Perturbative Renormalizability of Chiral Two Pion Exchange
in Nucleon-Nucleon Scattering: P- and D-waves
We study the perturbative renormalizability of chiral two pion exchange in nucleon-nucleon scattering for p- and d-waves within the effective field theory approach. The one pion exchange potential is fully iterated at the leading order in the expansion, a choice generating a consistent and well-defined power counting that we explore in detail. The results show that perturbative chiral two pion exchange reproduces the data up to a center-of-mass momentum of at next-to-next-to-leading order and that the effective field theory expansion convergences up to .
The nature and derivation of the nucleon-nucleon interaction is probably the central problem of nuclear physics. The modern point of view is that any serious theoretical formulation of the nuclear force should be grounded on quantum chromodynamics (QCD), the fundamental theory of the strong interactions. However direct calculations on the lattice are still unavailable at the physical pion mass, despite promising results which are beginning to appear in the two Beane:2006mx (); Ishii:2006ec () and three Doi:2011gq () nucleon sectors at larger pion masses. A different, more indirect path is provided by the effective field theory (EFT) formulation of the nuclear forces Beane:2000fx (); Bedaque:2002mn (); Epelbaum:2005pn (); Epelbaum:2008ga (); Machleidt:2011zz (), in which the pion exchanges are constrained by the requirement of broken chiral symmetry, the main low energy manifestation of QCD. This approach exploits the well-known separation of scales appearing in the two-nucleon system with the purpose of generating a systematic and model independent low energy expansion of the nuclear potential and the scattering amplitudes. In addition, nuclear EFT may potentially bridge current lattice QCD calculations with physical results by means of chiral extrapolations.
where represents the low-energy scales of the system, usually the momentum exchanged between the nucleons and the pion mass , while stands for the high-energy scales that are not explicitly taken into account in the theory, e.g. the rho meson mass . The full potential is then iterated into the Schrödinger or Lippmann-Schwinger equation Ordonez:1993tn (); Ordonez:1995rz (); Frederico:1999ps (); Eiras:2001hu (); Entem:2001cg (); Entem:2002sf (); Entem:2003ft (); Epelbaum:2003gr (); Epelbaum:2003xx (); Epelbaum:2004fk (); Nogga:2005hy (); PavonValderrama:2005gu (); PavonValderrama:2005wv (); PavonValderrama:2005uj (); Krebs:2007rh (); Higa:2007gz (); Valderrama:2008kj (); Valderrama:2010fb (); Entem:2007jg (); Yang:2007hb (); Yang:2009kx (); Yang:2009pn (); Albaladejo:2011bu (), or more recently within the nuclear lattice approach Borasoy:2006qn (); Borasoy:2007vy (); Borasoy:2007vi (); Borasoy:2007vk (); Epelbaum:2008vj (); Epelbaum:2009zsa (); Epelbaum:2009pd (); Epelbaum:2010xt (); Epelbaum:2011md () 111It should be noted that lattice EFT implementations of the Weinberg counting only iterate the order piece of the interaction, while the subleading pieces generally enter as perturbations. The short range operator is an exception as it is promoted from order to order to form an “improved” LO interaction., in order to obtain theoretical predictions. Regarding the notation, order will also be referred to as leading order (), order as next-to-leading order (), order as next-to-next-to-leading order (), and so on.
Among the advantages of the Weinberg prescription we can count that it observes the non-perturbative nature of the nuclear forces and naturally fits into the traditional nuclear physics paradigm of employing a potential as the basic calculational input. In particular, the construction of potential descriptions of the nuclear force is conceptually straightforward, as the procedure only involves the expansion of the chiral potential up to a given order and the determination of the free parameters of the theory by a fit to the available low-energy scattering data. Eventually, at high enough order, in particular , this procedure leads to chiral potentials that are able to reproduce the two-nucleon scattering data below a laboratory energy of with a Entem:2003ft (); Epelbaum:2004fk (), a remarkable degree of success which justifies the popularity of the Weinberg prescription.
However, the theoretical basis of the Weinberg counting is debatable and, as we will argue, far from robust. In particular we can identify two serious problems: (i) the counterterms included in the Weinberg counting are not enough as to renormalize the amplitudes (either at Eiras:2001hu (); Nogga:2005hy (); PavonValderrama:2005gu () or / PavonValderrama:2005wv (); PavonValderrama:2005uj ()) and (ii) it is not clear whether the scattering amplitudes obey the power counting of the chiral potential as a result of the full iteration process. In addition, there is the issue of the chiral inconsistency Kaplan:1996xu () which inspired the Kaplan, Savage and Wise (KSW) counting Kaplan:1998tg (); Kaplan:1998we (). Without these ingredients, renormalizability and power counting, the Weinberg prescription is merely reduced to a recipe for constructing nuclear potentials.
