Power Counting in Peripheral Partial Waves: The Singlet Channels

Power Counting in Peripheral Partial Waves: The Singlet Channels

M. Pavón Valderrama pavonvalderrama@ipno.in2p3.fr School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China Institut de Physique Nucléaire, CNRS-IN2P3, Univ. Paris-Sud, Université Paris-Saclay, F-91406 Orsay Cedex, France    M. Sánchez Sánchez sanchezmario@ipno.in2p3.fr Institut de Physique Nucléaire, CNRS-IN2P3, Univ. Paris-Sud, Université Paris-Saclay, F-91406 Orsay Cedex, France    C.-J. Yang yangjerry@ipno.in2p3.fr Institut de Physique Nucléaire, CNRS-IN2P3, Univ. Paris-Sud, Université Paris-Saclay, F-91406 Orsay Cedex, France    Bingwei Long bingwei@scu.edu.cn Department of Physics, Sichuan University, 29 Wang-Jiang Road, Chengdu, Sichuan 610064, China    J. Carbonell carbonell@ipno.in2p3.fr Institut de Physique Nucléaire, CNRS-IN2P3, Univ. Paris-Sud, Université Paris-Saclay, F-91406 Orsay Cedex, France    U. van Kolck vankolck@ipno.in2p3.fr Institut de Physique Nucléaire, CNRS-IN2P3, Univ. Paris-Sud, Université Paris-Saclay, F-91406 Orsay Cedex, France Department of Physics, University of Arizona, Tucson, AZ 85721, USA
July 15, 2019

We analyze the power counting of the peripheral singlet partial waves in nucleon-nucleon scattering. In agreement with conventional wisdom, we find that pion exchanges are perturbative in the peripheral singlets. We quantify from the effective field theory perspective the well-known suppression induced by the centrifugal barrier in the pion-exchange interactions. By exploring perturbation theory up to fourth order, we find that the one-pion-exchange potential in these channels is demoted from leading to subleading order by a given power of the expansion parameter that grows with the orbital angular momentum. We discuss the implications of these demotions for few-body calculations: though higher partial waves have been known for a long time to be irrelevant in these calculations (and are hence ignored), here we explain how to systematize the procedure in a way that is compatible with the effective field theory expansion.

Potential Scattering, Renormalization, Nuclear Forces, Two-Body System

The derivation of nuclear forces from first principles is one of the central problems of nuclear physics. In fact the nuclear potential – though not an observable – is the theoretical object from which we usually derive the properties of nuclei and calculate their reaction rates. With the recent onset of ab initio calculations, the direct link between nuclear forces and nuclear structure can be established for the first time, at least for light nuclei. Yet the understanding of nuclear forces from quantum chromodynamics (QCD) remains incomplete.

Nowadays, among the strategies to explain nuclear observables from QCD the most promising are lattice QCD and effective field theory (EFT). The first approach tries to calculate nuclear properties directly from QCD. Lattice QCD, though close to, is still not able to explain light nuclei in the physical world, i.e. for the physical pion mass. First calculations for pion masses larger than the physical one are promising Beane et al. (2011) and it is apparent that will be reached soon. The second approach – the EFT formulation – establishes the connection between nuclear physics and QCD indirectly, avoiding the requirement of complex lattice QCD calculations (see Refs. Bedaque and van Kolck (2002); Machleidt and Entem (2011) for reviews). Nuclear EFT is the renormalization group evolution of QCD at low energies and thus equivalent to QCD. The idea is to write down the most general Lagrangian involving the low-energy fields and respecting the low-energy symmetries of QCD, in particular chiral symmetry. In this regard nuclear EFT simply amounts to rewriting the QCD Lagrangian in terms of a different set of fields, nucleons and pions instead of quarks and gluons.

A key ingredient of nuclear EFT is the existence of a separation of scales. Pions and nucleons are no longer valid degrees of freedoms at momenta above a characteristic hard scale of the order of . Yet most nuclear processes of interest happen at a softer scale of the order of , which can be identified with the typical momenta of nucleons inside nuclei or with the pion mass. Moreover on a first approximation naive dimensional analysis (NDA) suggests that the dimensionful couplings of the EFT might be natural when expressed in units of the hard scale . From this point of view can be interpreted as the natural QCD mass scale for hadrons (of the order of the rho or the nucleon mass, not to be confused with the QCD perturbative scale ) or . As a consequence we can write the scattering amplitude in EFT as a power series in terms of the ratio between the soft-scale and the hard-scale :


This property goes under the name of power counting and is probably the most useful feature of EFT. Power counting implies that we can organize the calculations to achieve a given goal of accuracy and lies at the core of the systematics of the EFT approach.

