Renormalization of chiral two-pion exchange NN interactions. Momentum vs. coordinate space
D. R. Entem, E. Ruiz Arriola, M. Pavon Valderrama, R. Machleidt
aa r X i v : . [ nu c l - t h ] J a n Renormalization of chiral two-pion exchange NN interactions.Momentum vs. coordinate space.
D. R. Entem, ∗ E. Ruiz Arriola, † M. Pav´on Valderrama, ‡ and R. Machleidt Grupo de F´ısica Nuclear, IUFFyM, Universidad de Salamanca, E-37008 Salamanca,Spain Departamento de F´ısica At´omica, Molecular y Nuclear,Universidad de Granada, E-18071 Granada, Spain. Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany Department of Physics, University of Idaho, Moscow, Idaho 83844 § (Dated: November 12, 2018)The renormalization of the chiral np interaction in the S channel to N3LO in Weinberg countingfor the long distance potential with one single momentum and energy independent countertermis carried out. This renormalization scheme yields finite and unique results and is free of shortdistance off-shell ambiguities. We observe good convergence in the entire elastic range below pionproduction threshold and find that there are some small physical effects missing in the purely pionicchiral NN potential with or without inclusion of explicit ∆ degrees of freedom. We also studythe renormalizability of the standard Weinberg counting at NLO and N2LO when a momentumdependent polynomial counterterm is included. Our numerical results suggest that the inclusion ofthis counterterm does not yield a convergent amplitude (at NLO and N2LO). PACS numbers: 03.65.Nk,11.10.Gh,13.75.Cs,21.30.Fe,21.45.+vKeywords: NN interaction, One and Two Pion Exchange, Renormalization.
I. INTRODUCTION
The modern Effective Field Theory (EFT) analysis ofthe NN interaction using chiral symmetry as a constrainthas a recent but prolific history [1, 2] (for comprehen-sive reviews see e.g. Ref. [3, 4, 5]). Most theoreticalsetups are invariably based on a perturbative determina-tion of the chiral potential [6, 7, 8, 9, 10, 11, 12, 13, 14]and the subsequent solution of the scattering prob-lem [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Actu-ally, the theory encounters many problems in the lowpartial waves and in particular in the s − waves (see how-ever Ref. [26] for a more optimistic view). Indeed, thereis at present an ongoing debate on how an EFT pro-gram should be sensibly implemented within the NN con-text and so far no consensus has been achieved (see e.g.[27, 28, 29, 30]). The discussion is concerned with theissue of renormalization vs. finite cutoffs, a priori (powercounting) vs. a posteriori error estimates or the appli-cability of perturbation theory both on a purely shortdistance theory or around some non-perturbative dis-torted waves. At the moment, it seems fairly clear thatan EFT scheme with a cutoff-independent and system-atic perturbative power counting for the S-matrix (theso-called KSW counting) fails [31, 32, 33, 34]. On theother hand, the original EFT inspired scheme [1, 2] (theso-called Weinberg counting) has recently been shownto produce many results which turn out to be strongly ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] cutoff dependent [27, 35] and hence to be incompatiblewith renormalizability . Thus, some acceptable com-promise must be made. Actually, the chosen approachbetween this dichotomy depends strongly on the pursuedgoals and it is fair to say that any choice has both ad-vantages and disadvantages. In any case, the reason forboth failures can be traced back to the nature of the long-distance chiral potentials; while pion-exchange potentialsfall off exponentially at long distances they include strongpower-law singularities at short distances. Those sin-gularities become significant already at distances com-parable with the smallest de Broglie wavelength probedin NN scattering below pion production threshold. Ob-viously, any development of NN interactions based onchiral dynamics will presumably require a deeper under-standing and proper interpretation of the peculiarities ofthese highly singular chiral potentials. Although singu-lar potentials where first analyzed many years ago [36](for an early review see e.g. [37] and for a more updatedview within an EFT context see [38]), their short dis-tance singular character within the NN interaction hasseriously been faced more recently within a renormal-ization context for the one-pion exchange (OPE) poten-tial [27, 29, 39, 40, 41, 42, 43, 44, 45, 46] and the two-pionexchange (TPE) potential [35, 47, 48].In the np scattering problem the S channel is veryspecial since the scattering length is unnaturally largeas compared to the range of the strong interaction, α = − . ≫ /m π = 1 . In this paper we refer to renormalizability in the sense of param-eterizing the short distance physics by a potential which matrixelements in momentum space are a polynomial in the momenta. at zero energy the wave function probes relatively shortdistance components of the chiral potential [35] . Higherenergies become even more sensitive to short distance in-teractions. Consequently, this channel looks like an idealplace to learn about the size of the most relevant shortrange corrections to the NN force in the elastic scatter-ing region. Actually, in the S channel, most EFT in-spired schemes yield at leading order (LO) (which con-sists of OPE plus a nonderivative counterterm) an al-most constant phase of about 75 o around k = 250 MeV,and an effective range of r LO0 = 1 . r exp0 = 2 . It is quiteunbelievable that such large changes can be reliably ac-commodated by perturbation theory starting from thisLO result despite previous unsuccessful attempts treat-ing OPE and TPE perturbatively [31, 32, 33, 34]. Ac-tually, for the singlet channel case, short distance com-ponents are enhanced due to the large value of the scat-tering length and the weakness of the OPE interactionin this channel. This is why TPE contributions havebeen treated with more success in a non-perturbativefashion [2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].Despite phenomenological agreement with the data, theinclusion of finite cutoffs suggests that there might besome regulator dependence in those calculations.In a series of recent papers [35, 44, 48], two of us(M.P.V. and E.R.A.) have proposed not only to iteratebut also to renormalize to all orders the NN chiral po-tential within a long distance expansion. By allowing theminimal number of counterterms to yield a finite result,long-distance regulator-independent correlations are es-tablished. In practice, the potential must be computedwithin some power counting scheme. While the potentialis used within Weinberg’s power counting to LO, next-to-leading order (NLO), and next-to-next-to-leading or-der (N2LO), we only allow for those counterterms whichyield a finite and unique scattering amplitude. In the S np channel a single energy and momentum independentcounterterm C is considered which is determined by ad- This can be best seen by means of the effective range formula r = 2 Z ∞ dr " u ( r ) − „ − rα « where u ( r ) is the zero energy wave function, fulfilling the asymp-totic condition u ( r ) → − r/α . Most of the integrand is locatedin the region around r = 1fm which is in between OPE and TPEranges. Moreover, the low energy theorem of Ref. [35] allows towrite r = A + B/α + C/α which in the extreme limit α → ∞ yields r → A . Numerically it is found that A is far more de-pendent than B and C when being evaluated at LO, NLO andNNLO. In fact, these high quality potential models yield slightly smallervalues, r ≈ . justing the physical scattering length. In the derivationof this result the mathematical requirements of complete-ness and self-adjointness for the renormalized quantummechanical problem for a local chiral potential play a de-cisive role. This surprising result is in contrast to thestandard Weinberg counting where an additional coun-terterm C is included already at NLO. This C coun-terterm could, in fact, be determined by fitting the ex-perimental value of the effective range; the physics of C is to provide a short distance contribution to the ef-fective range in addition to the contribution from theknown long distance chiral potential. Within this con-text it is remarkable that according to Ref. [35], wheresuch a short distance contribution vanishes (or equiva-lently C = 0 when the cut-off is removed), rather ac-curate values are predicted yielding r NLO0 = 2 .
29 fm and r N2LO0 = 2 .
86 fm after renormalization . This latter valueis less than 3% larger than the experimentally acceptedvalue and it suggests that most of the effective rangeis saturated by N2LO TPE contributions and calls forpinning down the remaining discrepancy. This trend toconvergence and agreement is also shared by higher orderslope parameters in the effective range expansion with-out strong need of specific counterterms although there isstill room for improvement. Motivated by this encourag-ing result, one goal of the present paper is to analyze thesize of the next-to-next-to-next-to-leading order (N3LO)corrections to the results found in [35].The calculations in Ref. [35, 44, 48, 52] exploit explic-itly the local character of the chiral potential by conduct-ing the calculations in coordinate space which makes theanalysis more transparent. Many results, in particularthe conditions under which a renormalized limit exists,can be established a priori analytically. Moreover, thehighly oscillatory character of wave functions at shortdistances is treated numerically using efficient adaptive-step differential equations techniques. This situation con-trasts with momentum space calculations where, with theexception of the pion-less theory, there is a paucity ofanalytical results, and one must mostly rely on numeri-cal methods. Moreover, the existence of a renormalizedlimit is not obvious a priori and one may have to resortto some trial and error to search for counterterms. Fi-nally, renormalization conditions are most naturally for-mulated at zero energy for which the momentum spacetreatment may be challenging, at times. Of course, be-sides these technical issues, there is no fundamental dif-ference between proceeding in momentum or coordinatespace, particularly after renormalization, provided thesame renormalization conditions are specified, since dis-parate regulators stemming from either space are effec-tively removed. Indeed, we will check agreement for thephase-shifts determined in different spaces whenever sucha comparison becomes possible. This equivalence is in it-self a good motivation for renormalization.However, at N3LO some unavoidable non-localities ap-pear in the chiral long distance potential. Although theycould be treated in configuration space, we adopt here amomentum space treatment. This will also allow us toanswer an intriguing question which was left open in thecoordinate space analysis of previous works [35, 44, 48],namely, the role played by the conventional momentum-polynomial representations of the short distance interac-tion used in most calculations [15, 16, 18, 19, 20, 21, 22,23, 24, 25] in the renormalization problem. More specifi-cally, Ref. [35] showed that taking C = 0 was consistent,and a regularization scheme exists where a fixed C wasirrelevant, but could not discriminate whether C = 0was inconsistent as far as it was readjusted to the ef-fective range parameter for any cut-off value. In thisregard, the present paper yields a definite answer mak-ing the surprising agreement of the effective range foundin Ref. [35] an inevitable mathematical consequence ofrenormalization.The paper is organized as follows. The problem isstated in Sec. II where a general overview of the renor-malization problem is given both in momentum as wellas coordinate space. In Sec. II B, we particularize themomentum space formulation of the scattering problemwith counterterms for the S channel within a sharpthree-momentum cutoff scheme. Likewise, in Sec. II Cwe proceed similarly in the coordinate space formula-tion within a boundary condition regularization with ashort distance cutoff r c . In Sec. III, we discuss somefeatures of both coordinate and momentum space for-mulations in the pion-less theory and try to connect thehigh momentum cutoff Λ with the short distance radialcutoff r c . This allows a one-to-one mapping of coun-terterms in both spaces which will prove useful later onin the pion-full theory. The identification between thesharp momentum cutoff and the short distance radiusfound in the pion-less theory is discussed further in Ap-pendix A in the presence of a long distance potential inthe light of the Nyquist theorem. In Sec. IV, we come tothe central discussion on N3LO corrections to the phaseshifts when the scattering amplitude is renormalized withonly one short distance counterterm. A wider perspec-tive is achieved by further considering the role of explicit∆-excitations in intermediate states and the subsequentone-counterterm renormalization of the scattering ampli-tude. We also discuss the role of three pion exchange aswell as how the results depend on the renormalizationscheme used to compute the potential based on cut-offindependent counterterms. The mathematical justifica-tion for using just one counterterm is provided in Sec. Vwhere the standard Weinberg scheme is pursued both in We mean of course the case when the cut-off is being removed.The essential issue is whether or not one can fix by a short dis-tance potential which is a polynomial in the momenta the effec-tive range independently on the potential and remove the cut-offat the same time. Of course, the very definition of the poten-tial is ambiguous and requires a specific choice on the polynomialparts. Technically, we find that any fixed , cut-off indendependent C , becomes irrelevant in the limit Λ → ∞ (see below). momentum and coordinate space at NLO and N2LO andshown to potentially have some problems. Finally, inSec. VI, we summarize our main points. II. THE RENORMALIZATION PROBLEMA. General overview and main results
