1. Introduction
The derivation of effective interactions from NN dynamics has been a major task in Nuclear Physics ever since the pioneering works of Moshinsky [
1] and Skyrme [
2]. The use of those effective potentials, referred to as Skyrme forces, in mean field calculations can hardly be exaggerated due to the enormous simplifications that are implied as compared to the original many-body problem [
3,
4,
5,
6]. Similar ideas advanced by Moszkowski and Scott [
7] (see also [
8] for an udated view) have become rather useful in shell model calculations [
9,
10]. The Skyrme (pseudo)potential is usually written in coordinate space and contains delta functions and its derivatives [
2]. In momentum space, it corresponds to a power expansion in the CM momenta
and
, corresponding to the initial and final state, respectively. To second order in momenta, the potential reads
where
is the spin exchange operator with eigenvalues
for spin singlet
and
for spin triplet
states, and
and
are the Pauli matrices. For a local potential
, our convention is such that
. In practice, these effective forces are parameterized in terms of a few constants which encode the relevant physical information and should be deduced directly from the elementary and underlying NN interactions. Unfortunately, there is a huge variety of Skyrme forces depending on the fitting strategy employed (see e.g., [
11,
12]). This lack of uniqueness may indicate that the systematic and/or statistical uncertainties within the various schemes are not accounted for completely. Interestingly, the natural units for those parameters have been outlined in Ref. [
13,
14] yielding the correct order of magnitude. A microscopic basis [
15,
16] for the Density Functional Theory (DFT) approach has also been set up, but uncertainties still remain. Conversely, while this is a two-body interaction, the connection with the free NN-force is not obvious. Therefore, Skyrme forces, while extremely useful in practice are neither uniquely defined nor obviously related to NN-scattering. A recent work discriminates positively 16 out of 240 sets of Skyrme parameters from nuclear matter constraints [
17]. Energy density functionals implying density dependent parameters have been derived from chiral low two- and three nucleon interactions [
18,
19].
The Skyrme pseudo-potential in Equation (2) may and has been taken literally in mean field calculations, a procedure that makes sense due to the finite extension of the nucleus and, of course, under the assumption that such an extension is sufficiently large as not to unduly amplify short distance components of the pseudo-potential. However, its interpretation in the simplest two-body problem requires some regularization to give a precise meaning to the Dirac delta interactions. The standard view of a pseudo-potential (in the sense of Fermi) is that it corresponds to the potential,which, in the Born approximation, yields the real part of the full scattering amplitude. This is a prescription which implements unitarity of the S-matrix, but necessarily fails at low energies when the scattering length is unnaturally large as it is the case for NN interactions. The reason is that the Born approximation is no longer valid.
On the contrary, the Wilsonian viewpoint corresponds to a coarse graining of the NN interaction to a certain energy scale. There are several schemes to coarse grain interactions in Nuclear Physics. The traditional way has been by using the oscillator shell model, where matrix elements of NN interactions are evaluated with oscillator constants of about
[
10]. A modern way of coarse graining nuclear interactions is represented by the
method [
20] (for a review, see [
21,
22]) where all momentum scales above
are integrated out. The recent Euclidean Lattice Effective Field Theory (EFT) calculations (for a review see e.g., [
23]), although breaking rotational symmetry explicitly, provide a competitive scheme where coarse grained interactions allow
ab initio calculations combining the insight of EFT and Monte-Carlo lattice experience, with lattice spacings as large as
. These length scales match the typical inter-particle distance of nuclear matter
. Actually, the three approaches feature energy-, momentum- and configuration space coarse graining, respectively, and ignore explicit dynamical effects below distances
(These are qualitative relations. A more quantitative determination is discussed in Ref. [
24], where it is found that, at low energies, the CM cut-off Λ can be related with a short distance cut-off
by the relation
.), which advantageously sidesteps the problems related to the hard core and confirms the modern view that
ab initio calculations are subjected to larger systematic uncertainties than assumed hitherto. Clearly, any computational setup implementing the coarse graining philosophy yields by itself a
unique definition of the effective interaction. However, there is no universal effective interaction definition. For definiteness, we will follow here the
scheme to determine the effective parameters because, within this framework, some underlying old nuclear symmetries, namely those implied by Wigner and Serber forces, are vividly displayed [
25,
26,
27,
28].