We stress that the existence of a power counting depends on renormalizability. EFT expresses the interactions among its fields as a low-energy expansion. While at low momenta each term in this expansion is smaller than the previous one, at high momenta the contrary happens and eventually the EFT interactions diverge in the ultraviolet. This does not pose a conceptual problem because EFT is only expected to work at low energies. But it poses a technical one: the ultraviolet divergences threaten to destroy the low-energy EFT expansion via loop corrections. The way to deal with this issue is renormalization: we introduce a cut-off to regularize the loops, include counterterms to absorb the -dependence and finally check that the low-energy observables do not depend on the choice of . Cut-off independence, i.e. renormalizability, is therefore a necessary condition to have power counting in EFT. Renormalizability and power counting are intimately intertwined, a feature that is emphasized for instance in Wilsonian renormalization Wilson and Kogut (1974). If renormalizability is absent at a certain order we have to include new contact interactions until cut-off independence is restored, which in turn constrains the power counting. If for some reason our prejudices about power counting do not lead to a renormalizable EFT, they have to be revisited and amended.

Traditional EFT knowledge is perturbative, chiral perturbation theory (ChPT) being a good example. The understanding of the renormalization group analysis of non-perturbative physics – as is the case in nuclear physics – is recent in comparison. But the first formulation of nuclear EFT, the Weinberg counting Weinberg (1990, 1991), predates these developments. To deal with the non-perturbative features of nuclear physics, the Weinberg counting applies not directly to the scattering amplitude but rather to the nuclear potential:


The rationale is that, though power counting is not obvious for the scattering amplitude in the non-perturbative case, it is for the potential. Thus the idea is to iterate the EFT potential, or at least the leading order () piece, in the Lippmann-Schwinger (or Schrödinger) equation with the implicit expectation that the resulting -matrix follows the EFT expansion


Taking into account the simplicity of the approach outlined by Weinberg it comes as no surprise the enthusiasm with which it has been followed, leading to the development of two-nucleon potentials with  Entem and Machleidt (2002, 2003); Epelbaum et al. (2005).

Yet there is a fly in the ointment: it is not clear whether the scattering amplitudes follow a power counting in Weinberg’s proposal. In fact the situation is rather the contrary. Several works have shown once and again Kaplan et al. (1996); Beane et al. (2002); Nogga et al. (2005); Pavon Valderrama and Arriola (2006); Pavon Valderrama and Ruiz Arriola (2006); Entem et al. (2008); Yang et al. (2009a, b, c); Zeoli et al. (2013); Marji et al. (2013) that the Weinberg counting is not renormalizable. As the cut-off is increased above , the scattering amplitudes show all kinds of fascinating behavior with the notable exception of cut-off independence. The most natural interpretation of the previous results is that renormalization is lost in the Weinberg counting. Not only that, the connection with QCD disappears because this approach does not make for a proper EFT. We mention in passing that a few authors have proposed to reinterpret what renormalization means in order to accommodate the previous observations in the Weinberg counting Epelbaum and Meissner (2006); Epelbaum and Gegelia (2009). This is adequate as long as one does not insist on a clear power counting (beyond the vague impression that calculations get better at higher orders) and a direct connection to QCD. However this is not the solution for an appropriate nuclear EFT. Rather we should strive to get scattering amplitudes that are unambiguously renormalizable and have a power counting.

As a matter of fact there exists already a consistent version of nuclear EFT Nogga et al. (2005); Birse (2006); Valderrama (2011); Pavon Valderrama (2011); Long and Yang (2011, 2012a, 2012b). It is renormalizable, can describe the scattering amplitudes for , converges well and power counting is realized at the level of observables. Its foundation relies on a better understanding of the renormalization of non-perturbative physics and singular interactions Beane et al. (2001, 2002); Nogga et al. (2005); Pavon Valderrama and Arriola (2006); Pavon Valderrama and Ruiz Arriola (2006); Long and van Kolck (2008). The key improvements over the original Weinberg proposal are the non-perturbative renormalization of the amplitudes and the addition of subleading order contributions as perturbations. At the main difference with the Weinberg counting lies in the promotion of a series of - and -wave counterterms to in triplet partial waves for which the tensor force is attractive, a change proposed in Ref. Nogga et al. (2005). At subleading orders there are more counterterms than in Weinberg counting, for instance in the attractive triplets that already received a counterterm at . The convergence of the EFT expansion is acceptable and the description of the data too, but it has not achieved yet a as in the Weinberg approach. However this is expected if we take into account that the calculations of Refs. Valderrama (2011); Pavon Valderrama (2011); Long and Yang (2011, 2012a, 2012b) are still one order below the most advanced ones in the Weinberg approach Entem and Machleidt (2002, 2003); Epelbaum et al. (2005).

