1. Summary Talk PN Pairing and Quartet Correlations in Nuclei
Stuart Pittel

2. La mulţi ani y Feliz cumpleanos
For Nicu
La mulţi ani y Feliz cumpleanos

3. pn pairing
Quartets
Alpha Clusters

4. Why pn pairing?
Since there are two kinds of nucleons (n and p), nucleons can in principle form four distinct types of correlated (Cooper) pairs (nn, pp and pn) , each with net orbital angular momentum of zero and thus strongly correlated in space. np pairs can be either isoscalar (T=0, with S=1) or isovector (T=1, with S=0).
In finite nuclei, neutrons and protons must be in same major shell to strongly exploit pn pairing. Otherwise only identical nucleon nn and pp pairing important
This means N ~𝑍, which can now be
studied up to near A=100.

5. Why quartets Quoted from
Boson Mappings and four-particle correlations in algebraic neutron-proton pairing models, J. Dobes and S. Pittel; Phys. Rev. C57 (1998) 688.
“From these model calculations, we conclude that correlations involving pairs of fermion pairs, or alternatively quartets of fermions, are important in the regime of neutron-proton pairing. It is not sufficient only to pair nucleons. Whenever possible, two nucleon pairs will couple together to form a T=0 J=0 alpha-particle-like structure.
Qualitatively, this conclusion can be understood as follows. The smallest ‘‘cluster’’ that can simultaneously accommodate two-neutron pairing correlations, two-proton pairing correlations and neutron-proton pairing correlations is one that involves four nucleons—two neutrons and two protons. Of course, when there is an excess of particles of a given type, not all particles can form these maximally correlated alpha-like structures. Instead, they remain in like-particle and/or neutron-proton pairs, appended to the alpha-like condensate.”

6. From quartets to alpha clusters
Here the talks of Peter Schuck and Thomas Neff were particularly revealing to me.
Clusters with spatial correlations on a much smaller scale than in the usual shell model arise in the vicinity of thresholds and through the mixing of many many major shells (perhaps 60 or so).
Can be treated on the same footing as shell model configurations through the use of methods such as Fermion Molecular Dynamics.
The talk of Alexander Volya was also very instructive to me, as it showed how one can use the shell model, typically in a very large but of course still truncated space, and analyze the wave functions to study their cluster structure. Whether strong spatial clustering can emerge depends of course on the size of the full shell model space.

7. Methods for describing pn pairing correlations
QCM (Quartet Condensate Model) : Discussed by Nicu Sandulescu and
Danilo Gambacurta.
. Nicu discussed the main ideas and showed how well it works for
T=0, T=1 or mixed pairing. Method works very well, especially in
deformed nuclei where the calculation is carried out in Nilsson
or deformed HF basis. SU(4)?
. Danilo reported first principles analysis including deformation
arising from Skyrme HF treatment. Discussed, among many topics,
the suppression of T=0 pairing effects by spin-orbit interaction.
. Needed is projection on good J, preferably before variation, and self
consistency between the treatment of mean field and pairing effects.
Iteration?
. Also needed for the future is calculation of pn transfer probabilities.

8. - QM (Quartet Model) Discussed by Michelangelo Sambataro. Analogous to
a proposal of Arima-san many years ago for mixing of pair
configurations.
. Rather than treating a condensate of quartets, he chooses different quartets from the four-particle system(s). Get even better results than for QCM. Only a limited number of correlated quartets typically needed.
. If system is deformed and a description up to high-spin states is desired (into backbend region, for example), will need more quartets. Should be studied.
CI (Configuration Interaction) studies. Discussed by Dick Chasman.
. Got a simple and accurate prediction for Wigner effect (47/A) in terms of diagonal pairing matrix elements.
. Includes spin-aligned configurations in his iterative variational analysis.

9. Algebraic Approach : Jorge Dukelsky reviewed exactly-solvable pairing models, both for pn pairing and for other forms of pairing (constant G and Gogny-like) for like-particle systems.
. Showed that in the thermodynamic limit, pn pairing satisfies the pn BCS equation, making clear that pn pairing involves two-body correlations, even though exact solutions involve quartets. Quartets critical to get symmetry structure right, but most likely don’t add new correlations. -

10. More on theory of pn pairing
Isoscalar pairing in a single orbit: Piet Van Isacker focused on T=0 pairing, showing first that in a single orbit the aligned state is favored.
. He then discussed pair transfer to T=1 pairing vibrations, giving roughly analytical expressions.
. Finally he presented systematic picture of T=0 and T=1 deuteron transfer, likewise in terms of pair vibrations.
. Spin aligned pairs and np correlations: Chong Qi described np correlations from a shell model perspective.
. Talked about spin traps in various regions.
. Focused much discussion on mass regime, where quadrupole effects suppressed. Searched for minimal degrees of freedom needed to describe properties.
. g9/2 dominant, but can also have some population of other orbits.
. Spin-aligned pairs seem to be a crucial building blocks in this region, and perhaps elsewhere.

