spectrum spanning ±
10 MeV. Thiskind of error would be within the uncertainties of the best ab initio excited-state methods for light (p-shell)nuclei, such as Green’s function Monte Carlo [1] or large-basis no-core SM diagonalizations [12]. • If better results are desired, the program described here can be extended to include the implicit energy depen-dence. The expansion around an average energy E V E − QH QV = V (cid:20) E − QH − E − QH ( E − E ) 1 E − QH + · · · (cid:21) QV generates the correction linear in E that is seen numerically. This second term is clearly quite small, explicitlysuppressed by the ratio of scales discussed above. But, in any troublesome channel, the second term could be5represented by contact-gradient operators of low order, with the contribution suppressed by an overall factor of( E − E ). V. DISCUSSION AND SUMMARY
One of the important motivations for trying to formulate an effective theory for nuclei in a harmonic oscillatorbasis is the prospect of incorporating into the approach some of the impressive numerical technology of approacheslike the SM. Numerical techniques could be used to solve significant P -space problems, formulated in spaces, such ascompleted N ¯ hω bases, that preserve the problem’s translational invariance.My collaborators and I made an initial effort to construct a HOBET some years ago, using a contact-gradientexpansion modeled after EFT. We performed a shell-by-shell integration, in the spirit of a discrete renormalizationgroup method, but encountered several abnormalities connected with the running of the coefficients of the expansion.Subsequent numerical work in which we studied individual matrix elements revealed the problems illustrated in Fig.3. These problems – the difficulty of representing Q -space contributions that are both long range and short range –are not only important for HOBET, but also are responsible for the lack of convergence of perturbative expansions ofthe effective interaction. Fig. 1 provides one example. In other work [6] we have shown that convergent expansionsin the bare interaction (deuteron) or g-matrix ( He/ H) for H eff do exist, if the long-range part of this problem isfirst solved, as we have done here.Thus the current paper returns to the problem of constructing a contact-gradient expansion for the effective in-teraction, taking into account what has been learned since the first, less successful effort. This paper introduces aform for that expansion that eliminates operator mixing, simplifying the fitting of coefficients and guaranteeing thatcoefficients determined in a given order remain fixed when higher-order terms are added. Thus the N LO resultspresented here contain the results of all lower orders. The expansion is one in nodal quantum numbers, and is directlyconnected with traditional Talmi integral expansions, generalized for nonlocal interactions.Convergence does vary from channel to channel, but in each channel the order-by-order convergence is very regular.Each new order brings down the scale Λ at which deviations appear, and in each new order the Lepage plot steepens,showing that the omitted physics does have the expected dependence on higher-order polynomials in ( n (cid:48) , n ). Thechannel-by-channel variations in convergence reflect similar behavior seen in EFT approaches, where the need foralternative power counting schemes has been noted to account for this behavior. From a practical standpoint, however,the N LO results are effectively exact: in the most important difficult channel, S , measures of the quality of thematrix H eff yielded results on the order of (1-3) keV.The summation done over QT yields a simple result, but still one that is quite remarkable in that long-range physicsis governed by a single parameter κ , that depends on the ratio of | E | and ¯ hω . Despite all of the attractive analyticproperties of the HO as a basis for bound states, its