One prediction of the original Weinberg counting is that the one-pion-exchange (OPE) potential – the nuclear potential – is always non-perturbative. Yet this is in contradiction with common wisdom in nuclear physics. Pion exchanges have been known for a long time to be perturbative in peripheral partial waves Kaiser et al. (1997, 1998). While this is easy to understand in terms of the repulsive centrifugal barrier for high angular momenta (), the analogous explanation in terms of power counting has traditionally remained elusive. From time to time this has been discussed in the literature Nogga et al. (2005); Birse (2006); Pavon Valderrama (2011); Long and Yang (2012a), but rather as an afterthought. This peripheral demotion already happens for moderate angular momenta . Different arbitrary angular momentum cut-offs for when to consider OPE perturbative (or not) have been proposed, usually in the range . However, taking into account that the importance of this matter is rather tangential in two-body calculations, it comes as no surprise the disinterest for this particular problem 111 Actually power counting takes into account the peripheral suppression in the particular case of the contact-range interactions, where the -wave counterterms are less relevant than the -wave ones by a factor of in naive dimensional analysis.. The task of power counting should be to quantify the size of the peripheral wave suppression to systematically include it in EFT calculations.

The answer to this question is not only of academic interest, but has practical implications too. The main applications are few-body calculations, which usually require the inclusion of contributions arising from two-body partial waves up to a critical value of the orbital angular momentum (typically or in the three-nucleon system Witala et al. (2001)). Yet the choice of a maximum angular momentum is driven by numerical considerations rather than by the constraints that the EFT expansion imposes on the accuracy of physical observables.

The purpose of this manuscript is to translate the quantum-mechanical peripheral-wave suppression into a power-counting demotion of the OPE potential. This will allow to take the decisions regarding which two-body partial waves to ignore based on power-counting arguments. That means improving the systematics of these calculations or even simplifying them at the lowest orders where probably very few partial waves need to be included. Here we limit ourselves to the spin-singlet waves where OPE is not singular and thus can be defined without counterterms.

The manuscript is structured as follows. In Section I we will compare the non-perturbative OPE predictions for the phase shifts in the singlets with their perturbative expansion. This will allow us to check up to what point OPE is perturbative. In Section II we will find a power-counting explanation for the peripheral demotion of central OPE, which will later be checked against numerical calculations of the expansion parameter of central OPE. Finally we will present our conclusions in Section III.

I Perturbative Pions in Peripheral Singlet Waves

In this section we will analyze whether the OPE potential is perturbative in the peripheral singlet waves, that is, the , , , partial waves. The approach is straightforward: we will compare the amplitudes that we obtain after the full iteration of OPE with the perturbative ones. We will work out the phase shifts up to fourth order in perturbation theory. The outcome of these lengthy calculations is that the OPE potential is definitely perturbative in all the singlet waves with orbital angular momentum (the case will be dealt with in a separate manuscript). The interpretation in terms of traditional power counting is that the OPE potential is subleading (where is reserved for interactions that have to be iterated to reproduce the non-perturbative physics of the -waves).

i.1 Formalism

Here we will determine the perturbative character of OPE in the peripheral singlets by means of a direct calculation. In first place we will solve the Schrödinger equation with the OPE potential to obtain the non-perturbative phase shift , where indicates the orbital angular momentum. In a second step we will expand the phase shift perturbatively as


where the numbers in brackets indicate the number of insertions of the OPE potential. Comparing the right- and left-hand sides of the equation above, we can check the convergence of the perturbative series.

At this point we find it worth commenting that this kind of direct comparison can only be used in the singlets. As it is well-known, the OPE potential comprises central and tensor pieces. The former is regular, and hence the phase shifts and their perturbative expansion are well-defined. But the latter is singular: to determine whether the tensor piece is perturbative or not requires renormalization. In the singlet channels only the central piece of the OPE potential is involved in the calculations. This is what makes possible the simple analysis we pursue in this section.

The calculations are done as follows. For the phase shifts, we begin with the reduced Schrödinger equation:


where is the OPE potential. We solve it with a regular boundary condition at the origin, i.e. . We can extract the phase shifts from the asymptotic behavior of the wave function at large distances ():


where , are reduced spherical Bessel functions. For obtaining the non-perturbative phase shifts, we will match the asymptotic expression above to the reduced wave function at the infrared cut-off .

In contrast, if we are interested in the perturbative phase shifts, the asymptotic matching will not be the method of choice. We will find much more useful instead the following integral representation of the phase shift:


in which the wave function is normalized as in Eq. (6). The advantage of the integral formula above is that it is better suited for a perturbative calculation: with a wave function of order we can calculate the phase shift at order . A derivation of this formula can be found in Appendix A.

If the potential is weak enough, we can expand the reduced wave function perturbatively as follows


where the brackets indicate the number of OPE insertions. The differential equations followed by the ’s are then

for , (10)

which we obtain from plugging the perturbative expansion of the wave function into the reduced Schrödinger equation and rearranging the terms.

