Nuclear pairing from microscopic forces (with applications) Paolo Finelli Dept. of Physics and Astronomy, University of Bologna paolo. fj nelli@bo.infn.it Based on Phys. Rev. C 90, 044003 – Published 21 October 2014
Pairing time-reversal states J π = 0 + (a) +m -m k F − δ k F − δ K k F + δ L.N. Cooper considered the scattering of two particles which have an attractive k F + δ interaction in the presence of a Fermi sea (restricting the possible momenta where the k z (b) particles can scatter). He ~ k ↑ showed that even for arbitrarily ω small interactions, pairs will be formed in the system. Instability of the Fermi k y Filled Fermi sea sea: pair formation happens for any non-zero, attractive interaction. − ~ k x k ↓ L. Cooper, Phys. Rev. 104 (1956) 1189 D. Brink and R. Broglia, Nuclear Supe fm uidity , Cambridge Press
BCS gap equation • Quasi-particle energy h k | V | k 0 i ∆ ( k 0 ) X ∆ ( k ) = � E ( k ) 2 = ✏ ( k ) 2 + | ∆ ( k ) | 2 2 E ( k 0 ) • NN potential (partial waves expansion) k 0 X (2 L + 1) P L ( ˆ k · ˆ h k | V | k 0 i = 4 π k 0 ) V L ( k, k 0 ) we used the method suggested by Khodel L where the original gap equation is replaced by a coupled set of equations for the • Partial waves expansion of the gap function dimensionless gap function χ (p) de fj ned by r 4 π Δ (p) = Δ F χ (p) and a non-linear algebraic X 2 l + 1 Y LM ( ˆ ∆ ( k ) = k ) ∆ LM ( k ) equation for the gap magnitude Δ F = Δ (p F ) at L,M the Fermi surface. • Angle-average approximation V.V. Khodel, V.A. Khodel, J.W. Clark, NPA 679 (2000) 827 | ∆ ( k ) | 2 → D ( k ) 2 ≡ 1 1 Z k | ∆ ( k ) | 2 = X 2 L + 1 | ∆ LM ( k ) | 2 d ˆ 4 π L,M the di ff erent M-components become uncoupled and all equal. • Gap equation Z 1 ∆ L ( k ) = − 1 V L ( k, k 0 ) k 0 2 dk 0 ∆ L ( k 0 ) ✏ ( k 0 ) 2 + [ P p L 0 ∆ L 0 ( k 0 ) 2 ] ⇡ 0 No dependence on the quantum number M, but L components are coupled by tensor terms
Khodel’s method (I) vanishes at the The potential is written in the following way Fermi surface V ll 0 ( k, k 0 ) = v ll 0 φ ll 0 ( k ) φ ll 0 ( k 0 ) + W ll 0 ( k, k 0 ) , φ ll 0 ( k ) = V ll 0 ( k, k F ) /V ll 0 ( k F , k F ) v ll 0 = V ll 0 ( k F , k F ) The gap equation becomes d τ 0 W ll 0 ( k, k 0 ) ∆ l 0 ( k 0 ) Z X X ( − 1) Λ ∆ l ( k ) − E ( k 0 ) = D ll 0 φ ll 0 ( k ) l 0 l 0 d τ = k 2 dk/ π d τ φ ll 0 ( k ) ∆ l 0 ( k ) Z D ll 0 = ( − 1) Λ v ll 0 E ( k ) V.V. Khodel, V.A. Khodel, J.W. Clark, NPA 679 (2000) 827
Khodel’s method (II) X D l 1 l 2 χ l 1 l 2 ∆ l ( k ) = ( k ) scale factor l (calculations must be l 1 l 2 δ -independent) d τ 0 W ll 0 ( k, k 0 ) χ l 1 l 2 ( k 0 ) Z X χ l 1 l 2 ( − 1) Λ l 0 ( k ) − = δ ll 1 φ l 1 l 2 ( k ) l E ( k 0 ) l 0 χ l 1 l 2 ( k 0 ) Z d τ 0 W ll 0 ( k, k 0 ) X χ l 1 l 2 ( − 1) Λ l 0 ( k ) − ξ 2 ( k 0 ) + δ 2 = δ ll 1 φ l 1 l 2 ( k ) l p l 0 D d τ ∆ l 0 Z D ll 0 = ( − 1) Λ v ll 0 E φ ll 0 d τ 0 W ll 0 χ l 1 l 2 Z χ l 1 l 2 X ( − 1) Λ l 0 = δ ll 1 φ l 1 l 2 − l E l 0 χ Δ Self-consistent scheme X D l 1 l 2 χ l 1 l 2 ∆ l = l l 1 l 2