Let us define the scope and goals of the present work.The standard non-perturbative formulation of the renor-malization problem starts with an effective Lagrangianor Hamiltonian (see e.g. [1, 2] and [3, 4] and referencestherein), from which a certain set of irreducible Feynmandiagrams (usually up to a certain order) is calculated.These irreducible diagrams are defined to represent a po-tential V . The potential is then inserted into a scatteringequation where it is iterated infinitely many times or, inother words, re-summed non-perturbatively. In the CMframe, where the np kinetic energy is given by E = p /M ,with M = 2 µ np = 2 M n M p / ( M p + M n ), the scatteringprocess is governed by the Lippmann-Schwinger equation T = V + V G T , (1)with V the potential operator and G = ( E − H ) − theresolvent of the free Hamiltonian. The outgoing bound-ary condition corresponds to E → E + i0 + . Using thenormalization h ~x | ~k i = e i ~k · ~x / (2 π ) / one has h ~k ′ | T ( E ) | ~k i = h ~k ′ | V | ~k i + Z Λ d q h ~k ′ | V | ~q ih ~q | T ( E ) | ~k i E − ( q / µ ) . (2)Here Λ means a generic regulator and represents the scalebelow which all physical effects are taken into account explicitly . The degrees of freedom which are above Λare taken into account implicitly by including a suitablecutoff dependence in the potential. The precise equationgoverning this cutoff dependence was described and stud-ied in some detail in Ref. [53] with particular emphasison infrared fixed points. We will analyze the cutoff de-pendence below focusing on the ultraviolet aspects of theinteraction.Motivated by the low energy nature of the effectivetheory, the potential is usually separated into short andlong distance components in an additive form h ~k ′ | V | ~k i = V S ( ~k ′ , ~k ) + V L ( ~k ′ , ~k ) , (3)where the long distance contribution is usually given bysuccessive pion exchanges V L ( ~k ′ , ~k ) = V π ( ~k ′ , ~k ) + V π ( ~k ′ , ~k ) + . . . , (4)and the short distance component is characterized by apower series expansion in momentum V S ( k ′ , k ) = C + C ~k · ~k ′ + C ( ~k + ~k ′ ) + . . . , (5)where for simplicity we assume a spin singlet channel .Note that C and C contribute to s-waves while C con-tributes to p-waves, and so on. One should face the factthat, although the decomposition given by Eq. (3) is per-turbatively motivated and seems quite natural, the ad-ditivity between short and long range forces is actuallyan assumption which has important consequences, as wewill see. In principle, for a given regularization scheme charac-terized by a cutoff Λ, the counterterms C , C , C and soon are determined by fixing some observables. One nat-urally expects that the number of renormalization con-ditions coincides with the number of counterterms in away that all renormalization conditions are fully uncor-related. The statement of UV-renormalizability is thatsuch a procedure becomes always possible when the cut-off Λ is removed by taking the limit Λ → ∞ . This maynot be the case as there may appear redundant contribu-tions (see the discussion below in Sec. III) meaning thatone counterterm or counterterm combination can takeany value. Another possible situation is just the opposite;one may want to impose more renormalization conditionsthan possible. In this case some counterterms or coun-terterm combinations are forbidden. Rather than beingintricate mathematical pastimes, these features have al-ready been investigated recently [27, 35] for large cutoffscasting some doubt on the regulator independence of theoriginal proposal [1, 2].Even if one admits generically Eq. (3) as well as Eq. (4)and Eq. (5), it is not obvious how many terms should beconsidered and whether there is a clear way of defining aconvergence criterium or identifying a convergence pat-tern. It is fairly clear that, on physical grounds, oneshould consider an expansion of the potential that startsat long distance and decreases in range as the numberof exchanged particles increases. Note, however, that,while OPE is well-defined, the general form of the TPEpotential is not uniquely determined. Within this con-text, one of the main attractive features of the EFTapproach has been the definition of a power countingscheme which provides a hierarchy and an a priori cor-relation between long and short range physics. In thepresent paper, we will assume Weinberg power count-ing [6, 7, 8, 9, 10, 11, 12, 13, 14] (see below for a moreprecise definition), where the long distance potential isdetermined in a dimensional power expansion.The long distance component V L is obtained by par-ticle exchanges and, in some simple cases, depends only More detailed expressions including spin triplet channels can belooked up e.g. in Ref. [4]. Specifically, Eq. (3) does not foresee for instance terms of theform V ( p ′ , p ) C (Λ), i.e. terms which are not polynomial but in-fluence the renormalization process. Of course, once additivityis relaxed there are many possible representations, in particularfor the short distance components. on the momentum transfer . In such a case, if we for-mally take a Fourier transformation of the long distancepotential V L ( ~x ) = Z d p (2 π ) V L ( ~p ) e i~p · ~x , (6)and take the limit Λ → ∞ one has the standardSchr¨odinger equation in coordinate space, − M ∇ Ψ k ( ~x ) + V ( ~x )Ψ k ( ~x ) = E Ψ k ( ~x ) , (7)where the coordinate space potential is V ( ~x ) = V L ( ~x ) + C δ (3) ( ~x ) + C ~ ∇ δ (3) ( ~x ) ~ ∇ + C h ∇ δ (3) ( ~x ) + δ (3) ( ~x ) ∇ i + . . . . (8)The whole discussion, which has been carried on foryears now, concerns the precise meaning of these deltaand derivatives of delta interactions, particularly whena long distance potential is added to the short distanceone. A crucial finding of the present paper based on a di-rect analysis in momentum space is that non-perturbativerenormalization imposes restrictions on the number ofterms and form of the short distance potential whichdepend also on the particular long distance potential.Some of these restrictions were discussed in previousworks [27, 35, 44]. Remarkably these new renormaliz-ability restrictions were conjectured in coordinate spacein Ref. [28, 35, 44] for the S channel based on self-adjointness and completeness of states and apply to theTPE potential; a single C = 0 counterterm is allowedwhile two counterterms, C = 0 and C = 0, are forbid-den . B. Momentum space formulation
In the S channel the scattering process is governedby the Lippmann-Schwinger equation T ( k ′ , k ) = V ( k ′ , k ) + Z Λ0 dqV ( k ′ , q ) q Mp − q + i + T ( q, k ) , (9)where T ( k ′ , k ) and V ( k ′ , k ) are the scattering amplitudeand the potential matrix elements, respectively, betweenoff-shell momentum states k and k ′ in that channel andthe sharp three-momentum cutoff Λ represents the scalebelow which all physical effects are taken into account This assumption will be relaxed immediately below when dis-cussing the Weinberg counting in the S channel. Again, we mean cut-off dependent counterterms designed to fitphysical observables. explicitly . From the on-shell scattering amplitude thephase shift can be readily obtained T ( p, p ) = − πM p e iδ sin δ ( p ) . (10)The short range character of the nuclear force impliesthat at low energies one has the effective range expansion(ERE) p cot δ ( p ) = − M π Re (cid:20) T ( p, p ) (cid:21) = − α + 12 r p + v p + v p + . . . (11)where α is the scattering length, r the effective rangeand v , v etc. are slope parameters.In the S channel, the potential is decomposed as thesum of short and long range pieces V ( k ′ , k ) = V S ( k ′ , k ) + V L ( k ′ , k ) . (12)In the standard Weinberg counting, the short distancecontribution is written as follows V S ( k ′ , k ) = C (Λ) + ( k + k ′ ) C (Λ)+ C ′ (Λ) k k ′ + C (Λ)( k + k ′ ) + . . . (13)where the counting is related to the order of the momen-tum which appears explicitly. The long distance compo-nent of the potential is taken to be the sum of explicitpion exchanges V L = V π + V π + V π + . . . (14)where [1] V π = V (0)1 π + V (2)1 π + V (3)1 π + V (4)1 π + . . .V π = V (2)2 π + V (3)2 π + V (4)2 π + . . .V π = V (4)3 π + . . . (15)using dimensional power counting. Ideally, one shoulddetermine the physically relevant long range regulator in-dependent correlations, i.e., long distance effects of simi-lar range. This would amount to consider all nπ exchangeeffects on the same footing, since they yield a long dis-tance suppression ∼ e − nm π r modulo power corrections.At present, the only way how these long distance poten-tials can be systematically computed is by dimensionalpower counting in perturbation theory, as representedschematically in Eqs. (14) and (15).In the standard Weinberg counting one has V LO = V (0) S + V (0)1 π V NLO = V LO + V (2) S + V (2)1 π + V (2)2 π V N2LO = V NLO + V (3)1 π + V (3)2 π V N3LO = V N2LO + V (4) S + V (4)1 π + V (4)2 π + V (4)3 π (16) Note that this counting involves both unknown short-distance physics and chiral long-distance physics in a un-correlated way . Note also that there is no first order con-tribution and that there is no third order contribution tothe short distance potential.Regarding Eq. (12) one should stress that the separa-tion between long and short range contributions to thepotential is not unique. In fact, there is a polynomial am-biguity in the long range part which can freely be trans-ferred to the short distance contribution. However, thenon-polynomial part is unambiguous as it is directly re-lated to the left hand cut of the partial wave amplitudewhich for nπ exchange is located at p = inm π / | p | > m ρ / C. Coordinate space formulation
In coordinate space, the problem in the S channelis formulated as follows [47, 52]. Assuming a local longdistance potential V L ( r ) one has to solve the Schr¨odingerequation − u ′′ p ( r ) + U L ( r ) u p ( r ) = p u p ( r ) , r > r c , (17)where U L ( r ) = 2 µ np V L ( r ) is the reduced potential (infact, the Fourier transformation of V L ( q ) ) and u p ( r ) thereduced wave function for an s-wave state. Here r c isthe short distance cutoff and the reduced wave functionis subject to the boundary condition at r = r c and thestandard long distance free particle behaviour u ′ p ( r c ) u p ( r c ) = p cot δ S ( p ) , (18) u p ( r ) → sin( pr + δ ( p ))sin δ ( p ) . (19)where δ S ( p ) is the short distance phase-shift encoding thephysics for r < r c . In the case of a vanishing long rangepotential U L ( r ) = 0 the phase shift is given by δ S ( p, r c ).On the other hand, if we take δ S ( p ) = 0 we get a standardproblem with a hard core boundary condition, u p ( r c ) = 0which for r c → full phase-shift δ ( p ) and the short distance phase-shift δ S ( p ) can be de-scribed by some low energy approximation, like e.g., aneffective range expansion, p cot δ S ( p ) = − α ,S + 12 r ,S p + . . . (20) p cot δ ( p ) = − α + 12 r p + . . . (21) The best way to recognize the ambiguity is in terms of the spec-tral function representation of the potential [7], where the sub-traction constants can be fixed arbitrarily. In coordinate spacethe non-ambiguous part corresponds to the potential V ( r ) forany non-vanishing radius (see e.g. the discussion in Ref. [35, 48].) where α ,S is the short range scattering length, r ,S theshort range effective range, and α and r the full ones .If we also make an expansion at low energies of the re-duced wave function u p ( r ) = u ( r ) + p u ( r ) + . . . (22)we get the hierarchy of equations − u ′′ ( r ) + U ( r ) u ( r ) = 0 , (23) α ,S u ′ ( r c ) + u ( r c ) = 0 ,u ( r ) → − rα , and − u ′′ ( r ) + U ( r ) u ( r ) = u ( r ) , (24) α ,S u ′ ( r c ) + u ( r c ) = 12 r ,S α ,S u ( r c ) ,u ( r ) → r α (cid:0) r − α r + 3 α r (cid:1) , and so on. The standard way to proceed would be tointegrate the equations for u ( r ), u ( r ), etc. from infinitydownwards, with a known value of α , using Eq. (23) toobtain α ,S and then one can use Eq.(17) together withEq. (19) and Eq. (20) to compute δ ( k ) for any energy witha given truncated boundary condition. This procedureprovides by construction the low energy parameters westarted with and takes into account that the long rangepotential determines the form of the wave function atlong distances. The only parameter in the procedure isthe short distance radius r c , which is eventually removedby taking the limit r c → energy depen-dent but real boundary condition at short distances whicheventually violates self-adjointness. On the other hand,the momentum space formulation allows the discussionof nonlocal long distance potentials and the renormaliza-tion is done in terms of a momentum dependent shortdistance polynomial potential. Although this looks like aself-adjoint problem, we will see that in this formulationthe counterterms may in fact become complex. III. THE RENORMALIZATION PROBLEM FORTHE PION-LESS THEORY
The renormalization of the pion-less theory, i.e., a setof pure contact interactions, has been treated with greatdetail in the literature [54, 55, 56, 57, 58, 59, 60, 61, This is not the only possible short distance representation [52].See Sec. III B for a further discussion on this.
