Short-range correlations in nuclei with similarity renormalization group transformations
Realistic nucleon-nucleon interactions induce short-range correlations in nuclei. To solve the many-body problem unitary transformations like the similarity renormalization group (SRG) are often used to soften the interactions.
Two-body densities can be used to illustrate how the SRG eliminates short-range correlations in the wave function. The short-range information can however be recovered by transforming the density operators.
The many-body problem is solved for He in the no core shell model (NCSM) with SRG transformed AV8’ and chiral N3LO interactions. The NCSM wave functions are used to calculate two-body densities with bare and SRG transformed density operators in two-body approximation.
The two-body momentum distributions for AV8’ and N3LO have similar high-momentum components up to relative momenta of about , dominated by tensor correlations, but differ in their behavior at higher relative momenta. The contributions of many-body correlations are small for pairs with vanishing pair momentum but not negligible for the momentum distributions integrated over all pair momenta. Many-body correlations are induced by the strong tensor force and lead to a reshuffling of pairs between different spin-isospin channels.
When using the SRG it is essential to use transformed operators for observables sensitive to short-range physics. Back-to-back pairs with vanishing pair momentum are the best tool to study short-range correlations.
pacs:21.60.De, 21.30.Fe, 05.10.Cc, 25.30.-c
Realistic nucleon-nucleon () interactions are fitted to scattering data up to the pion production threshold. Therefore their properties at short distances and also their off-shell behavior is not completely constrained. All realistic interactions include pions to describe the long- and medium-range parts of the potential. The short-range part is parameterized phenomenologically Wiringa et al. (1995), by the exchange of heavy mesons Machleidt (2001), or in chiral effective field theory by (regularized) contact-terms Entem and Machleidt (2003); Epelbaum et al. (2005). At short distances the different interactions induce typical short-range correlations due to short-range repulsion and the tensor force which is reflected in a depopulation of one-body momentum distributions below the Fermi momentum and an enhancement at high momenta when compared to single-particle mean-field occupation probabilities Benhar et al. (1994); Pandharipande et al. (1997); Müther and Polls (2000); Neff and Feldmeier (2003); Dickhoff and Barbieri (2004); Rios et al. (2009).
However short-range correlations are two-body correlations and therefore two-body densities in coordinate and momentum space provide the best tool to study these correlations. Experimentally short-range correlations have been studied in inclusive, semi-inclusive and triple-coincidence reactions, see the reviews Frankfurt et al. (2008); Arrington et al. (2012) and references therein. The most direct information about two-body correlations can be obtained in triple-coincidence experiments where one knocks out a nucleon pair with protons Tang et al. (2003) or electrons Shneor et al. (2007); Subedi et al. (2008); Baghdasaryan et al. (2010); Korover et al. (2014) at high-momentum transfer. In these experiments one found a dominance of over pairs at high relative momenta, that indicated the importance of short-range tensor correlations. Tensor correlations also appear to play a major role in reactions at high proton energies Miki et al. (2013); Ong et al. (2013).
Recent theoretical studies of two-body momentum distributions can explain these observations. In a simple picture only close nucleons found in relative -wave pairs are affected by short-range correlations and in a first approximation high-momentum components are generated by pairs in deuteron-like configurations Vanhalst et al. (2012). The two-body momentum distributions at high momenta can also be connected to the nuclear contacts Weiss et al. (). Detailed few- and many-body calculations almost exclusively use the Argonne (AV18) or Argonne (AV8’) interactions Schiavilla et al. (2007); Wiringa et al. (2008, 2014); Alvioli et al. (2012, 2013a). Compared to the Argonne interactions, interactions derived in chiral effective field theory Entem and Machleidt (2003) are regularized with a relatively low momentum cut-off and one might expect noticeable differences for the short-range correlations.
