Adiabatic approach to large-amplitude collective motion with the higher-order collective-coordinate operator

We propose a new set of equations to determine the collective Hamiltonian including the second-order collective-coordinate operator on the basis of the adiabatic self-consistent collective-coordinate (ASCC) theory. We illustrate, with the two-level Lipkin model, that the collective operators including the second-order one are self-consistently determined. We compare the results of the calculations with and without the second-order operator and show that, without the second-order operator, the agreement with the exact solution becomes worse as the excitation energy increases, but that, with the second-order operator included, the exact solution is well reproduced even for highly excited states. We also reconsider which equations one should adopt as the basic equations in the case where only the first-order operator is taken into account, and suggest an alternative set of fundamental equations instead of the conventional ASCC equations. Moreover, we briefly discuss the gauge symmetry of the new basic equations we propose in this paper.

We propose a new set of equations to determine the collective Hamiltonian including the second-order collective-coordinate operator on the basis of the adiabatic self-consistent collective-coordinate (ASCC) theory. We illustrate, with the two-level Lipkin model, that the collective operators including the second-order one are self-consistently determined. We compare the results of the calculations with and without the second-order operator and show that, without the second-order operator, the agreement with the exact solution becomes worse as the excitation energy increases, but that, with the second-order operator included, the exact solution is well reproduced even for highly excited states. We also reconsider which equations one should adopt as the basic equations in the case where only the first-order operator is taken into account, and suggest an alternative set of fundamental equations instead of the conventional ASCC equations. Moreover, we briefly discuss the gauge symmetry of the new basic equations we propose in this paper.

Introduction
In recent papers, we elucidated the relation among the higher-order collective operators, the a † a terms, and the gauge symmetry and its breaking in the adiabatic self-consistent collective-coordinate (ASCC) theory [1][2][3]. The ASCC method [4] is a practical method of describing the large-amplitude collective motion, which is an adiabatic approximation to the SCC method [5] and can be regarded as an advanced version of the adiabatic time-dependent Hartree-Fock(-Bogoliubov)[ATDHF(B)] theory. Although several versions of the ATDHF theory had been proposed so far, they encountered difficulties such as non-uniqueness of the solution (See Refs. [6][7][8] for a review). As for the ATDHFB theory, which is an extension of the ATDHF theory including the pairing correlation, Dobaczewski and Skalski attempted to develop an ATDHFB theory assuming the axial quadrupole deformation parameter β as a collective coordinate [9]. Recently, Li et al also attempted to construct the five-dimensional collective Hamiltonian on the basis of the ATDHFB theory [10]. However, the extension of the ATDHF theory to the ATDHFB theory is not straightforward, because one needs to decouple the pair-rotational mode from the collective mode of interest.
In the ASCC method, Matsuo et al [4] first assumed the gauge angle dependence of the state vector in the form |φ(q, p, ϕ, n) = e −iϕN e iĜ(q,p,n) |φ(q) . (1.1) Here, (q, p) are collective coordinate and conjugate momentum. n = N − N 0 is the particle number measured from a reference value N 0 , and ϕ is the gauge angle conjugate to n. With this form of the state vector, the collective Hamiltonian is independent of the gauge angle ϕ, and it is guaranteed that the expectation value of the particle number is conserved.
Furthermore, there appears no gauge-angle degree of freedom in the equations of motion. In Ref. [4], Matsuo et al considered the expansion ofĜ(q, p, n) up to the first orderĜ(q, p, n) = pQ(q) + nΘ(q) and derived the moving-frame HFB (Hartree-Fock-Bogoliubov) & QRPA (quasiparticle random phase approximation) equations, which are equations of motion in the ASCC theory.