62, 63, 64, 65, 66] although without much considera-tion on how this problem might be embedded into thewider and certainly more realistic situation where thefinite range and short distance singular chiral NN po-tentials are present. In fact, much of the understandingof non-perturbative renormalization within the modernNN context has been tailored after those and furtherstudies based on the non-singular OPE singlet S po-tential [63, 67, 68, 69] plus the standard perturbativeexperience. In previous [35, 43, 44, 47, 48] and in thepresent work, we pursue exactly the opposite goal: wewill only consider renormalization procedures which candirectly be implemented in the presence of long distancepotentials since, after all, contact NN interactions arealways assumed to approximate truly finite range inter-actions in the long wavelength limit. Thus, it is useful toreview here those developments with an eye on the newingredients which appear in the non-perturbative renor-malization of singular pion exchange potentials as ana-lyzed in later sections. In addition, the deduced runningof the counterterms in the contact theory in the infrareddomain serves as a useful starting point when the longdistance pion exchange potential is switched on. Finally,we will also discuss the size of finite cutoff corrections tothe renormalized result depending both on the particularregularization as well as the corresponding representationof the short distance physics. A. Momentum space
Although the previously described momentum spaceframework has extensively been used in the past to de-scribe successfully the data [15, 16, 18, 19, 21, 22, 23, 24,25] with a finite cutoff
Λ it is worth emphasizing somepuzzling features regarding the off-shell ambiguities ofthe short distance potential when finite range corrections,encoded in the C , C etc. counterterms, are included.In momentum space, the pion-less theory correspondsto taking V L ( k ′ , k ) = 0. In such a case the Lippmann-Schwinger equation reduces to a simple algebraic equa-tion [58, 64]. At very small values of the cutoff Λ < m π / V S ( k ′ , k ) = C (Λ), the Lippmann-Schwinger equation (LSE) may be directly solved. Usingthe basic integral J = Z Λ0 dq q p − q + i + = − Λ − i πp p p Λ − p , (25) The result for a different momentum cutoff scheme such as V ( k ′ , k ) → g ( k ′ , Λ) V ( k, k ′ ) g ( k, Λ) corresponds to making the re-placement R Λ0 dq → R dqg ( q, Λ) . In dimensional regularization(minimal subtraction scheme) the integral is just the unitaritypiece, J = − i πp . for a sharp momentum cutoff Λ and 0 ≤ p ≤ Λ, oneobtains for the phase shifts p cot δ ( p ) = − M πC − π + pπ log Λ + p Λ − p . (26)At zero energy, T (0 ,
0) = 2 α /M π and, thus, the runningof C is given by M Λ C (Λ) = − α α − π . (27)In this case, the phase shift is given by p cot δ ( p ) = − α + pπ log Λ + p Λ − p = − α + O ( 1Λ ) , (28)which corresponds to an ERE with r = v = · · · = 0 inthe limit Λ → ∞ . Note that finite cutoff corrections scaleas 1 / Λ. This indicates a relatively slow convergence to-wards the infinite cutoff limit and hence that finite cutoffeffects are quantitatively important and might even be-come a parameter of the theory. Actually, one mightdetermine Λ by fixing the effective range from the first ofEq. (28), r (Λ) = 4 / ( π Λ) = 2 . . T ( k ′ , k ) = T ( p, p ).The running given by Eq. (27) must be used for any cutoff Λ if we want to renormalize in the end. However,thinking of the more general case where finite range cor-rections are relevant such a running is only reliable forvery small cutoffs Λ ≪ π/ α . If we consider also a C (Λ)coefficient in the potential, the corresponding LSE can besolved with the ansatz T ( k ′ , k ) = T ( p ) + T ( p )( k + k ′ ) + T ( p ) k k ′ , (29)which yields a set of three linear equations for T ( p ), T ( p ) and T ( p ). After some algebraic manipulation, thefinal result for the phase shift can then be written in theform p cot δ ( p ) = 10( C M Λ + 3) / ( M π )9( C M Λ − C ) − C ( C M Λ + 6) p − π + pπ log Λ + p Λ − p . (30)Matching at low energies to the ERE, Eq. (11), we getthe running of C and C − α = 10( C M Λ − M π ( − C M Λ + 5 C ) − π r = 50 C (cid:0) C M Λ (cid:1) (cid:0) C M Λ (cid:1) π ( − C + C Λ M ) + 2 π Λ . (31) The first equation allows to eliminate uniquely C infavour of α and C , but as we see there are two branchesfor the solutions. However, we choose the branch forwhich C decouples in the infrared domain, i.e. fulfills C → →
0. In fact, at small cutoffs, one gets forthis branch
M C (Λ)Λ = 2 α Λ π + (cid:18) α Λ π (cid:19) + 23 (cid:18) α Λ π (cid:19) + . . .M C (Λ)Λ = − (cid:18) α Λ π (cid:19) + . . . (32)The factor 2 / C differs already from the coefficient in the case C = 0.Eliminating C and C in favour of α and r , the phaseshift becomes p cot δ ( p ) = − πα ( π − α ) π − α ) + α p ( r π Λ − − π + pπ log Λ + p Λ − p = − α + 12 r p + O ( 1Λ ) . (33)Note that the finite cutoff corrections are, after fixing r ,again O (Λ − ). So, fixing more low energy constants inthe contact theory does not necessarily imply a strongershort distance insensitivity, as one might have naturallyexpected . In other words, the inclusion of a higherdimensional operator such as C does not improve theultraviolet limit, at least in the polynomial representa-tion given by Eq. (13). In Sec. IV we will show, however,that with just one counterterm C the inclusion of pionexchange long distance contributions generates a muchfaster convergence towards the renormalized limit as an-ticipated in Refs. [35, 44, 48] (see also Ref. [52] for aquantitative estimate). In Sec. V we will also show thatwhen a C counterterm is added this scaling behaviour isnot only broken but also the phase shift fails to convergein the limit Λ → ∞ .Thus, we see that one can establish a one-to-one map-ping between the counterterms C , C and the thresholdparameters α and r . Nevertheless, this is done at theexpense of operator mixing, i.e., both C and C are in-tertwined to determine both the scattering length andthe effective range. In other words the cutoff dependenceof C is different depending on the presence of C . As wehave seen this is not a problem since for small cutoffs weexpect the running of C to be fully independent of C We have in mind dispersion relations where any subtraction atzero energy and derivatives thereof of the dispersive part improvethe high energy behaviour and become more insensitive in theultraviolet. As we see this is not the case in the contact pion-lesstheory. and hence on r . However, unlike the one countertermcase, where C = 0, the solutions of Eq. (32) may becomecomplex when α r π Λ − α Λ + 12 α π Λ − π ≤ . (34)For the physical S threshold parameters this happensalready for Λ > Λ c = 382MeV (the other two roots arecomplex). Above this critical value the potential violatesself-adjointness. For r → c → / ( πr ) →∞ . Thus, the cutoff can only be fully removed with aself-adjoint short distance potential if r = 0. This isconsistent with the violation of the Wigner causality con-dition reported in [56, 57, 58, 59]. Note that the violationof self-adjointness is very peculiar since once C and C have been eliminated the phase-shift (33) remains real .One feature in the theory with two counterterms C and C is that the off-shell T − matrix becomes on-shellonly in the infinite cutoff limit, T ( k ′ , k ) = T ( p, p ) + O (cid:0) Λ − (cid:1) . (35)This is unlike the theory with one counterterm C wherethere is no off-shellness at any cutoff. Thus, finite cutoffeffects are also a measure of the off-shellness in this par-ticular problem. This will have important consequencesin Sec. V when attempting to extend the theory with twocounterterms in the presence of the long distance pionexchange potentials since the off-shellness of the shortdistance contribution of the potential becomes an issuein the limit Λ → ∞ .The situation changes qualitatively when the fourthorder corrections depending on two counterterms C and C ′ are considered. Obviously, we cannot fix both C and C ′ simultaneously by fixing the slope parameter v of the effective range expansion, Eq. (11). Clearly, oneexpects some parameter redundancy between C and C ′ or else an inconsistency would arise since a sixth orderparameter in the effective range expansion v should befixed. The situation worsens if higher orders in the mo-mentum expansion are considered due to a rapid pro-liferation of counterterms while there is only one morethreshold parameter for each additional order in the ex-pansion. This required parameter redundancy is actuallya necessary condition for consistency which is manifestlyfulfilled within dimensional regularization but not in thethree-momentum cutoff method . Moreover, it was re-alized some time ago [57, 58] that the finite cutoff regu-larization and dimensional regularizarion in the minimal Nonetheless, off-shell unitarity deduced from sandwiching the re-lation T − T † = − πiT † δ ( E − H ) T between off-shell momentumstates, is violated, since the Schwartz’s reflection principle fails ˆ T ( E + i0 + ) ˜ † = T ( E − i0 + ). This would also have far reachingconsequences for the three body problem, since three body uni-tarity rests on two-body off-shell unitarity and self-adjointnessof three body forces. This operator redundancy has also been discussed on a La-grangean level [70] based on equations of motion and in the ab-sence of long distance interactions ( see also [71]). subtraction scheme yielded different renormalized ampli-tudes for a truncated potential. This non-uniqueness inthe result due to a different regularization happens whena non-vanishing C counterterm is considered. In anycase, the dimensional regularization scheme has neverbeen extended to include the long range part of the TPEpotential which usually appear in the present NN con-text. Thus, for the momentum space cutoffs which havebeen implemented in practice the short distance repre-sentation is somewhat inconsistent at least for a finitevalue of the cutoff Λ.Alternatively, one may choose an energy dependentrepresentation of the short distance physics as V S = C + 2 p C + p (2 C + C ′ ) + . . . (36)In this case the correspondence between countertermsand threshold parameters α , r , v , etc. is exactly one-to-one, and the parameter redundancy is manifest, sincethe on shell T -matrix depends only on the on-shell po-tential. Actually, under dimensional regularization therepresentations of the potential Eq. (13) and Eq. (36)yield the same scattering amplitude. Although this on-shell equivalence is certainly desirable it is also unnatu-ral, if the long distance potential is energy independent.We will nevertheless analyze such a situation in the nextsubsection in coordinate space.The previous discussion highlights the kind of undesir-able but inherent off-shell ambiguities which arise whenfinite range corrections are included in the short distancepotential . In our view these are unphysical ambiguitieswhich have nothing to do with the unambiguous off-shelldependence of the long distance potential. Of course,one way to get rid of the ambiguities is to take the limitΛ → ∞ which corresponds to the case where a truly zerorange theory is approached. However, even for a finitecutoff there is a case where one is free from the ambi-guities, namely when the short distance potential is both energy and momentum independent for s-wave scattering V S ( k ′ , k ) = C (Λ) . (37)The key point is that we allow only this counterterm to becutoff dependent and real, as required by self-adjointness.Of course, the discussion above for the contact theorysuggests the benefits of using just one C countertermbut does not exactly provide a proof that one must takefurther counterterms such as C to zero. The extensionof this analysis to the case of singular chiral potentials inSec. V will yield the definite conclusion that renormaliz-ability is indeed equivalent to take C = 0. This fact becomes more puzzling if the potential V = C ( k + k ′ − p ) is considered. It vanishes on the mass shell k = k ′ = p but nonetheless generates non trivial on shell scattering for thethree-momentum cutoff. B. Coordinate space