The renormalizability problem is a consequence of the appearance of singular interactions in the chiral expansion of the potential: the order -th contribution to the potential behaves as for . In this regard the theory of non-perturbative renormalizability developed in Refs. Beane:2000wh (); Nogga:2005hy (); PavonValderrama:2005gu (); PavonValderrama:2005wv (); PavonValderrama:2005uj () states that, if the finite range potential is singular and attractive, counterterms need to be included for removing the cut-off dependence. This result implies in particular the non-renormalizability of the Weinberg counting at any order, as there is always an infinite number of partial waves for which the chiral potential is singular and attractive. Two possible solutions are (i) to treat the chiral potential, in particular one pion exchange (OPE), perturbatively in partial waves with sufficiently high angular momentum, as advocated in Refs. Nogga:2005hy (); Birse:2005um (), or (ii) to realize that the infinite number of counterterms is partly an artifact of the partial wave projection that can be avoided by correlating the short-range operators of different channels Valderrama:2010fb (). In the present work we will follow the first solution, which is the simplest and most natural one from the EFT perspective.
On the contrary, a repulsive singular interaction is completely insensitive to the value of the counterterms. This condition leads to unexpected consequences, in particular the deuteron disaster PavonValderrama:2005wv (): at the chiral potential turns out to be repulsive in the triplet channel leading to the disappearance of the deuteron as the cut-off is removed, in clear disagreement with the experimental evidence. In addition, the eventual change of sign of the chiral potential at certain orders prevents the formulation of a sensible power counting in a fully non-perturbative set-up, as has been recently stressed by Entem and Machleidt Machleidt:2010kb ().
However, the difficulties associated with non-perturbative renormalizability are not the only reason to avoid the iteration of chiral two pion exchange (TPE). A different, but also potentially serious issue is the possible mismatch between the power counting in the potential and the physical observables. As is well known, the Weinberg prescription defines power counting in terms of the potential, which is not an observable. The iteration of the potential can alter the relative importance of each component of the interaction, specially if the cut-off is not soft enough Yang:2009kx (); Yang:2009pn (); Lepage:1997cs (); Epelbaum:2009sd (), and thus destroy the original power counting at the level of observables. In this respect the preliminary results of Ref. Valderrama:2010aw () for the singlet channel indicate that this may be happening for the typical values of the cut-off employed in the Weinberg scheme, calling for a perturbative reanalysis of the Entem:2001cg (); Epelbaum:2003gr (); Epelbaum:2003xx () and Entem:2003ft (); Epelbaum:2004fk () results.
As argued in the previous paragraphs, the combined requirement of renormalizability and power counting imposes stringent constraints on the set of theoretically acceptable EFT formulations of the nuclear force. At leading order the non-perturbative features of the two-nucleon interaction requires the iteration of certain pieces of the interaction, minimally a contact operator in the and channels Kaplan:1998tg (); Kaplan:1998we (); vanKolck:1998bw (), and optimally the OPE potential in s- and p-waves (and eventually d-waves), plus the extra counterterms required to renormalize the scattering amplitude Nogga:2005hy (). In order to avoid counterterm proliferation, we should be conservative with respect to what pieces of the chiral potential we iterate. Finally, to guarantee the existence of a power counting and prevent the renormalization issues we have mentioned, the subleading pieces of the interaction should be perturbative corrections over the results. The modified Weinberg scheme of Ref. Nogga:2005hy () fulfills these conditions and is in addition a phenomenologically promising approach, as can be inferred from its lowest order results. However, as we are dealing with a posteriori power counting, the previous is not necessarily the only possible realization of a consistent and phenomenologically acceptable nuclear EFT, and there may be other frameworks or variations (see Refs. Beane:2001bc (); Beane:2008bt () for two examples) that may eventually work as well.
The present work explores the perturbative renormalizability of chiral TPE at and assuming the non-perturbative treatment of OPE at . This formulation is to be identified with the proposal of Nogga, Timmermans and van Kolck Nogga:2005hy () for constructing a consistent power counting for nuclear EFT. Regarding the formal aspects of the theory, we elaborate upon the perturbative renormalization framework of Ref. Valderrama:2009ei () and determine the general conditions that ensure the renormalizability of the subleading corrections to the scattering amplitudes. That is, we deduce the power counting of the contact operators. As expected the scaling of the counterterms is very similar to the one previously obtained by Birse Birse:2005um (). As an application of the formalism, we compute the p- and d-wave phase shifts resulting in a phenomenologically acceptable description which improves over the Weinberg counting at the same order Epelbaum:2003xx ().