unphysical binding at large r has been viewed as a shortcoming.But the ladder properties of the HO in fact allow an exact summation of QT . It seems unlikely that any otherbound-state basis would allow the coupling of P and Q by T to be exactly removed. That is, the HO basis may bethe only one that allows the long-range physics in Q to be fully isolated, and thus subtracted systematically. In thissense it may be the optimal basis for correctly describing the asymptotic behavior of the wave function. Note, inparticular, that the right answer is not going to result from using “improved” single-particle bases: κ depends of | E | ,not on single-particle energies of some mean field.The effects associated with κ are large, typically shifting edge-state matrix elements by several MeV, and alteringspectral measures, like the first energy moment of H eff , by similar amounts. This dependence, if not isolated, destroysthe systematic order-by-order improvement important to HOBET, as Fig. 3 clearly illustrates.The explicit energy dependence captured in κ accounts for almost all of the energy dependence of H eff ( | E | ). Inmore complicated calculations this dependence, in the BH formulation used here, generates the state-dependence thatallows ET wave functions to have the proper relationship to the exact wave functions, namely that the former arethe P -space restrictions of the latter. While in principle additional energy dependence important to this evolutionresides in V ( E − QH ) − QV and thus in the coefficients of the contact-gradient expansion, in practice this residualimplicit energy dependence was found to be very weak. This dependence was examined channel by channel, andits impact on global properties of H eff ( | E | ) was determined for the most troublesome channel, S . Even in thischannel, the impact of the remaining implicit energy dependence on H eff ( | E | ) spectral properties such as the firstmoment, eigenvalue spacing, and eigenvalue overlaps, was found to be quite small compared to the explicit dependenceisolated in κ . This is physically very reasonable: QT generates nearest-shell couplings between P and Q , so thatexcitation scales are comparable to typical nuclear binding energies. Thus this physics, extracted and expressed as afunction of κ , should be sensitive to binding energies. In contrast, V ( E − QH ) − QV involves large scales associatedwith the hard core, and thus should be relatively insensitive to variations in | E | . In the S channel, the explicitdependence captured in κ is about 20 times larger than the implicit energy dependence buried in the contact-gradient6coefficients. Numerically, the latter could cause drifts on the order of 100 keV over 10 MeV intervals. Thus, to anexcellent approximation, one could treat these coefficients as constants in fitting the properties of low-lying spectra.Alternatively, the HOBET procedure for accounting for this implicit energy dependence has been described, and couldbe used in any troublesome channel.The weakness of the implicit energy dependence will certainly simplify future HOBET efforts to determine H eff ( E, b, Λ P ) directly from data (rather than from an NN potential like v ). Indeed, such an effort will be the nextstep in the program. The approach outlined here is an attractive starting point, as it can be shown that the states | (cid:101) α ( κ ) (cid:105) become asymptotic plane-wave states, when E is positive. Thus the formalism relates bound and continuumstates through a common set of strong-interaction coefficients operating in a finite orthogonal basis.The relationship between current work and some more traditional treatments of the H eff for model-based ap-proaches, like the SM, should be mentioned. Efforts like those of Kuo and Brown are often based on the division H = H + ( V − V ), where H is the HO Hamiltonian [13]. Such a division would allow the same BH reorganizationdone