Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 53

Renormalizing random-phase approximation by using exact pairing†L. Tan Phuc,∗1 N. Quang Hung,∗1 and N. Dinh Dang∗2
1.2

The random-phase approximation (RPA) is a popular theoretical method to study the low-lying excitations and high-lying giant resonances in nuclei. The
excited states of RPA are built on the particle-hole(ph) components, which are represented by the ph pairs†. Because of the unknown
operators: Bph and Bphof the RPA ground state |RP A⟩, the quasi-boson approximation (QBA), which assumes a ph pair as a boson and |RP A⟩ ∼ |HF ⟩, is used to derive the RPA
expectation value, namely ⟨RP A|[Bph , Bp†′ h′ ]|RP A⟩ ∼†∗1
∗2Condensed from the article in Phys. Rev. C 99, 064322 (2019)
Duy Tan University
RIKEN Nishina Center- 68 -B IV (E1)(e 2fm2)

RPA
HFEPRPAHFEPRPA-SC

0.8
0.6
0.4
0.2051015E (MeV)

201015E (MeV)

201015E (MeV)

20660b) Ni
B IV (E1)(e 2fm2)

54321058c)

790Zr

6B IV (E1)(e 2fm2)⟨HF |[Bph , Bp†′ h′ ]|HF ⟩ = δpp′ δhh′ .1) Using QBA within
RPA leads to a basic drawback that is the violation ofPauli principle because the nucleons are fermions insteadof boson as the assumption of QBA.
To overcome this problem, the ground-state correlations (GSC) factor Dph are taken into account to renormalize the residual interaction of
RPA (RRPA). The RPA expectation value becomes⟨RP A|[Bph , Bp†′ h′ ]|RP A⟩ ≃ Dph ≡ fh − fp , where fpand fh are p and h occupation numbers.2) Although theRRPA restored the Pauli principle, the energy-weighted
sum rule (EWSR) was violated.3) Therefore, anothertechnique was developed to ﬁx this drawback, namely
the pp and hh correlations were added in the phRRPA
matrix to fulﬁll the EWSR.3) However, this way doubled the matrix size of RPA equations and lead to timeconsuming. On the other hand, the RRPA does notinclude superﬂuid pairing. Therefore, we propose amethod to overcome all these shortcomings. In particular, by using the Hartree-Fock mean ﬁeld plus exact
pairing solution (HFEP),4) we can supply a good setof initial inputs for RPA. The GSC factor Dph , which
is used to renormalize the residual interactions, is calculated directly from exact pairing (EP). These are expected to preserve both Pauli principle and EWSR without adding any pp and hh correlations that make timeconsuming.
The proposed approach is applied for the dipole
case in several light-, medium-, and heavy-mass
Thenuclei, namely 22, 24, 28 O, 60 Ni, and 90 Zr.
calculations are performed within the conventional
RPA, HFEP+RPA (HFEPRPA) and self-consistent
HFEPRPA (SC-HFEPRPA). The E1 EWSR will becompared with the model independent sum rule suchas the Thomas-Reich-Kuhn (TRK) sum rule. Our approach shows that the drawback of the phRRPA is removed, namely the EWSR are fulﬁlled without adding
any pp and hh conﬁgurations, hence the extensionof RPA matrices and time-consuming calculations areavoided. As compared to the RPA results, the eﬀects22a) O
15432105Fig. 1. The Isovector dipole transition probabilities of 22 O(a), 60 Ni (b) and 90 Zr (c) obtained within diﬀerent approaches. The sticks and lines represent to the probabilities B(E1, 0 → 1− ) and strength functions S(E).

of GSC and EP in the renormalization are signiﬁcant inlight nuclei and small in medium and heavy nuclei (seethe shift of S(E) in Fig. 1(a)–(c)). The antipairing eﬀect
is observed for the ﬁrst time within the SC-HFEPRPA,
which reduces the pairing energy from more than 10%up to around 30% in the neutron-rich 22 O nucleus. Thisshows the contribution from the mutual eﬀect of theshort-range pairing and long-range ph residual interaction to the mean ﬁeld. The PDR is also found to beenhanced in neutron-rich nuclei because of the pairingeﬀect.
References

1) P. Ring, P. Schuck, The Nuclear Many-Problem (Springer,2004).
2) F. Catara, N. D. Dang, M. Sambataro, Nucl. Phys. A
579, 1 (1994).
3) N. Q. Hung, N. D. Dang, T. V. N. Hao, L. T. Phuc, Phys.
Rev. C 94, 064312 (2016).
4) L. T. Phuc, N. Q. Hung, N. D. Dang, Phys. Rev. C 97,024331 (2018).