The general approach we follow and the perturbative techniques we employ, which are based on distorted wave Born approximation (DWBA), are equivalent to the momentum space formulation of Refs. Long:2007vp (); Long:2011qx (). In particular, the recent exploration of the partial wave by Long and Yang Long:2011qx () leads to conclusions that are equivalent to the ones obtained in the present work for the aforesaid wave. The role of perturbative chiral TPE has also been studied in the “deconstruction” approach of Refs. Birse:2007sx (); Birse:2010jr (); Ipson:2010ah (), which analyzes the scaling of the short range interaction for most of the uncoupled s-, p- and d-waves within the power counting derived from Ref. Nogga:2005hy (). In addition deconstruction provides interesting insights on important aspects of the theory such as the determination of the expansion parameter of the EFT, which we will employ in the present work.
The article is structured as follows: in Sect. II we review the modifications induced by the requirement of renormalizability to the Weinberg power counting at leading Nogga:2005hy () and subleading Valderrama:2009ei () orders. In Sect. III we explore in detail the divergences related to the perturbative treatment of chiral TPE within the DWBA to determine the power counting of the subleading contact operators at and . The p- and d-waves phase shifts resulting from this renormalization scheme are presented in Sect. IV. Finally, we discuss the results and present our conclusion in Sect. V. In the appendix A we explain the technical details behind the non-perturbative renormalization.
Ii Modifications to the Weinberg Power Counting
In this section we want to explain in an informal manner why renormalizability requires certain modifications to the Weinberg counting. The starting point is the work of Nogga, Timmermans and van Kolck Nogga:2005hy (), which pointed out, by means of a thorough numerical exploration, that cut-off independence of the leading order nucleon-nucleon scattering amplitude requires new counterterms in partial waves for which the tensor force is attractive. This precise pattern is naturally explained within the non-perturbative renormalization framework developed in Refs. PavonValderrama:2005gu (); PavonValderrama:2005wv (); PavonValderrama:2005uj (), which proved that the inclusion of a specific number of counterterms is a necessary and sufficient condition for the renormalizability of attractive singular potentials. Curiously, the failure of the Weinberg scheme is not accidental: the existence of singular interactions in the EFT expansion of the chiral potential is just a consequence of power counting itself. We will illustrate this point in Sect. II.1. For exemplifying the ideas behind the non-perturbative renormalization of singular potentials and the modifications of the power counting Nogga:2005hy (); PavonValderrama:2005gu (); PavonValderrama:2005wv (); PavonValderrama:2005uj (), we will consider the particular case of the partial wave in Sect. II.2.
Nevertheless, the purpose of the present work is to explore the extension of the Nogga et al. Nogga:2005hy () counting to subleading orders. As already conjectured in Ref. Nogga:2005hy (), the most adequate approach for formulating a canonical EFT expansion of the amplitudes is the perturbative treatment of the higher order corrections to the chiral nuclear potential and the addition of a certain number of counterterms to guarantee the renormalizability of the approach. The power counting for the subleading contact interactions, that is, the order of the promoted counterterms, was constructed for the first time by Birse in Ref. Birse:2005um () (see also Ref. Birse:2009my () for a more accessible presentation). More recently, Ref. Valderrama:2009ei () proved the viability of the previous proposal and found some deviations from the original power counting of Ref. Birse:2005um (). Incidentally, the subleading power counting deviates from the original Weinberg prescription even in partial waves for which the leading order counting remained unaltered in the Nogga et al. proposal Nogga:2005hy (). The best example is provided by the partial wave: although at both Weinberg and Nogga et al. predict the same number of counterterms, at and the counterterms predicted by Weinberg are not enough as to renormalize the scattering amplitude in this partial wave. Based on the related discussion of Ref. Valderrama:2010aw (), we will explain the power counting of the singlet channel in Sect.II.3.
ii.1 Singular Potentials
The appearance of singular interactions in the chiral potential is the fundamental reason which requires the power counting modifications suggested by Nogga et al. Nogga:2005hy (). Curiously, singular potentials are a consequence of the existence of a power counting in the potential itself. The argument for the previous statement is relatively straightforward. We first consider the scaling of the order -th contribution to the chiral potential, which according to power counting is given by
where represents the light scales of the system, like the pion mass or the relative momentum of the nucleons, while stands for the heavy scales, for example the mass of the meson.
It is interesting to notice that the two aforementioned light scales play a very different role: while the pion mass is fixed, the momentum of the nucleons can vary, and in particular it can take values that are much larger than the pion mass, that is, . In this regime the momentum space representation of fulfills the approximate scaling law
which is just a restatement of Eq. (2). After Fourier transforming, this translates into the following scaling relation in coordinate space
which is valid for .