11. - Alex Gezerlis spoke about pairing in several different types of systems. Cold atoms in particular might provide a laboratory in which to explore pairing effects in a wide variety of interesting scenarios outside the usual nuclear scenario.
. In nuclei, he reported detailed and systematic calculations using pn pairing hamiltonians, with HFB plus the gradient method. Unfortunately deformation wasn’t included.
.Key conclusion was that in 136Gd, there seems to be a mixed phase with both T=1 pairing and T=0 pairing coexisting. Whether this will hold up when deformation is included and whether it will be possible to experimentally access this system are issues still to be addressed.
. Nonetheless, potentially very exciting.

12. Spin aligned vs Cooper pairs (Frauendorf)
. Emphasized that one really needs correlations to call it pairing.
. Emphasized that I=2j pairs don’t correlate.
. Gave a description of Wigner effect as spontaneous breaking of isospin symmetry in analogy to deformation deriving from spontaneous breaking of rotational symmetry. Showed that one gets good (and robust) description of Wigner energy either thru IV pairing or with some IS pairing and a renormalized IV component.
. Though expressing doubt that we’ll ever see an IS condensate, he did note, however, that there could be some enhancement in GT to IS 1+ state .
. Also talked a bit about 92Pb, noting that alignment mechanism might be coexisting with IV correlation, and that they are definitely not orthogonal.

13. .
- Three-body calculations for N=Z o-o nuclei: Tanimura-san reported
systematic 3-body (core +2 particles) calculations for o-o nuclei, some
with spherical cores and some deformed.
. Focused on GT properties, getting results in qualitative agreement with
data.
. For nuclei near 100Sn, used relativistic Skyrme HF for core and pairing.
. Relativistic analysis led to near degeneracy between d5/2 and g7/2 , as
known. A nice example of pseudo-spin symmetry.
. Near degeneracy critical to get reasonable GT results.

14. Pair Transfer Reaction theory
Andrea Vitturi discussed how to extract meaningful info on pair transfer from experimental data.
. Discussed the formalism for pair transfer, using a sequential framework.
. Emphasized the need to include carefully info on radial degrees of freedom in pair transfer density. .
. Most importantly, not enough to just get 𝜎 0+ 𝜎 to determine relative collectivity, since each may be enhanced differently relative to uncorrelated transfer.
. Also, may be different angular dependence of enhancements.

15. Pair transfer experiments
Marlene Assie described impressive experimental program to measure pair transfer enhancements, mostly in sd shell but a bit in fp.
Results seem to have correct behavior as we pass thru the shell, albeit with the caveats raised by Andrea still to be considered.
Emphasized by Augusto Machiavelli in his Colloquium and throughout discussions that pn transfer experiments seem like most promising probe to see pn condensate.

16. Interesting shell model developments
Calvin Johnson reported large scale shell model calculations and what you can learn from them by expansion in symmetry bases.
. Found, e.g., that while SU(3) is broken in 48Cr, there seems to be a good quasi-SU(3) symmetry.
. Talked about importance of using Abelian quantum numbers and the possible use of entanglement in analysis of results.
Alexander Volya talked about how to analyze wave functions from shell model calculations to isolate possible clustering features and to study interplay with pairing.
. Can decompose wave functions into products of different substructures of the system, of relevance not only to possible spatial clustering but also to get appropriate spectroscopic transfer form amplitudes even when less spatially correlated.

17. Clustering Theory
Peter Schuck discussed alpha condensation in finite nuclei and elsewhere.
. Discussed variational ansatz in terms of product of clustering states, fully antisymetrized and with distinct size parameters for each cluster.
. Very interesting results for 12C. Ground state has all clusters of roughly the size of the nucleus. Second excited 0+ state has an alpha with size roughly of a free alpha. This is the Hoyle state and comes out naturally from formalism.
- Nick Manton showed how Skyrmions can be used to build solutions for finite nuclei. Showed impressive results for a wide variety of fairly light systems, including 16O, with more promised in future.

18. Thomas Neff first discussed short range correlations in nuclei, their treatment with ab initio approaches and impressive comparison with JLab data.
. Then described the Fermion Molecular Dynamics Approach. Uses products of Gaussian wave packets with full antisymmetrization and minimizes.
. Discussed importance of parity projection to get meaningful results.
. For light enough systems can use VAP.
. Presented impressive results for 8Be, 12C and Ne isotopes.
. Showed that in 12C, g.s. only requires about 9 oscillator levels whereas Hoyle state needs about 60.
. Emphasized importance of thresholds to get clustering.
- Rafael-David Lasseri discussed influence of pairing effects in several systems. Used relativistic HF and then implemented pairing. To treat pairing correctly, he had to use quartet approach with full restoration of isospin and found that it had a significant impact.

