Shortrange correlations in nuclei with similarity renormalization group transformations
Abstract
 Background

Realistic nucleonnucleon interactions induce shortrange correlations in nuclei. To solve the manybody problem unitary transformations like the similarity renormalization group (SRG) are often used to soften the interactions.
 Purpose

Twobody densities can be used to illustrate how the SRG eliminates shortrange correlations in the wave function. The shortrange information can however be recovered by transforming the density operators.
 Method

The manybody 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 twobody densities with bare and SRG transformed density operators in twobody approximation.
 Results

The twobody momentum distributions for AV8’ and N3LO have similar highmomentum components up to relative momenta of about , dominated by tensor correlations, but differ in their behavior at higher relative momenta. The contributions of manybody correlations are small for pairs with vanishing pair momentum but not negligible for the momentum distributions integrated over all pair momenta. Manybody correlations are induced by the strong tensor force and lead to a reshuffling of pairs between different spinisospin channels.
 Conclusions

When using the SRG it is essential to use transformed operators for observables sensitive to shortrange physics. Backtoback pairs with vanishing pair momentum are the best tool to study shortrange correlations.
pacs:
21.60.De, 21.30.Fe, 05.10.Cc, 25.30.cI Introduction
Realistic nucleonnucleon () interactions are fitted to scattering data up to the pion production threshold. Therefore their properties at short distances and also their offshell behavior is not completely constrained. All realistic interactions include pions to describe the long and mediumrange parts of the potential. The shortrange part is parameterized phenomenologically Wiringa et al. (1995), by the exchange of heavy mesons Machleidt (2001), or in chiral effective field theory by (regularized) contactterms Entem and Machleidt (2003); Epelbaum et al. (2005). At short distances the different interactions induce typical shortrange correlations due to shortrange repulsion and the tensor force which is reflected in a depopulation of onebody momentum distributions below the Fermi momentum and an enhancement at high momenta when compared to singleparticle meanfield 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 shortrange correlations are twobody correlations and therefore twobody densities in coordinate and momentum space provide the best tool to study these correlations. Experimentally shortrange correlations have been studied in inclusive, semiinclusive and triplecoincidence reactions, see the reviews Frankfurt et al. (2008); Arrington et al. (2012) and references therein. The most direct information about twobody correlations can be obtained in triplecoincidence 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 highmomentum transfer. In these experiments one found a dominance of over pairs at high relative momenta, that indicated the importance of shortrange 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 twobody momentum distributions can explain these observations. In a simple picture only close nucleons found in relative wave pairs are affected by shortrange correlations and in a first approximation highmomentum components are generated by pairs in deuteronlike configurations Vanhalst et al. (2012). The twobody momentum distributions at high momenta can also be connected to the nuclear contacts Weiss et al. (). Detailed few and manybody 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 cutoff and one might expect noticeable differences for the shortrange correlations.
From the perspective of nuclear manybody calculations shortrange correlations are not a desired feature but pose a severe problem. One way to address this problem is to include the shortrange 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 phaseshift 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 twobody, or for state of the art SRG calculations, in threebody approximation Jurgenson et al. (2009); Hebeler (2012); Roth et al. (2014). Even if we only have a twobody interaction at the beginning the unitary transformation will induce three and higherbody 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 twobody level will be a good approximation. The remaining dependence of observables on the unitary transformation can be used to analyze the nature of missing manybody 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 longrange operators like radius or electromagnetic transition operators has been studied in Schuster et al. (2014) in two and threebody approximation. As one might expect the effect on these longrange operators is not very large. On the other hand we expect large effects for shortrange or highmomentum observables. Within the UCOM approach we investigated these effects in twobody approximation and with simple trial wave functions for the onebody momentum distribution Neff and Feldmeier (2003) and for twobody 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 highmomentum observables. These ideas were extended for Fermi gases in Bogner and Roscher (2012).
In this paper we investigate twobody 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 twobody 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 twobody approximation and use the flowdependence of the results to test the quality of the twobody approximation. We will show that for pairs with vanishing pair momentum the contributions of threebody correlations are negligible.
After describing in Sec. II the used methods, twobody densities in coordinate and momentum space are presented in Sec. III where we also discuss the role of manybody correlations for different observables. Summary and conclusions follow in Sec. IV. Technical details about the calculation of translational invariant twobody densities in the NCSM framework are presented in the Appendix.