We solve the set of differential equations iteratively, starting with for which we take , i.e. the free solution. If we now expand the integral expression for the phase shifts (Eq. (7)) perturbatively, we end up with the Born approximation


For the only subtlety is finding a suitable boundary condition for . Actually, the easiest thing to do is to integrate from infinity to the origin. The reason is that the asymptotic form of is readily available from expanding Eq. (6) perturbatively. At first order we find the asymptotic boundary condition


from which we can integrate for arbitrary . Then we can calculate the second order contribution to the phase shifts by inserting in the perturbative expansion of Eq. (7), yielding


For the higher orders we simply repeat the previous steps. The asymptotic boundary conditions and the expressions for the contributions to the phase shifts read:


Expressions for orders can be obtained analogously.

i.2 Perturbative OPE

Now we evaluate the previous perturbative differential equations and integrals to calculate the phase shifts. We remind that the form of the central piece (the only one we need in the singlets) of the OPE potential is given by


where ( for the singlets), , is the nucleon mass and is the characteristic momentum scale of OPE, defined as


with and . By characteristic scale we mean the following: we naively expect the OPE to behave perturbatively for momenta , while for higher momenta it becomes non-perturbative Kaplan et al. (1998a, b); Fleming et al. (2000). As we will see, this is not the case in the peripheral singlets.

We have previously commented that the perturbative expansion is finite and well-defined in the singlets. Yet we will keep a finite cut-off in the calculations. In particular we regularize the OPE potential as follows


which simply amounts to changing the lower limit of the perturbative integrals from to . The reasons for having a cut-off are: i) in actual EFT calculations (including counterterms and the two-pion exchange potential, which we have ignored in this work) a finite cut-off is a must, ii) the computational requirements of doing perturbation theory to high order decrease significantly with a finite cut-off 222We are using a variable step integration method for the set of coupled differential equations that define the perturbative expansion, Eqs. (9) and (10). Owing to the number of differential equations involved it is increasingly difficult to solve it for small cut-offs when i) we increase the perturbative order for which we integrate and ii) we increase the angular momentum, particularly at low momenta. The chosen cut-off is actually on the limit of what we can compute at fourth order, yet it is more than enough for a nuclear EFT calculation.. We have chosen as a reference value for the cut-off. For this cut-off the perturbative calculations with OPE have already converged (as a matter of fact, there are only tiny differences in the results for ). Besides, is also significantly below the standard cut-off ranges employed in previous EFT calculations in -space Valderrama (2011); Pavon Valderrama (2011). For an -wave the equivalence of the previous -space cut-off with a sharp -space cut-off is , where we have used the relationship . This relationship can be obtained by comparing the running of the -wave counterterm in a pionless theory for a delta-shell -space regulator and a sharp cut-off -space regulator (see Ref. Entem et al. (2008) for details). The extension of the previous relationship for peripheral partial waves is trivial, yielding the numerical values for a wave, for a wave and higher values for . For checking purposes we explicitly show the cut-off dependence of the isoscalar partial waves in Fig. 1, where it can be appreciated that the more peripheral the wave the weaker the cut-off dependence. We have chosen to show the isoscalar channels because it is for these channels that central OPE is strongest, yielding more cut-off dependence. Yet, with the exception of the channel for (already a relatively large cut-off), the cut-off dependence ranges from rather mild () to negligible ().

We obtain the phase shifts shown in Figure 2, where we can appreciate that the perturbative expansion is converging extremely quickly even for the wave, the lowest partial wave considered. The perturbative series is more convergent the higher the partial wave: the tree level (that is, the Born approximation) phase shifts already match the non-perturbative ones with a precision of a fraction of a degree, with the exception of the wave. All this indicates that the convergence parameter of the perturbative expansion is certainly smaller (i.e. a faster convergence) than that of the EFT for the particular case of the singlet waves.

Figure 1: Cut-off dependence of the non-perturbative phase shifts for the OPE potential in the singlet peripheral waves , and . We show the isoscalar partial waves because for them the OPE potential is three times stronger than in the isovector partial waves. Hence the cut-off dependence is also expected to be more evident in the isoscalar channels. We show the cut-off range for the c.m. momenta and .
Figure 2: Convergence of the perturbative expansion of the phase shifts for the OPE potential in the singlet peripheral partial waves , , , , and . We compare the full, non-perturbative OPE phase shifts – the black solid line in the figures – with the perturbative ones computed at increasing orders, where “T” stands for “tree level” (i.e. the Born approximation) and “1L”, “2L”, “3L” for the one-, two- and three-loop calculation (i.e. second, third and fourth order perturbation theory). The calculations have been performed with a finite cut-off of .

Ii The Peripheral Wave Demotion

In this section we will discuss the role of angular momentum for the power counting of the singlets. The aim is to explain the fast convergence rate of the perturbative series for central OPE in the singlets. As already seen, the iteration of the OPE potential is suppressed in the peripheral waves with respect to the expectations of standard power counting. However, the problem is how to incorporate the peripheral suppression into the EFT expansion. For that we have to quantify the size of this effect. We will discuss a series of ideas to find the factor by which the iteration of the OPE potential is suppressed in the higher partial waves.

ii.1 Quantum Mechanical Suppression

We begin by considering the peripheral suppression of a finite-range potential in standard quantum mechanics. The arguments are straightforward and relatively well-known, but we find it convenient to repeat them for illustrative purposes. This type of suppression is only apparent at momenta far below the inverse of the range of the potential, i.e. for nuclear forces. This means that it will help to explain the power counting in pionless theories. However, the application to pionful theories will be limited to very peripheral waves.