On the basis of Ref. [4], Hinohara et al performed a numerical calculation and encountered a difficulty of finding the solution due to the numerical instability [11]. They found that this instability was caused by the symmetry under the following transformation under which the basic equations are invariant if the collective-coordinate and particle number operators commute, i. e. [Q,N ] = 0.Q As this transformation changes the phase of the state vector, they called it the "gauge" transformation and proposed a gauge-fixing prescription to remove the redundancy associated with the gauge symmetry. Their prescription for the gauge fixing is as follows. While the moving-frame HFB & QRPA equations are invariant under the above transformation at the HF equilibrium point ∂ q V = 0, it is not the case at non-equilibrium points unless [Q,N ] = 0. Therefore, they first require the commutativity of the collective-coordinate and particle-number operators [Q,N ] = 0 for the gauge symmetry of the moving-frame HFB & QRPA equations, and then fix the gauge. With this prescription, they succeeded in obtaining the solution and applied the one-dimensional (1D) ASCC method to the multi-O(4) model [11] and the oblate-prolate shape-coexistence phenomena in the proton-rich Se and Kr isotopes [12,13] (Here, we mean by the D−dimensional ASCC method that the dimension of the collective coordinate q is D.) After the successful application of the 1D ASCC method, an approximate version of the 2D ASCC method, the constrained HFB plus local QRPA method, was proposed and applied to large-amplitude quadrupole collective dynamics [14][15][16][17][18][19][20][21]. Moreover, the 1D ASCC method without the pairing correlation was also applied to nuclear reaction [22,23] (In these calculations, the so-called curvature term was neglected.) However, little progress had been made in understanding of the gauge symmetry, until very recently we analyzed the gauge symmetry and its breaking under more general gauge transformations in the ASCC theory, on the basis of the Dirac-Bergmann theory of the constrained systems [1,2]. There, it was shown that the gauge symmetry in the ASCC method is broken by the adiabatic approximation, and that the gauge symmetry is partially retained by containing the higher-order collective operators in the adiabatic expansion.
According to the generalized Thouless theorem (Refs. [5,[24][25][26][27]),Ĝ in Eq. (1.1) can be written in terms of only a † a † and aa terms. We shall call a † a † and aa terms A-terms and 2/30 operator which are written in terms of (multiple) commutators ofQ A andQ B can be taken into account in Approach B. In this paper, we employ Approach A.
In the conventional ASCC theory without the pairing correlation, only the first-order operators consisting of A-terms are taken into account, and the contributions from the higher-order operators to the equations of motion and the inertial mass are missing. However, it cannot be neglected from a simple order counting in the adiabatic expansion. In Refs. [22,23], Wen and Nakatsukasa successfully reproduced the inertial mass against the translational motion without including the higher-order collective operators. Actually, for the mode with ∂ q V = 0, the higher-order operators and B-terms do not contribute to the inertial mass. The reason for this is explained in Ref. [3] from the viewpoint of Approach B, but it also applies to Approach A straightforwardly. For general collective modes of interest, however, ∂ q V = 0 is not satisfied, and a theory with which one can correctly evaluate the contribution from the higher-order operators is necessary. In Ref. [23], Wen and Nakatsukasa failed to find the solution at a certain point on the collective path and pointed out a possibility to solve this problem by including the pairing correlation. Depending on the particle number, or on the collective coordinate even in one nucleus, the pairing gap changes and can vanish. Therefore, a theory is favorable with which one can treat the cases with and without the pairing correlation on an equal footing. In this paper, we first consider the case without the pairing correlation on the basis of Approach A including the second-order collectivecoordinate operatorQ (2) and propose a new set of fundamental equations to determine the collective operators. The set of fundamental equations for the case with the pairing correlation included is derived in a straightforward way.