The previous renormalization scheme is the momen-tum space version corresponding to the coordinate spacerenormalization adopted in a previous work by two of us(MPV and ERA) [35, 44, 48]. Actually, in the pure con-tact theory, we can relate the renormalization constantwith the momentum space wave function explicitly. Atlarge values of the short distance cutoff r c , the zero en-ergy wave function reads, u ( r c ) = 1 − r c α . (38)Thus, the following relation holds α α − r c = 1 − r c u ′ ( r c ) u ( r c ) . (39)Comparing with Eq. (27), we get M Λ C (Λ) = r c u ′ ( r c ) u ( r c ) (cid:12)(cid:12)(cid:12) r c = π/ − , (40)where the momentum cutoff Λ and the short distancecutoff r c are related by the equationΛ r c = π . Note that for the standard regular solu-tion u ( r ) ∼ r one has a vanishing counterterm C = 0.In contrast, C = 0 for the irregular solution. In the caseof the singular attractive potentials the solution is reg-ular but highly oscillatory and the C takes all possiblevalues for r c →
0. A more detailed discussion on theseissues can be seen in Refs. [43, 47, 72]. Of course, strictlyspeaking both Eq. (40) and Eq. (41) are based on a zeroenergy state, and in the finite energy case we will assumethese relations having the limit r c → → ∞ inmind.Let us now deal with finite energy scattering states.Since there is no potential, U L ( r ) = 0, for r > r c we havethe free wave solution u p ( r ) = sin( pr + δ ( p ))sin δ ( p ) . (42)In the theory with one counterterm, we fix the scatteringlength α by using the zero energy wave function andmatching at r = r c so we get u ′ ( r c ) u ( r c ) = u ′ p ( r c ) u p ( r c ) = p cot( pr c + δ ( p )) , (43) This relation will be shown to hold also in the presence of a localpotential, see Appendix A. yielding 1 r c − α = p cot( pr c + δ ( p )) , (44)and thus p cot δ ( p ) = − p − p ( α − r c ) tan( pr c ) p ( α − r c ) + tan( pr c )= − α + O ( r c ) . (45)This is in qualitative agreement with the momentumspace result when the cutoff is being removed, Eq. (28)and, as we can see, the approach to the renormalizedvalue is similar if the identification r c = π/ (2Λ) is made.Note further that since the boundary condition is energyindependent the problem is self-adjoint and hence orthog-onality between different energy states is guaranteed.The theory with two counterterms where both α and r are fixed to their experimental values opens up a newpossibility, already envisaged in Ref. [52], related to thenon-uniqueness of the result both for a finite cutoff aswell as for the renormalized phase-shift. As pointed outabove, this non-uniqueness was noted first in momen-tum space Ref. [57, 58] when using a finite three dimen-sional cutoff or dimensional regularizarion (minimal sub-traction). Remarkably, within the boundary conditionregularization we will be able to identify both cases asdifferent short distance representations.Actually, when fixing α and r we are led to u ′ p ( r c ) u p ( r c ) = d p ( r c ) = u ′ ( r c ) + p u ′ ( r c ) u ( r c ) + p u ( r c ) + O ( p ) . (46)where u and u are defined in Sec. II C. Note that nowself-adjointness is violated from the beginning due to theenergy dependence of the boudary condition. Within thesecond order approximation in the energy the neglectedterms are O ( p ), so any representation compatible to thisorder might in principle be considered as equally suitable.The close similarity to a Pad´e approximant suggests tocompare the following three possibilities for illustrationpurposes d I p = u ′ ( r c ) + p u ′ ( r c ) u ( r c ) + p u ( r c ) ,d II p = u ′ ( r c ) u ( r c ) + p (cid:20) u ′ ( r c ) u ( r c ) − u ′ ( r c ) u ( r c ) u ( r c ) (cid:21) ,d III p = u ′ ( r c ) u ( r c ) u ′ ( r c ) + p [ u ( r c ) u ′ ( r c ) − u ( r c ) u ′ ( r c )] , (47)and study what happens as the cutoff is removed, r c →
0. Note that all three cases possess by construction thesame scattering length α and effective range r and no0potential for r > r c . Straightforward calculation yields p cot δ ( p ) = − α + 12 r p + O ( r c ) (I) p cot δ ( p ) = − α + 12 r p + O ( r c ) (II) p cot δ ( p ) = − α − α r p + O ( r c ) (III)(48)As we see, all three representations provide the same threshold parameters, but do not yield identical renor-malized amplitude for finite energy. Actually, cases I andII coincide with the three-dimensional cutoff regulariza-tion method (see Sec. III A), whereas case III correspondsto dimensional regularization (MS). Moreover, the finitecutoff corrections to the renormalized result are, gener-ally, O ( r c ) while the rational representation yields correc-tions O ( r c ). These observations survive at higher orderswhen v , v , etc. threshold parameters are further takeninto account. This indicates that not all short distancerepresentations are equally “soft” in the UV-cutoff. Thegeneralization of these results to the case of singular TPEchiral potentials was studied in Ref. [52] and will be alsore-analyzed in Sec. V while discussing the consistency ofthe standard Weinberg’s power counting.In any case, when finite range corrections are consid-ered within the boundary condition regularization, thereare two possible renormalized solutions depending on theparticular parameterization of short distance physics. Anice feature of this regularization is that they can be iden-tified with similar results found already in the momentumspace analysis of Ref. [57, 58] when confronting three-momentum cut-off and dimensional regularization. Wenote also here that no ambiguity arises when the bound-ary condition is assumed to be energy independent, inwhich case self-adjointness is guaranteed. IV. RENORMALIZATION OF PIONEXCHANGES WITH ONE COUNTERTERM
The study of the contact theory in Sec. III providessuggestive arguments why it is highly desirable to carryout a regularization with a single counterterm in the S channel by adjusting it to the physical scattering lengthfor any cutoff value. In this section we want to extendthat study when the long distance chiral potential orga-nized according to Weinberg power counting enters thegame and the cutoff is removed. By taking the cutoff toinfinity, we are actually assuming that all degrees of free-dom not included in the present calculation become in-finitely heavy. This way we expect to learn about missingphysics in a model and regularization independent fash-ion. The traditional strategy of adjusting an increasingnumber of counterterms may obscure the analysis. Inother words, by using this minimal number of countert-erms, we try not to mock up what might be still missing -100-80-60-40-20 0 20 40 0 100 200 300 400 500 600 C ( Λ ) ( G e V - ) Λ (MeV) OPENLON2LON3LOcontact
FIG. 1: LO, NLO, N2LO and N3LO running of the coun-terterm (in GeV − as a function of the cutoff Λ in the S channel for small cutoffs Λ ≤ α = − . V L . in the long range description.In this context there is of course the question of con-vergence or cutoff insensitivity of the phase shift, whenΛ → ∞ , provided we keep at any rate the scatteringlength α to its physical value by suitably adjusting theunique counterterm C (Λ). In coordinate space this isa rather trivial matter if the long distance potential islocal [35, 44, 48], as it happens in the LO, NLO, andN2LO Weinberg counting. The analysis in momentumspace involves detailed large momenta behaviour of theLippmann-Schwinger equation and one must resort totrial and error. Indeed, the N3LO case analyzed belowincludes nonlocalities and it turns out to provide conver-gent results.For numerical calculations, we take the values for the c i and d i parameters appearing in the pion exchange po-tential V π used in Ref. [24], which do a good job for pe-ripheral waves, where re-scattering effects are suppressedand where one is, thus, rather insensitive to cutoff ef-fects. We will only consider TPE contributions to theN3LO potential. A. Renormalized N3LO-TPE
To determine the running of the counterterm we startfrom low cutoffs Λ ≪ m π since the long range part ofthe potential is suppressed and Eq. (27) may be used.Actually, the analytical result is well reproduced by thenumerical method used to solve the Lippmann-Schwinger1 δ ( T l a b = M e V ) Λ (MeV) OPENLON2LON3LO 0 20 40 60 80 100 120 140 160 0 1000 2000 3000 4000 δ ( T l a b = M e V ) Λ (MeV) OPENLON2LON3LO-20 0 20 40 60 80 100 120 140 0 1000 2000 3000 4000 δ ( T l a b = M e V ) Λ (MeV) OPENLON2LON3LO -100-80-60-40-20 0 20 40 60 80 100 0 1000 2000 3000 4000 δ ( T l a b = M e V ) Λ (MeV) OPENLON2LON3LO
FIG. 2: LO, NLO, N2LO, and N3LO convergence of the phase shifts as a function of the momentum cutoff Λ for fixed LABenergies, T LAB = 15 , , ,
300 MeV. The renormalization counterterm C (Λ) is always determined by fixing the scatteringlength to its experimental value α = − . equation in the pure contact theory with C . Once thisidentification has been done, the value of the cutoff isincreased steadily so that the scattering length is alwaysfixed to the experimental value, α = − . .The running of the counterterm C (Λ) is depicted forthe LO, NLO, N2LO, and N3LO potentials in Fig. 1 inthe low cutoff region. As expected, the deviations fromthe simple result of the pure contact theory, Eq. (27),start at Λ ∼ m π for LO because of the 1 π exchange po-tential. For higher cutoffs the NLO, N2LO, and N3LOcounterterm displays a cycle structure very similar towhat has been observed in coordinate space [47].The convergence of the S phase shift for fixed val-ues of the Lab energy, T LAB = 15 , , ,
300 MeV isdisplayed in Fig. 2. Of course, one observes a faster con-vergence for small energies. For the maximal value of T LAB = 300 MeV, cutoff values Λ ∼ o .In agreement with the analytical estimates of Ref. [52], In general, there may appear many solutions for C (Λ) fitting α . They are physically unacceptable unless they behave as MC (Λ) → α /π for Λ → the convergence of the regulated phase-shifts towardstheir renormalized values follows a computable power likepattern, δ ( k ) − δ ( k, Λ) = O (Λ − n/ − ). The more singu-lar the potential at large momenta the faster the con-vergence. Thus, the expected increased insensitivity atshort distances is indeed confirmed.Finally, the renormalized S phase shift is presentedin Fig. 3 for LO, NLO, N2LO, and N3LO. As a check ofthe present calculation in momentum space, let us men-tion that we reproduce the coordinate space renormalizedphase shifts of [35] at LO, NLO, and N2LO. For instance,at the maximal CM momentum of p = 400 MeV, themaximal discrepancy between the coordinate space andmomentum space phase shifts is less than half a degreewhen r c = 0 . T LAB = 300MeV,one has δ LO = 72 . o , δ NLO = 6 . o , δ N2LO = − . o ,and δ N3LO = − . o . However, there is still a discrep-ancy with the Nijmegen PWA result which at this energyis δ Nijm = − . ± . o for np scattering.To have an idea on the uncertainty of the calcu-lated phase shift, we vary the scattering length α = − . g πNN = 13 . g A =1 . g A only. In addition, forthe chiral constants we take the central values used in[24] which provided a good description for the peripheral2 -40-20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ ( Λ = G e V ) p (MeV) OPENLON2LON3LONIJ