There are two kinds of solution to the scaling relations induced by the power counting of the potential: (i) zero range and (ii) finite range solutions. The zero (or contact) range solutions are constructed from the scaling properties of the three dimensional Dirac delta, that is
which corresponds to a solution of the scaling equation for , yielding
where the subscript indicates the contact range nature of the contribution. Higher order contact range solutions can be easily obtained by adding derivatives to the Dirac delta, although due to parity conservation only an even number of derivatives can appear. That is, new contact terms enter only at at even orders, , as is well known.
The finite range solutions of the scaling relations are fairly straightforward and take the form
where the subscript is used to indicate the finite range. This form corresponds to the observed degree of divergence of the pion contributions to the chiral potentials 222 The inclusion of light but static degrees of freedom, like the isobar in the small scale expansion Hemmert:1997ye (), can alter however the previous scaling.. As we will see, the renormalizability problems arise from the non-perturbative interplay between contact and finite range operators, which modifies their scaling properties at the level of observables, thus changing the power counting of the operators.
ii.2 Leading Order Modifications of the Counting
Weinberg power counting implicitly assumes that the counterterms appearing at each order in the chiral expansion are capable of renormalizing the scattering amplitude. However, this is not the case, as has been repeatedly discussed in the literature Kaplan:1996xu (); Nogga:2005hy (); PavonValderrama:2005wv (); PavonValderrama:2005uj (). In particular, Nogga et al. Nogga:2005hy () found numerically that even at leading order the counterterms of the Weinberg counting do not render the phase shifts unique: several counterterms should be added in the , channels and eventually the channel, depending on the cut-off region under consideration (see also the related comments of Ref. Epelbaum:2006pt ()) 333 Notice however that the recent application of the method to nuclear EFT Albaladejo:2011bu () does not require the inclusion of an additional counterterm in the or the partial waves..
The channel probably provides the best example to exemplify the ideas behind the Nogga et al. proposal Nogga:2005hy (). We start by consider this partial wave in the Weinberg counting, where we can explicitly check the appearance of a strong cut-off dependence of the phase shift at . In the Weinberg scheme, the wave function in this channel can be described by the following Schrödinger equation
where is the order chiral potential. As the potential diverges as at short distances, we include a regularization scale which serves as a separation scale between the unknown short range physics and the known long range physics. In particular, we only consider the chiral potential to be valid for . For radii below the regularization scale, , we do not know how the system can be described, but it is reasonable to assume that at short enough distances the behaviour of the wave function will be dominated by the centrifugal barrier, that is,
for . The previous assumption can be effectively translated into a logarithmic boundary condition for the Schrödinger equation at
corresponding to the radial regulator employed in Ref. PavonValderrama:2005uj (). By integrating the Schrödinger equation from to with the previous initial integration condition, we can extract the phase shifts by matching to the asymptotic form of the wave function at large distances, which is given by
where is the phase shift, and , , with and the Spherical Bessel functions.
The results for the phase shifts in the previous regularization scheme are depicted in Fig. (1). The phase shifts, which have been computed for a series of cut-off radii ranging from to , show a remarkable cut-off dependence not anticipated by the Weinberg counting. The formal reason for this cut-off dependence lies in the details of the short range () solution to the Schrödinger equation Eq. (8), which reads
where is a length scale related to the strength of the tensor force in this channel. As can be seen, the determination of the wave function near the origin requires the determination of the phase . The boundary condition, Eq. (10), just chooses a different, non-converging value of for each , and therefore does not yield unique results. In particular, there is not a well defined limit of for : the value of simply oscillates faster and faster on the way to the origin.
From the EFT point of view, the previous cut-off dependence means that the a priori estimation of the strength of the contact term for the channel was not correct. Otherwise, the cut-off dependence would have not appeared until radii of the order of the breakdown scale of the theory . However, as can be seen in the left panel of Fig. (1), the cut-off dependence manifest for scales of the order of . This indicates the necessity of including a new short range operator in this channel at .
By including the new counterterm, the previous uncontrolled cut-off dependence of the phase shift disappears, as can be appreciated in the right panel of Fig. (1). We have employed the regularization scheme of Refs. PavonValderrama:2005gu (); PavonValderrama:2005wv (); PavonValderrama:2005uj (), in which to unambiguously determine the phase shifts in the partial wave we fix the value of the scattering length 444 Of course, using the term scattering length for a -wave is language abuse. The proper name is scattering volume, as it has dimensions of . A similar comment would apply in the d-wave case, in which the so-called scattering length has dimension of and henceforth is a (5-dimensional) hypervolume. However, for notational simplicity we will always use the term scattering length for the coefficient related to the low energy behaviour of the -wave phase shift, . , see the appendix A for the details. Of course, there is still a residual cut-off dependence that is however mostly harmless owing to the existence of a clear convergence pattern in the limit. That is, we can interpret the residual cut-off dependence as a higher order effect. For the regularization employed in this case, the cut-off dependence of the phase shifts can be explicitly computed by employing the methods of Ref. PavonValderrama:2007nu (), yielding the result
which serves as a formal confirmation of the existence of the limit.