First we begin by considering the matrix element of a central potential in the partial wave , which is given by

where is the standard spherical Bessel function and the potential in coordinate space. In the Born approximation the size of this matrix element for is proportional to the scattering amplitude. If we compare


for different angular momenta we can obtain a baseline estimation of the peripheral suppression factor. We can do this by calculating the ratio of against a reference partial wave . A natural choice is the wave as it is the smallest angular momenta considered in this work. There is the complication that even (odd) partial waves are isoscalars (isovectors). This can be circumvented by taking into account the isospin factors into the definition of the ratio


where , , and or depending on the total isospin of a particular channel. We have calculated the inverse of this ratio in Figure 3, where the choice of the inverse is for illustrating the suppression in a more obvious way. As can be seen as the angular momentum increases the suppression becomes much bigger, especially at low energies.

Figure 3: Inverse of the ratio of the the diagonal potential in momentum space between the -wave and the -wave. (see the main text for a detailed explanation of the ratio).

The next step is to quantify this effect by means of a more concrete argument. For instance we can expand Eq. (LABEL:eq:V-central) at low momenta. For we can expand as


If we take into account that for radii the OPE potential falls off exponentially, then for momenta :


where we have plugged the expansion of the spherical Bessel function into the integral. As can be seen, the power law piece of the matrix element is in fact consistent with the behavior of the lowest-order -wave counterterm in pionless theory (), in agreement with naive expectations. In addition, if we take into account the exponential tail of OPE it is apparent that


which means that every and within the expansion of the potential is divided by a . That is, we find the expected breakdown scale of the pionless power counting.

This begs the question about whether the previous argument can be used to justify the peripheral demotion of OPE, and in principle the answer is yes: if the argument applies over all the range of validity of the pionful theory. Using this happens for , for which a demotion of OPE similar to the one found in pionless EFT will begin to show up. However this is of little value for practical purposes. Though we can show that a peripheral demotion indeed exists it is only apparent for extremely high angular momenta, far beyond where the partial-wave expansion is truncated in three-body calculations. This means that we have to resort to a different type of argument for analyzing the demotion at moderate angular momenta. We will do this in the next section.

ii.2 Power Counting Suppression

Here we consider the peripheral wave suppression from the power-counting perspective. The idea is to find a relationship between the scales of a two-body system and the angular momentum. The arguments we present are in principle tailored for the particular case of the OPE potential in the singlet channels, in which case the potential is regular and we do not have to take into account the issue of cut-off dependence.

We begin by writing the OPE potential in the form


which underlines the scales that enter into the problem: is the natural momentum scale of OPE and we factor out the that always appears in non-relativistic scattering. In the formula above is a dimensionless function defined as


where – the central piece of the OPE potential after removing the factor – is given by


The scale determines the strength of the OPE potential in the singlets, modulo the spin and isospin factors which we have included in the dimensionless function . How to count determines whether OPE is perturbative or not. If we iterate the OPE potential, we can argue on dimensional grounds that


where stands for the -wave projection of the OPE potential and is either the external momentum or the pion mass . From this estimation, we can see that the decision of iterating OPE or not depends on the dimensionless ratio .

At this point we have the choices (Weinberg) and (Kaplan-Savage-Wise, or KSW). The first requires the iteration of OPE at LO and the second not. The numerical value of lies in between of what one could consider a soft and a hard scale. As a matter of fact, none of the previous counting conventions works for all partial waves: on one hand we have the and triplets where OPE is thought to be non-perturbative, while on the other we have the peripheral singlets where the explicit calculations of Section I show that OPE is perturbative and probably demoted even with respect to the scenario.

The previous mismatch between scaling expectations and numerical calculations lies in the dimensionless functions and numerical factors conforming the potential. Usually we naively assume that these dimensionless factors are of and do not affect the counting. But if that were the case OPE would be either perturbative or non-perturbative in all partial waves. If we take into account these factors, we can have a clearer idea of what soft and hard means for .

To be more specific, let us consider the -wave -matrix of the OPE potential in the absence of contact-range physics. If we define the on-shell -matrix as:


then – according to the naive analysis of Eq. (30) – we should have the expansion


where is either or and refers to the number of loops. We expect the coefficients/functions to be of . On dimensional grounds the are functions of , as can be deduced from Eq. (27). The dependence on the angular momentum quantum number enters via the specific value of the coefficients. As a consequence, provided that the naturalness hypothesis for is correct, the convergence radius of the series is independent of .

On the contrary, if the convergence of perturbative OPE depends on the partial wave, then the form of the loop expansion must take a different form


where, as in the previous case, refers to the number of loops and the coefficients are . However we include now a factor to account for the different expansion parameter and convergence radius in each partial wave . This factor means that is not the ratio to check when discussing the power counting of OPE, but rather .