From the perspective of nuclear many-body calculations short-range correlations are not a desired feature but pose a severe problem. One way to address this problem is to include the short-range correlations explicitly, as in the correlated basis function theory Fabrocini et al. (2000) or in variational Monte Carlo (VMC) calculations Schiavilla et al. (2007); Wiringa et al. (2008, 2014). Another approach is to use soft phase-shift equivalent effective interactions that are obtained from the bare interactions by means of unitary transformations like Bogner et al. (2003), the unitary correlation operator method (UCOM) Feldmeier et al. (1998); Neff and Feldmeier (2003); Roth et al. (2010) or the similarity renormalization group (SRG) Bogner et al. (2007, 2010); Roth et al. (2010). In real life the unitary transformations are performed in an -body approximation. This usually means in two-body, or for state of the art SRG calculations, in three-body approximation Jurgenson et al. (2009); Hebeler (2012); Roth et al. (2014). Even if we only have a two-body interaction at the beginning the unitary transformation will induce three- and higher-body terms. If the unitary transformation acts mainly at short distances and the density of the nuclear system is low enough, so that the probability to find three nucleons simultaneously close together is small, transformations on the two-body level will be a good approximation. The remaining dependence of observables on the unitary transformation can be used to analyze the nature of missing many-body terms.
In a consistent calculation it is not enough to transform the Hamiltonian, all operators have to be transformed. Within the SRG approach the transformation of long-range operators like radius or electromagnetic transition operators has been studied in Schuster et al. (2014) in two- and three-body approximation. As one might expect the effect on these long-range operators is not very large. On the other hand we expect large effects for short-range or high-momentum observables. Within the UCOM approach we investigated these effects in two-body approximation and with simple trial wave functions for the one-body momentum distribution Neff and Feldmeier (2003) and for two-body coordinate and momentum space distributions Feldmeier et al. (2011). The SRG operator evolution for the deuteron was studied extensively in Anderson et al. (2010) and analyzed in terms of a factorization for high-momentum observables. These ideas were extended for Fermi gases in Bogner and Roscher (2012).
In this paper we investigate two-body densities for the ground state of He in coordinate and momentum space for SRG transformed AV8’ Wiringa et al. (1995) and chiral N3LO Entem and Machleidt (2003) interactions using the no core shell model (NCSM). For the bare AV8’ interaction the NCSM results are not converged and we use in this case the method of correlated Gaussians Feldmeier et al. (2011); Suzuki and Horiuchi (2009); Mitroy et al. (2013). We will discuss the two-body densities obtained with both, bare and transformed density operators. The SRG transformation depends on the flow parameter that controls the softness of the transformed interaction. We perform the SRG transformation in two-body approximation and use the flow-dependence of the results to test the quality of the two-body approximation. We will show that for pairs with vanishing pair momentum the contributions of three-body correlations are negligible.
After describing in Sec. II the used methods, two-body densities in coordinate and momentum space are presented in Sec. III where we also discuss the role of many-body correlations for different observables. Summary and conclusions follow in Sec. IV. Technical details about the calculation of translational invariant two-body densities in the NCSM framework are presented in the Appendix.
The SRG flow equation for the Hamiltonian and the corresponding transformation matrix are given by
with the generator taken to be the commutator of the intrinsic kinetic energy and the evolved Hamiltonian:
In this paper the Hamiltonian is transformed in two-body approximation:
The evolution (1) can therefore be performed for the relative motion in two-body space. The flow equations are solved on a momentum space grid with relative momenta going up to for the AV8’ interaction. In the end the momentum space matrix elements are evaluated in the harmonic oscillator basis to be used in the NCSM. We will present results for flow parameters of , (a typical value used in many-body calculations), and corresponding to a very soft effective Hamiltonian .
The many-body problem for He is then solved with the SRG transformed two-body Hamiltonian using the shell model code Antoine Caurier and Nowacki (1999):
The results for the ground state energy of He and the two-body distributions in coordinate and momentum space discussed in this paper are well converged within the model space (, oscillator parameter ). The bare AV8’ interaction () can however not be converged in this space and we use here the results obtained with the correlated Gaussian method Feldmeier et al. (2011); Suzuki and Horiuchi (2009); Mitroy et al. (2013).