ii.3 Subleading Order Modifications of the Counting
The modifications of the power counting induced by renormalizability are not confined to leading order. Further deviations occur at subleading orders, which can be identified by analyzing the cut-off dependence of the phase shifts, even though these orders are included as perturbations. In this case, the most direct example is the channel, which follows the Weinberg counting at , but diverts from it when subleading corrections are included. At orders and the Weinberg counting dictates a total of two counterterms for the channel, which in momentum space take the form
In terms of renormalization, this counterterm structure is equivalent to fixing two observables, for example the scattering length and the effective range of the channel. For making such a calculation we will follow the formalism of Ref. Valderrama:2009ei (), but adapting the number of counterterms to two. In such a case, we obtain the phase shifts of Fig. (2).
As can be appreciated, there is a strong cut-off dependence in the results of Fig. (2). The reason can be found by analyzing the cut-off behaviour of the perturbative corrections to the phase shifts. According to Ref. Valderrama:2009ei (), the contribution to the phase shift is proportional to the integral
where is the wave function in the Weinberg counting, which can be described by adapting the Schrödinger equation for the channel, Eq. (10), to the case, and is the order contribution to the chiral potential. It should be noted that scales as , even though nominally it is of order . The degree of divergence of can be easily determined by considering the Taylor expansion of the wave function PavonValderrama:2005wv (), that is
where the behaviour of the different terms at short enough distances is given by , , , and so on. With the previous information, and the additional fact that , we find that for the particular case the integral in Eq. (15) diverges as
with the length scale governing the divergences and where , and are dimensionless numbers that are expected to be of order one, while the terms in the dots are already finite. In Fig. (2) we have only fixed the and behaviour for reproducing the low energy phase shifts, that is, the scattering length and the effective range. This means that there is still a contribution that diverges as explaining the failure of the Weinberg counting in the perturbative context.
Again, the solution is to modify the counting in such a way as to include a new counterterm. This means that the correct and contact interaction should be 555We have ignored the operator, as it is redundant Beane:2000fi (). However, the equivalence of this operator with only shows up either in dimensional regularization or once the floating cut-off is removed. For finite cut-off calculations, the difference between the and operators is presumably a higher order effect. On the practical side the inclusion of this operator will surely be advantageous in momentum space treatments as it may help to reduce the cut-off dependence of the scattering observables.
instead of the form dictated by Weinberg dimensional counting in Eq. (14). The additional counterterm is able to absorb the divergence in the phase shift. In such a case, the new cut-off dependence is the one given by Fig. (2), in which there is still a residual cut-off dependence of . However, the same comments apply as in the channel at : the residual cut-off dependence is a higher order effect and can be safely ignored.
Of course, the power counting modification of Eq. (18) is well-known to arise in two-body systems with a large value of the scattering length Kaplan:1998tg (); Kaplan:1998we (); Birse:1998dk (); Barford:2002je (), where the renormalization group analysis of the operator indicates that its actual scaling is rather than the naive dimensional expectation . In this regard, it should be stressed that the phase shifts cannot be properly renormalized at / without the inclusion of the operator. The previous is not merely a statement about the existence of the limit in the calculation, but also about the accuracy of the theoretical description of the system. The renormalized calculation is accurate up to () at (). On the contrary, if we do not to include the operator, the error of the calculation will be , much bigger than expected. This may prove to be a significant drawback for the nuclear EFT lattice approach Borasoy:2007vi (); Borasoy:2007vk (); Epelbaum:2008vj (); Epelbaum:2009zsa (); Epelbaum:2009pd (); Epelbaum:2010xt (); Epelbaum:2011md (), which does not include the aforementioned operator in the channel.