Microscopic forces The nuclear force at large distances is governed by the exchange of one or multiple pions. LO ( Q/ Λ χ ) 0 NLO ( Q/ Λ χ ) 2 … l a Short-range i t r n e t o P Medium-range Long-range
Microscopic forces LO The nuclear force at large distances is governed ( Q/ Λ χ ) 0 by the exchange of one or multiple pions. NLO LO ( Q/ Λ χ ) 2 ( Q/ Λ χ ) 0 … ��������������������������������������� NLO ����������������� The short-range part ( Q/ Λ χ ) 2 of the nuclear force is ������������������������������������� driven by physics not … l ������������������������������������� a resolved explicitly in Short-range i t reactions with typical r n e nucleon momenta of t o the order of M π c. It can P Medium-range be mimicked by zero- ������������ ���������������� range contact Long-range interactions with an increasing number of derivatives. � -symm. constrained parametrized � ������������������������������������������ ���������� ������� ��� �������� ����
Microscopic forces ( ) contact interactions LO + ... and 1 π -exchange … ( ) including + NLO + ... 2 π -exchange … + N2LO Short-range + ... ( ) including + N3LO + ... 3 π -exchange
NN potential (2-body) R. Machleidt, D.R. Entem / Physics Reports 503 (2011) 1–75 Phase shifts of np scattering as calculated from NN potentials at di ff erent orders of ChPT LO N2LO NLO N3LO (black dots are experimental data) see also Epelbaum, Hammer and Meissner, RMP 81 (2009) 1773
Microscopic forces V = V 2 B + V 3 B ' V 2 B + V eff 2 B ( ρ ) R. Machleidt, D.R. Entem / Physics Reports 503 (2011) 1–75 3 body ➜ 2 body density dependent Holt et al., [PRC 81 (2010) 024002] In-medium NN interaction generated • by the two-pion 2 3 exchange component ( c 1 , c 3 , c 4 ) 4 5 6 • and by the one-pion in-medium nucleon propagator exchange ( c D ) and short- range component ( c E )
How to build a chiral potential Machleidt (N3LO) Epelbaum (N3LO) Lippmann-Schwinger cuto ff V ( k , k 0 ) ! V ( k , k 0 ) f Λ ( k, k 0 ) f Λ = exp − ( k 0 / Λ ) 2 n − ( k/ Λ ) 2 n � � with n = 2 , 3 cuto ff the short-range part of the 2PE contribution dimensional spectral regularization function { Λ , ˜ Λ } { 450 , 500 } , { 450 , 700 } , = Λ = 450 , 500 , 600 { 550 , 600 } , { 600 , 600 } , { 600 , 700 }
How to renormalize NN forces k ′ k ′ V srg V lowk k k Features One can use RG transformations to evolve to lower Λ 1. Decoupling while preserving the truncation error of the original 2. EFT Hamiltonian ➩ eliminate coupling between high- and low-momenta components (k F ≤ 2 fm − 1 ) 3. Universality Renormalization Group could help for 4. Perturbativeness 1. Strong short-range repulsion ( hard core ) 5. Many-body 2. Iterated tensor ( S 12 ) interaction 6. Cutoff-dependence 3. Near zero-energy bound states