All two-body information contained in the many-body states can be expressed in terms of the two-body density matrix that is obtained by integrating over all coordinates besides the pair position and the relative coordinates of the pair. The two-body density as calculated in the NCSM is not translationally invariant due to the localization of the NCSM wave function in the origin of the coordinate system. However the translational invariant two-body density can be obtained from the two-body density in the laboratory system by a linear transformation as discussed in Appendix D. In the harmonic oscillator basis we express the two-body density matrix with two-body basis states
Here summarizes the harmonic oscillator quantum numbers. stand for the relative coordinates, and give the total spin and isospin of the pair, respectively, and summarizes the harmonic oscillator quantum numbers for the pair coordinate. The oscillator parameters for the relative and the pair motion are given by
A compact notation can be obtained by defining the two-body density operator, which acts in two-body space, as
The expectation value of any ‘bare’ two-body operator can then simply evaluated as
where denotes the trace in two-body space. If one evolves the observable in the same way as the Hamiltonian, namely , the expectation value
does not depend on because . However, as for the Hamiltonian, the evolved observable is in general no longer a two-body operator and contains induced higher-body operators.
The two-body approximation consists in calculating
where the two-body operator is SRG transformed in two-body space. It should be noted that acts only on the relative coordinate part of or and leaves the center of mass motion of the pair unchanged.
If the two-body approximation was exact (for both the Hamiltonian and the observables ) would be -independent. The remaining dependence therefore indicates the size of the neglected contributions from the induced three- and higher-body terms.
With the many-body states obtained in the NCSM various bare and SRG transformed two-body quantities will be studied in the following. With bare operators we can investigate how the properties of the eigenstates change with increasing flow parameter. This will reflect the increasing ‘softness’ of the transformed Hamiltonian . On the other hand properties calculated with transformed operators should be independent from the flow parameter if the two-body approximation is justified. However one has to be careful when drawing conclusions about omitted higher-order terms in the transformed operators. In this paper the two-body approximation is employed both for the transformed Hamiltonian and for the transformed operators. Even if the two-body approximation is perfect for the transformed operators we would still expect a dependence on the flow parameter due to the two-body approximation for the transformation of the Hamiltonian.
It might be possible however that particular components of the wave function are less sensitive to higher-order terms in the transformed Hamiltonian and can be obtained reliably within the two-body approximation. This appears to be the case for the pairs with small pair momentum as will be discussed in Sec. III.5.
In two-body approximation is exactly unitary in two-body space, but in many-body space unitarity is only approximate. Therefore the energy eigenvalue depends on and this dependence can be taken as a measure for the induced three- and four-body interactions (see Eq. (3)) that are neglected when calculating the He ground state energy in two-body approximation. As seen in Fig. 1 the total energy varies by about 10% indicating that the contribution of three- and four-body terms are small. In Fig. 1 we also show the individual contributions from the four channels to the total ground state energy of He. These can be calculated by using Eq. (10)
where the operator projects on the spin-isospin channel . The total energies for the AV8’ and N3LO interactions are very similar (although the individual kinetic and potential contributions are quite different) and in both cases the dominant contribution to the binding energy is coming from pairs with a large contribution from the tensor force. The channel gives a much smaller attractive contribution whereas the channel provides repulsion and the contribution from the channel is negligible. With increasing flow parameter we observe a reduction of the attractive contribution of the channel and the contribution becomes less repulsive. These changes are related to changes in the occupation of the different channels that will be discussed in Sec. III.4.
iii.2 Relative density distribution in coordinate space
The repulsive nature of the nuclear interaction is easily seen in the the relative density distribution that gives the probability to find a pair of nucleons at a distance . It is calculated by replacing in Eq. (8) with the two-body operator
where we sum over all quantum numbers except the radial distance .