Ii Method
The SRG flow equation for the Hamiltonian and the corresponding transformation matrix are given by
(1) 
with the generator taken to be the commutator of the intrinsic kinetic energy and the evolved Hamiltonian:
(2) 
Please note that used here and in Ref. Roth et al. (2010) corresponds to in Refs. Bogner et al. (2007, 2010).
In this paper the Hamiltonian is transformed in twobody approximation:
(3) 
The evolution (1) can therefore be performed for the relative motion in twobody 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 manybody calculations), and corresponding to a very soft effective Hamiltonian .
The manybody problem for He is then solved with the SRG transformed twobody Hamiltonian using the shell model code Antoine Caurier and Nowacki (1999):
(4) 
The results for the ground state energy of He and the twobody 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 twobody information contained in the manybody states can be expressed in terms of the twobody density matrix that is obtained by integrating over all coordinates besides the pair position and the relative coordinates of the pair. The twobody 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 twobody density can be obtained from the twobody density in the laboratory system by a linear transformation as discussed in Appendix D. In the harmonic oscillator basis we express the twobody density matrix with twobody basis states
(5) 
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
(6) 
respectively.
A compact notation can be obtained by defining the twobody density operator, which acts in twobody space, as
(7) 
The expectation value of any ‘bare’ twobody operator can then simply evaluated as
(8) 
where denotes the trace in twobody space. If one evolves the observable in the same way as the Hamiltonian, namely , the expectation value
(9) 
does not depend on because . However, as for the Hamiltonian, the evolved observable is in general no longer a twobody operator and contains induced higherbody operators.
The twobody approximation consists in calculating
(10) 
where the twobody operator is SRG transformed in twobody 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 twobody 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 higherbody terms.
With the manybody states obtained in the NCSM various bare and SRG transformed twobody 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 twobody approximation is justified. However one has to be careful when drawing conclusions about omitted higherorder terms in the transformed operators. In this paper the twobody approximation is employed both for the transformed Hamiltonian and for the transformed operators. Even if the twobody approximation is perfect for the transformed operators we would still expect a dependence on the flow parameter due to the twobody approximation for the transformation of the Hamiltonian.
It might be possible however that particular components of the wave function are less sensitive to higherorder terms in the transformed Hamiltonian and can be obtained reliably within the twobody approximation. This appears to be the case for the pairs with small pair momentum as will be discussed in Sec. III.5.
Iii Results
iii.1 Energies
In twobody approximation is exactly unitary in twobody space, but in manybody 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 fourbody interactions (see Eq. (3)) that are neglected when calculating the He ground state energy in twobody approximation. As seen in Fig. 1 the total energy varies by about 10% indicating that the contribution of three and fourbody 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)
(11) 
where the operator projects on the spinisospin 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 twobody operator
(12) 
where we sum over all quantum numbers except the radial distance .
The shortrange 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 meanfield wave function. Thus, the SRG transformation brings us from a highly correlated system to a simple shell model or meanfield 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 twobody approximation would be unitary in fourbody space and would not depend on . The remaining dependence indicates that the induced three and fourbody 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:
(13) 
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 twobody operator
(14) 
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 shortrange 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 twobody 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 shortrange tensor correlations play an important role as will be discussed in more detail later. With increasing flow parameter the highmomentum 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 fourbody terms are also small in momentum space. The most visible dependence on occurs for the AV8’ interaction around . This dependence is related to threebody correlations induced by the twobody 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 threebody correlations are lost in twobody 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 threebody 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.
(0,0)  (0,1)  (1,0)  (1,1)  

AV8’, bare  0.008  2.572  2.992  0.428 
AV8’,  0.008  2.708  2.992  0.292 
AV8’,  0.007  2.821  2.993  0.179 
AV8’,  0.005  2.925  2.995  0.075 
N3LO, bare  0.009  2.710  2.991  0.290 
N3LO,  0.007  2.745  2.992  0.255 
N3LO,  0.006  2.817  2.994  0.183 
N3LO,  0.004  2.921  2.995  0.079 
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 meanfield 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 twobody approximation does not connect different channels.
The reshuffling of probability between the different spinisospin channels also tells us that the omitted manybody terms in and the transformed Hamiltonian will have a nontrivial spin and isospindependence.
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 manybody correlations.