here: QT and Q ( H − V ) are clearly equivalent. But in practice terms are, instead, organized in perturba-tion theory according to H , i.e., so that Green’s functions involve single-particle energies. This would co-minglethe long- and short-range physics is a very complicated way. In addition, often the definitions of Q and P used innumerical calculations are not those of the HO: instead, a plane-wave momentum cut is often used, which simplifiesthe calculations but introduces uncontrolled errors. Either this approximation (plane waves are diagonal in T ) or theuse of perturbation theory (because of the co-mingling) would appear to make it impossible to separate long- andshort-range physics correctly, as has been done here.Another example is V low k , in which a softer two-nucleon potential is derived by integration over high-momentumstates [14]. This is a simpler description of Q than arises in bound-state bases problems, like those considered here: thedivision between P and Q is a specified momentum, and T is diagonal. There would be no analog of the κ -dependencefound in HOBET. However, HOBET and V low k may have an interesting relationship. Effective operators for HOBETand for EFT approaches (which also employ a momentum cutoff) agree in lowest contributing order. When there aredifferences in higher order, it would seem that these differences must vanish by taking the appropriate limit, namelythe limit of the HOBET Q ( b, Λ P ) where b → ∞ while Λ P /b is kept fixed. This keeps the average (cid:104) p (cid:105) of the lastincluded shell fixed, while forcing the number of shells to infinity and the shell splitting to zero. Numerically it wouldbe sufficient to approach this limit, so that Q resembles the plane-wave limit over a distance characteristic of thenuclear size. It is a reasonable conjecture that V low k would emerge from such a limit of the HOBET H eff ( b, Λ P ).It would follow that all κ dependence should vanish in that limit. It would be interesting to try to verify theseconjectures in future work, and to study the evolution of the HOBET effective interaction coefficients as this limit istakenThe state-dependence of effective interactions is sometimes treated in nuclear physics by the method of Lee andSuzuki. One form of Lee-Suzuki produces a Hermitian energy-independent interaction. While it is always possibleto find such an H to reproduce eigenvalues, it is clear that basic wave function requirements of an effective theory –that the included wave functions correspond to restrictions of the true wave functions to P – are not consistent withsuch an H .Another form produces an energy-independent but nonHermitian H . This can be done consistently in an effectivetheory. However, the results presented here make it difficult to motivate such a transformation. It appears that thestate-dependence is almost entirely attributable to the interplay between QT and QV , removed here analytically interms of a function of one parameter κ , which relates the bound-state momentum scale (in ¯ hω ) to a state’s energy.There is no obvious benefit in obscuring this simple dependence in a numerical transformation of the potential, giventhat the Lee-Suzuki method is not easy to implement. The physics is far more transparent in the BH formulation, andthe self-consistency required in BH makes the use of an energy-dependent potential as easy as an energy-independentone. More to the point, the necessary κ dependence is already encoded in the potential for a single state of definiteenergy | E | – thus no additional complexity is posed by the state-dependence of the potential.I thank Tom Luu and Martin Savage for helpful discussions. This work was supported in part by the Office ofNuclear Physics and SciDAC, US Department of Energy. VI. REFERENCES [1] S. C. Pieper and R. B. Wiringa,
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VII. APPENDIX: EFFECTIVE INTERACTION MATRIX ELEMENTS
