Medium polarization effects inneutron-proton pairing

Outline: The puzzle of the missing neutron-proton pairing in nuclei Self-energy effects on n-p (3SD1) pairing energy gap Induced interaction theory (Landau limit) Vertex corrections in pairing interaction in 3SD1 two-body channel Nuclear matter vs. finite nuclei

The neglect of the neutron-proton interaction is the major weakness of the pairing force theory. This interaction is just as strong as that between a pair of like nucleons. In fact in the T=0 state is stronger.( A.M. Lane, Nuclear Theory, Benjamin 1964)

Later this statement received support by numerical estimates of np pairing gap in nuclear matter with realistic interactions ( 8-12 MeV).(for a review see U. Lombardo, Superfluidity in nuclear matter, World Sci. 1999,Ed. M.Baldo)

Despite the numerical predictions by A. Goodman (PRC 60,1999), no clear evidence for n-p pairing has so far been found in nuclei.

Recently, first Bertsch et al. (PRC 2010) and later Sagawa et al. (Phys.Scr. 2016) studied the competion between spin triplet and spin singlet pair correlations in nuclei Bertsch predicted a transition from spin singlet (nn) to spin triplet (np) pairing in large N=Z nuclei (A=130-140) . But the suppression of the np pairing in nuclei has not yet been completely understood. Possible candidates are many body effects, including self-energy and core polarization. It is already established the role of self-energy effects (effective mass and quasi-particle strength).The effects of core polarization have been predicted for nn and pp pairing (S=0,T=1) within the induced interaction theory either in nuclei ( Milano group) and in nuclear matter(Catania-Orsay Coll.). The extension of such analysis to np pairing is now on the way.

The puzzle of neutron-proton pairing in nuclei

np pairing in SD-channel (spin-triplet)

free sp spectrum

Comparison with nn pairing (spin-singlet)

> nuclear surface >

Rios et al arXiv:1707.04140

BARE FORCE

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np vs. nn pairing

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Collaboration

• Guo W.M., Lombardo U. ,INFN (LNS, Catania)• Schulze H.-J. : INFN (sez . Catania)• Schuck P. : U. Paris Sud (IPN, Orsay)

BCS model for coupled channels (3SD1 np pairing):

)

 

effective mass: E(k)2 = (k 2/ 2m* - kF

2/2m*)2

+ Δ2

Gap Eqs with self-energy corrections

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neutron-proton SD pairing

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degeneracy Z-factors inasymmetric nuclear matter

Quasi-Degenerate Fermi System

Self-energy k()

Dong, 2015: 4th order

Z

Degenerate Fermi System

(2p-1h) (2h-1p)

+ + +…+

degeneracy Z-factors inasymmetric nuclear matter

1

( )

( , )( ) 1

( ) 1 ( , ) ( )

k

F BHF

kZ k

Z k k k

�� � � ��� �

� �

ppHF hh

1

( )

( , )( ) 1

( ) 1 ( , ) ( )

k

F BHF

kZ k

Z k k k

�� � � ��� �

� �

1

( )

( , )( ) 1

( ) 1 ( , ) ( )

k

F BHF

kZ k

Z k k k

�� � � ��� �

� �

Self-Energy corrections

Full self-energy corrections

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Self-Energy corrections

Full self-energy corrections

Δnp/ Δnn 2

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np vs. nn pairing

Core polarization : ph excitationswithin the BandB induced interaction theory

�= Gph + �i

ph = +G��

ph G = BHF G-matrix (LNS)

� = induced interaction

Λ= dressed polarization

propagator(DPP) particle-hole coupling

particle-particle coupling

spin-singlet (e.g.,neutron-neutron 1S0 channel)

4(�i)01pp ΣT [(�i)0T

ph – 3(�i)1Tph ] = + ) 3 + ) 

( 0 ≤ q ≤ 2pF)

spin-triplet (e.g.,neutron-proton 3SD1 channel)

(�i)10pp (pF , pF , q)

4(�i)10pp ΣT[(-)T(2T+1) ((�i)0T

ph + (�i)1Tph ) ]= ) +(3  

Landau limit : : p= p’ = pF

main drawback : missing off-diagonal m.e.

It turns out tio be repulsive at any density, quite small in comparison wih bare interaction

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spin-triplet Induced Interaction

S and D partial waves

q= |p - p’| = 2pF sin θ/2

partial wave expansion

(�i)ppL = ½

 

p

P’θ

Fermi surface

We need to understand how the induced interactionaffects the pairing gap, that can be done incorporating the off-diagonal m.e. of the bare interaction (low and high momentum transitions) into the pairingInteraction at the Fermi surface.

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((J i) 10

pp (

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BCS model for coupled channels (3SD1 np pairing):

)

Renormalization of the pairing interaction:

)

=

Two main advantages: The constant gap approximation is a good approximation in the case of weak coupling( Δ≪�F ) . In the strong coupling limit it can be applied if w is small enough.The window gap equation makes it more unstandable the role of off-diagonal matrix elements of the pairing interaction vs. , in particular

 

Momentum space I = P + Q

Numerical procedure

For w<<kf we can assume Δ(k) Δ(kf) and the gap eqs. go over into a coupledsystem of (non-linear) algebraic equations:

where E2(k)= (�k f) � 2 + () ≡ (�k f) � 2 + . It can be numericallysolved along with equations for the renormalized interaction by means of linearization and iteration.

 

Off-diagonal matrix elements V(kf,k)

ratio

Kf=1.2 fm-1

<kf| VR|kf>

0.2 0.4 0.6 Window/kf

(�i)10pp << <kf |VR ||kf >

No appreciable effect expected!

Conclusions:

There is a general consensus that self-energy corrections conspire againstall kinds of pair correlations, including the neutron proton 3SD1 pairing.In nuclear matter they severely reduce the pairing gaps and shink the pairing domain to lower densities, typical of nuclear surface

The vertex corrections could provide a screening to the pair interactionand could be good candidates to solve the puzzle of the missing neutron-protongaps in nuclei . We found in fact that , in the deuteron channel, they have a repulsive effect in contrast with what happens in the case of 1S0 channel. However we found a negligible effect in the approximation adopted in the present calcs.

Approaches in finite nuclei attribute the missing np pairing to the spin-orbit splittingbut for N=Z very heavy nuclei a crossover from nn (pp) pairing to np pairing is predicted . This result is consistent with the present calculation.

Isospin splitting of potential energy (BHF: Bonn+3BF)

E/A= K + UaS (ρ)=U(ρ ,β=1)- U(ρ ,β=0)

Strong compensation for SLy4 Tensor component U10 the largest ChEFT: U10≈USD≈-17 MeV (Kohno,2013)

Bertsch et al. PRC 81 (2010)

Correlation energies EcS=1/Ec S=0

Sagawa et al. Phys.Scr.91,2016

Ec(S=1) > Ec(S=0)

Ec(S=1) < Ec(S=0)

Xavier,Welcome into the club of die-hard physicists!

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