The only problem left is to determine the expansion parameter of the power series for . From complex analysis we know that the radius of convergence of this series is given by the pole of that is closest to the threshold. However the OPE potential is relatively weak, which means that the poles of the -matrix will be far from threshold and not easy to find. With this in mind we can resort to a trick to circumvent this difficulty. Instead of finding the poles of for the physical value of , we can change the scale up to the point that there is a pole at , i.e. a bound state at threshold.

We will give the name to the critical value of that generates a bound state at threshold in the -wave. For the perturbative series at (the location of the threshold bound state) diverges:


where the notation indicates that this is the -matrix for . This means that the expansion parameter is one for the previous series


Consequently, for the physical we have at :


from which it is clear that the expansion parameter is simply .

For finite momenta the analysis can be extended easily, though the conclusions are not as clear-cut as for . We begin by disentangling the power series explicitly in the perturbative expansion of the -matrix


where are coefficients that differentiate the contributions coming from powers of and . The point is that we can rearrange the previous expansion as


where the new coefficients are defined as


If the coefficients are of order one, the expansion parameter of the perturbative series will still be for . Meanwhile for the expansion parameter is of the order , which amount to assuming that and play the same role in the expansion. However we cannot discard the possibility of relative numerical factors between the expansions in powers of and . For instance, it could happen that the or that , giving a different convergence radius in terms of than in terms of . Though this makes no difference at the conceptual level, this effect could have a moderate impact when estimating the peripheral demotion of OPE. We will briefly discuss this at the end of the section, but we can advance that the impact is going to be negligible. Part of the reason lies in the fact that the OPE for is very similar to the Coulomb potential, which happens to be always perturbative except for low energies. In terms of the perturbative expansion this translates into the coefficients having a behavior with respect to that is extremely suppressed (e.g. ).

For quantizing the power counting demotion, we compare the expansion parameter of perturbative OPE with the expansion parameter of nuclear EFT,


Putting the pieces together the actual order of OPE in the -wave singlet is , instead of as in Weinberg or as in KSW countings. It is important to mention here that the existence of a peripheral demotion hangs on the fact that the scale separation in nuclear EFT is imperfect. That is, the clear peripheral demotion we observe depends on the fact that the separation of scales in nuclear EFT is not particularly good. If we use the values and , the expansion parameter lies theoretically between and . However, we mention in passing that concrete calculations Valderrama (2011); Pavon Valderrama (2011); Long and Yang (2011, 2012a, 2012b) and analyses Birse (2007, 2010); Ipson et al. (2010) in pionful EFT suggest an expansion parameter closer to than .

In Fig. 4 we calculate the ratio for the peripheral singlets as a function of the angular momentum . For convenience we consider the angular momentum as a continuous variable 333That is, we take real – instead of integer – and determine the location of the threshold bound states from the asymptotic condition . In practice we take an infrared cut-off of , though the results are stable for . and include the actual peripheral waves as particular points along this curve. In the isoscalar partial waves the ratio is negative, the reason being that the OPE potential is repulsive in isoscalar singlets. The specific values of the ratios leading to a bound state at threshold are shown in Table 1, where we also list the effective order at which OPE enters in each peripheral singlet. The power counting is normalized in agreement with the conventions of the previous discussion: corresponds to a potential that has to be iterated to all orders (for instance, the lowest-order contact interaction in the partial wave) and to the size of the OPE potential in the KSW counting. Even for the partial wave OPE is slightly demoted with respect to KSW. Table 1 only shows partial waves for which OPE is not much demoted beyond (considering the average demotion). The reason for this is because contributions above this order are unlikely to enter in any practical EFT calculation in the near future. The chiral nuclear potential has not been used beyond leading three-pion exchange (or subsubleading TPE) in full EFT calculations 444Though recently the EFT potential has been calculated one order further and used in first-order perturbation theory for peripheral nucleon-nucleon scattering, see Ref. Entem et al. (2015).. This piece of the potential enters at in a Weinberg-inspired counting 555This corresponds to in the traditional notation used in Ref. Epelbaum et al. (2005), which skips one order because the contribution vanishes. and at in a KSW-inspired one (for which the convergence is expected to be slower, leading to a stronger peripheral demotion). Thus is does not seem necessary to go beyond that point.

The spread in the demotion depends on the expansion parameter taken: for the demotion is relatively mild – the lowest estimation in Table 1 –, while for it is much more obvious. For instance, in the case corresponds to and to . If we take into account the observation that the actual expansion parameter seems to be closer to , then the larger estimations for the demotion are expected to be more accurate than the lower ones. However overestimating the demotion can lead to the underestimation of the theoretical errors in a calculation. In this sense it might be more cautious to use a value in the middle.