This appendix provides details on the evaluation of the modified HO states EE − QT | n l (cid:105) . (A-1)for nodal quantum numbers n = 1 , , , ... , and for corresponding contact-gradient effective interaction matrixelements between such states. Closed-form expressions allow these matrix elements to be evaluated quickly to anyorder. Here two alternative evaluations are provided, one based on a harmonic oscillator expansion and one on thefree Green’s function. Harmonic Oscillator Expansion:
The harmonic oscillator Green’s function expansion is EE − QT | n l (cid:105) = ∞ (cid:88) i =0 (cid:101) g i ( − κ ; n, l ) | n + i l (cid:105) (A-2)where the (cid:101) g i are determined by a set of continued fractions generated from the ladder properties of the operator QT .In practice the sum can be truncated: numerical convergence is discussed below, in comparison with the Green’sfunction approach. While this paper focuses on the simple example of the deuteron, the approach is more general :the relationship between the relative kinetic energy and the three-dimensional harmonic oscillator can be extendedto the n-dimensional harmonic oscillator, with the hyperspherical harmonics replacing the spherical harmonics aseigenfunctions of the kinetic energy. The corresponding ladder properties for the harmonic oscillator in hypersphericalcoordinates are known [15]: this is the essential requirement for the expansion.As discussed in the text, it is convenient to modify the usual contact-gradient expansion, to remove operator mixingand create an expansion in nodal quantum numbers. Each term O in the usual contact-gradient expansion is replacedby O → ¯ O ≡ e r / Oe r / (A-3)where r is the dimensionless Jacobi coordinate | (cid:126)r − (cid:126)r | / ( b √ O are also defined in terms ofthis dimensionless coordinate.In each partial wave the lowest contributing operators are based on gradients, maximally coupled, acting on wavefunctions, with the result evaluated at (cid:126)r = 0. For example, for S , P , and D states e r / R nl ( r ) Y l (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = δ l π (cid:20) n + 1 / n − (cid:21) / −→∇ e r / R nl ( r ) Y l (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = δ l (cid:114) n + 1)3 R n ( r ) Y (Ω r ) (cid:12)(cid:12) r =0 = δ l (cid:20) (cid:21) / π (cid:20) n + 3 / n − (cid:21) / −→∇ ⊗ (cid:126) ∇ ) e r / R nl ( r ) Y l (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = δ l (cid:114) n + 1)(2 n + 3)15 R n ( r ) Y (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = δ l (cid:20) (cid:21) / π (cid:20) n + 5 / n − (cid:21) / (A-4)As the gradients are maximally coupled, all coupling schemes are equivalent. If one defines by ( −→∇ q ) q the expressionswith q gradients maximally coupled, the results of Eq. (A-4) are examples of the more general formula( −→∇ q ) q e r / R nl ( r ) Y l (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = δ lq l (cid:20) l !(2 l + 1)!! (cid:21) / π (cid:20) n + l + 1 / n − (cid:21) / . (A-5)A form of this equation that will be used below is( −→∇ q ) q e r / R n − p l ( r ) Y l (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = δ lq l (cid:20) l !(2 l + 1)!! (cid:21) / π (cid:20) n + l + 1 / − p ]( n − − p )! (cid:21) / = (cid:20) ( n − n + l + 1 / − p ]( n − − p )! Γ[ n + l + 1 / (cid:21) / δ lq l (cid:20) l !(2 l + 1)!! (cid:21) / π (cid:20) n + l + 1 / n − (cid:21) / (A-6)Contact-gradient operators beyond the lowest contributing order involve −→∇ acting on harmonic oscillator wavefunctions. One can quickly verify −→∇ e r / R nl ( r ) Y lm (Ω r ) = − (cid:112) ( n − n + l − / e r / R n − l ( r ) Y lm (Ω r ) (A-7)so that ( −→∇ ) p e r / R nl ( r ) Y lm (Ω r ) = ( − p (cid:20) ( n − n + l + 1 / n − − p )! Γ[ n + l + 1 / − p ] (cid:21) / e r / R n − p l ( r ) Y lm (Ω r ) . (A-8)Thus by first using Eq. (A-8) and then applying Eq. (A-6), one finds the general expression for a contact-gradientoperator of arbitrary order acting on a harmonic oscillator state( −→∇ ) p ( −→∇ q ) q e r / R nl ( r ) Y lm (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = δ lq ( − p ( n − n − − p )! (cid:32) l (cid:20) l !