Finally, the observation that the contact interaction of Eq. (18) renormalizes the scattering amplitude can be employed to extract the breakdown scale of the EFT in the singlet channel, as has been shown by the deconstruction approach of Refs. Birse:2007sx (); Birse:2010jr (). The idea is to employ the phenomenological phase shifts as the input of a calculation for determining the form of the contact range interaction up to an error of . This represents in fact the complementary process of fitting the parameters to the phase shifts. By considering the on-shell form of the contact range potential, , Ref. Birse:2010jr () is able to determine that (i) the form of the short range interaction complies with Eq. (18) and (ii) the hard scale governing is . This figure is much lower than the expected and translates into a slow convergence rate for the singlet channel, , emphasizing the importance of the proper renormalization of the scattering amplitude to minimize the theoretical uncertainty. The breakdown scale can in fact be appreciated in the right panel of Fig. (2), where the cut-off dependence of the renormalized results starts to become sizable at , in agreement with the estimates of Ref. Birse:2010jr () 666A similar comment applies for the phase shifts computed in the Weinberg counting: if we inspect the left panels of Figs. (1) and (2), the cut-off dependence is already conspicuous at , suggesting a relatively soft breakdown scale for both the and partial waves. In contrast, for the channel in the Nogga et al. counting Nogga:2005hy () the cut-off dependence of the right panel of Fig. (1) does not provide a clear indication of the breakdown scale. In this case the determination of the range of applicability of the theory is better done by analyzing the size the subleading order corrections to the phase shift. .
Iii Perturbative Renormalizability of Singular Potentials and Power Counting
In this section we study the power counting that arises when one pion exchange is iterated to all orders and chiral two-pion exchange is treated as a perturbation. We follow a more formal and detailed approach than in the previous section. We start by defining the power counting conventions in Sect. III.1. We derive the power counting by requiring the perturbative corrections to remain finite in the limit, first for the uncoupled channels in Sect. III.2 and then we extend the results to the coupled channels in Sect. III.3. We informally comment on the convergence rate and the expansion parameter of the resulting EFT in Sect. III.4. Finally, we discuss the perturbative power counting in Sect. III.5. The results of this section are briefly summarized in Table 1, where we can check the number of counterterms required in the lower partial waves from up to .
We assume that the chiral potential is expanded in terms of a power counting as follows
where () is the soft (hard) scale of the system. The expansion starts at order , and we only consider the corrections to the chiral potential up to a certain order ( in the present paper). The T-matrix follows the same power counting expansion as the potential, that is
The scattering equations can be obtained by reexpanding the Lippmann-Schwinger equation
in terms of the power counting of the potential, taking into account that the resolvent operator is formally of order . In this convention, only contributions to the chiral potential which are of order should be iterated, yielding the equation
The corresponding dynamical equation of the higher order contributions to the T-matrix are increasingly more involved.
In principle, in the Weinberg scheme the power counting expansion starts at order
Within the power counting conventions we follow, the previous means that all pieces of the potential should be treated as perturbations. However, it is clear that needs to be iterated in certain partial waves. This requires a promotion from order to , that is
at least in the partial waves in which either the contact operator or OPE are non-perturbative. From a purely quantum-mechanical point of view the iteration is necessary if the potential is not weak. In terms of power counting, a large coupling constant can be accounted for by identifying it with , which is a large number. Alternatively, we can try to explicitly identify the low energy scale which requires the potential to be iterated. For the contact operator the additional low energy scale is well known to be the inverse of the large scattering length of the and channels, as has been extensively discussed in the literature Kaplan:1998tg (); Kaplan:1998we (); Gegelia:1998gn (); Birse:1998dk (); vanKolck:1998bw (). For justifying the iteration of the OPE potential Birse has proposed Birse:2005um () the identification of
as the new low energy scale. However, this choice does not explain the fact that OPE remains perturbative in the singlet channels even if iterated Fleming:1999ee (). A different possibility may be to treat the inverse of the length scale governing the strength of the tensor force as the missing low energy scale, giving a that is in general a fraction of .
After the promotion of potential from order to order , we encounter a remarkable simplification in the dynamical equations describing the and contributions to the T-matrix, namely
where, due to the absence of contributions to the finite range chiral potential from order to order , second order perturbation theory is not needed up to order . That is, the equation above is valid for .
iii.2 Uncoupled Channels
We consider in the first place the power counting expansion of the -wave phase shift
The phase shift is calculated non-perturbatively by solving the reduced Schrödinger equation with the chiral potential, see Appendix A for details. On the other hand, we compute the order -th contribution to the phase shift, , in the distorted wave Born approximation (DWBA), that is
where is the leading order phase shift, the reduced mass of the two nucleon system, the center-of-mass momentum, the angular momentum of the system, a normalization factor and the perturbative integral. The perturbative integral is defined as
where is the wave function, which we assume to be asymptotically () normalized to
where , , with and the spherical Bessel functions. For the purpose of analyzing the renormalization of the perturbative phase shifts, we will assume that the normalization factor is defined in such a way that the wave function is energy-independent at , that is
a prescription that will make relatively easy the identification of the necessary number of counterterms. However, in practical calculations we will simply take
which is easier to implement than Eq. (31).