A particular implementation of the coarse graining idea [
29] has facilitated benchmarking partial wave analyses of the NN interaction and has also provided an alternative way to define the effective couplings as a function of the maximal fitting CM momentum
[
30], including different One Pion Exchange (OPE) and Chiral Two Pion Exchange (
χTPE) [
31,
32] and an evaluation of uncertainties of both statistical and systematic origin [
33].
In the present paper, we want to show that, in fact, these parameters can uniquely be determined from known NN scattering threshold parameters by rather simple calculations where the interaction is just coarse grained over all wavelengths larger than the typical ones occurring in finite nuclei. As we will show, this introduces a momentum scale Λ in the nine effective parameters
,
and
(
), which allow for connecting the two body problem to the many body problem. Of course, for a finite nucleus, higher order corrections to the two body interaction as well as few body forces will be needed, and just stopping to second order in momenta will not be sufficient (see e.g., Ref. [
34] for a calculation to
). However, going beyond Equation (2) requires further information than just two-body low energy scattering, in particular knowledge about three and four body forces and their scale dependence consistently inherited from their NN counterpart. The finite
situation relevant for heavy nuclei and nuclear matter involves mixing between operators with different particle numbers and, in principle, could be conveniently tackled with the method outlined in Ref. [
35], where the lack of genuine medium effects is manifestly built in.
The method we will be using is the implicit renormalization approach described already in Ref. [
36,
37,
38], which has been positively tested for a simple toy model with just
S-waves with the Block Diagonal (BD) formulation of the Similarity Renormalization Group (SRG) [
39], which is an upgraded version of the
-approach (see also [
40] for a review). In previous work [
36,
37,
38], it has explicitly been shown that the scale dependence deduced for the parameters of Equation (2) is not expected to change when higher orders in the expansion are included in a wide range of cut-offs Λ. This is our motivation to pursue the present analysis.
The paper is organized as follows. In
Section 2, we review the
approach, with a particular stress on the low energy expansion and the connection to the threshold parameters in the relevant partial waves. In
Section 3, we clarify the important distinction between pesudopotentials and
potentials as well its significance in coordinate space. In
Section 4, we provide the pertinent partial wave decomposition of the Skyrme interaction, Equation (2), which becomes suitable to simplify the solution of the scattering problem. The analysis of the counterterms, and hence on the dependence of the Skyrme force parameters on the scale, is carried out in
Section 4.2. Finally, in
Section 5, we come to the conclusions. Further results are presented in the
Appendix.
2. The Approach
For completeness, we review here the
approach [
21] in a way that our points can be easily stated. The starting point is a
given phenomenological NN potential,
V, and usually denominated
bare potential, whence the scattering amplitude or
T matrix is obtained as the solution of the Lippmann–Schwinger (LS) coupled channel equation in the CM system
where
J is the total angular momentum,
S the total spin and
are orbital angular momentum quantum numbers,
are CM momenta,
is the Nucleon mass, and
is the CM energy. Solutions can be obtained just from the half off-shell T-matrix, taking, for instance,
.
For later reference, we also quote our convention for the relation with a local potential
with the momentum space case,
where
are spherical Bessel funcions. Using the Bessel function expansion for small argument
, we get a low momentum expansion of the potential matrix elements. We keep up to total order
corresponding to
S-,
P- and
D-waves as well as S-D and P-F mixing parameters,
To facilitate comparison, we will use below the usual spectroscopic NN notation
for diagonal channels and
for off-diagonal channels. We will call the coefficients in the expansion counterterms, although, properly speaking, the name is justified when the potential
is used to solve the problem in a restricted Hilbert space
, which means, in particular, fitting scattering data up to CM momentum
[
36] and providing a prescription to ensure hermiticity of the interaction. If
a is the range of the interaction, only under these conditions is a truly universal behavior of the counterterms guaranteed for
as will be shown below.
