Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 53

Double charge-exchange phonon states†
X. Roca-Maza,∗1,∗2 H. Sagawa,∗3,∗4 and G. Colò∗1,∗2The possibility of inducing double charge-exchange
excitations by heavy-ion beams at intermediate energies has recently fostered interest in new collective excitations such as double isobaric analog states (DIAS)
and double Gamow-Teller giant resonances (DGT). We
present some formulas to evaluate diﬀerent combinationsof the average excitation energies of DIAS and DGTRby using commutator relations.
The expectation value for the energy of the DIAS
(DGT) is deﬁned as
EDIAS(DGT) ≡ ⟨DIAS(DGT)|H|DIAS(DGT)⟩− ⟨0|H|0⟩.

(1)

Here |0⟩ represents the ground state, and|DIAS(DGT)⟩ ≡O− |IAS(GT)⟩⟨IAS(GT)|O+ O− |IAS(GT)⟩1/2(2)is the deﬁnition of the DIAS (DGT) state in terms ofthe IAS (GT), which can be written as
O− |0⟩|IAS(GT)⟩ ≡.
⟨0|O+ O− |0⟩1/2(3)∑A
The operators O± = T± ≡i t± (i) are the isospin
raising and lowering operators∑
for IAS and DIAS. ForAGT and DGT, O± = G± ≡i σz (i)t± (i). Startingfrom Eq. (1), one may write the excitation energy of theDIAS (DGT) asEDIAS(DGT) =22, [H, O−]]|0⟩⟨0|[O+,22⟨0|O+ O− |0⟩(4)

assuming that the ground state has good isospin, thatis, there is no isospin mixing and T+ |0⟩ = 0. Within thesame approximation, EIAS(GT) reads
EIAS(GT) = ⟨IAS(GT)|H|IAS(GT)⟩ − ⟨0|H|0⟩=⟨0|[O+ , [H, O− ]]|0⟩.
⟨0|O+ O− |0⟩(5)

EDIAS(DGT) = 2EIAS(GT)⟨0|[O+ , [O+ , [[H, O− ], O− ]]]|0⟩.
2(N − Z)(N − Z − 1)

(6)

The denominator of Eq. (4) is evaluated to be
2 2⟨0|O+O− |0⟩ = 2(N − Z)(N − Z − 1). The second term
†
∗1
∗2
∗3
∗4

H = H0 + V + VC + VISB ,(7)where V is the spin- and isospin-dependent interaction,
VC is the Coulomb interaction, and VISB is an ISB effective interaction originating from the nuclear strongforce. From Eq. (6), we can derive the relation betweenthe DGTR and DIAS energies as,
EDGTR − EDIAS =⟨0|[G2+ , [V, G2− ]]|0⟩,2(N − Z)(N − Z − 1)(8)since [G2+ , [VC + VISB , G2− ]] = [T+2 , [VC + VISB , T−2 ]] . In
order to evaluate the energy diﬀerence between EDGTRand EDIAS , we adopt the separable interaction1)
V =A
∑i+A

κls l(i) · s(i) +1 κστ
2 AA∑i̸=j1 κτ ∑τ (i) · τ (j)2 Ai̸=j(σ(i) · σ(j))(τ (i) · τ (j)),(9)

where κls is the one-body spin-orbit coupling strength,
while κτ and κστ are the coupling strengths of the residual two-body interactions in the isospin and spin-isospin
channels, respectively. The average energy of the GTR
minus that of the IAS is expressed as⟨0|[G+ , [V, G− ]]|0⟩
EGT − EIAS =(N − Z)
� ⟩⟨ � A��
4 κlsN −Z�∑�=−0�.
l(i) · s(i)� 0 +2(κστ −κτ )��
3 N −ZAi

One can write the energies for DIAS and DGT as+on the right-hand side in Eq. (6) could be diﬀerent fromzero only for the isospin symmetry breaking (ISB) termsin H because they contribute to EIAS [cf. Eq. (5)]. In
other words, the IAS and DIAS energies are a special ﬁlter for the terms in the Hamiltonian that break isospin
symmetry (Coulomb and the small contributions fromthe strong force), while the isospin-conserving part of H
does not contribute.
In order to evaluate the quartic and double commutators, we take the Hamiltonian

Condensed from the article in Phys. Rev. C 101, 014320(2020)
Dipartimento di Fisica “Aldo Pontremoli,” Università degliStudi di Milano
INFN, Sezione di Milano
RIKEN Nishina Center
Department of Physics, Omsk University- 74 -The energy diﬀerence between DGTR and DIAS (8) isexpressed in a similar manner as)(N −Z(EGT −EIAS )EDGTR −EDIAS − 1+N −Z −1⟩⟨ ∑A4 κls 0| i l(i) · s(i)|01 N −Z−6(κστ −κτ ).
=3 (N −Z)(N −Z −1)
A N −Z −1
Quantitative discussions of the above formulas are reported elsewhere.
Reference1) A. Bohr, B. R. Mottelson, Nuclear Structure Vol.I (WorldScientific, 1975).