Apart from the uncertainty in there is a second source of error in Table 1, namely the interplay between the and expansions that we have previously discussed at the qualitative level. Addressing this problem formally is beyond the scope of this manuscript and in fact it has never been done in the literature for a pionful EFT expansion. So instead of analyzing in detail the perturbative expansion, we will explore the demotion with an alternative definition of , since this scale is up to a certain extent condition-dependent. Previously we defined as the for which there is a bound state at threshold. As a consequence, the ratio corresponds to the expansion parameter of the -matrix at zero energy. However, the existence of a low-lying virtual state or resonance also calls for the iteration of the potential, the required strength of the potential being weaker than in the bound state case. Therefore we release the definition of the scale as the value of for which there is a pole at (either a bound/virtual state or a resonance, which means that is a complex value), this new scale is related to the convergence of the -matrix for the physical value of at the location of the pole

In turn we could have used as the basis of an alternative definition of the peripheral demotion


In terms of the scale , if we choose a bound state with we will get a smaller (i.e. more strength for the OPE potential) and more demotion (). On the contrary the virtual state / resonance condition – let’s say at a momentum – entails a bigger , i.e. a smaller ratio, and less peripheral demotion (). Actually we are more interested in the possibility that we might have been overestimating the demotion, which means that we only have to consider the virtual-state/resonance case. We have performed the calculations in Appendix B and checked that the effect of a change in conditions from a threshold bound state to a resonance is actually tiny for , usually of the order of . If we compare this change to the uncertainty related to , which lies in the range , we see that corrections to the threshold bound state condition can be safely ignored in most partial waves.

Figure 4: Ratio between the actual value of appearing in the OPE potential and the critical generating a bound state at threshold for -wave. The ratio is calculated for isoscalar and isovector channels. In the isoscalar case the ratio is negative as the OPE potential is actually repulsive in the isoscalar channels , , and so on.
Table 1: Power counting of OPE for peripheral singlet partial waves. We show the critical value of that renders the central potential non-perturbative in each of the singlets. The actual power counting assignment for OPE in each of the partial waves depends on the expansion parameter of nuclear EFT, which is not known exactly. However assuming the baseline values of and , we expect the expansion parameter to be between and . Using this range of values, we calculate the OPE demotion in each partial wave.

ii.3 The Peripheral Perturbative Expansion Revisited

Now that we have a power-counting argument for the centrifugal suppression of the singlets, we want to check how it stands against concrete calculations. The approach we find most convenient is the comparison of multiple iterations of the OPE potential. We will see how at low energies the expansion parameter of perturbative OPE is indeed set by . The ratio of iterated versus non-iterated diagrams has been already used in the past as a tool for determining the convergence of the EFT series Fleming et al. (2000), but calculations have been usually limited to just a few iterations of OPE. While this might not be a drawback in -wave scattering, peripheral waves require the evaluation of higher orders of perturbation theory in order to be able to estimate the expansion parameter.

We will begin with the standard ratio of once iterated OPE versus tree level for the -wave, which we define as


where we define the matrix elements above as


which roughly corresponds to the diagram ratio already evaluated in Ref. Fleming et al. (2000) for the waves. The ratio also coincides with the ratio of the one-loop over the tree level phase shifts


Once we consider the peripheral demotion factor , we expect to behave as


However there is a catch: the ratio can actually be computed analytically for the OPE potential at low energies and it turns out to be much smaller than expected from the previous argument. For showing this we write the perturbative expansion of the reduced wave functions in terms of a Green’s function

where the Green’s function is defined by


with and the reduced spherical Bessel functions we already introduced in Section I. If we take into account that and expand the reduced Bessel functions in powers of , we can evaluate yielding


that is, much more suppressed than expected (owing to ).

Does mean that we have underestimated the peripheral demotion? Not necessarily: the ratio , and in general lower-order perturbation theory, exaggerates the effect of the centrifugal barrier. The reasons for this mismatch are not completely clear. It might be related to the interplay of the regular () and irregular () components of the wave function at low energies () and intermediate distances (), with the irregular piece giving a larger contribution and only appearing at higher order perturbation theory. Be it as it may, the bottomline is that we have to go to higher-order perturbation theory to probe the expansion parameter.

For exploring higher-order perturbation theory, we define the matrix element of the -iterated OPE potential as


We can now define a generalized :


We expect the scaling


for large . This expectation is robust and well-grounded as a consequence of the potential only having a specific fraction of the strength necessary to bind.

We present the calculations of the ratios of the -th versus the -th order of perturbation theory in Fig. 5. For convenience we have chosen the inverse of the ratio . The reason is the inverse of the expansion parameter is a more natural indication of how well the perturbative expansion converges: the bigger the more convergent. As a matter of fact, for we have


which when evaluated yield the values of the second column of Table 1, i.e. , , and so on. As can be seen in Fig. 5, does converge to the predicted value at momenta below the pion mass (), where we have extended the previous calculations to seventh order in perturbation theory to be able to appreciate the convergence pattern. For momenta of the order of the pion mass the expansion parameter is not so good as for but compatible with the EFT expectations, where we have to take into account that the expansion contains powers of both and . At larger momenta – and even as we approach , the expansion still works rather well. This might be puzzling from the EFT perspective but has a natural explanation in terms of the form of the central OPE potential. For large momenta the exponential decay of central OPE is irrelevant and we are left with a Coulomb-like potential (). Coulomb is perturbative at momenta above the inverse Bohr radius. In the case of central OPE the equivalent is , which is actually smaller than . Yet this is a particular feature of central OPE that is not expected to happen for other contributions of the EFT nuclear potential.