We might expect that backtoback pairs with are less affected by manybody 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 manybody 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 twobody operator
(15) 
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 manybody correlations are not very important, as anticipated in the discussion above. The pairs are therefore the best candidates for experimental studies of shortrange twobody 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 shortrange 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 longrange 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 shortrange 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 manybody 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 manybody 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
(16) 
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 midmomentum region the number of pairs in the channel is comparable to those in the even channels. This is the region where manybody correlations have the largest effect. For small relative momenta the relative probabilities are dominated by the meanfield, for relative momenta above about in case of the AV8’ and about in case of the N3LO interaction the relative probabilities are dominated by shortrange correlations. This influence of manybody 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 manybody 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 twobody space that measures the probability to find a pair of two protons (: ) or a protonneutron pair (: ) at relative momentum and pair momentum as
(17) 
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
(18) 
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 twonucleon 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 twobody approximation. Twobody densities in coordinate and momentum space calculated with the unevolved density operators illustrate how shortrange correlations are eliminated by the SRG evolution. With increasing flow parameter the interaction gets ‘softer’ and the wave functions become essentially uncorrelated meanfield wave functions without correlation holes and highmomentum components. The shortrange or highmomentum information can be recovered by calculating twobody densities with the SRG evolved density operators, again in twobody approximation. Using these effective density operators we see a dependence of the twobody densities on the flow parameter . This dependence is due to missing contributions from three and fourbody terms in the effective operators, that are omitted in the twobody approximation. We find that not all components of the twobody density are equally affected by manybody correlations. The momentum distributions for pairs with pair momentum show only a very weak dependence and therefore provide direct access to twobody shortrange 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 threebody correlations induced by the strong tensor force as the main contributor to these manybody 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 MonteCarlo and fewbody approaches. We confirm the important role of tensor correlations that explain the experimentally observed dominance of over pairs in exclusive twonucleon knockout 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 twonucleon 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 shortrange 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 shortrange 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 threebody 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 threebody interaction. We would expect that threebody interactions would not significantly change the shortrange twobody correlations in the pairs, they may however have a significant effect on the manybody correlations and therefore change the results for the momentum distributions integrated over all pair momenta.
Appendix A TalmiMoshinsky transformation
For an orthogonal transformation of the coordinates (, ) into (, ) with the mass ratio
(19) 
the product of harmonic oscillator wave functions in coordinates and can be expressed with the TalmiMoshinsky brackets in the new coordinates as:
(20) 
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 twobody densities in the intrinsic system will be expressed in Jacobi coordinates. We follow Navrátil (2004) and use massscaled Jacobi coordinates that fulfill the orthogonality condition (19):
(21)  
(22)  
(23)  
(24) 
The transformation from the coordinates to is orthogonal. Translational invariant onebody densities can be expressed in these Jacobi coordinates. For the twobody 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
(25) 
(26) 
Appendix C Twobody densities in the laboratory system
In the NCSM the wave function is expanded in a harmonic oscillator singleparticle basis with oscillator parameter
(27) 
The NCSM uses second quantization techniques and the twobody density in the harmonic oscillator basis can be expressed as
(28) 
For the discussion of twobody correlations it is natural to transform from singleparticle to pair coordinates. In the laboratory system we define relative and pair coordinates as
(29) 
and the conjugate coordinates in momentum space as
(30) 
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 twobody density (28) can be transformed into .
Using this new basis
(31) 
we can define the density operator in twobody space as
(32) 
allows to easily express the twobody density in coordinate space as a function of relative distance and pair position
(33) 
or as function of relative momentum and pair momentum
(34) 
This can be trivially extended to calculate offdiagonal densities or densities for pairs of a given spin and isospin.
These densities can also be expressed as
(35) 
and can be obtained from the wave functions by integrating out coordinates (we have omitted spin and isospin indices for brevity)
(36) 
where we have used the antisymmetry of the wave function.
Appendix D Twobody densities in the intrinsic system
The twobody 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 centerofmass wave function in the ground state of the harmonic oscillator Navrátil et al. (2000)
(37) 
This allows us to relate the twobody density in the laboratory system with the twobody density in the intrinsic system. The derivation is lengthy but straightforward and follows the derivation of the translational invariant onebody density in Ref. Navrátil (2004).
One starts with the twobody density in the laboratory system as given from the wave functions written in the coordinates and :
(38) 
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
(39) 
In the next step a second orthogonal coordinate transformation from to is performed employing the TalmiMoshinsky 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:
(40) 
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 ClebschGordan coefficients to obtain the relation between the density matrices in the laboratory (38) and the intrinsic system (40)
(41) 
with the transformation matrix
(42) 
given by the TalmiMoshinsky brackets with the mass ratio . This reflects the distribution of the oscillator quanta among the nucleon pair and the remaining nucleons. The translationally invariant twobody density averaged over the orientations of the pair momenta is then obtained from the twobody density matrix in the laboratory system by inverting Eq. (41).