FIG. 3: LO, NLO, N2LO, and N3LO renormalized phaseshifts in the S channel as a function of the CM momen-tum compared to the Nijmegen Partial Wave Analysis [49].Only one counterterm is used and fixed by the physical valueof the scattering length. waves and for the uncertainties we assume as an educatedguess those from the πN study [73]. The only exceptionis c , for which the error from πN [73] is much smallerthan the systematic discrepancy with the NN determina-tion [24]. So it is more realistic to take the systematicdiscrepancy as the error. Thus we take c = − . c = 2 . c = − . ± . c = 4 . ± . d + d = 3 . d = − . d − d = − . c dominates the total error. At T LAB = 300 MeV, one has δ LO = 75 . ± . o , δ NLO = 6 . ± . o , δ N2LO = − ± , δ N3LO = − ± o . As we see the statistical uncertaintystemming from the input parameters increases with theorder. If we take the difference δ N3LO − δ N2LO ∼ o asan estimate of the systematic error, adding in quadraturestatistical and systematic errors we have, ∆ δ N3LOTOT ∼ o ,still a smaller quantity than the discrepancy to the Ni-jmegen phase shift. This may suggest that after renor-malization there are some physical effects missing evenat the N3LO level beyond pure TPE. B. Inclusion of Delta
In the previous section we have seen that all TPE ef-fects included to N3LO display a convergent pattern afterrenormalization, but there is still some missing physics.Note that while at LO and NLO the only parameters are g A , m π , M N and f π , at N2LO and N3LO there appearnew low energy constants (the c i and d i , respectively) which can be related to πN scattering and encode shortrange physics not considered explicitly. The ∆ resonanceis an outstanding feature of πN scattering and explainsa great deal of the low energy constants. Thus, it is in-teresting to analyze the role of explicit ∆ excitations asintermediate states in the N N potential. The importanceof explicit ∆ degrees of freedom has been emphasized onpower counting grounds in several previous works withfinite cutoffs where the N ∆ splitting is regarded as asmall parameter ∼ m π [2, 16, 27, 74]. The crucial roleplayed in the renormalization problem has been stressedin Ref. [35]. In this section, we analyze the NN poten-tial with the NLO terms of [7] together with the 1∆ and2∆ in the box diagrams as computed in Ref. [8]. Oneadvantage of such an approach is that, as compared tothe standard ∆-less theory, there only appears the N ∆splitting as a parameter.The renormalization of the S channel proceeds alongthe lines discussed in the ∆-less theory. The results forLO, NLO, NLO+1∆, and NLO+2∆ renormalized phaseshifts in the S channel are plotted and compared tothe Nijmegen Partial Wave Analysis [49] as a function ofthe CM momentum in Fig. 4. The most striking result isthe very strong resemblance between the N2LO ∆-less vsNLO+1∆ and the N3LO ∆-less vs NLO+2∆. These re-sults sustain the treatment of the N ∆ splitting as a smallparameter ∼ m π corresponding to the ∆-counting NLO ∆ = NLO + 1∆+ 2∆. The fact that all the contributionsfall off at large distances as ∼ e − m π r suggests that de-spite the ability to mimic higher order corrections in theWeinberg counting for the long distance potential in the S channel there is still missing shorter range physicsbeyond TPE. C. Three-pion exchange contributions
At N3LO, three-pion exchange occurs for the firsttime. These contributions have been calculated byKaiser [10, 11] and found to be small, which is whypresent N3LO NN potentials omit these contributionswhen renormalization is not implemented. However, itshould be noted that, for small distances, the 3 π dia-grams are proportional to r − and, thus, will ultimatelydominate at short distances. A rough estimate of the re-sults published in Refs. [10, 11] suggests that the sum ofall 3 π graphs is attractive. Thus, we infer from there thatthe scattering length will still be a free input parameter.On the other hand the 3 π -exchange contribution falls offas ∼ e − m π r at long distances so it becomes active atrather short distances, and so the effect is expected tobe small because of the short distance suppression (mod-ulo oscillations) of the wave function u ( r ) ∼ r / typicalof potentials with a short distance power like singular-ity [35, 44, 48]. This agrees with the rule that the moresingular the potential the more convergent is the calcula-tion, as we have extensively discussed above. Therefore,in a complete and renormalized N3LO calculation of the3 ∆ α ∆ g A ∆ c ∆ c ∆ c ∆ c ∆( d + d ) ∆ d ∆ d ∆( d − d ) TOTAL∆ δ LO δ NLO δ N2LO δ N3LO S phase shift (in degrees) at T LAB = 300 when the input parameters are varied. The sign“–” means that there is no contribution to the variation and a zero, “0”, means that the change is ∆ δ < .
1. The total resultis obtained by summing the partial contributions in quadrature, ∆ δ TOTAL = pP i (∆ δ i ) S phase shifts, we expect the 3 π effects not to be large,although the predictions may be closer to the empiri-cal values than in Fig. 3. An accurate investigation ofthe impact of 3 π exchange at N3LO on NN phase shiftsrepresents an interesting and challenging project for thefuture. D. Irrelevance of a fixed C counterterm As we have mentioned, the very definition of the po-tential is ambiguous as it requires fixing the polynomialterms in the momentum. In the calculations presentedabove, we have taken the renormalization scheme for thepotential where the fixed and cut-off independent choice C = 0 for the potential is made. We have also analyzedthe situation when a different renormalization scheme istaken, namely a non-vanishing arbitrary C coefficientwhich does not run with Λ. However, we allow C (Λ) torun in a way that the scattering length α is reproduced.This generates a different renormalization trajectory for C (Λ) as compared to the case C = 0. We find by actualcalculations that this fixed C coefficient is irrelevant,i.e. the renormalized phase shift does not depend on thisfixed value in the limit Λ → ∞ . Roughly speaking, thereason is that while the polynomial combination C q islarge at large momenta, the pion exchange part behavesas q log ( q ) with an additional logarithm and thus dom-inates for fixed C . This irrelevance of C was highlightedin the coordinate space analysis of Ref. [35] where the reg-ularization based on a radial cut-off r c would provide acompact support not sensing the details of the distribu-tional contributions in Eq. (8), regardless on how manyderivatives of delta’s are included. A quite different situ-ation arises when C depends on the cut-off in a way thatthe effective range is fixed as we discuss in the next Sec-tion V. There, it will be shown that if C (Λ) is relevantthen the phase shift is not convergent. -40-20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ ( Λ = G e V ) p (MeV) OPENLONLO+ ∆ NLO+ ∆∆ N2LON3LONIJ
FIG. 4: LO, NLO, NLO+1∆, and NLO+2∆ renormalizedphase shifts in the S channel as a function of the CMmomentum compared to the Nijmegen Partial Wave Anal-ysis [49]. Only one counterterm is used and fixed by thephysical value of the scattering length. V. WEINBERG’S COUNTING ANDRENORMALIZATION WITH TWOCOUNTERTERMS.A. Momentum space
In previous sections, we have seen that the LO, NLO,N2LO, and N3LO chiral potentials can be renormalizedwhen one counterterm C (Λ) is determined for any valueof the cutoff Λ by fixing the scattering length to its ex-perimental value, α = − . → ∞ is subsequently taken. This agrees with the observationof Refs. [35, 44, 48] that attractive singular potentialscan be renormalized with a single counterterm. However,Weinberg counting requires further counterterms in theshort distance potential, see Eq. (13). For instance, bothat NLO and N2LO a C counterterm should be included.In this section, we discuss whether the S scatteringamplitude is renormalizable, i.e. whether the scatter-4ing amplitude has a well defined limit, when both the C (Λ) and C (Λ) counterterms are included and fixedby fitting the scattering length α = − .
74 fm as wellas the effective range r = 2 .
77 fm for any value of thecutoff Λ and the limit Λ → ∞ is pursued. Actually,when the NLO (N2LO) long distance potential, Eq. (15)is used, this way of proceeding corresponds to renormal-izing the NLO (N2LO) approximation of the S channelin the standard Weinberg counting. The method we fol-low in practice is a straightforward extension of the casewith only just one counterterm C . In the case when C = 0, we get a finite (renormalized) effective range r NLO0 = 2 . C = 0 onegets r N2LO0 = 2 .
86 fm. This last value is so close to theexperimental one that one would not expect big changeswhen a C is added to exactly fit the experimental r .Thus, it makes sense to investigate what would happenif one uses a non-vanishing C to account for the miss-ing, and in principle tiny, contribution to the effectiverange. The surprissing result, to be discussed below indetail, is that trying to fit the discrepancy in the effec-tive range with a C counterterm is incompatible withrenormalizability.In the contact theory we found that for Λ > C and C diverge before becomingcomplex numbers. Thus, the corresponding Hamiltonianbecomes non-self-adjoint, in harmony with the Wignercausality bound violations unveiled in Ref. [56, 75].When the NLO contribution to the potential is included,the critical value of the cutoff is slightly shifted towardsthe higher values Λ c = 480 − ∼ E LAB = 105MeV and Λ ∼ E LAB = 300MeV.On the other hand, let us note that a plateau region notalways appears, and when it does the residual contri-bution from pion-exchange effects is less important andthe counterterms may dominate the calculation. In otherwords, if the cutoff was too small, one would be drivenback to an effective range expansion with no visible con-tribution from chiral potentials whatsoever. Thus, any fi-nite cutoff calculation within a higher cutoff regime turnsout to be strongly cutoff dependent, or else the cutoffmust be fine tuned to intermediate energy data, hencebecoming an esssential, and not an auxiliary, parameterof the theory. Note that all the problems are triggered -60-40-20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ p (MeV)
480 MeV1010 MeV2000 MeV2780 MeV-60-40-20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ p (MeV)
480 MeV1010 MeV
FIG. 5: NLO (upper pannel) and N2LO (lower pannel) con-vergence of the phase shifts as a function of the CM momen-tum for some fixed values of the cutoff Λ. The renormalizationcounterterms C (Λ) and C (Λ) are always determined by fix-ing the scattering length α and the effective range r to theirexperimental values α = − .