19. Josef Cseh discussed many aspects of clustering from a semi-microscopic perspective.
. Most interesting to me was his description of the Multi Channel Dynamical Symmetry Method. Though only semi-microscopic, and sticking with the SU(3) limit, he was able to use method to simultaneously describe low-lying states of 28Si and 12C+16O cluster states. Fit to low-lying states and then prediction of cluster states. Impressive!
Taniguchi-san described method to study influence of cluster states of rotational properties of deformed nuclei.
. Used AMD (much like FMD used by Neff) and GCM.
. Showed impressive results on how cluster configurations affect rotational properties of excited deformed (ND) and (SD) bands in 35Cl, 28Si and 40Ca. Didn’t seem to find clear threshold effect for location of cluster configurations, as sometimes two different clusters with different thresholds contributed to same system.
. Interesting result that cluster components with oblate shapes affect triaxiality of deformed excited states.

20. Theory of alpha decay
Roberto Liotta described a microscopic procedure for treating alpha decay.
. Showed that you cannot reproduce data with ordinary shell model treatment of formation amplitude. Need to include a cluster component, obtained by adding a pocket (first proposed by Delion) for the alphas just outside the nucleus. Pocket gives small size for alpha, as needed to reproduce data.
Virgil Baran described more recent efforts to build alpha decay amplitudes, but with same basic physics ingredients as in Liotta work. Described implementation of approach to 104Te decay.

21. Double Beta Decay and pn pairing
Peter Vogel reviewed the current status of efforts to determine the Nuclear
Matrix elements appropriate to double beta decay, both two neutrino and
neutrinoless.
. The QRPA analysis makes use of a pn pairing interaction, for which the
results are extremely sensitive to its strength. β+ contribution to matrix
element strongly Pauli blocked, which is why results so sensitive
. Peter compared SM and IBM results with the QRPA results for the various
double beta decay candidate systems.
. Roughly a factor of 3 overall sensitivity to the method at present.

22. High spin signature of pn pairing
- Giacomo de Angelis described the current status of measurements of high-spin
phenomena in N~Z nuclei.
. He discussed possible signatures of important pn pairing effects, e.g. delayed
spin alignment and a spin aligned mode. He then presented the results of Large Scale Shell Model studies of deformed nuclei to explore these possible effects. Unfortunately since the calculation could not readily separate the T=0 part of the quadrupole force from the T=0 pairing force, he could not draw meaningful conclusions on the precise role of pn pairing on these rotational properties.
- Since we (with Nicu and Alfredo Poves) have carried out some calculations of relevance to
high spin states and pn pairing and since we do better separate the various pierces of the force, I’ll show you some of our selected results.

23. Reference: Y. Lei, SP, N. Sandulescu, A. Poves, B. Thakur and Y. M
Reference: Y. Lei, SP, N. Sandulescu, A. Poves, B. Thakur and Y.M. Zhao, Phys. Rev. C84 (2011)
x = MeV
a= 12
b= 12
α= 20
Acceptable for all including,
rotational band in 48Cr with
backbend.

24. How do the number of isoscalar and isovector pairs evolve as function of angular momentum in the YRAST band?
Backbend

25. My favorite description of the Wigner Energy
. A subject that was discussed/debated extensively during the workshop concerned
the origin of the Wigner energy behavior.
. My favorite was the picture put forward by Stefan, that it arises from spontaneously breaking isospin symmetry in the presence of an isovector interaction. Not only does it have a simple analogy to how rotations are generated in nuclei by spontaneously
breaking rotational symmetry, but it seems to apply equally well in a scenario dominated by IV pairing or one with some IS pairing as well, as long as the isovector strength is correspondingly reduced. Whether the Wigner energy is precisely of a T(T+1) form doesn’t bother me, since as we know there are typically corrections to the J(J+1) behavior in rotational nuclei, e.g. in the Harris model where higher powers of J(J+1) enter..

26. Possible fingerprints of pn pairing on various observables (from Gilles)
Rotational properties  G. de Angelis
Delayed alignments
T=0/T=1 band crossings
Deformation/shape coexistence
Pair/alpha transfer  A. Vitturi, M. Assié, P. Van Isacker
Reaction model dependent
Alpha-decay/cluster: Te-Xe N=Z. Role of T=0 np? (N. Sandulescu, R. Liotta)
Seniority breaking/spin aligned configuration  C. Qi and P. Van Isacker
92Pd, 96Cd experiments
Binding energies
Precise mass measurements. Many new trap developments.
Radii
Effect of T=0
B(E2), Q0 and magnetic moments?
B(E2) and Q0. LoI S3.
Magnetic moments of 7+ isomer in 94Ag?
Beta-decay: low-lying GT strength and T=0? (Y. Tanimura). n-less double beta-decay (P. Vogel)
Pair vibrations/GPV?
Vocabulary: is SA pairing? Counting the number of pairs