The short-range correlations induced by the nuclear interaction are reflected in the relative density distributions shown in the upper part of Fig. 2. They are calculated with the eigenstates of the bare and SRG evolved Hamiltonians according to Eq. (8).
For the bare interactions the relative density distributions show the typical correlation hole at short distances. Due to the repulsive core of the interaction the probability to find a pair of nucleons at a distance between their centers is depleted at distances below . The chiral N3LO interaction is obviously not as repulsive as the AV8’ interaction at short distances, as the correlation hole is less pronounced for the N3LO interaction.
One should keep in mind that the term ‘correlation hole’ is misleading as for the nucleon densities are already strongly overlapping so that there is no hole in the baryonic matter density. On the contrary at short distances one encounters locally large baryonic matter densities and expects strongly polarized nucleons.
With increasing flow parameter the correlation hole disappears more and more. Furthermore the density distributions calculated for the evolved AV8’ and N3LO interactions become increasingly similar, cf. Fig. 2 (a), (b). For the largest flow parameter the relative density distribution is essentially of Gaussian shape — as would be expected for an uncorrelated mean-field wave function. Thus, the SRG transformation brings us from a highly correlated system to a simple shell model or mean-field like situation.
On the other hand the density distributions shown in the lower part of Fig. 2, which are obtained with the SRG transformed density operators according to Eq. (10), are all very similar to those obtained for the bare Hamiltonians . Without employing the two-body approximation would be unitary in four-body space and would not depend on . The remaining dependence indicates that the induced three- and four-body terms are small but not completely negligible.
The probability densities obtained with both bare and transformed density operators are normalized to the number of pairs:
The spatial distributions shown in Fig. 2 may help our intuition but can not easily be related to experiment. Therefore momentum distributions and momentum correlations will be addressed in the following.
iii.3 Relative momentum distributions
The relative momentum distribution, i.e., the probability to find a nucleon pair with relative momentum , relative orbital angular momentum , spin , and isospin is obtained by replacing in Eq. (8) with the two-body operator
where denotes the spherical momentum space representation of the relative motion.
The relative momentum distributions for pairs with relative momentum , irrespective of their orbital angular momentum , spin and isospin , , and its SRG transformed partner are shown in Fig. 3 for the two interactions and different flow parameters.
The short-range repulsive correlations, which manifest themselves in coordinate space as the correlation holes, show up as tails in the momentum distributions that reach out to large relative momenta . Note however that the momentum distribution is not the Fourier transform of the diagonal two-body density in coordinate space.
Whereas the tail of the momentum distribution shows an exponential behavior at large relative momenta for the AV8’ interaction, the N3LO relative momentum distribution reflects the momentum space regulator that cuts off high momenta beyond about . For both interactions short-range tensor correlations play an important role as will be discussed in more detail later. With increasing flow parameter the high-momentum components are more and more reduced until the probability distribution of the relative momentum assumes a Gaussian shape for both evolved interactions, cf. Fig. 3 (a) and (b), corresponding to an uncorrelated wave function.
Fig. 3 (c) and (d) show that the density distributions obtained with the SRG transformed density operators are again all very similar to those obtained for the bare Hamiltonian, indicating that the induced three- and four-body terms are also small in momentum space. The most visible dependence on occurs for the AV8’ interaction around . This dependence is related to three-body correlations induced by the two-body tensor force to be discussed in the following subsection.