(2 l + 1)!! (cid:21) / π (cid:20) n + l + 1 / n − (cid:21) / (cid:33) (A-9)Equation (A-9) defines the needed matrix elements, evaluated below for each partial-wave channel contributingthrough N LO. [Note that if one wanted to write a potential, as opposed to partial-wave matrix elements of thatpotential that are given here, suitable projection operators could be inserted as needed. For example, the l = 0 tripletand singlet channels could be distinguished by introducing the projection operators P ( S ) = (3 + (cid:126)σ · (cid:126)σ ) / (cid:126)S / − (cid:126)σ · (cid:126)σ ) / − (cid:126)S /
2, respectively; the three triplet l = 1 channels could be distinguished from the singlet l = 1 channel and from each other by the projectors P ( P ) = ( − (cid:126)l · (cid:126)S ) ) (cid:126)S P ( P ) = (2 − (cid:126)l · (cid:126)S − ( (cid:126)l · (cid:126)S ) ) (cid:126)S P ( P ) = (2 + 3 (cid:126)l · (cid:126)S + ( (cid:126)l · (cid:126)S ) ) (cid:126)S
12 (A-10)and so on.] The matrix elements of Table I, which all are scalar products of spin-spatial tensor operators, are of twotypes. One is diagonal in l , where O = ←− O Ll · δ ( (cid:126)r ) −→ O Rl , with ←− O Ll and −→ O Rl spatial tensors. In this case (cid:104) n (cid:48) ( lS ) JM T M T | O | n ( lS ) JM T M T (cid:105) = (cid:104) n (cid:48) lm l = 0 |←− O Ll δ ( (cid:126)r ) −→ O Rl | nlm l = 0 (cid:105) (A-11)The second type of operator, O = ←− O Ll · δ ( (cid:126)r ) [ −→ O Rl +2 ⊗ [ σ (1) ⊗ σ (2)] ] l , enters for the off-diagonal triplet S − D , P − F ,and D − G cases, (cid:104) n (cid:48) ( lS = 1) J = l + 1 M T M T | O | n ( l + 2 S = 1) J = l + 1 M T M T (cid:105) =2 (cid:114) l + 12 l + 3 (cid:104) n (cid:48) l m l = 0 |←− O Ll δ ( (cid:126)r ) −→ O Rl +2 0 | n l + 2 m l = 0 (cid:105) (A-12)9As both of these expressions reduce the matrix element to a product of terms like Eq. (A-9), the needed N LOmatrix elements follow directly. For the S ↔ S or S ↔ S channels, (cid:104) n (cid:48) ( l = 0 S ) JM T M T | e r / (cid:20) a S LO δ ( (cid:126)r ) + a S NLO ( ←−∇ δ ( (cid:126)r ) + δ ( (cid:126)r ) −→∇ ) + a S , NNLO ←−∇ δ ( (cid:126)r ) −→∇ + a S , NNLO ( ←−∇ δ ( (cid:126)r ) + δ ( (cid:126)r ) −→∇ )+ a S , N LO ( ←−∇ δ ( (cid:126)r ) −→∇ + ←−∇ δ ( (cid:126)r ) −→∇ ) + a S , N LO ( ←−∇ δ ( (cid:126)r ) + δ ( (cid:126)r ) −→∇ ) (cid:21) e r / | n ( l = 0 S ) JM T M T (cid:105) =2 π (cid:20) Γ[ n (cid:48) + 1 / n + 1 / n (cid:48) − n − (cid:21) / (cid:18) a S LO − n (cid:48) −
1) + ( n − a S NLO ++16 (cid:2) ( n (cid:48) − n − a S , NNLO + (( n (cid:48) − n (cid:48) −
2) + ( n − n − a S , NNLO (cid:3) − (cid:2) (( n (cid:48) − n (cid:48) − n −
1) +( n (cid:48) − n − n − a S , N LO + (( n (cid:48) − n (cid:48) − n (cid:48) −
3) + ( n − n − n − a S , N LO (cid:3)(cid:19) ; (A-13)For the S ↔ D channel, recalling −→ D ≡ (( −→∇ ⊗ −→∇ ) ⊗ ( σ (1) ⊗ σ (2)) ) , (cid:104) n (cid:48) ( l = 0 S = 1) J = 1 M T M T | e r / (cid:20) a SDNLO ( δ ( (cid:126)r ) −→ D + ←− D δ ( (cid:126)r )) + a SD, NNLO ( ←−∇ δ ( (cid:126)r ) −→ D + ←− D δ ( (cid:126)r ) −→∇ )+ a SD, NNLO ( δ ( (cid:126)r ) −→∇ −→ D + ←− D ←−∇ δ ( (cid:126)r )) + a SD, N LO ( ←−∇ δ ( (cid:126)r ) −→ D + ←− D δ ( (cid:126)r ) −→∇ ) + a SD, N LO ( ←−∇ δ ( (cid:126)r ) −→∇ −→ D + ←− D ←−∇ δ ( (cid:126)r ) −→∇ ) + a SD, N LO ( δ ( (cid:126)r ) −→∇ −→ D + ←− D ←−∇ δ ( (cid:126)r )) (cid:21) e r / | n ( l = 2 S = 1) J = 1 M T M T (cid:105) =815 √