The unitary (coupled channel) S-matrix is obtained as usual:
Using the matrix representation
with
, a hermitian coupled channel matrix (also known as the K-matrix), at low energies, the effective range theory for coupled channels reads
which, in the absence of mixing and using
, reduces to the well-known expression
An extensive study and determination of the low energy parameters for all partial waves has been carried out in Ref. [
41] for both the NijmII and the Reid93 potentials [
42] and also the AV18 [
43] and the six modern Granada potentials in Ref. [
33], yielding similar numerical results. Dropping these coupled channel indices for simplicity, the
potential is then defined by the equation:
where
. We use here a sharp three-dimensional cut-off Λ to separate between low and high momenta since results are not sensitive to the specific form of the regularization. Note, however, that the sharp function separates explicitly the model space from the rest in terms of orthogonal projection operators
P and
Q fulfilling
and
and
, which is
only fullfilled by the step functions
and
, and hence the total Hilbert space separates as
. Moreover, this definition does not provide a hermitean
potential, and usually a specific choice is made in order to restore hermiticity. Note that, in this context, such a prescription is equivalent to fulfill off-shell unitarity for the two body problem, a condition which proves essential for the three-body unitarity. However, there is generally an ambiguity in
defining a restricted model space potential. The lack of uniqueness is not sufficiently emphasized in most
works. The Block Diagonal formulation of the Similarity Renormalization group (BD-SRG) [
39] provides a suitable implementation of hermiticity at any stage of the calculation. In our case, and to the level of approximation of Equation (2), we will explicitly see that there is no ambiguity. Thus, eliminating the
T matrix, we get the equation for the effective potential, which evidently depends on the cut-off scale Λ and corresponds to the effective interaction which nucleons see when all momenta higher than the momentum scale Λ are integrated out. It has been found [
21] that high precision potential models,
i.e., fitting the NN data to high accuracy and incorporating One Pion Exchange (OPE) at large distances and describing the deuteron form factors, collapse into a unique self-adjoint nonlocal potential for
. This is not an unreasonable result since all the potentials provide a rather satisfactory description of elastic NN scattering data up to the pion production threshold
. Note that this universality requires a marginal effect of off-shell ambiguities (beyond OPE off-shellness), which is a great advantage as this is a traditional source for uncertainties in nuclear structure.
Actually, in the extreme limit when
, one is left with zero energy
on-shell scattering yielding
. Moreover, for sufficiently small Λ, the potential which comes out from eliminating high energy modes can be accurately represented as the sum of the truncated original potential and a polynomial in the momentum [
44]. However, as discussed in [
26], a more convenient representation is to separate off all polynomial dependence explicitly from the original potential:
with
, so that, if
contains up to
, then
starts off at
,
i.e., the next higher order. This way, the departures from a pure polynomial may be viewed as true and explicit effects due to the potential, and, more precisely, from the logarithmic left cut located at CM momentum
at the partial wave amplitude level due to particle exchange with mass
m. Thus, the coefficients in Equation (5) universally include all contributions to the effective interaction at low energies. Although we cannot calculate them
ab initio, we may relate them to low energy scattering data for any value of Λ, in harmony with the expectation that off-shell effects are marginal in this energy regime. Not surprisingly, the physics encoding the effective interaction in Equation (5) will be related to the threshold parameters defined by Equation (6). Thus, the relevance of specific microscopic nuclear effective (coarse grained) forces has to do with the extent to which these threshold parameters are described by the underlying forces and not so much with their detailed structure. We will discuss below the limitations to this universal pattern.
In Ref. [
45], an interesting study was conducted regarding the saturation of the short distance contributions due to heavy resonance exchange, namely
. While this approach is, in principle, very appealing, it does not specify what the relevant scale is. Quite generally, the coefficients are scale dependent and this separation is scale dependent. In fact, the method used in Ref. [
45] retains only the leading perturbative contribution in the resonance exchange. This procedure would be legitimate for peripheral waves, but certainly not for central S-waves where non-perturbative effects become crucial. As we will see, the Wilsonian renormalization point of view befits the situation in a more satisfactory manner.
A further objection to attempt the separation explicitly concerns the long distance pionic physics (see also [
25,
26,
27,
28]). Actually, we may separate the pionic contributions as follows:
There is the subtle issue on how to define numerically the pionic contribution. For instance, in Refs. [
31,
32], this separation is made in coordinate space as a function of the short distance cut radius
in the range where the pionic contributions are proven to be indispensable when the fit is undertaken up to a maximum CM momentum,
. Thus, the separation is done as
The potentials diverge like powers at large momenta, and, therefore, they become singular at short distances. Besides these extreme cases, the region around
and
does not map so easily. In fact, the numerical values quoted in [
31,
32,
45] are not very similar when separations implied by either Equation (
10) or Equation (
11) are invoked, respectively.