iii.4 relative momentum distributions
In order to get a deeper understanding of the nature of the correlations we separate the momentum distribution according to Eq. (14) into its parts coming from the different channels and display the results in Fig. 4. The first observation is that the channel, in which the tensor force is strongest, shows only a weak dependence. In contrast to that there is strong dependence in the channel for momenta around and also in the channel. With increasing strength moves from the odd to the even channel. This effect has been discussed in detail in Ref. Feldmeier et al. (2011). In a simplified picture one may consider the situation of a localized pair with a third nucleon of different flavor not too far away. Additional binding from the strong tensor force acting between the third nucleon and a nucleon of the pair can be obtained by flipping the spin of one nucleon in the pair and thus converting with some probability the pair into an pair. It is energetically more favorable to break some pairs and loose their binding if more binding from the tensor interaction with the third particle can be gained. These genuine three-body correlations are lost in two-body approximation and introduce an -dependence. With increasing the tensor part of the SRG transformed Hamiltonian is weakened while the central part is strengthened. With the weakening of the tensor force the effect of the three-body correlations will be reduced. This explains the reduction of the number of pairs with increasing and the -dependence of the pairs at momenta around where the relative importance of the tensor force is most pronounced.
This transfer of probability between the even and the odd channel can be seen in Table 1 where the number of pairs, given by the integrals over the momentum distributions, are listed as a function of . With increasing flow parameter the occupation numbers approach the limit of the mean-field or independent particle model with 3 pairs in both even channels and 0 pairs in the odd channels. The number of pairs and obtained with bare and SRG transformed density operators are identical as in two-body approximation does not connect different channels.
The reshuffling of probability between the different spin-isospin channels also tells us that the omitted many-body terms in and the transformed Hamiltonian will have a non-trivial spin- and isospin-dependence.
iii.5 Relative momentum distributions for pairs
Up to now we have investigated relative momentum distributions for all pairs, indiscriminate of the pair momentum . It has been found that the relative momentum distributions depend quite significantly on the pair momentum Wiringa et al. (2008); Alvioli et al. (2013b). In the context of this paper it is interesting to see how this is related to many-body correlations.
We might expect that back-to-back pairs with are less affected by many-body correlations than pairs with a large pair momentum . In a pair with large relative momentum both nucleons have large individual momenta. For pairs with large pair momentum however there is a high probability that one of the nucleons has a momentum less than or close to Fermi momentum. We would therefore expect that these nucleons are interacting more strongly with other nucleons and therefore are susceptible to many-body correlations.
In order to study this we investigate the more exclusive joint probability to find a nucleon pair with spin and isospin at relative momentum and total pair momentum . It is calculated with the two-body operator
where denotes the spherical momentum representation of the relative momentum of the pair with respect to the remaining nucleons. The factor originates from the transformation from Jacobi coordinates to the coordinates and , see Sec. D. In this paper we consider only pair momentum for which the sum over reduces to .
The momentum distributions , obtained by summing over , are displayed in Fig. 5. One sees that in all channels the results are essentially independent on . We can also observe that with increasing flow parameter the momentum distributions , as shown in Fig. 4, become more and more similar to the momentum distributions . This consistently tells us that for pairs with total momentum many-body correlations are not very important, as anticipated in the discussion above. The pairs are therefore the best candidates for experimental studies of short-range two-body correlations. Similar considerations can be found in Refs. Wiringa et al. (2008); Alvioli et al. (2013b); Arrington et al. (2012).
We can also notice significant differences between the two even channels. In the channel the momentum distribution has a node at relative momenta of about . This is very different in the channel. Here the momentum distribution does not show a minimum and the number of pairs for relative momenta above about is significantly larger. This difference is due to short-range tensor correlations that only contribute in the channel. To illustrate this we show in Fig. 7 the momentum distributions in the channel decomposed into their contributions from different for the AV8’ and N3LO interactions. For the distributions we do not have the results for the bare AV8’ interaction and used the SRG transformed densities for instead. Because of the very weak dependence this should be equivalent to the exact result for the bare AV8’ interaction. The distributions in this channel are of particular interest, because they show the dominant contribution from pairs with relative angular momentum in the momentum region from to almost for the AV8’ and from to about for the N3LO interaction. These -wave pairs are directly reflecting the correlations induced by the tensor force. In the AV8’ case the tensor correlations are present at all relative momenta and dominate over the contribution for momenta above . For N3LO the regularization cuts them off at higher momenta so that they dominate only between and . It is important to note that the tensor correlations contribute also at low momenta. There is no natural scale in the tensor correlations that would allow to separate low and high momentum regions.