The paper is organized as follows. Sect. 2 describes the formulation. In Sect. 2.1, we propose a set of the basic equations including the second-order collective-coordinate operatorQ (2) . In Sect. 2.3, we introduce the two-level Lipkin model, and it is shown in Sect. 2.3 that the basic equations proposed in Sect. 2.1 are reduced to one differential equation in the case of the two-level Lipkin model. We give the collective Schrödinger equation in Sect. 2.4, and the solution to the conventional ASCC equations withoutQ (2) is given in Sect. 2.5. The numerical results are shown in Sect. 3. We compare the calculations with and without Q (2) employing the Lipkin model. The numerical results show that, for low-energy states, both of the calculations reproduce the exact solution well, but that, with increasing the excitation energy, the agreement with the exact solution becomes worse when the secondorder collective operatorQ (2) is neglected. WithQ (2) , the exact solution is well reproduced even for higher excited states. In Sect. 4, we consider which equations should be adopted as the basic equations of motion when only the first-order operatorQ (1) is included, taking the Lipkin model as a simple example. We propose an alternative set of the basic equations including an equation introduced in Sect. 2 instead of the conventional moving-frame RPA equation of O(p 2 ). In Sect. 5, we briefly discuss the gauge symmetry of the basic equations derived in Sect. 2 with the pairing correlation included. Concluding remarks are given in Sect. 6. In Appendix A, the expressions of the derivatives of the collective operators are given in terms of quasispin operators in the Lipkin model. In Appendix B, the potential curvature C in the q space with B(q) = 1 is given in the case where only the first-order operator is taken into account. 4/30 2. Formulation 2.1. Basic equations with the second-order operatorQ (2) First, we consider the equations of motion without pairing correlation. In preceding papers [2,3], we considered the ASCC theory with the second-and third-order collective-coordinate operators. Here we adopt the approach with the higher-order operators consisting of only A-terms, i. e., Approach A. The state vector in the ASCC theory is given by |φ(q, p) = e iĜ(q,p) |φ(q) . (2.1) Here,Ĝ is expanded asĜ 3) The equations of motion in the ASCC theory is derived from the invariance principle of the Schrödinger equation which can be rewritten into the equation of collective submanifold (CS): with (P ,Q) := (i∂ q , −i∂ p ). The collective Hamiltonian is given by One can see that the second-order operatorQ (2) contributes to the inertial mass, and that one cannot neglect the term involvingQ (2) in B(q) from a simple order counting, as both of the first and second terms of the right-hand side of Eq. (2.8) are O(p 2 ) terms. By substituting the state vector (2.1) into Eq. (2.5) and expanding in powers of p, the moving-frame HF & RPA equations, which are the equations of motion in the ASCC theory (without the pairing), are obtained as 5/30 Moving-frame HF equation Moving-frame RPA equations Here we expandĜ up toQ (2) and omitQ (i) (i ≥ 3).Q (3) contributes to the moving-frame RPA equation of O(p 2 ) (2.11) but does not contribute to the equations of motion of O(1) and O(p) and the inertial mass. When the expansion up toQ (2) is taken, there are one more unknown operators than in the conventional ASCC equations taking up toQ (1) . Thus, one more equation is necessary to determine the collective operators and the state vector self-consistently. The moving-frame equations are derived by expanding the equation of CS (2.5) in powers of p. The O(p 3 ) equation of CS is a possible candidate for such an equation to add to the three moving-frame equations. However, as easily confirmed, there appearŝ Q (4) as well asQ (3) in the O(p 3 ) equation of CS, so one needs to make an approximation neglectingQ (3) andQ (4) to close the set of the equations. Therefore, we adopt another equation, that is, the q-derivative of the moving-frame HF equation. As mentioned above, the moving-frame RPA equations of O(p 2 ) (2.11) is derived from the second-order equation of CS and the q-derivative of the moving-frame HF equation [2,4]. Here we consider the following equations as a set of the basic equations in the case whereQ (2) is included. (2) |φ(q) = 0, (2.14) (1) ,Q (2) ]|φ(q) = 0, (2.16) where the covariant derivative ofQ (1) is given by (1) andQ (2) .
In the above equations, there appears the covariant derivative ofQ (1) . In actual calculations, one can choose a coordinate system with B(q) ≡ 1 by the scale transformation of q.
Here we mean by the symbol ≡ that the inverse inertial mass B(q) is unity everywhere along the collective path. Then, the covariant derivative reduces to the ordinary partial derivative, and the equations may be solved by approximating ∂ qQ (1) (q) by finite difference. The simplest scheme of the finite difference is ∂ qQ where the moving-frame Hamiltonian is given bŷ withÑ is the particle-number operator measuring from N 0 ,Ñ =N − N 0 . In Sect. 5, we shall give a brief consideration on the gauge symmetry of these equations.