3. vs. Potentials
The most astonishing feature of the original approaches, which made them so popular, was the universality of all potentials which have an OPE potential tail and simultaneously fit the phase-shifts up to pion production threshold with a . It is important to address here why the approach works, i.e., why there is a universal and approximately scale independent behavior at . As we see, such a situation corresponds to a model space where particles do not interact, i.e., where , for CM momenta whence the original Lippmann–Schwinger equation directly reduces to the equation. If we think of the channel, the traditional interpretation is that the vanishing of the phase shift at MeV is an indication of the core of the interaction. Of course, an additional interpretation of a vanishing phase is that the interaction is weak. As already mentioned, in the BD-SRG version of the approach, the Hilbert space is separated into the P and Q orthogonal subspaces, and hence both components of the Hilbert space are kept and thus phase equivalence is preserved. However, if we choose Λ such that and neglect the Q space, we expect some stability against variations in Λ. As a matter of fact, much of the usefulness of the approach has to do with the relative insensitivity to the change of the separation scale between the P and the Q spaces.
While renormalization issues are often posed in momentum space, one can provide an equivalent and insightful point of view in coordinate space [
46] keeping the Wilsonian spirit (see also [
47] for a nice presentation within the context of power counting). Actually, these features were anticipated in coordinate space long ago by the separation method of Moszkowski and Scott [
7] (see a contextualized view in Ref. [
8]). Indeed, while the standard view is that NN potentials present a repulsive core below a certain distance,
, the basic observation is that there is a given length scale, the separation distance,
below which one can instead replace the NN-interaction by a small potential. The typical values that one finds are in the range
. Let us see how this happens in the particular case of the
-channel where the core effects become more visible. Higher partial waves are already suppressed due to the centrifugal barrier, so we expect them to become less sensitive to short range repulsion of the central force.
In coordinate space, one has to look for regular solutions at the origin of the Schrödinger equation for positive energy states with scattering boundary conditions:
where
is the reduced NN-potential,
is the range of the interaction and
is a constant which is fixed by the long distance normalization. Let us consider first the zero energy wave function, which corresponds to take
and
, fulfilling
As is well-known, the scattering length is defined by the intersection with the
x-axis of the asymptotic zero energy wave function. For an interaction which is attractive at long distances,
, we have that
, meaning that the function is concave in the attractive regime. The regular non interacting solution at the origin behaves as
, and it may happen that there is a distance
d where the integrated-in full wave function matches a free solution, which means
If this
d exists, one can replace the zero energy scattering problem by the truncated original potential, which, following [
25,
26,
27], we denote as the
potential. The corresponding reduced potential,
, reads
The situation is illustrated in
Figure 1 for the Granada Gauss–OPE potential [
48]. As we see, the inflexion point
corresponds to the vanishing potential value
,
fm, before the short distance repulsion sets in. Of course, this will generally not be true for finite momenta, and the picture gets somewhat modified. We will analyze, for illustration purposes, two possible schemes that we label as
and
and treat the short distances differently.
The simplest possibility is to make the separation distance
d, below which the interaction can be taken to be zero, depending on
p. This can be done by means of the modified relation:
where we have taken, up to an irrelevant normalization,
. Actually, we expect this change to be very small for
, as can be seen in
Figure 2 (left panel). The resulting Vhighr potential becomes
As we see, the for the critical value , which is the value where the phase vanishes.
In another alternative scheme, one may keep the value of
d at zero energy and add a (small) constant short distance square well potential
where we have the condition
. The result is shown in
Figure 2 (right panel), where we compare the computed values with a fit
(V in MeV, p in fm
). As we see from
Figure 2, the short distance contribution of the effective potential
remains moderate over the values of
p compared to the core displayed by the original potential (see
Figure 1). Note that the two different representations of the
interaction, Equations (18) and (21), produce exactly the
same phase-shifts by construction.