74 fm and r = 2 .
77 fm. ForΛ > Λ c ∼ C (Λ) and C (Λ) become complexwhile the phase shifts remain real. by insisting on fitting the effective range parameter tothe experimental value by introducing the C countert-erm required by Weinberg power counting on the shortdistance interaction. In contrast, if C = 0, as advocatedin Refs. [35, 44, 48] and Sec. IV above, not only is thephase shift convergent in the limit Λ → ∞ (in practiceΛ > .Although we have checked that these results are nu-merically robust within the LSE by increasing the num- Actually, in the limit α → ∞ and under the assumption of Vanver Waals dominance one gets at N2LO the analytical result (see B. Coordinate space
Let us compare the previous findings in momentumspace with related investigations in coordinate space [52].A thorough study in coordinate space has revealed thatthe very existence of the Λ → ∞ limit may actually de-pend on the specific representation of the short distancephysics. This analytical result has been verified by nu-merical calculations and will become extremely helpfulin analyzing the momentum space calculations presentedabove. Thus, for the sake of completeness it is worthreviewing the emerging pattern from Ref. [52].As we have discussed in Sec. II C, within the bound-ary condition regularization the unknown short distanceinteraction is represented by the logarithmic derivativeof the wave function at the short distance cutoff radius r c , u ′ k ( r c ) /u k ( r c ). In an energy expansion of the wavefunction at short distances, one has u k = u + k u + . . . and its logarithmic derivative for which a continuity con-dition is required. This introduces an energy dependencewhich eludes the Wigner causality bound discussed inRefs. [56, 75], since self-adjointness is broken from thestart. As we discussed previously in the contact theorydescribed in Sec. III B, at second order in the energy theneglected terms are O ( p ) and any of the three represen-tations displayed by Eq. (47) might be equally accept-able. Actually, we found that renormalized amplitudesfall into two classes and that not all of them approachthe renormalized limit in the same way, see Eq. (48).Here, we will extend that study by inquiring what hap-pens when the same three representations displayed byEq. (47) are used and the long distance potential V L ( r )is taken to be the NLO and N2LO for r > r c as the cut-off is removed, r c →
0. Note that all three cases possess by construction the same scattering length α and effec-tive range r and the same long distance potential for r > r c . Thus, any difference is clearly attributable to thedifferent short distance representation. The results aredisplayed in Fig. 8 for fixed short distance cutoff values r c as a function of the CM momentum p . For finite valuesof r c we see a difference which can naturally be explainedby the different off-shell behaviour of the short distance Ref. [35] for details), r = 16Γ(5 / π " g A π f π ` − g A + 24 c M − c M ´ which yields r ∼ . -100-80-60-40-20 0 20 40 60 80 100 0 1000 2000 3000 δ ( T l a b = M e V ) Λ (MeV) NLON2LO-100-80-60-40-20 0 20 40 60 80 100 0 1000 2000 3000 δ ( T l a b = M e V ) Λ (MeV) NLON2LO
FIG. 6: NLO and N2LO convergence of the phase shifts asa function of cutoff Λ for fixed values of the LAB energy, T LAB = 105 MeV (upper pannel) and T LAB = 300 MeV(lower pannel). The renormalization counterterms C (Λ) and C (Λ) are always determined by fixing the scattering length α and the effective range r to their experimental values α = − . r = 2 . > Λ c ∼ C (Λ) and C (Λ) become complex while the phase shiftsremain real. physics. As we see, the difference persists as the cutoff isbeing removed and in fact is magnified. As was alreadypointed out in a previous work [52], only the case I rep-resenting a rational function turns out to yield a uniqueand well defined finite value for the phase shift .This fact is clearly seen from inspection of Fig. 7 wherewe plot the phase shifts for a fixed value of the CM mo-mentum p = 300 MeV as a function of the equivalent Note that in the contact theory, this rational representation pro-vides the softest regulator, i.e. cutoff effects scale quadraticallyand not linearly as in all others, see Eq. (47). -100-80-60-40-20 0 20 40 60 80 100 0 1000 2000 3000 δ ( T l ab = M e V ) Λ = π / (2 r c ) [MeV] d k = d /(1 - k d /d ) NLONNLO -100-80-60-40-20 0 20 40 60 80 100 0 1000 2000 3000 δ ( T l ab = M e V ) Λ = π / (2 r c ) [MeV] d k = d + k d NLON2LO -100-80-60-40-20 0 20 40 60 80 100 0 1000 2000 3000 δ ( T l ab = M e V ) Λ = π / (2 r c ) [MeV] d k = (du + k du )/(u + k u )NLONNLO FIG. 7: NLO and N2LO running of the phase shifts as a function of the equivalent momentum cutoff Λ = π/ (2 r c ) (in MeV)with r c the short distance cutoff, for fixed values of the LAB energy T LAB = 300 MeV and for several parameterizations of theshort distance energy dependent logarithmic derivative (see Eq. (47). In all three cases, the chiral potential for r > r c is thesame and the scattering length α and the effective range r are both fixed to their experimental values α = − .
74 fm and r = 2 .
77 fm. -20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ [ deg ] k c.m. [MeV] 1 S NLOr c = 0.5 fm NijmIIPaded = d + k d d = d /(1 - k d /d ) -20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ [ deg ] k c.m. [MeV] 1 S NNLOr c = 0.5 fm NijmIIPaded = d + k d d = d /(1 - k d /d )-20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ [ deg ] k c.m. [MeV] 1 S NLOr c = 0.2 fm NijmIIPaded = d + k d d = d /(1 - k d /d ) -20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 δ [ deg ] k c.m. [MeV] 1 S NNLOr c = 0.2 fm NijmIIPaded = d + k d d = d /(1 - k d /d ) FIG. 8: NLO (left panel) and N2LO (right panel) convergence of the phase shifts as a function of the CM momentum for fixedvalues of the short distance cutoff r c for several parameterizations of the short distance energy dependent logarithmic derivative(see Eq. (47). In all three cases, the chiral potential for r > r c is the same and the scattering length α and the effective range r are both fixed to their experimental values α = − .
74 fm and r = 2 .
77 fm. π/ (2 r c ) derived for the pion-less theory in Sec. III. The striking similarity between themomentum space calculation presented in Fig. 6 and thecoordinate space calculation displayed in Fig. 7 for finitecutoffs is noteworthy although not completely surprisingin the light of the analysis of Appendix A. There, thefinite cutoff equivalence, Λ = π/ (2 r c ), deduced in thecontact theory is shown to hold also in the presence of alocal potential.Of course, all these features depend on the singularcharacter of the interaction at short distances and donot depend on the specific form of the potential, so weexpect them to hold also when ∆ degrees of freedom areexplicitly taken into account.Although we cannot prove it analytically in momentumspace, the remarkable similarity of the coordinate spaceanalysis with the present momentum space calculationsstrongly suggests that the standard polynomial represen-tation of the short distance potential is incompatible withrenormalization. Of course, this does not preclude thepossible existence of a suitable potential representationof the short distance interaction, most likely with com-plex counterterms, such that both low energy parameterscan be fixed and the cutoff can at the same time be re-moved. We leave such an interesting investigation for thefuture. VI. CONCLUSIONS AND OUTLOOK
In the present paper, we have analyzed the renormal-ization of the singlet np phase shift in the S chan-nel incorporating one- and two-pion exchange effects.For the long distance potential, the standard Weinbergscheme based on dimensional power counting is adoptedto N3LO. However, the short distance physics is param-eterized in terms of one unique energy and momentumindependent counterterm whose cutoff dependence is de-termined by adjusting the scattering length to its phys-ical value. The present analysis is carried out in mo-mentum space in a somewhat complementary manner aspreviously done in coordinate space [43]. Actually, wehave analyzed and reproduced those results directly inthe more popular momentum space after the cutoff hasbeen effectively removed. In order to stress the equiva-lence of both approaches, we have displayed many resultsin parallel. This is not just a stylistic matter of presen-tation; besides numerical simplicity, much understandingof the renormalization problem has been achieved by adirect analysis in coordinate space at least for local po-tentials. The present work provides a further example inthis respect.The momentum space framework allows a direct exten-sion to include N3LO contributions which include non-local pieces in the long distance chiral potential. Themain outcome of such a calculation is that the N3LOchiral potential induces rather small corrections as com-pared to the N2LO results in all of the elastic scattering region. As anticipated previously [35], this happens to beso even when the N3LO potential is more singular thanthe N2LO one at short distances. Moreover, the analyti-cal scaling behaviour for large cutoffs Λ of the scatteringamplitude predicted in a previous coordinate space anal-ysis [52] is confirmed qualitatively by the present mo-mentum space calculations; increasing the order in theexpansion of the long distance potential suppresses morethe finite cutoff dependence. In summary, the schemeis convergent, but differs from the expected phase shiftsobtained from partial wave analyses. An error analysis ofthe results shows that although there are some uncertain-ties induced by the input parameters, the correspondingerror bands are not large enough to be compatible withthe Partial Wave Analysis of the Nijmegen group.The N3LO approximation to the long distance chiralpotential is computed within a heavy baryon expansionand assuming only explicit nucleon degrees of freedomin the NN potential. This implies in particular thatthe N ∆ splitting is considered a non-small parameter.However, this number is about twice the pion mass, soit is not clear whether such an assumption is fully jus-tified. Therefore, we have analyzed the NN scatteringproblem when explicit ∆ intermediate state excitationsare included in the potential. The net result is that, afterrenormalization, the 1∆ and 2∆ box diagrams contribu-tions mimic extremely accurately the ∆-less N2LO andN3LO potentials, respectively, and thus fail to describethe higher energy region of the S phase shift. Giventhe fact that these contributions also fall off at large dis-tances as ∼ e − m π r , this result reinforces the conclusionthat there is some shorter range missing physics beyondthat provided by TPE.Further, it is interesting to compare the present renor-malized results with those obtained for a two pion ex-change potential computed within a relativistic baryonframework which sums up all heavy nucleon compo-nents [76] thus contains a full determination of TPE con-tributions assuming all other degrees of freedom (includ-ing explicit ∆’s) are infinitely heavy. There, all np partialwaves with j ≤ S channel the results are rather similar to thosefound here. This, again, suppports the present conclusionthat, presumably, the components of the NN potentialwith a shorter range than TPE might finally provide themissing repulsion needed to reduce the 3% overshootingof the S effective range as well as the too low value ofthe phase shift in the region with CM momenta p > m π .We have also investigated the convergence of the stan-dard Weinberg counting for both the short distance aswell as the long distance potential. At NLO and N2LOthis corresponds to including a polynomial momentumdependence in the short distance interaction by meansof two counterterms which may then be fixed by adjust-ing the scattering length and the effective range to theirexperimental values. We find that, with these two renor-malization conditions, the counterterms turn out to becomplex for not too large cutoffs (around 500MeV), sig-8naling the breakdown of self-adjointness of the potential,and more generally suggesting a violation of Wigner’scausality