The Lipkin model
In this section, we introduce the Lipkin model with two N -fold degenerate levels [30]. We follow the formulation of Ref. [31] by Holzwarth. We label the states in the upper and lower level by p = 1, 2, ..., N and −p, respectively. In this model, the Hamiltonian is given by The quasispin operators satisfy We assume that the system contains N particles. If there is no interaction V = 0, the lower levels are fully occupied and the upper levels are completely empty in the ground state. We denote the ground state in the non-interacting case by |0 . Then, we have We define the particle/hole creation and annihilation operators a ±p as follows: Since a ±p |0 = 0, |0 is the vacuum with respect to a ±p . a ±p satisfies the canonical commutation relations of fermions. With a ±p , the quasispin operators are written aŝ Following Ref. [31], we introduce the "deformed" state as |a = exp(aK + )|0 , (2.32) 8/30 and the "deformed" operators as With the deformed operators, we shall define the new quasispin operators: which are connected with the undeformed quasispin operators by The inverse transformation is given by As shown in Ref. [31], |a is not normalized, and the normalized deformed state is given by As in Ref. [31], we set ψ = 0 and denote the quasispin operators asĴ i (=Î i | ψ=0 ). Then the Hamiltonian can be written aŝ We shall defineJ and rewrite the Hamiltonian as follows.
As easily confirmed, the last four terms in Eq. (2.42) are normally-ordered quartic terms of (α, α † ). Here, V (φ) corresponds to the collective potential, and is given by The reader should not confuse V (φ) with the interaction parameter V . The Hartree-Fock equations is given by the condition for V (φ) to take an extremum, which is equivalent to the condition for the third term in Eq. (2.42) to vanish. Examples of the collective potential for χ ≶ 1 are depicted in Figs. 1 and 2. In all the numerical calculations in this paper, we set the energy splitting between the two levels ǫ = 1. In other words, we measure energy in units of ǫ.

Moving-frame equations with the Lipkin model
We shall consider the basic equations (2.13)-(2.16) employing the Lipkin model. In general cases, the basic equations may be solved by approximating ∂ qQ (1) (q) with the finite difference. With this simple model, the basic equations reduce to one first-order differential equation of Q (1) (φ) as shown below. Following Ref. [31], we assume that any collective operator is written in terms of the quasispin operatorsĴ ± andĴ 0 . In Approach A, the collective operators are written in terms of only A-terms, which implies that they are written in terms ofĴ ± . Noting the Hermiticity and the time-reversal symmetry of the collective operators, one findsQ (1) Here, Q (1) (q), BP (q) and Q (2) (q) are real numbers. WhileQ (1) is time-even,P and Q (2) are time-odd. The canonical-variable condition [4] determines the normalization of 10/30 (Q (1) (q), P (q)) as follows.