10 2 π (cid:20) Γ[ n (cid:48) + 1 / n + 5 / n (cid:48) − n − (cid:21) / (cid:18) a SDNLO − (cid:2) ( n (cid:48) − a SD, NNLO + ( n − a SD, NNLO (cid:3) +16 (cid:2) ( n (cid:48) − n (cid:48) − a SD, N LO + ( n (cid:48) − n − a SD, N LO + ( n − n − a SD, N LO (cid:3)(cid:19) ; (A-14)For the D ↔ D or D J ↔ D J channels, recalling −→ D M ≡ ( −→∇ ⊗ −→∇ ) M , (cid:104) n (cid:48) ( l = 2 S ) J = 1 M T M T | e r / (cid:20) a D NNLO ←− D · δ ( (cid:126)r ) −→ D + a D N LO ( ←− D ←−∇ · δ ( (cid:126)r ) −→ D + ←− D · δ ( (cid:126)r ) −→∇ −→ D ) (cid:21) e r / | n ( l = 2 S ) J = 1 M T M T (cid:105) = 3215 2 π (cid:20) Γ[ n (cid:48) + 5 / n + 5 / n (cid:48) − n − (cid:21) / (cid:18) a D NNLO − (cid:2) ( n (cid:48) −
1) + ( n − (cid:3) a D N LO (cid:19) ; (A-15)For the D ↔ G channel, recalling −→ G M ≡ (cid:104) (( −→∇ ⊗ −→∇ ) ⊗ ( −→∇ ⊗ −→∇ ) ) ⊗ ( σ (1) ⊗ σ (2)) (cid:105) M , (cid:104) n (cid:48) ( l = 2 S = 1) J = 3 M T M T | e r / a DFN LO ( ←− D · δ ( (cid:126)r ) −→ G + ←− G · δ ( (cid:126)r ) −→ D ) e r / | n ( l = 4 S = 1) J = 3 M T M T (cid:105) =512315 √
15 2 π (cid:20) Γ[ n (cid:48) + 5 / n + 9 / n (cid:48) − n − (cid:21) / a DFN LO ; (A-16)For the P ↔ P or P J ↔ P J channels, (cid:104) n (cid:48) ( l = 1 S ) JM T M T | e r / (cid:20) a P NLO ←−∇ · δ ( (cid:126)r ) −→∇ + a P NNLO ( ←−∇←−∇ · δ ( (cid:126)r ) −→∇ + ←−∇ · δ ( (cid:126)r ) −→∇ −→∇ )+ a P , N LO ←−∇←−∇ · δ ( (cid:126)r ) −→∇ −→∇ + a P , N LO ( ←−∇←−∇ · δ ( (cid:126)r ) −→∇ + ←−∇ · δ ( (cid:126)r ) −→∇ −→∇ ) (cid:21) e r / | n ( l = 1 S ) JM T M T (cid:105) =43 2 π (cid:20) Γ[ n (cid:48) + 3 / n + 3 / n (cid:48) − n − (cid:21) / (cid:18) a P NLO − n (cid:48) −
1) + ( n − a P NNLO ++16 (cid:2) ( n (cid:48) − n − a P , N LO + (( n (cid:48) − n (cid:48) −
2) + ( n − n − a P , N LO (cid:3)(cid:19) ; (A-17)For the P ↔ F channel, recalling −→ F M ≡ (cid:104) ( −→∇ ⊗ ( −→∇ ⊗ −→∇ ) ) ⊗ ( σ (1) ⊗ σ (2)) (cid:105) M , (cid:104) n (cid:48) ( l = 1 S = 1) J = 2 M T M T | e r / (cid:20) a P FNNLO ( ←−∇ · δ ( (cid:126)r ) −→ F + ←− F · δ ( (cid:126)r ) −→∇ ) + a P F, N LO ( ←−∇←−∇ · δ ( (cid:126)r ) −→ F +0 ←− F · δ ( (cid:126)r ) −→∇ −→∇ ) + a P F, N LO ( ←−∇ · δ ( (cid:126)r ) −→∇ −→ F + ←− F ←−∇ · δ ( (cid:126)r ) −→∇ ) (cid:21) e r / | n ( l = 3 S = 1) J = 2 M T M T (cid:105) =3235 √
14 2 π (cid:20) Γ[ n (cid:48) + 3 / n + 7 / n (cid:48) − n − (cid:21) / (cid:18) a P FNNLO − (cid:2) ( n (cid:48) − a P F, N LO + ( n − a P F, N LO (cid:3)(cid:19) ; (A-18)And finally, for the F ↔ F or F J ↔ F J channels, recalling −→ F M ≡ ( −→∇ ⊗ ( −→∇ ⊗ −→∇ ) ) M , (cid:104) n (cid:48) ( l = 3 S ) JM T M T | e r / (cid:20) a F N LO ←− F · δ ( (cid:126)r ) −→ F (cid:21) e r / | n ( l = 3 S ) JM T M T (cid:105) π (cid:20) Γ[ n (cid:48) + 7 / n + 7 / n (cid:48) − n − (cid:21) / a P FN LO . (A-19)In each case the general matrix element of the effective interaction, for edge amd nonedge states, can then beexpanded in terms of Eqs. (A-13-A-19), (cid:104) n (cid:48) ( l (cid:48) S ) JM T M T | EE − T Q ¯ O EE − QT | n ( lS ) JM T M T (cid:105) = (cid:88) i,j =0 (cid:101) g j ( − κ ; n (cid:48) , l (cid:48) ) (cid:101) g i ( − κ ; n, l ) (cid:104) n (cid:48) + j ( l (cid:48) S ) JM T M T | ¯ O | n + i ( lS ) JM T M T (cid:105) . (A-20) Free Green’s Function Results:
An equivalent set of results can be derived by making use of Eq. (36) (cid:104) (cid:126)r | EE − QT | n l m l (cid:105) ≡ (cid:104) (cid:126)r | (cid:101) α nlm l (cid:105) = − Y lm l (Ω r ) (cid:20) √ r I l +1 / ( κr ) (cid:90) ∞ r r (cid:48) dr (cid:48) √ r (cid:48) K l +1 / ( κr (cid:48) ) (cid:104) (cid:126)r (cid:48) | α nlm l > + 1 √ r K l +1 / ( κr ) (cid:90) r r (cid:48) dr (cid:48) √ r (cid:48) I l +1 / ( κr (cid:48) ) (cid:104) (cid:126)r (cid:48) | α nlm l > ≡ (cid:101) R αnl ( r ) Y lm l (Ω r ) (A-21)where the source term in the Green’s function is | α nlm l (cid:105) ≡ (cid:2) − κ − (2 n + l − / − (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 / (cid:3) | nlm l > − (cid:112) ( n − n + l − / | n − lm l (cid:105)− (cid:112) n ( n + l + 1 / P [ n + 1 , l ] | n + 1 lm l (cid:105) . (A-22)The inclusion of the last term makes this equation valid for non-edge as well as edge states: P [ n + 1 , l ] = 1 if andonly if | n + 1 l (cid:105) belongs to P . The reproduction of simple HO nonedge states is a helpful numerical check. Here (cid:104) (cid:126)r | α nlm l (cid:105) ≡ R αnl ( r ) Y lm l (Ω r ).The first task is to evaluate ( −→∇ ) p e r / on this wave function. The inclusion of e r / – which eliminates mixingamong HO states and simplifies other HO expressions – is a bit of an annoyance in the Green’s function case, generatinga series of surface terms. One finds the generic result, analogous to Eq. (A-7),( −→∇ ) p e r / (cid:101) R αnl ( r ) Y lm l (Ω r ) = e r / Y lm l (Ω r ) × (cid:26)(cid:20) − f pI ( κ , r ) 1 √ r I l +1 / ( κr ) − f pI (cid:48) ( κ , r )2 r ddr (cid:18) ( 1 √ r I l +1 / ( κr ) (cid:19)(cid:21) (cid:90) ∞ r x dx √ x K l +1 / ( κx ) R αnl ( x )+ (cid:20) − f pK ( κ , r ) 1 √ r K l +1 / ( κr ) − f pK (cid:48) ( κ , r )2 r ddr (cid:18) ( 1 √ r K l +1 / ( κr ) (cid:19)(cid:21) (cid:90) r x dx √ x I l +1 / ( κx ) R αnl ( x ) (cid:27) + f pα ( κ , r ) e r / R αnl ( r ) Y lm l (Ω r ) + f pα (cid:48) ( κ , r )2 r ddr (cid:0) e r / R αnl ( r ) Y lm l (Ω r ) (cid:1) + f pα (cid:48)(cid:48) ( κ , r ) −→∇ (cid:0) e r / R αnl ( r ) Y lm l (Ω r ) (cid:1) + f pα (cid:48) ( κ , r )2 r ddr −→∇ (cid:0) e r / R αnl ( r ) Y lm l (Ω r ) (cid:1) + f pα (cid:48) ( κ , r ) −→∇ (cid:0) e r / R αnl ( r ) Y lm l (Ω r ) (cid:1) + f pα (cid:48) ( κ , r )2 r ddr −→∇ (cid:0) e r / R αnl ( r ) Y lm l (Ω r ) (cid:1) + ... (cid:27) (A-23)where each f ( κ , r ) is a polynomial that can be evaluated using standard gradient formulas for spherical harmonics.At N LO f pα (cid:48) is the highest contributing surface term. This form allows one to use Eq. (A-7) to evaluate repeatedoperations of −→∇ . One can show, by expanding this expression around r = 0, that the leading order terms, which come1from the second line above and from the surface terms in line four and the following lines, are proportional to the solidharmonics r l Y lm (Ω r ), with corrections involving additional powers of r . As the lowest order contributing operators,( −→∇ q ) q , can annihilate only the former at the origin, in fact this expression effectively simplifies for contact-gradientpurposes( −→∇ ) p e r / (cid:101) R αnl ( r ) Y lm l (Ω r ) eff. −→ − r l Y lm l (Ω r )(2 l + 1)!! × (cid:26)(cid:20) f pI ( κ ,
0) + 2 lf pI (cid:48) ( κ , (cid:21)(cid:114) π (cid:18) κ l +1 / (cid:90) ∞ x dx √ x K l +1 / ( κx ) R αnl ( x ) (cid:19) + 2 l +1 (cid:20) n + l + 1 / π ( n − (cid:21) × (cid:20)(cid:20) f pα ( κ ,
0) + 2 lf pα (cid:48) ( κ , (cid:21)(cid:18) κ + 3 n + l − / (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 /
2) + ( n + l + 1 / P [ n + 1 , l ] (cid:19) − (cid:20) f pα (cid:48) ( κ ,
0) + 2 lf pα (cid:48) ( κ , (cid:21)(cid:18) ( n − (cid:2) κ + 3 n + l − / (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 / (cid:3) + n ( n + l + 1 / P [ n + 1 , l ] (cid:19) +16 (cid:20) f pα (cid:48) ( κ ,
0) + 2 lf pα (cid:48) ( κ , (cid:21) ( n − n − (cid:2) κ + 3 n + l − / (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 / (cid:3) + n ( n − n + l + 1 / P [ n + 1 , l ] (cid:21)(cid:27) (A-24)For the cases of interest here (through N LO), the required nonzero polynomials are f pI ( κ ,
0) + 2 lf pI (cid:48) ( κ ,
0) = p =03 + κ + 2 l p =115 + 10 κ + κ + 4 l ( l + 1) + 2 l (6 + 2 κ ) p =2105 + 105 κ + 21 κ + κ + 4 l ( l + 1)(13 + 3 κ ) + 2 l (45 + 30 κ + 3 κ + 4 l ( l + 1)) p =3 f p =1 α ( κ ,
0) + 2 lf p =1 α (cid:48) ( κ ,
0) = f p =2 α (cid:48) ( κ ,
0) + 2 lf p =2 α (cid:48) ( κ ,
0) = f p =3 α (cid:48) ( κ ,
0) + 2 lf p =3 α (cid:48) ( κ ,
0) = 1 f p =2 α ( κ , lf p =2 α (cid:48) ( κ ,
0) = 7+ κ +2 l f p =3 α (cid:48) ( κ , lf p =3 α (cid:48) ( κ ,
0) = 11+ κ +2 lf p =3 α ( κ ,
0) + 2 lf p =3 α (cid:48) ( κ ,
0) = 57+18 κ + κ +4 l ( l +1)+4 l (6+2 κ )The analog of Eq. (A-9) then becomes( −→∇ ) p ( −→∇ q ) q e r / R αnl ( r ) Y lm l (Ω r ) (cid:12)(cid:12) (cid:126)r =0 = − δ lq (cid:115) l !4 π (2 l + 1)!! × (cid:26)(cid:20) f pI ( κ ,