It is also interesting to compare the He momentum distributions in the channel with those of the deuteron shown in Fig. 7. Whereas the momentum distributions in the low momentum region up to about are noticeably different, reflecting the differences in the long-range parts of the wave functions for the loosely bound deuteron and the strongly bound He, the momentum distributions are almost indistinguishable for momenta above . This is consistent with the observed universality of short-range correlations discussed in Ref. Feldmeier et al. (2011).
If we compare the momentum distributions for the pairs integrated over all pair momenta as shown in Fig. 7 (a) and (b) with those for the corresponding pairs we notice that the node for the -wave pairs has vanished and that we find additional contributions from higher relative orbital angular momenta (-wave pairs). This is consistent with our expectation that many-body correlations play a greater role for pairs with large pair momentum . This is further confirmed by a strong dependence for these momentum distributions (not shown here). Alvioli et al. Alvioli et al. (2013b) found that the AV8’ momentum distributions factorize in a relative and a pair distribution for pair momenta up to about , whereas for higher pair momenta the momentum distributions no longer factorize and depend on the relative orientation of and .
iii.6 Relative probabilities for pairs
The strength of two- and many-body correlations strongly depends on the relative momenta of the pairs. To highlight the relative importance of these correlations it is advantageous to look at the relative probabilities
to find pairs in a given channel as a function of relative momentum . In Fig. 8 we show these relative probabilities for the bare AV8’ and N3LO interactions. If we look at the relative probabilities for all pairs with relative momentum it is obvious that the pairs contribute significantly, especially around . The total number of pairs is much smaller than the number of pairs in the even channels as shown in Tab. 1, but in this mid-momentum region the number of pairs in the channel is comparable to those in the even channels. This is the region where many-body correlations have the largest effect. For small relative momenta the relative probabilities are dominated by the mean-field, for relative momenta above about in case of the AV8’ and about in case of the N3LO interaction the relative probabilities are dominated by short-range correlations. This influence of many-body correlations is also related to a strong dependence on . With increasing flow parameter the relative probabilities for all pairs become more and more similar to the relative probabilities of the pairs shown in Fig. 8 (c) and (d). As there is no significant dependence for the momentum distributions, the relative probabilities for the pairs are independent from as well and are therefore not sensitive to many-body correlations.
It is interesting to note that the relative probabilities for the pairs are quite similar for the AV8’ and N3LO interactions, even for very large relative momenta, whereas the absolute values of the momentum distributions are very different. Differences in the relative probabilities between AV8’ and N3LO reflect differences in the relative importance of tensor correlations for the two interactions due to differences in the regularization of the tensor force.
iii.7 Relative probabilities for and pairs
In an experiment one measures protons and neutrons and not pairs. Therefore we define the operator in two-body space that measures the probability to find a pair of two protons (: ) or a proton-neutron pair (: ) at relative momentum and pair momentum as
In the case of He one pair corresponds to one -pair, and one pair to of a , of a and of a pair.
The corresponding relative probabilities for pairs
are shown in Fig. 9. The first observation is that the relative probabilities are rather similar for the AV8’ and N3LO interactions. At low momenta both show a ratio close to to find versus pairs. This is to be expected because an uncorrelated system of two protons and two neutrons can form one -pair, one -pair but four -pairs.
Around the minimum in the channel (see Fig. 5) together with the contribution from the tensor interaction in the channel (see Fig. 7) enhances the relative probability to find a pair to almost 100%. This dominance of pairs has been observed in exclusive two-nucleon knockout experiments Tang et al. (2003); Subedi et al. (2008). Recently the to ratio has been measured for relative momenta from up to almost Korover et al. (2014) showing an increase in the / ratio. Within the experimental uncertainties the data agree with our results for both AV8’ and N3LO interactions.