As shown in Appendix A, the derivatives of (Q (1) ,P ) are written as In the ASCC method, as only the variation of A-type, a † a † |φ(q) , is taken, so here we take the variation of the formĴ + |φ(q) . We shall give the expression of the inertial mass B(q) first. It is given by Eq. (2.8), and using Eqs.(2.42), (2.45), and (2.46), we have Although we choose the coordinate system with B(q) ≡ 1, we keep B(q) explicitly in the expressions below. One can always choose such a coordinate system with B = 1 by the scale transformation of the collective coordinate q. Note that Q (1) and Q (2) are rank-1 and rank-2 contravariant tensors, respectively. B(q) is a rank-2 contravariant tensor, and P is a rank-1 covariant tensor (vector). Let us move on to the equations of motion. The moving-frame HF equation (2.13) reads By comparing this with Eq. (2.52), we find The last equality follows from the canonical-variable conditions (2.48). Let us consider the rest of the basic equations. First, we shall see the case where ∂ q V = 0. Eqs. (2.14) and (2.15) reduce to the RPA equations: The RPA equations can be written as Here, BC plays a role of the eigenfrequency squared ω 2 and is given by The RPA solution reads We have used the canonical-variable condition (2.48) for the normalization. Here φ in the above expression is a solution to the HF equation where ∂ q V = 0. One easily sees that φ = 0 and φ = π are always solutions to the HF equation (2.44), and that for χ > 1 there is another solution φ 0 := cos −1 (1/χ). For χ > 1, the RPA solution at φ = φ 0 is given by The solutions at φ = 0 and φ = π are where + and − correspond to φ = 0 and φ = π, respectively. Next, noting that [Q (1) ,Q (2) ] is a B-term, one finds that Eq. (2.16) is independent of ∂ q V . From Eq. (2.16), we obtain (1) . This equation gives the relation between ∂ q Q (1) and Q (2) . (At this point, the value of ∂ q Q (1) at the HF equilibrium point is unknown, so is Q (2) .) The partial derivative ∂ q Q (1) can be rewritten as using Eq. (2.54), so if we choose the coordinate system with B(q) ≡ 1 and can express Q (2) in terms of Q (1) , then Eq. (2.64) reduces to a differential equation of Q (1) with respect to φ. We shall move on to the case where ∂ q V = 0. Then, as long as Eq. (2.16) is adopted as one of the basic equations, the set of Eqs. (2.14) and (2.15) are equivalent to the moving-frame 12/30 RPA equations and leads to We shall express BC = ω 2 as a function of Q (1) Note that this is the indeterminate form of 0/0 when ∂ q V = 0. Eq .(2.69) can be also obtained from the expression of the inertial mass (2.51). By substituting Eq. (2.69) into Eq. (2.64), and setting B ≡ 1, we obtain the differential equation of Q (1) below.
Now all one has to do is to integrate the above differential equation as an initial-value problem. We choose φ = 0 as the initial point. From the symmetry of the system, we can assume that Q (1) (φ) is symmetric under the reflection about φ = 0 and differentiable at φ = 0, and then ∂ φ Q (1) (φ = 0) = 0. From the O(p 2 ) equation of CS (2.64) and ∂ φ Q (1) (φ = 0) = 0, it follows that Q (2) (φ) vanishes at φ = 0. The relation between q and φ can be also obtained from Eq. (2.54) and we find We have chosen the origin q = 0 such that φ = φ 0 = cos −1 (1/χ) at q = 0. Before ending this subsection, we shall give a remark. We have seen that the q-derivative which can be simplified with B ≡ 1 and the canonical-variable condition (2.48) as below.
The above equation (

Collective mass and Schrödinger equation
When we choose the coordinate system such that B(q) = 1, the collective Hamiltonian reads We shall rewrite the kinetic energy in terms of φ.
Here, the inertial mass M (φ) can be written as where we have used Eq. (2.54). We solve the collective Schrödinger equation in the φ space rather than the q space because it is more convenient for the comparison between the calculations with and withoutQ (2) . The classical collective Hamiltonian is written in terms of φ as The quantized Hamiltonian is given bŷ 14/30 2.5. Solution in the case withoutQ (2) In the ASCC theory withoutQ (2) , the equations of motion are given by the moving-frame HF equation (2.52) and the moving-frame RPA equations with Q (2) omitted, From Eq. (2.80) and the canonical-variable condition (2.48), we obtain which coincides with the ATDHF mass (41) and the GCM mass (72) in Ref. [31]. When χ < 1, B is positive for sufficiently small φ ∈ [0, π], but B becomes 0 at a certain point, beyond which (Q (1) ,P ) is no longer Hermitian and B < 0. We shall denote the point at which B = 0 by φ max ∈ (0, π). We solve the collective Schrödinger equation with the boundary condition for the collective wave functions to vanish outside the region (−φ max , φ max ), because the potential energy is sufficiently high there. When χ > 1, B is always positive. Then, we solve the collective Schrödinger equation with the periodic boundary condition. We employ these boundary conditions similarly in the case with Q (2) included. The eigenfrequency ω 2 = BC of the moving-frame RPA equations withoutQ (2) is obtained from Eq. (2.81) as BC = ǫ cos φ + ǫχ(1 + sin 2 φ) ǫ cos φ + ǫχ(sin 2 φ − cos 2 φ) and ∂ q V is also obtained analytically (2.85)

Numerical results
We compare the numerical results obtained by solving Eq. (2.70) with those obtained from the conventional ASCC equations withoutQ (2) . For the symmetry of the system, we solve Eq. (2.70) in the region [0, 180 • ] as an initial-value problem, starting from φ = 0.