The twobody density in the Jacobi coordinates and is given by
(43) 
The massscaled Jacobi coordinates have technical advantages, however we prefer to express the twobody densities in the more intuitive coordinates
(44) 
that are related to the Jacobi coordinates by
(45) 
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 centerofmass system this is the same as the momentum of the pair).
(46) 
The twobody density in these coordinates is then
(47) 
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 twobody space
(48) 
with
(49) 
The twobody densities in coordinate space can then be expressed as
(50) 
and in momentum space as
(51) 
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.References
 R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995).
 R. Machleidt, Phys. Rev. C 63, 024001 (2001).
 D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003).
 E. Epelbaum, W. Glöckle, and U.G. Meißner, Nucl. Phys. A 747, 362 (2005).
 O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Nucl. Phys. A 579, 493 (1994).
 V. R. Pandharipande, I. Sick, and P. K. A. d. Huberts, Rev. Mod. Phys. 69, 981 (1997).
 H. Müther and A. Polls, Prog. Part. Nucl. Phys. 45, 243 (2000).
 T. Neff and H. Feldmeier, Nucl. Phys. A 713, 311 (2003).
 W. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004).
 A. Rios, A. Polls, and W. H. Dickhoff, Phys. Rev. C 79, 064308 (2009).
 L. Frankfurt, M. Sargsian, and M. Strikman, Int. J. Mod. Phys. A 23, 2991 (2008).
 J. Arrington, D. Higinbotham, G. Rosner, and M. Sargsian, Prog. Part. Nucl. Phys. 67, 898 (2012).
 A. Tang, J. W. Watson, J. Aclander, J. Alster, G. Asryan, Y. Averichev, D. Barton, V. Baturin, N. Bukhtoyarova, A. Carroll, S. Gushue, S. Heppelmann, A. Leksanov, Y. Makdisi, A. Malki, E. Minina, I. Navon, H. Nicholson, A. Ogawa, Y. Panebratsev, E. Piasetzky, A. Schetkovsky, S. Shimanskiy, and D. Zhalov, Phys. Rev. Lett. 90, 042301 (2003).
 R. Shneor et al. (Jefferson Lab Hall A Collaboration), Phys. Rev. Lett. 99, 072501 (2007).
 R. Subedi, R. Shneor, P. Monaghan, B. D. Anderson, K. Aniol, J. Annand, J. Arrington, H. Benaoum, F. Benmokhtar, W. Boeglin, J.P. Chen, S. Choi, E. Cisbani, B. Craver, S. Frullani, F. Garibaldi, S. Gilad, R. Gilman, O. Glamazdin, J.O. Hansen, D. W. Higinbotham, T. Holmstrom, H. Ibrahim, R. Igarashi, C. W. de Jager, E. Jans, X. Jiang, L. J. Kaufman, A. Kelleher, A. Kolarkar, G. Kumbartzki, J. J. LeRose, R. Lindgren, N. Liyanage, D. J. Margaziotis, P. Markowitz, S. Marrone, M. Mazouz, D. Meekins, R. Michaels, B. Moffit, C. F. Perdrisat, E. Piasetzky, M. Potokar, V. Punjabi, Y. Qiang, J. Reinhold, G. Ron, G. Rosner, A. Saha, B. Sawatzky, A. Shahinyan, S. Širca, K. Slifer, P. Solvignon, V. Sulkosky, G. M. Urciuoli, E. Voutier, J. W. Watson, L. B. Weinstein, B. Wojtsekhowski, S. Wood, X.C. Zheng, and L. Zhu, Science 320, 1476 (2008).
 H. Baghdasaryan et al. (CLAS Collaboration), Phys. Rev. Lett. 105, 222501 (2010).
 I. Korover et al. (Jefferson Lab Hall A Collaboration), Phys. Rev. Lett. 113, 022501 (2014).
 K. Miki, A. Tamii, N. Aoi, T. Fukui, T. Hashimoto, K. Hatanaka, T. Ito, T. Kawabata, H. Matsubara, K. Ogata, H. Ong, H. Sakaguchi, S. Sakaguchi, T. Suzuki, J. Tanaka, I. Tanihata, T. Uesaka, and T. Yamamoto, FewBody Syst. 54, 1353 (2013).
