Ⅱ-2. Nuclear Physics (Theory)
RIKEN
Accel. Prog. Rep. 56 (2023)

RIKEN Accel. Prog. Rep. 56 (2023)

Does the second-order operator in the adiabatic expansion contributeto the collective mass?
K. Sato∗1,∗2
The adiabatic self-consistent collective coordinate(ASCC) method1) is a practical method for describing
large-amplitude collective motion in atomic nuclei withsuperﬂuidity and an advanced version of the adiabatictime-dependent Hartree-Fock-Bogoliubov (HFB) theory. According to the generalized Thouless theorem,
the state vector in the ASCC theory can be written inthe form

We consider the case where the pairing correlation
is not taken into account, and the Hamiltonian does
not include the three-body interaction. Then, we shallshow that the second-order collective operator Q̂(2)does not directly contribute to the collective mass if
the second-order canonical-variable condition⟨ϕ(q)|[Q̂(1) , Q̂(2) ]|ϕ(q)⟩ = 0

is imposed.2,3) Noting that Q̂(2) is written in terms ofA-terms only and that the Hamiltonian can be written
in terms of A-terms, B-terms, and normally ordered
quartic terms, one can easily see that|ϕ(q, p)⟩ = eiĜ(q,p) |ϕ(q)⟩,where Ĝ(q, p) is a linear combination of the a† a† and aaterms. We shall call the a† a† and aa terms as A-terms,
and the a† a and aa† terms as B-terms. Recently, theASCC theory including a second-order collective operator has been proposed.2) There, Ĝ(q, p) is expanded
as1Ĝ(q, p) = pQ̂(1) (q) + p2 Q̂(2) (q).
2

⟨ϕ(q)|[Ĥ, iQ̂(2) ]|ϕ(q)⟩ = ⟨ϕ(q)|[ĤA , iQ̂(2) ]|ϕ(q)⟩. (4)Here, ĤA denotes the A-part of Ĥ.
From the moving-frame Hartree-Fock equation, weobtain(1)

ĤA = ∂q V Q̂(1) .
⟨ϕ(q)|[Ĥ, iQ̂(2) ]|ϕ(q)⟩= i∂q V ⟨ϕ(q)|[Q̂(1) , Q̂(2) ]|ϕ(q)⟩ = 0∗2(6)if the second-order canonical-variable condition inEq. (3) is met. Then, the inertial mass B(q) isB(q) = −⟨ϕ(q)|[[Ĥ, Q̂(1) ], Q̂(1) ]|ϕ(q)⟩.

(7)

The second-order collective operator Q̂(2) does not contribute to the inertial mass directly, but it can contribute only through the equations of motion. Q̂(2)contributes to the ﬁrst- and second-order movingframe equations of motion, which may aﬀect the statevector |ϕ(q)⟩.
Here, we have concentrated on the case without thepairing correlation. When the pairing correlation istaken into account, the Q̂(2) term in the inertial massin Eq. (2) does not vanish even if the condition inEq. (3) is imposed. Before ending this report, we add
one remark. It is not trivial whether one should/caninclude the second-order canonical-variable condition
in Eq. (3) in the set of fundamental equations of theASCC theory. If it is imposed in addition to the ﬁrstorder canonical-variable conditions, there may be too
many conditions to determine the unknown quantities,
and the problem may be overdetermined. This point
will be investigated and reported in a future publication.

(2)

It is worth mentioning that the second term on theright-hand side gives a contribution of the same orderas the ﬁrst term.
The fundamental equations in the ASCC theory consist of the moving-frame HFB and quasiparticle random phase approximation (QRPA) equations and thecanonical-variable conditions, which are derived fromthe invariance principle of the Schrödinger equationand the canonicity conditions, respectively. The former is the equation of motion, and the latter the conditions for the collective variables to be canonical. In theconventional ASCC theory, only the canonical-variableconditions of O(1) and O(p) have been taken into account, while the equations of motion up to O(p2 ) aresolved to determine the state vector and collective operators, from which the collective Hamiltonian is calculated. In Ref. 3), the ASCC theory including thesecond-order collective operator Q̂(2) was successfully
applied to the Lipkin model, and it was shown thatthe inclusion of Q̂(2) improves the agreement with theexact solution.
∗1(5)

By substituting Eq. (5) into Eq. (4), we obtainIn the conventional ASCC theory, only the ﬁrst-ordercollective operator Q̂(1) is included. However, as shown
in Refs. 2) and 3), the second-order collective operator
Q̂(2) is involved in the moving-frame equations of motion. Moreover, Q̂(2) contributes to the collective massas
B(q) = −⟨ϕ(q)|[[Ĥ, Q̂(1) ], Q̂(1) ]|ϕ(q)⟩+⟨ϕ(q)|[Ĥ, iQ̂(2) ]|ϕ(q)⟩.

(3)

References
1) M. Matsuo et al., Prog. Theor. Phys. 103, 959 (2000).
2) K. Sato, Prog. Theor. Exp. Phys. 2018, 103D01 (2018).
3) K. Sato, Prog. Theor. Exp. Phys. 2017, 033D01 (2017).

Faculty of Education, Kochi University
RIKEN Nishina Center- 41 -