The renormalizability of the perturbative phase shifts can be easily analyzed in terms of the short range behaviour of the reduced wave function and the contribution to the chiral potential, which determine whether the perturbative integral is divergent or not. In particular, for , we have the following
where describes the power-law behaviour of the wave function at short distances. For the perturbative integral we have
which diverges if . The necessary number of counterterms can be determined by considering the Taylor expansion of the perturbative integral in terms of
in which each new term is less singular than the previous one as a consequence of the related expansion for the reduced wave function
where for small radii, with . We can see now that the reason for choosing the energy-independent normalization of Eq. (31) is to have additional power law suppression at short distances in the expansion of the wave function. In terms of the expansion of the perturbative integral, we obtain
which converges for . Eventually, as increases, we will have well-defined integrals. In this regard, we can define the number of subtraction/counterterms as the smallest value of for which
It should be noted that refers to the total number of subtraction/counterterms required at a given order. For example, if the counterterm only affects the scattering length of the -th order of the phase shift. If the scattering length was already fixed at a lower order, this just indicates that a perturbative correction must be added to the lower order counterterm.
To summarize, we have that the perturbative integral can be divided into a divergent and a regular piece
where the divergent piece can be expanded in powers of the squared momentum ,
with the number of counterterms that renders the perturbative phase shifts finite. The perturbative integral can be regularized in any specific way which we find convenient. A particularly simple and straightforward method is to add free parameters to the original integral as follows
where the parameters are to be fitted to the experimental phase shifts. The previous parameters, which we will informally call -parameters, can be related to the more standard counterterms if we consider a concrete representation of the short range physics, for example
where is a factor which ensures that the matrix elements of the previous potential in momentum space behaves as
for . If we have chosen to work in the practical normalization of Eq. (32), the following relationship is obtained between the fitting parameters and the usual counterterms
For other representations of the short range physics, the relationship will take a more complex form.
We can distinguish four cases: (i) the irregular solution of a non-singular potential (), (ii) the regular solution of a non-singular potential (, ), (iii) the general solution of an attractive singular potential (, ) and (iv) the regular solution of a repulsive singular potential (). In the following paragraphs we will discuss each of these cases in detail.
iii.2.1 Non-Singular Potential
Non-singular (or regular) potentials are potentials for which the regularity condition is fulfilled
The regularity condition implies in turn that the short range behaviour of the wave function is determined by the centrifugal barrier, which overcomes the potential at short distances. For , we can define a regular and irregular solutions which behave as
If there is no short range physics at , the regular solution is chosen. On the contrary, if a contact operator is included at , the reduced wave function will be a superposition of the regular and irregular solutions.
The regular solution gives rise to the Weinberg power counting unaltered. If we consider the expansion of the regular solution, Eq. (36), we have
that is, and . The convergence of the perturbative integral requires in this case
where stands for the number of counterterms in a given partial wave. Therefore new counterterms appear as expected in naive dimensional analysis, that is
For the particular case of the channel, the first counterterm () is required at , a second () at , and so on.
In contrast, the occurrence of the irregular solution at will change the power counting. The expansion of the wave function yields in this case
which implies and . The finiteness condition reads
which implies a significant deviation from naive dimensional analysis. In fact the scaling of the counterterms is given by
For the channel, the previous requires that the first perturbative correction to the counterterm enters at order , while the counterterms (with ) are required at order . It should be noticed that the order assignment to the first perturbative correction to the counterterm is merely formal: this correction actually enters at the same order as the first perturbative correction to the finite range piece of the potential, which happens at order . In fact, the scaling is rather an indication of the fine-tuning required for having an unnatural scattering length. In terms of Birse’s renormalization group analysis (RGA) Birse:1998dk (), the eigenvalue is a reflection of the unstable nature of the infrared fixed point associated with two-body systems with a large scattering length.
It is interesting to notice that the extension of the power counting for irregular solutions to -waves leads to the same conclusions as p-wave Halo EFT Bertulani:2002sz (), namely, that the first two contact operators should be iterated. The previous arguments imply that for a -wave, the first counterterm will be of order , the second , and the third . This has two interpretations: first, that the unstable infrared fixed point is even more fine-tuned than in -waves, and second, that it may be necessary to iterate two counterterms instead of only one. However, for the particular case of nucleon-nucleon scattering, we do not encounter this situation.
iii.2.2 Singular Potential
The appearance of power-law singular interactions at entails substantial changes to the power counting of the short range operators. In particular we are interested in the effect of tensor OPE, which at short distances () behaves as
where is the reduced mass of the two-nucleon system and a length scale that governs the strength of the tensor force. We have chosen a convention in which and where the attractive or repulsive character of the potential is indicated by the explicit sign. The renormalization of singular potentials implies that attractive singular interactions require the inclusion of a counterterm, while repulsive singular interactions do not PavonValderrama:2005gu (); PavonValderrama:2005wv (); PavonValderrama:2005uj (), a feature which will be reviewed in the following paragraph.