condition. Nonetheless, despite phase shifts re-maining real, a unique renormalized limit does not exist.This lack of convergence agrees with similar findings incoordinate space anticipated in Ref. [52]. The physicalunderstanding of the situation is as follows. On the onehand the long range physics is fixed, thus, the off-shellpropagation within the long range region is unambigu-ous. On the other hand, the short range potential mustbe adjusted to provide the threshold parameters such asthe scattering length, the effective range etc. If the shortrange potential is just determined from fixing the scatter-ing length only, this is at zero energy and becomes zerorange when the cutoff is removed, so there is no off-shellambiguity. In contrast, fixing further the effective rangerequires non-zero energy and there are in fact infinitelymany ways of parameterizing this, with different off-shellbehaviour even when the cutoff is removed. In otherwords, fixing a finite range and removing the cutoff de-pends on details of the method. These conflicts betweenoff-shellness and finiteness are not new in field theory.Green’s functions which are renormalized on-shell do notnecessarily provide finite off-shell amplitudes. The stan-dard Weinberg parameterization as a polynomial in mo-menta is one choice which may not turn out to be con-sistent with renormalizability. In this regard, it is inter-esting to mention that coordinate space detailed studiesbased on renormalization group properties [52] suggestthe existence of a suitable short distance representationyielding convergent results in theories with more thanone counterterm. The generalization of those results tomomentum space would require an in-depth study of therenormalization group in the presence of eventually non-local but singular potentials. We note that the energydependence of the coordinate space solution violates self-adjointness explicitly and we expect that most likely anenergy independent non-polynomial momentum space so-lution would spoil self-adjointness as well.It is important to note that with a finite cutoff and fourcounterterms the S phase shift has been successfully de-scribed within the standard Weinberg counting [24, 25].In this regard, it is natural to question the usefulness oftaking the infinite cutoff limit and to carry out a renor-malization process. From a physical point of view thelimit Λ → ∞ corresponds to consider other degrees offreedom than those not considered explicitly to be in-finitely heavy. Thus, the aim of the renormalizationprogram is far more stringent than previous finite cut-off calculations. Within such a framework any failurecan be unambiguously attributed to missing physical in-formation on the long distance potential, and the renor-malization process highlights this in a rather vivid man-ner as one clearly sees when going from OPE to TPEpotentials. From a mathematical perspective the renor-malizability requirement imposes tight constraints on theadmissible forms of the short distance physics for somegiven long distance interactions. These are powerful con- ditions which have traditionally been the great strengthof the renormalization ideas to avoid unnecessary prolif-eration of interactions. Since the power counting of longdistance potentials is not uniquely determined yet (onemay e.g. include or not explicit ∆’s), it would be veryhelpful to see what kind of short distance constraints onthose potentials might be imposed as well on the basis ofrenormalizability or other principles.The problems with the Weinberg counting for the shortdistance polynomial form in the momenta of the interac-tion with renormalization found in this paper for NLOand N2LO is to be added to the other ones noted in pre-vious works [27, 35]. On the mathematical side, it isnoticeable that the coordinate space analysis of Ref. [35]conjectured this result by exploiting the compelling re-quirement of completeness and self-adjointness for therenormalized quantum mechanical problem for a localchiral potential which at first sight may seem completelygermane concepts to the EFT machinery. On the phe-nomenological side, it should be noted that this incon-sistency result does not explain why the renormalizedphase shift with just one counterterm comes out reason-ably close to the accepted ones, but certainly makes thisunique and finite prediction more credible and inevitablefrom a theoretical perspective if self-adjointness is main-tained, and provides further confidence on the virtues ofa renormalization principle within the chiral approach tothe NN interaction. Acknowledgments
One of us (E.R.A.) thanks Diego Pablo Ruiz Padillo(Departamento de F´ısica Aplicada, Universidad deGranada) for patient instruction on the virtues ofNyquist theorem some years ago. We also thank EvgenyEpelpaum for a critical reading of the manuscript andDaniel Phillips for remarks on the paper.The work of D. R. E. has been partially funded bythe Ministerio de Ciencia y Tecnolog´ıa under contractNo. FPA2004-05616, the Junta de Castilla y Le´on undercontract No. SA016A07. The work of E.R.A. is sup-ported in part by funds provided by the Spanish DGIand FEDER funds with grant no. FIS2005-00810, Juntade Andaluc´ıa grants no. FQM225-05, EU Integrated In-frastructure Initiative Hadron Physics Project contractno. RII3-CT-2004-506078. M. P. V. has been fundedby the Deutsche Forschungsgemeinschaft (SFB/TR 16),Helmholtz Association (contract number VH-NG-222).R. M. has been supported in part by the U. S. NationalScience Foundation under Grant No. PHY-0099444.9
APPENDIX A: MOMENTUM VS COORDINATESPACE FOR FINITE CUTOFFS AND LOCALPOTENTIALS
In this appendix we discuss further the relation be-tween momentum and coordinate space for finite cut-offs and local potentials. We will show that solving theLippmann-Schwinger equation for a long range poten-tial with a sharp cutoff Λ is equivalent to the solutionof the Schr¨odinger equation for the same potential in adiscretized grid with ∆ r = π/ Λ. This equivalence re-minds the previous identification between sharp momen-tum cutoff Λ and the short distance cutoff r c found in thecontact theory in Sec. III, Λ = π/ (2 r c ). We can choosethe initial grid point at r c = π/ (2Λ), thus recovering thepreviously mentioned equivalence. Actually, by invokingNyquist theorem on the cut-off LS equation, we deducea discretized version of the variable phase equation [77]which allows to discuss both the renormalization as wellas the decimation problem of the NN-force based on chi-ral potentials.
1. The cutoff Lippmann-Schwinger equation
Let us consider as a starting point the Lippmann-Schwinger equation for s wave scattering, written in theform T ( k ′ , k ) = V ( k ′ , k ) + M Z ∞ dqV ( k ′ , q ) q p − q T ( q, k ) , (A1)where T ( k ′ , k ) are the matrix elements of the T − matrixbetween initial and final CM momentum states k and k ′ respectively and the corresponding potential matrixelement is given by V ( k ′ , k ) = 2 π Z ∞ drj ( k ′ r ) j ( kr ) V ( r ) r , (A2)for a local potential. Since we are integrating the in-termediate momentum up to infinity, we are implicitlyassuming that V ( k ′ , k ) is a regular potential. The sin-gular potential case will be discussed later. Now, if wecutoff the potential in momentum space , we get theregularized potential V Λ ( k ′ , k ) = θ (Λ − | k ′ | ) θ (Λ − | k | ) V ( k ′ , k ) , (A3) Cutting off the high momentum states is not exactly the sameintegrating out the high energy states, which produces a low mo-mentum effective energy dependent “optical” potential. We arefocusing on the long range potential here. The short missing dis-tance piece could be included by a Taylor expansion in momentaor energies (see Sec. A 7). where θ ( x ) is the Heaviside step function. In coordinatespace the cutoff potential becomes V Λ ( r ′ , r ) rr ′ = 2 π Z Λ0 k dk Z Λ0 k ′ dk ′ j ( kr ) j ( kr ′ ) V ( k, k ′ ) → δ ( r − r ′ ) rr ′ V ( r ) , (A4)which is obviously nonlocal and becomes local only whenΛ → ∞ (second line). The cutoff LSE, where | q | ≤ Λ, canbe solved by standard means. In the spirit of an EFT,based on the idea that low energy dynamics does notdepend on the details at short distances, it may perhapsbe appropriate to proceed a bit differently. Actually, ifhigh momentum states are cutoff from the theory thesame idea should apply to small resolution scales, i.e.regardless whether they are short or long. That meansthat using too much information on the potential V ( r )even if it is exactly known point-wise may be illusory atwave-lengths longer than a given resolution, ∆ r , sothat one can sample V ( r ) with some resolution ∆ r ∼ / Λ. This obviously reduces the number of coordinatemesh points in the integration.
2. Nyquist theorem
In the present context, Nyquist theorem [78] is remark-ably useful because it provides an optimal way of sam-pling signals which have a bandwidth in Fourier space,i.e., functions for which F ( k ) = Z ∞−∞ e ikx f ( x ) dx = 0 for | k | ≤ Λ , (A5)then, for the original function we have f ( x ) = Z Λ − Λ e − ikx F ( k ) dk π . (A6)If we define the sampling function f S ( x ) of the function f ( x ) at the equidistant points x n = n ∆ x , with the opti-mal ∆ x = π/ Λ separation, f S ( x ) = ∞ X n = −∞ f ( x n )∆ xδ ( x − x n ) , (A7)and compute its Fourier transform we get F S ( k ) = ∆ x ∞ X n = −∞ f ( x n ) e ikx n . (A8) For example, if a potential V ( r ) is supplemented by highly os-cillatory ripples at short resolution scales, ∆ r the phase shiftsshould be insensitive to them at λ ≫ ∆ r . F S ( k ) = F ( k ) .Inverting the Fourier transform we get¯ f S ( x ) = Z Λ − Λ e − ikx F S ( k ) dk π = ∞ X n = −∞ f ( x n ) sin [Λ( x − x n )] π ( x − x n ) ∆ x . (A9)Thus, if we sample the function according to Eq. (A7),the following identity holds at the sampling points¯ f S ( x n ) = f ( x n ) . (A10)Hence, there is no loss of information on the samplingpoints if the sampling is done equidistantly with the op-timal Nyquist frequency, ∆ x = π/ Λ. In particular, itdoes not really make sense to sample the function withsmaller ∆ x . In the next section we apply this samplingprinciple to the potential.
3. The optimal grid for sampling the potential incoordinate space
Nyquist theorem also applies for the special case ofthe Lippmann-Schwinger equation in s-wave scattering,although the derivation is different in some details to theone presented in the previous section. For clarity, wepresent here these details.We will consider first the general case of a non-localpotential, for which the Schr¨odinger equation in s-wavereads − u ′′ ( r ) + M Z ∞ dr ′ V ( r, r ′ ) u ( r ′ ) = k u ( r ) . (A11)We can sample this non-local potential as V S ( r, r ′ ) = (∆ r ) ∞ X m,n =0 V ( r m , r n ) δ ( r − r m ) δ ( r − r n ) . (A12)For the momentum space representation of the sampledpotential we get V S ( k, k ′ ) = (∆ r ) π ∞ X m,n =0 ˆ V m,n j ( kr m ) j ( k ′ r n ) , (A13) As the original F ( k ) is bandwidth limited to the [ − Λ , Λ] interval,it can be expressed as a Fourier sum F ( k ) = ∞ X n = −∞ a n e ikx n where x n = nπ/ Λ. Thus it is trivial to see that F S ( k ) = F ( k )when the sampling is done with ∆ x = π/ Λ. where ˆ V m,n = r m r n V ( r m , r n ). As a consequence of thecut-off Λ, the potential V ( k, k ′ ) can be expressed as asum of spherical bessel functions V ( k, k ′ ) = ∞ X m,n =0 a n,m j ( kr n ) j ( k ′ r m ) , (A14)where r n = nπ/ Λ. Then by taking ∆ r = π/ Λ, the sam-pled potential recovers the original one, i.e. V S ( k, k ′ ) = V ( k, k ′ ), for k, k ′ < Λ. By Fourier-transforming V S ( k, k ′ )back to coordinate space, it can be checked that it repro-duces the original sampling points, i.e.¯ V S ( r n , r m ) = ( π Λ ∆ r ) V ( r n , r m ) = V ( r n , r m ) (A15)for the Nyquist sampling frequency.In the case of a local potential, the one which interestsus most, we sample the following way V S ( r ) = ∆ r ∞ X n =0 V ( r n ) δ ( r − r n ) . (A16)After double Fourier-transforming, we get¯ V S ( r n , r m ) = V ( r n ) Λ π δ nm , (A17)which makes the potential local for the grid points. It isin fact a finite cut-off version of V ( r, r ′ ) = V ( r ) δ ( r − r ′ )once we notice that Λ π δ nm → δ ( r − r ′ ).