How to renormalize NN forces k ′ k ′ V srg V lowk k k d d Λ T Λ ( p, q ; q 2 ) = 0 The SRG is based on a continuous sequence of unitary transformations that suppress o ff - diagonal matrix elements , driving the Hamiltonian towards a band-diagonal form . Bogner, Kuo and Schwenk, PR 386 (2003) 1 Bogner, et al. , PPNP 65 (2010) 94-147
Effective mass Z 1 V ( k, k 0 ) ∆ ( k 0 ) ∆ ( k ) = − 1 dk 0 k 0 2 ( ✏ ( k ) − ✏ ( k F )) 2 + ∆ ( k ) 2 p ⇡ 0 E ff ective mass E ff ective nucleon mass M*/M ✏ ( k ) − ✏ ( k F ) = k 2 − k 2 F ✏ ( k ) − ✏ ( k F ) = k 2 − k 2 2 M N F 2 M ∗ 1 N effective nucleon mass 0.9 Substantial agreement with * ( ρ ) / M non relativistic Skyrme 0.8 phenomenology at saturation M density (0.7 < M*/M < 1). 0.7 0.6 0 0.05 0.1 0.15 0.2 Holt, Kaiser and Weise, EPJ A 47 (2011) 128 -3 ] ρ [fm
nuclear 1 S 0 matter only 2B neutron 1 S 0 matter FINITE TEMPERATURE • fj rst calculations at fj nite T, still in progress • relevant for astrophysics and neutron stars S. Maurizio, J. W. Holt and Paolo Finelli, Phys. Rev. C 90 (2014) 044003
nuclear neutron 3 SD 1 3 PF 2 matter matter SRG evolutions S. Maurizio, J. W. Holt and Paolo Finelli, Phys. Rev. C 90 (2014) 044003
� � � � � � � � � � Extension to N4LO Phase shifts and mixing angles � � � � � � Epelbaum, HK, Meißner, arXiv: 1412.4623 0 0 20 20 60 60 3 D 1 3 D 1 1 D 2 1 D 2 1 S 0 1 S 0 3 P 0 3 P 0 20 20 15 15 -10 -10 � [ deg ] � [ deg ] 40 40 10 10 10 10 20 20 -20 -20 0 0 5 5 � � � 0 0 -30 -30 -10 -10 � � -20 -20 0 0 30 30 10 10 0 0 30 30 3 D 2 3 D 2 3 P 2 3 P 2 1 P 1 1 P 1 3 P 1 3 P 1 0 0 � [ deg ] � [ deg ] 20 20 -10 -10 -10 -10 20 20 -20 -20 -20 -20 10 10 10 10 � � � -30 -30 -30 -30 0 0 0 0 180 180 10 10 � � 3 S 1 3 S 1 � 1 � 1 � 2 � 2 3 D 3 3 D 3 0 0 10 10 � [ deg ] � [ deg ] 120 120 5 5 -1 -1 5 5 � � -2 -2 0 0 0 0 60 60 � � � � -3 -3 -5 -5 -5 -5 0 0 -4 -4 -10 -10 -4 -4 -10 -10 0 0 100 100 200 200 300 300 0 0 100 100 200 200 300 300 0 0 100 100 200 200 300 300 0 0 100 100 200 200 300 300 E lab [MeV] E lab [MeV] E lab [MeV] E lab [MeV] E lab [MeV] E lab [MeV] E lab [MeV] E lab [MeV] R = 0.9 fm NLO N 2 LO N 3 LO N 4 LO � New renormalisation technique in the coordinate space ⇣ r ◆� 6 − r 2 ✓ ⌘ f = 1 − exp with the cuto ff R being chosen in the range of R = 0.8 . . . 1.2 fm . R R 2 For contact interactions, they use a non- local Gaussian regulator in momentum space with the cuto ff Λ = 2R -1 � � � � �
Extension to N4LO neutron nuclear 1 S 0 3 SD 1 matter matter momentum cuto ff momentum cuto ff (450-550 MeV) (450-550 MeV) coordinate cuto ff (0.8-1.2 fm) momentum cuto ff 3 PF 2 (450-550 MeV) neutron matter Paolo Finelli, unpublished, Ramanan et al. 1606.09053v2, Drischler et al. 1610.05213
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