The existence of this separation scale and the smooth dependence on the energy at low energies is ubiquitous for NN interactions, but not a general feature of
any potential; it depends on whether Equation (
17) admits a solution. Integrating in from large distances to short distances, one finds for the
channel and different potentials, the following separation distances:
As a matter of fact, the coarse grained potentials used to make the most recent combined pp+np partial wave analyses allow taking vanishing potentials at distances numerically around these
d values [
49,
50,
51].
Once we have the
interaction, we can compute the integrals and compare Equation (
3) with the low momentum expansion, Equation (27) for
,
and similarly for the
state. We get the values
and
in fair agreement with the results of
Table 1. For the coefficients
and
, the
contribution is negligible. As we see, at the separation scale, the Wigner symmetry is well reproduced as a long distance effect,
i.e., for
. This is a general pattern which has been found in recent fits [
31] in the Skyrme parameters and the counterterms contributions for distances larger than
. This pattern will reappear in our
analysis below, where
roughly marks the onset of the reported universality regime unveiled in the early works [
21].
5. Conclusions
In the present paper, we have provided a Wilsonian renormalization scale argument on how the much used effective interaction parameters of the venerable Skyrme interaction can be understood as coarse grained NN interactions over the suitable wavelengths. A mapping of these scale dependent parameters to counterterms usually employed in the NN interaction scattering analyses becomes possible. This view befits the idea that, even in the lightest nuclei, where density effects are negligible, nucleons interact with each other in an average fashion, sampling only the low energies relevant for nuclear binding or equivalently resolving only the physical effects for distances above the corresponding de Broglie wavelength. The typical momentum scales increase as one goes to heavier nuclei and nuclear and neutron matter where density effects become more visible. We have seen that, regarding the renormalization group, when running the parameters, there is scale saturation, i.e., above a certain scale, the effective interaction parameters do not depend strongly on the scale in a regime where most of the strength can be deduced from low energy NN scattering information regardless of the underlying forces. The fulfillment of the previously reported and well fulfilled Wigner and Serber symmetries in the effective interactions is reassuring and indicates that many of the main features of the effective interaction are kept. Of course, for heavier nuclei, many-body effects set in, and the number of terms in the effective interaction, and hence the number of parameters, grow rapidly. At the same time, explicit finite range effects such as pion exchange corrections will become relevant for higher dimensional terms beyond the second order in momenta analyzed here. The weak scale dependence not only provides a rationale for fitting strategies in mean field nuclear structure calculations, but also shows that, for , one should distinguish between density effects due to many body forces and scale dependence due to finite wavelength resolution. We expect that a more clear disentanglement of both properties might provide further insight into the modern DFT approach to nuclear physics.
Our results can be improved and extended in several ways. Within the low energy expansion, we have neglected terms
, which correspond to
P-waves and
S-wave range corrections. In configuration space, this corresponds to a dimensional expansion, since
and
,
. Within such a scheme, going to higher orders also requires including three-body interactions,
. Actually, at the two-body level, there are more potential parameters than low energy threshold parameters. For instance, in the
channel, one has two independent hermitean operators,
and
(which are on-shell equivalent), but only one
threshold parameter in the low energy expansion (see Equation (
6)). As it was shown in Ref. [
67] (see also Ref. [
68]), these two features are interrelated since this two body off-shell ambiguity is cancelled when a three body observable, like e.g., the triton binding energy, is fixed. An intriguing aspect of the present investigation is the modification induced by potential tails due to e.g., pion exchange, which cannot be represented by a polynomial since particle exchange generates a cut in the complex energy plane. The important issue, however, is that the low scale saturation unveiled in the present paper works accurately just to the second order as long as the low energy parameters determined from on-shell scattering are properly reproduced. Along these lines, we also expect that higher order two-body derivative terms accounting for higher energies will become more sensitive to the long range features of the interaction, and more specifically to pion exchanges. Finally, it would be very interesting to check this behavior of the counterterms by the explicit renormalization method based on the Block-Diagonal Similarity Renormalization Group, as was done in a toy model in Ref. [
36,
37], as the problem becomes extremely stiff and computationally expensive for
. There, it was found that the simple analytical formulas overlap in the physically interesting
region and allow for sidestepping the stiff equations for lower values. Work along these lines is in progress.