0) + 2 lf pI (cid:48) ( κ , (cid:21)(cid:18)(cid:114) π κ l +1 / (cid:90) ∞ x dx √ x K l +1 / ( κx ) R αnl ( x ) (cid:19) + 2 l +1 (cid:20) n + l + 1 / π ( n − (cid:21) × (cid:20)(cid:20) f pα ( κ ,
0) + 2 lf pα (cid:48) ( κ , (cid:21)(cid:18) κ + 3 n + l − / (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 /
2) + ( n + l + 1 / P [ n + 1 , l ] (cid:19) − (cid:20) f pα (cid:48) ( κ ,
0) + 2 lf pα (cid:48) ( κ , (cid:21)(cid:18) ( n − (cid:2) κ + 3 n + l − / (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 / (cid:3) + n ( n + l + 1 / P [ n + 1 , l ] (cid:19) +16 (cid:20) f pα (cid:48) ( κ ,
0) + 2 lf pα (cid:48) ( κ , (cid:21) ( n − n − (cid:2) κ + 3 n + l − / (cid:101) g ( − κ ; n, l ) (cid:112) n ( n + l + 1 / (cid:3) + n ( n − n + l + 1 / P [ n + 1 , l ] (cid:21)(cid:27) (A-25)2where ( −→∇ q ) q r l Y lm l (Ω r )(2 l + 1)!! (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)r =0 = δ lq (cid:115) l !4 π (2 l + 1)!! (A-26)has been used, and where the remaining integral can be evaluated using (cid:114) π κ l +1 / (cid:90) ∞ x dx √ x K l +1 / ( κx ) R αnl ( x ) = − (cid:112) n − n + l + 1 / n (cid:88) m =0 ( − m m !( n − m )!Γ[ l + 3 / m ] × (cid:20) ( κ + 3 n + l − m − / (cid:101) g ( − κ ; nl ) (cid:112) n ( n + l + 1 / n − m ) + n ( n + l + 1 / P [ n + 1 , l ] (cid:21) × l (cid:88) i =0 ( √ κ ) l − i i ( l + i )! i !( l − i )! (cid:20) −√ κ Γ[ m + 3 / l − i ) / F [ m + 3 / l − i ) /
2; 3 / κ / m + 1 + ( l − i ) / F [ m + 1 + ( l − i ) /
2; 1 / κ / (cid:21) . (A-27)With the effects of contact-gradient operators on wave functions of the form of Eq. (A-1) thus determined, theseresults can be plugged into Eqs. (A-11) and (A-12) to produce the needed expressions for matrix elements.The contact-gradient matrix elements needed through N LO have been evaluated using both the HO expansion andthe free Green’s function. The resulting agreement is a nice check. The sum over HO excitations is truncated at some N : at N LO the choice N =400 will give results accurate to at least 0.01% for the most sensitive N LO operators,which depend on higher derivatives at the origin. The Mathematica script for this summation is simple and efficient,so there is no practical limit to the N s that can be handled. Similarly, the Green’s function expressions can beevaluated very easily: this is the recommended scheme for evaluating edge-state matrix elements for contact-gradientoperators.Eq. (A-21) is also an efficient way to generate the wave function at all values of r , which is needed for evaluationsof edge-state matrix elements of the bare V . One finds (cid:101) R nl ( r ) = (cid:112) n − n + l + 1 / n (cid:88) m =0 ( − m m !( n − m )!Γ[ l + 3 / m ] × (cid:20)(cid:0) n − m (cid:1)(cid:18) κ + 3 n − m + l − / (cid:101) g ( − κ ; nl ) (cid:112) n ( n + l + 1 / (cid:19) + n ( n + l + 1 / P [ n + 1 , l ] (cid:21) × (cid:20) √ r I l +1 / ( κr ) G [ κ, l, m, r ] + 1 √ r K l +1 / ( κr ) G [ κ, l, m, r ] (cid:21) (A-28)where G [ κ, l, m, r ] = e − κr − r / √ π l (cid:88) i =0 ( l + i )! i !( l − i )! 2 m +( l − i ) / (2 κ ) i +1 / m +1+ l − i (cid:88) j =0 (2 m + 1 + l − i )!(2 m + 1 + l − i − j )! j ! ( r √ m +1+ l − i − j × (cid:104) −√ κ + r )Γ[1 + j/ F [1 + j/
2; 3 /
2; ( κ + r ) /
2] + Γ[(1 + j ) / F [(1 + j ) /
2; 1 /
2; ( κ + r ) / (cid:105) (A-29)and G [ κ, l, m, r ] = 1 √ π l (cid:88) i =0 ( l − i )! i !( l − i )! 2 m +( l − i ) / (2 κ ) i +1 / (cid:20) (( − i + ( − l ) κ √ m + 3 / l − i ) / F [ m + 3 / l − i ) /
2; 3 / κ / − i − ( − l )Γ[ m + 1 + ( l − i ) / F [ m + 1 + ( l − i ) /
2; 1 / κ / − e − r / m +1+ l − i (cid:88) j =0 (2 m + 1 + l − i )!(2 m + 1 + l − i − j )! j ! ( r √ m +1+ l − i − j (cid:18) ( − i e κr √ κ − r ) Γ[1 + j/ F [1 + j/
2; 3 /
2; ( κ − r ) / − l e − κr √ κ + r ) Γ[1 + j/ F [1 + j/
2; 3 /
2; ( κ + r ) / − i e κr Γ[(1 + j ) / F [(1 + j ) /
2; 1 /
2; ( κ − r ) / − ( − l e − κr Γ[(1 + j ) / F [(1 + j ) /
2; 1 /
2; ( κ + r ) / (cid:19)(cid:21) ..