4. The Lippmann-Schwinger equation with acut-off
For a Lippmann Schwinger equation with a finite cut-off Λ all the matrix elements become a linear combinationof separable terms. Thus the LS equation becomes alinear matrix equation which can be solved by standardtechniques by writing T ( k, k ′ ) = X ij T ij j ( kr i ) j ( kr j ) , (A18) V ( k, k ′ ) = X ij V ij j ( kr i ) j ( kr j ) , (A19)and defining propagator matrix elements G ij = Z Λ0 dq M q p − q j ( qr i ) j ( qr j ) . (A20)Therefore we get T ij = V ij + X lm V il G lm T mj . (A21)This equation can be reduced to a finite-dimensional N × N linear algebra problem by cutting the sums to i, j = N .1The effect of this simplification can be seen by sampling V ij in coordinate space (see Eqs. (A7) and (A16)) V ( r ) → ∆ r N X n =0 V ( r n ) δ ( r − r n ) , r n = nπ Λ , (A22)from which we can check that cutting the sum is equiv-alent to introducing the (harmless) infrared cut-off r N .Note that for local potentials, the sampled potential ma-trix elements at the grid points are diagonal V ij = 2 π r i V ( r i )∆ r δ ij = V i δ ij . (A23)Thus Eq. (A21) becomes T ij = V i δ ij + X m V i G im T mj , (A24)which looks like a multiple scattering equation, with on-shell propagation between delta-shell scatterers. Aftermatrix inversion, the on-shell solution is then given by T ( p ) = X ij T ij ( p ) j ( pr i ) j ( pr j ) . (A25)
5. The discretized Schr¨odinger equation
In the previous section we have seen that for theLippmann-Schwinger equation with a cut-off Λ, wecan either use the original momentum space potential V ( k, k ′ ) and solve by standard means, or expand thispotential, i.e. use the sampled potential V S ( k, k ′ ), andsolve as a linear algebra problem (as they are both thesame potential for momenta below the cut-off).Alternatively we can directly solve the Schr¨odingerequation for the sampling potential V S ( r ). This proce-dure will give an excellent approximation to the solu-tion of the LS equation (although not the exact solution,as explained at the end of this section) but at a muchsmaller computational cost. For this purpose we makethe replacement V ( r ) → V S ( r ) = N X i =0 ∆ rδ ( r − r i ) V ( r i ) , (A26)which is a superposition of equally spaced delta-shell po-tentials. It should be noted that the point r = 0 doesnot contribute, as it lies at the integration boundary ofthe Schr¨odinger equation. Then, we have∆ r = π Λ r N ≃ N ∆ r . (A27)Thus, for a potential of size a where we do not want todescribe energies higher than Λ we should do with aninfrared cut-off larger than the potential’s length scale, r N ≫ a , or equivalently N ≫ Λ a/π . We can solve theSchr¨odinger equation piecewise, u ( r ) = A i sin( kr + δ i − / ) , (A28) r i − ≤ r ≤ r i , where A i is the amplitude and δ i − / can be understoodas the accumulated phase-shift due to adding a new deltashell at r i chosen to be located at the midpoint r i + (for reasons to become clear soon). Note that with thischoice the lowest possible location of the phase-shift, δ / corresponds to the point r c = ∆ r π , (A29)which we may identify with a short distance (ultraviolet)cutoff. This identification between the momentum-spacecutoff agrees with the one obtained for the pure shortrange theory by solving the LSE without any discretiza-tion (see Sec. III).It should be noted that the previous method for solv-ing the Schr¨odinger equation does not really generate theexact solution of the LS equation with a cut-off, but onlya close approximation. Although V S ( r ) is the best sam-pling function for V Λ ( k, k ′ ), it is not the same quantummechanical potential. The solution to the LS equationwith V Λ ( k, k ′ ) can be exactly reproduced by solving thenon-local Schr¨odinger equation, Eq. (A11), with the po-tential ¯ V S ( r, r ′ ) which comes from inverse Fourier trans-forming V S ( k, k ′ ). The Schr¨odinger equation should besolved with trivial initial conditions, u (0) = 0, as all thephysically relevant information is included in ¯ V S ( r, r ′ ).On the contrary, if we solve V S ( r ), a sum of delta shells,we must include a non trivial initial condition , evenin the absence of any short range physics. If we solvethe discretized Schr¨odinger equation with a trivial initialboundary condition, δ / = 0, then a certain error will beincluded in the final solution. In the worst case we canexpect an error of order ∆ r . By means of the variablephase equation [77] we can perform a better assessmentof the error , yielding, for example, an O ((∆ r ) ) errorfor a Yukawa potential or O ((∆ r ) ) for a square well.From the previous discussion, it is apparent how toinclude a counterterm in the computation. It entersthrought the initial sampling point of V S , which maps Specially since we are effectively ignoring the r = 0 samplingpoint. For small enough ∆ r , the variable phase equation will yield δ ≃ − k Z ∆ r M V ( R ) dR . As a curiosity, we can see that δ / will scale as an inverse powerof ∆ r for a singular potential, thus signaling the need of a coun-terterm. V S ( k, k ′ ) . Thus it is ignoredwhen solving the Schr¨odinger equation for V S , and it en-ters through the initial condition δ / . Similar remarkscan be made for singular potentials, in which a countert-erm must be included in order to obtain a stable resultfor ∆ r →
6. The discrete variable phase equations
The previous discussion can be elaborated further toreach interesting results. Matching the wave functions atthe points where the delta shells are located, r = r i , wesimply get k cot( kr i + δ i +1 / ) − k cot( kr i + δ i − / ) = ∆ rU ( r i ) . (A30)where U ( r ) = 2 µV ( r ) = M V ( r ) is the reduced poten-tial. This is a recursion relation for the phase-shift atthe interval midpoint r i + = ( i + ) π/ Λ. Unlike the ma-trix equation which has traditional storage limitationsfor large number of grid points N , this equation does notposses this shortcoming allowing for rather large N val-ues. Similar equations were deduced many years ago [79]as a practical tool to attack the inverse scattering prob-lem and to determine the NN potential on the grid points.Defining the discretized effective range function, M i = k cot δ i , (A31)we get M i + k cot kr i − k M i + + k cot kr i − M i − k cot kr i − k M i − + k cot kr i = ∆ rU ( r i )(A32)which can be rewritten as a continuous fraction. Notethat for the cutoff theory both Eqs. (A30) and (A32) arealmost exact. The only approximation comes from the finiteness of the cutoff Λ. An important property whichwill be used later on is the reflection property, namelythe symmetry under the replacement∆ r → − ∆ r , δ i + → δ i − (A33)which means that on the grid running the relation up-wards or downwards are inverse operations of each other.Obviously this property may fail in practice only due toaccumulation of computer round-off errors over large evo-lution distances. This is also the reason why we choosethe midpoint r i +1 / for the accumulated phase shift: inany other case we would lose the reflection property.The regular solution at the origin reads δ ( k ) = 0 δ N + ( k ) = δ ( k ) (A34)If we take the limit Λ → ∞ we can define δ ( k, r i ) = δ i ( k ),to get dδ ( k, R ) dR = − k U ( R ) sin ( kR + δ ( k, R )) + O (∆ r ) . (A35)which is the variable phase equation [77] up to finitegrid corrections and can be interpreted as the changein the accumulated phase when a truncated potential ofthe parametric form U ( r ) θ ( R − r ) is steadily switchedon as a function of the variable R . Eq. (A30) is thus adiscretized variable phase equation, corresponding to adiscretized potential sampled at the optimal Nyquist fre-quency. This equation and its generalization to coupledchannels has extensively been used to treat the renormal-ization problem in NN scattering in Refs. [43, 72].The discrete equations for the low energy parameterscan be easily computed from the low energy expansion k cot δ i ( k ) = − α ,i + 12 r ,i k + v ,i k · · · (A36)which yields For example, the C counterterm when projected to the s-wavetakes the form C δ ( r )4 πr → C π ∆ r r (when discretizing).Thus, if we write the sampled potential V S ( k, k ′ ) = P ij V ij j ( kr i ) j ( kr j ), and take into account Eq. (A23), we seethat C maps only onto V = C π , giving a constant contribu-tion in momentum space, as expected. r i − α ,i + − r i − α ,i − = ∆ r U ( r i ) (A37)6 α ,i + r i − r i + α ,i + (cid:16) − r i + 3 r ,i + (cid:17) (cid:16) α ,i + − r i (cid:17) = 6 α ,i − r i − r i + α ,i − (cid:16) − r i + 3 r ,i − (cid:17) (cid:16) α ,i − − r i (cid:17) (A38)and similar equations for higher low energy parameters, v , v , v , . . . . Note that the potential only enters ex-3plicitly in the equation involving the discretized scatter-ing length. One of the nice features of this equation isthe way how it handles the case of singular points, when α ( R ) or other parameters diverge, since standard inte-gration methods for the corresponding differential equa-tion are based on the smoothness of the solution andhence fail.The way to proceed in practice is quite simple. Fora given number of grid points N we take the scatteringlength, α ,N +1 / = α , the effective range, r ,N +1 / = r and run Eq. (A37) and Eq. (A38) to determine α , / = α ( π/ r , / = r ( π/ M / = − /α , / + r , / k / M = k cot δ = M N +1 / . Due tothe reversibility of the algorithm one exactly has M →− /α + r k / k → r (modulo computer arithmetic round-off errors).In this way we can fix the initial boundary conditions toexactly reproduce any given scattering length, effectiverange, etc.This method was used successfully [80] to extract lowenergy threshold parameters in all partial waves with j ≤ α , / = 0, r , / = 0, etc, and obtain thepotential’s threshold parameters as α = α ,N +1 / , r = r ,N +1 / , etc.As we have said the discretized running parameters lo-cated at the lowest possible radius r c = π/
2Λ correspondto the short distance interactions. Nowhere in the equa-tions does the potential at the origin appears explicitly,since according to Eq. (A32), one starts with U ( π/ Λ).From this point of view the treatment of regular and sin-gular potentials at the origin is on equal footing.
7. The decimation process in momentum space
The previous equations provide the accumulated phaseshifts due to the addition of equidistant delta-shells sam-pling the original potential in coordinate space. We wantto show that they actually provide a solution of the dec- imation problem of the LSE where the low energy statesare cutoff. Assuming that we have the LSE with a givencutoff Λ related to potential, we ask how does the physi-cal phase-shift change when we make the transformationΛ → Λ /
2, in the potential fixed. Applying the process ex-plained above based on Nyquist theorem we see that thiscorresponds to double the grid resolution ∆ r → r .Obviously, by repeating the process we may effectivelyhave ∆ r ≫ a (being a the range of the potential) andhence the potential never contributes and δ N +1 / = δ / .Thus, for Λ a/π ≪
1, the short range theory is recovered.In the opposite limit Λ a/π ≫ N ∆ r = N π/
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