 H. Ong, I. Tanihata, A. Tamii, T. Myo, K. Ogata, M. Fukuda, K. Hirota, K. Ikeda, D. Ishikawa, T. Kawabata, H. Matsubara, K. Matsuta, M. Mihara, T. Naito, D. Nishimura, Y. Ogawa, H. Okamura, A. Ozawa, D. Pang, H. Sakaguchi, K. Sekiguchi, T. Suzuki, M. Taniguchi, M. Takashina, H. Toki, Y. Yasuda, M. Yosoi, and J. Zenihiro, Phys. Lett. B 725, 277 (2013).
 M. Vanhalst, J. Ryckebusch, and W. Cosyn, Phys. Rev. C 86, 044619 (2012).
 R. Weiss, B. Bazak, and N. Barnea, arXiv:1503.07047 [nuclth] .
 R. Schiavilla, R. B. Wiringa, S. C. Pieper, and J. Carlson, Phys. Rev. Lett. 98, 132501 (2007).
 R. B. Wiringa, R. Schiavilla, S. C. Pieper, and J. Carlson, Phys. Rev. C 78, 021001 (2008).
 R. B. Wiringa, R. Schiavilla, S. C. Pieper, and J. Carlson, Phys. Rev. C 89, 024305 (2014).
 M. Alvioli, C. Ciofi degli Atti, L. P. Kaptari, C. B. Mezzetti, H. Morita, and S. Scopetta, Phys. Rev. C 85, 021001 (2012).
 M. Alvioli, C. Ciofi degli Atti, L. P. Kaptari, C. B. Mezzetti, and H. Morita, Int. J. Mod. Phys. E 22, 1330021 (2013a).
 A. Fabrocini, F. Arias de Saavedra, and G. Co’, Phys. Rev. C 61, 044302 (2000).
 S. Bogner, T. Kuo, and A. Schwenk, Phys. Rep. 386, 1 (2003).
 H. Feldmeier, T. Neff, R. Roth, and J. Schnack, Nucl. Phys. A 632, 61 (1998).
 R. Roth, T. Neff, and H. Feldmeier, Prog. Part. Nucl. Phys. 65, 50 (2010).
 S. K. Bogner, R. J. Furnstahl, and R. J. Perry, Phys. Rev. C 75, 061001 (2007).
 S. Bogner, R. Furnstahl, and A. Schwenk, Prog. Part. Nucl. Phys. 65, 94 (2010).
 E. D. Jurgenson, P. Navrátil, and R. J. Furnstahl, Phys. Rev. Lett. 103, 082501 (2009).
 K. Hebeler, Phys. Rev. C 85, 021002 (2012).
 R. Roth, A. Calci, J. Langhammer, and S. Binder, Phys. Rev. C 90, 024325 (2014).
 M. D. Schuster, S. Quaglioni, C. W. Johnson, E. D. Jurgenson, and P. Navrátil, Phys. Rev. C 90, 011301 (2014).
 H. Feldmeier, W. Horiuchi, T. Neff, and Y. Suzuki, Phys. Rev. C 84, 054003 (2011).
 E. R. Anderson, S. K. Bogner, R. J. Furnstahl, and R. J. Perry, Phys. Rev. C 82, 054001 (2010).
 S. K. Bogner and D. Roscher, Phys. Rev. C 86, 064304 (2012).
 Y. Suzuki and W. Horiuchi, Nucl. Phys. A 818, 188 (2009).
 J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Rev. Mod. Phys. 85, 693 (2013).
 E. Caurier and F. Nowacki, Acta Phys. Pol. B 30, 705 (1999).
 T. Neff, H. Feldmeier, W. Horiuchi, and D. Weber, (2015), arXiv:1503.06122 [nuclth] .
 M. Alvioli, C. Ciofi degli Atti, L. P. Kaptari, C. B. Mezzetti, and H. Morita, Phys. Rev. C 87, 034603 (2013b).
 K. Wendt, B. Carlsson, and A. Ekström, (2014), arXiv:1410.0646 [nuclth] .
 E. Epelbaum, H. Krebs, and U. G. Meißner, (2014), arXiv:1412.4623 [nuclth] .
 M. M. Sargsian, Phys. Rev. C 89, 034305 (2014).
 O. Hen et al. (Jefferson Lab CLAS Collaboration), Science 346, 614 (2014).
 G. Kamuntavičius, R. Kalinauskas, B. Barrett, S. Mickevičius, and D. Germanas, Nucl. Phys. A 695, 191 (2001).
 P. Navrátil, Phys. Rev. C 70, 014317 (2004).
 P. Navrátil, G. P. Kamuntavičius, and B. R. Barrett, Phys. Rev. C 61, 044001 (2000).