Iv Summary and Conclusions
In this paper we applied the SRG formalism for the calculation of relative density and momentum distributions of He. The He ground state wave functions are calculated in the NCSM with the SRG evolved AV8’ and N3LO interactions in two-body approximation. Two-body densities in coordinate and momentum space calculated with the unevolved density operators illustrate how short-range correlations are eliminated by the SRG evolution. With increasing flow parameter the interaction gets ‘softer’ and the wave functions become essentially uncorrelated mean-field wave functions without correlation holes and high-momentum components. The short-range or high-momentum information can be recovered by calculating two-body densities with the SRG evolved density operators, again in two-body approximation. Using these effective density operators we see a dependence of the two-body densities on the flow parameter . This dependence is due to missing contributions from three- and four-body terms in the effective operators, that are omitted in the two-body approximation. We find that not all components of the two-body density are equally affected by many-body correlations. The momentum distributions for pairs with pair momentum show only a very weak dependence and therefore provide direct access to two-body short-range correlations. The momentum distributions integrated over all pair momenta on the other hand have a sizeable dependence. This dependence is particularly strong in the and channels for momenta around . We identified three-body correlations induced by the strong tensor force as the main contributor to these many-body correlations.
Our results for the AV8’ momentum distributions agree with previous results Schiavilla et al. (2007); Wiringa et al. (2008, 2014); Alvioli et al. (2013b) using variational Monte-Carlo and few-body approaches. We confirm the important role of tensor correlations that explain the experimentally observed dominance of over pairs in exclusive two-nucleon knock-out with large momentum transfer. We also show that the chiral N3LO interaction provides very similar results for the momentum distributions at least up to relative momenta of . This includes the role of tensor correlations that are very similar for both interactions. At larger relative momenta the N3LO momentum distributions fall off much faster than the AV8’ momentum distributions as would be expected from the relatively soft cutoff employed in the regularization of the N3LO interaction. These differences are however mostly hidden in the ratios of versus pairs as a function of relative momenta as investigated in two-nucleon knockout experiments. Recently new chiral interactions with different regularization schemes have been proposed Wendt et al. (2014); Epelbaum et al. (2014). It will be interesting to see how these affect the short-range behavior and momentum distributions.
In this paper we investigated the He nucleus as it allowed us to compare SRG evolved with bare interactions. SRG evolved soft Hamiltonians however will allow us to study heavier nuclei in the -shell using the NCSM. One interesting question is the isospin dependence of the short-range correlations Sargsian (2014); Hen et al. (2014) in asymmetric nuclei. Neutron halos or skins might be a useful laboratory for the study of neutron matter. The present study also did not include three-body forces. Wiringa et al. Wiringa et al. (2014) find only small differences in the proton momentum distributions obtained with AV18 alone and AV18 together with an UX three-body interaction. We would expect that three-body interactions would not significantly change the short-range two-body correlations in the pairs, they may however have a significant effect on the many-body correlations and therefore change the results for the momentum distributions integrated over all pair momenta.
Appendix A Talmi-Moshinsky transformation
For an orthogonal transformation of the coordinates (, ) into (, ) with the mass ratio
the product of harmonic oscillator wave functions in coordinates and can be expressed with the Talmi-Moshinsky brackets in the new coordinates as:
All harmonic oscillator wave functions have the same oscillator parameter . See Ref. Kamuntavičius et al. (2001) for further properties.