For a tensor force, the terms of the expansion of the wave function behave for as
If the tensor force is attractive, the function takes the form
with and where the specific linear combination of sine and cosine factors is determined by the counterterm. In contrast, for a repulsive tensor force we have a regular and irregular solution. We have indeed that
where the solution is regular at the origin, while the solution diverges exponentially for . For the repulsive case we will always chose the regular solution at .
In the attractive case, the convergence of the perturbative integrals is solely determined by the power-law behaviour of the reduced wave functions: the sine and cosine factors do not play any role on the finiteness of the perturbative corrections. Renormalizability requires
which in turn implies that the first perturbative correction to the counterterm scales as
that is, the first correction to enters at order , the counterterm at order , and so on. It is interesting to notice that (i) the power counting of the counterterms does not depend on the partial wave considered, (ii) the first perturbative correction to is , which means that the infrared fixed point is stable, and (iii) the appearance of new counterterms is delayed in comparison with the non-singular potentials.
In contrast, the analysis for repulsive tensor force is puzzling. The decreasing exponential behaviour of the wave function at short distances is able to render the perturbative integral finite, independently of the degree of singularity of he -th order potential. This feature can be easily appreciated by considering the integral
which is always finite. However, for a consistent power counting to emerge we need new counterterms at higher orders. Two solutions are possible: (i) follow the power counting rules for attractive singular interaction, that is, , (ii) follow naive dimensional analysis, i.e. Weinberg counting, leading to .
The first solution was proposed by Birse in Ref. Birse:2005um (). It only considers as relevant to the power counting the power-law behaviour of the wave function, but not the sine, cosine or exponential factor. This expectation is consistent with the fact that the exponentials are just a sine or cosine function with an imaginary argument. However, it is also natural to expect that short distances do not play a significant role if the particles cannot approach each other due to the repulsive potential barrier.
This observation leads to the second solution, namely to follow naive dimensional analysis, which implicitly assumes that the repulsive channel can be treated as a perturbation 777 Alternatively, we can try to analyze the scaling of the wave function at distances of the order of the breakdown scale of the theory, . If the wave function behaves as (instead of the singular ), naive dimensional expectations are fulfilled and the use of Weinberg counting is justified. Unfortunately, for the particular case of the channel the singular behaviour is achieved at distances of about (as in this case), so the wave function scaling argument cannot be applied. The non-perturbative wave function scaling at the breakdown radius does not imply however the failure of the perturbative treatment of OPE. The reason is that the non-perturbative contribution (i.e. from ) to the phase shifts is particularly small, as can be explicitly checked from applying the methods of Ref. PavonValderrama:2007nu () to this case, leading to The vanishing exponential ensures that, once the non-perturbative regime is achieved, the corresponding contributions to the total phase shift will be negligible. . For the particular case of the channel, this is not an unreasonable prospect in view of the analysis of perturbative OPE by Birse Birse:2005um (), which suggests a critical momentum around above which the perturbative treatment will fail. In contrast, the critical momentum for the attractive channel is around , requiring the non-perturbative treatment of the OPE tensor force if we want to describe the phase shifts up to . In addition, the critical momenta quoted above were obtained in the limit. For finite pion masses we expect the critical momenta to rise, as the behaviour is strongly suppressed for . In the repulsive case, the potential barrier prevents the two nucleons to explore in detail the region, where the singular nature of the intermediate range tensor force manifests, meaning that the critical momenta are expected to rise significantly with respect to the limit. On the contrary, in the attractive case, the two nucleons are driven to the region, and thus the critical momentum will only rise weakly.
However, there is still the problem of what happens if the repulsive singular interaction is non-perturbative. It is clear that the current understanding of the power counting of repulsive singular potentials is incomplete. Even though in the particular case of the partial wave we can overcome the problem by invoking the perturbative treatment of tensor OPE in this channel, the issue may eventually have important consequences for the power counting of coupled channels. For the moment we will simply assume that attractive and repulsive singular interactions follow the same power counting, unless we expect the singular potential to behave perturbatively. It may be possible that the extension of the methods of Long and Yang Long:2011qx () from the to the channel will provide new insight into the power counting of repulsive singular interactions.
iii.3 Coupled Channels
For analyzing the perturbative renormalization of chiral TPE in the coupled channel case, we employ the eigen representation of the phase shifts PhysRev.86.399 (). In this parametrization the DWBA formulas take their simplest form. The eigen phase shifts are expanded according to the power counting as follows
and the phase shifts are computed non-perturbatively as explained in Appendix A. The order -th contribution to the eigen phase shifts is given by
where the perturbative integrals , and are defined as