In the calculations of Fig. 5 we have reached seventh-order perturbation theory for checking the expansion parameter. Yet this is numerically demanding – more so for low momenta and high partial waves – and we limit the analysis to for , to for and to for . Though this is a limitation, the calculations are still very meaningful. For the ratios do converge to the expansion parameter predicted in Table 1 with an unrelated method, providing a cross-check of the calculation. For the situation is less clear – we are limited to higher momenta and apparently seventh order perturbation theory is not enough to stabilize the ratios – but nonetheless the ratios still seem to converge to the predicted value. One interesting feature that is evident from the calculations of Fig. 5 is that the low orders of perturbation theory converges actually faster than the high orders. The practical implication of this phenomenon is that results at tree level are more accurate than what is to be expected from the expansion parameter of the series. This might in turn point out towards choosing the higher-order estimates for the demotion.

Figure 5: Inverse ratios between the -th order over the -th order perturbation theory for OPE and the singlet partial waves with (See the main text for a detailed explanation of the ratios.) The theoretical value to which the ratios should converge is shown in the black solid line.

ii.4 Beyond One-Pion Exchange

At this point a question arises: how do we extend these ideas to TPE? NDA predicts leading TPE to be less important than OPE by . That is, if OPE is leading TPE is  666 Here we follow consistency instead of convention: traditionally leading is labeled as because in Weinberg’s counting there is no contribution to the nuclear potential that is -times smaller than OPE. If we are accounting for peripheral suppression, which in general are not integer powers of , this convention is no longer useful. . But we have seen here that OPE is probably demoted by a larger factor for and higher waves. It is natural to expect that a similar demotion will happen for TPE. The size of this effect will naturally depend on the partial wave and it might very well happen that in exceptional cases TPE might enter before OPE. This might explain why there are a few peripheral waves for which TPE is required to explain the phase shifts deduced from experimental data, such as , or  Pavon Valderrama and Ruiz Arriola (2006).

For OPE we have developed a power counting argument for the peripheral demotion. But this idea depends on an especial feature of the OPE potential in singlet waves: this is a regular potential that does not require regularization. For TPE the situation is different, it is a badly divergent potential at short distances and requires regularization. Exactly the same happens with OPE in the triplets. As a consequence the argument we have developed here cannot be applied either for TPE or for OPE in the triplet channels. There are strategies to cope with this, though they will require serious scrutiny to check whether they work. The most obvious one is to renormalize these partial waves: if we add a contact-range interaction we might be able to apply the previous argument. The drawback of this idea is that it mixes short- and long-range physics. The factor by which we have to rescale TPE for having a bound state at threshold depends on the scattering volume of the channel (before rescaling TPE), which fixes the contact-range coupling. The meaning of this is that the rescaling factor is contaminated by the physical scattering length, which is undesirable. It might happen though that the effect is negligible, as happened with the bound-state versus resonance condition in the previous discussion.

If we strive for a solution that is manifestly independent of the existence of short-range physics two possibilities come to mind. The first is Birse’s approach to tensor OPE Birse (2006), which adapts a series of techniques from molecular physics to study when tensor OPE is perturbative. A limitation of this approach is that it is formulated in the chiral limit, where the range of the OPE potential becomes infinite and it is thus similar to the potentials found in atomic physics. A possibility is to study the cut-off at which deeply bound states happen with TPE. Deeply bound states are non-physical bound states that occur when we have attractive singular interactions, such as TPE. They are of no consequence because their binding momenta are beyond the range of applicability of EFT. In addition techniques have been developed to get rid of them Nogga et al. (2005). The point is that the more peripheral the wave the harder the cut-off for which deeply bound states appear. This might in turn gives us information on the quantitative nature of the partial wave suppression.

Iii Conclusion

In this work we have analyzed from the EFT perspective – and in the particular case of the spin-singlet channels – the common wisdom observation that pion exchanges are perturbative in peripheral waves. For this we have studied the convergence of the perturbative expansion of the phase shifts numerically up to fourth order in perturbation theory. This calculation – which has been done for the first time up to such an order in this work – indicates that pions are perturbative in the singlets. In fact, the multiple iterations of one-pion-exchange potential turn out to be much more suppressed than expected even in a power counting (Kaplan-Savage-Wise) in which one-pion-exchange potential is treated as subleading.

To understand this pattern we have made use of a power counting argument to determine the actual demotion of one-pion-exchange potential with respect to leading order. The idea is to rescale the strength of the one-pion-exchange potential up to the point in which a bound state is generated at threshold. This critical strength can be translated into a critical