Appendix B Jacobi coordinates
The wave function and two-body densities in the intrinsic system will be expressed in Jacobi coordinates. We follow Navrátil (2004) and use mass-scaled Jacobi coordinates that fulfill the orthogonality condition (19):
The transformation from the coordinates to is orthogonal. Translational invariant one-body densities can be expressed in these Jacobi coordinates. For the two-body densities we prefer however a different set of Jacobi coordinates where we have the distance between the nucleons and the distance of the center of mass of the pair with respect to the rest of the nucleus as coordinates. This can be achieved by an additional orthogonal transformation from the coordinates into the coordinates
Appendix C Two-body densities in the laboratory system
In the NCSM the wave function is expanded in a harmonic oscillator single-particle basis with oscillator parameter
The NCSM uses second quantization techniques and the two-body density in the harmonic oscillator basis can be expressed as
For the discussion of two-body correlations it is natural to transform from single-particle to pair coordinates. In the laboratory system we define relative and pair coordinates as
and the conjugate coordinates in momentum space as
These coordinates differ from the ones in (19) by factors of and so that the oscillator parameters for the relative and the pair motion have to changed to and respectively. Using (20) and coupling orbital angular momenta, spins and isospins the two-body density (28) can be transformed into .
Using this new basis
we can define the density operator in two-body space as
allows to easily express the two-body density in coordinate space as a function of relative distance and pair position
or as function of relative momentum and pair momentum
This can be trivially extended to calculate off-diagonal densities or densities for pairs of a given spin and isospin.
These densities can also be expressed as
and can be obtained from the wave functions by integrating out coordinates (we have omitted spin- and isospin indices for brevity)
where we have used the antisymmetry of the wave function.
Appendix D Two-body densities in the intrinsic system
The two-body density is calculated in the harmonic oscillator basis of the NCSM localized at the origin of the coordinate system and is therefore not translationally invariant. However the wave function in the NCSM factorizes into an intrinsic wave function only depending on relative coordinates and the total center-of-mass wave function in the ground state of the harmonic oscillator Navrátil et al. (2000)
This allows us to relate the two-body density in the laboratory system with the two-body density in the intrinsic system. The derivation is lengthy but straightforward and follows the derivation of the translational invariant one-body density in Ref. Navrátil (2004).
One starts with the two-body density in the laboratory system as given from the wave functions written in the coordinates and :
We now perform an orthogonal coordinate transformation from to and correspondingly for the primed coordinates. Next one uses the properties of the delta function to obtain delta functions in the Jacobi coordinates and in . The delta function in is expanded in the harmonic oscillator basis
In the next step a second orthogonal coordinate transformation from to is performed employing the Talmi-Moshinsky transformation for the harmonic oscillator wave functions, rewriting products with linear combinations of products .
If one uses (37) one can now integrate over and to express the density matrix in the laboratory system with the intrinsic wave functions so that the density matrix in the laboratory system can be related to the density matrix in the intrinsic system:
In this paper we only discuss scalar densities in the pair coordinates (integrating over all pair momenta or ). Then and and we can use the completeness relations for the Clebsch-Gordan coefficients to obtain the relation between the density matrices in the laboratory (38) and the intrinsic system (40)
with the transformation matrix
given by the Talmi-Moshinsky brackets with the mass ratio . This reflects the distribution of the oscillator quanta among the nucleon pair and the remaining nucleons. The translationally invariant two-body density averaged over the orientations of the pair momenta is then obtained from the two-body density matrix in the laboratory system by inverting Eq. (41).
The two-body density in the Jacobi coordinates and is given by
The mass-scaled Jacobi coordinates have technical advantages, however we prefer to express the two-body densities in the more intuitive coordinates
that are related to the Jacobi coordinates by
The conjugate variables to and are the relative momentum of the nucleons in the pair and the relative momentum of the pair with respect to the rest of the nucleus (in the center-of-mass system this is the same as the momentum of the pair).
The two-body density in these coordinates is then
Note that the oscillator parameter for the pair motion is different than in the laboratory system, whereas the oscillator parameter is the same. The factor is the Jacobian of the nonorthogonal transformation from the coordinates and to and in (45).
As in the laboratory system a convenient notation is obtained by defining the density operator in two-body space
The two-body densities in coordinate space can then be expressed as
and in momentum space as
Acknowledgements.We acknowledge support by the ExtreMe Matter Institute EMMI in the framework of the Helmholtz Alliance HA216/EMMI and by JSPS KAKENHI Grant Number 25800121.
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