As an example of the calculations for χ > 1, we show the calculated results for χ = 1.8, N = 10, ǫ = 1 in Fig. 3. We depict (a) Q (1) , (b) Q (2) , (c) ∂ q V , (d) ∂ q V Q (2) , (e) ω 2 and (f) C as functions of φ. When Q (2) is not included, Q (1) , shown in Fig. 3(a), is obtained analytically 15 16/30 as seen in the previous section. It coincides with the RPA solutions at the potential extrema φ = 0, φ 0 and π. One can see that, also withQ (2) , the calculated result coincides with the RPA solutions at the potential extrema. (In Fig. 4, the magnified figure around φ = φ 0 = 56.3 • is shown). It is also noteworthy that, for φ φ 0 , both of the calculations give similar results, but for φ φ 0 , the deviation between the two becomes larger. [We shall give a minor remark. At φ = 180 • , the differentiability of Q (1) seems to be broken if we assume B(φ) is periodic. However, this is not a serious problem because the potential energy is sufficiently high and the collective wave function almost vanishes there. We also integrated Eq. (2.70) from φ = 180 • with the boundary condition ∂ φ Q (1) = 0 as in the case from φ = 0. The obtained solution coincides with the RPA solutions at the potential extrema, but in turn the differentiability at φ = 0 is broken. Therefore, we adopt here the solution obtained by the integration from φ = 0. ] In Fig. 3(b), Q (2) is plotted. It vanishes at φ = 0 and rapidly decreases as φ approaches to 180 • . However, it does not diverge there but converges to a finite value. Q (2) is involved in the equations of motion in the form of ∂ q V Q (2) , so we plot ∂ q V Q (2) in Fig. 3(d). As ∂ q V = 0 at the potential extrema φ = 0 • , φ 0 , and 180 • and Q (2) is always finite, their product vanishes at φ = 0, φ 0 , and 180 • . Thus, the equations of motion reduce to the HF & RPA equations there. [Note that, φ = 0, 180 • are (local) potential maxima, although the RPA equations are usually solved at the potential minimum.] Figs. 3(e) and 3(f) display the eigenfrequency squared ω 2 and the potential curvature C as functions of φ, respectively. In the calculation withQ (2) included, ω 2 , which is calculated through Eq. (2.68), coincides with the product of the curvature C and the inverse inertial mass B, i. e., ω 2 = BC. However, it is not the case whenQ (2) is ignored. (It is seen that the position of the zero of ω 2 around 120 • is different from that of C.) This will be investigated in the next section.
The inverse inertial mass B(φ) calculated with Q (2) is shown in Fig. 5, in comparison with that calculated without Q (2) . In Fig. 6, the ratio of the inverse inertial mass calculated withoutQ (2) to that withQ (2) , is plotted. Reflecting the difference in Q (1) , while for φ φ 0 , the difference between the two is not so large, it becomes more significant for φ φ 0 . There, the inverse inertial mass witĥ Q (2) is larger than that withoutQ (2) , and depending on φ or equivalently on the collective coordinate q, the difference can be by a factor of 3 approximately. The difference of the inertial masses becomes more important as the excitation energy increases, because the component of the collective wave function increases in the region with φ φ 0 , where the potential energy is high and the mass difference becomes larger. Table 1 displays the comparison of the excitation energies calculated includingQ (2) with those calculated withoutQ (2) and the exact solution for N = 10, χ = 1.8. For a first few excited states, the difference between the two calculated results is small (by several percents), and both of the calculations are in good agreement with the exact solution, although the excitation energy of the first excited state is somewhat overestimated. With increasing the excitation energy, the difference between the two calculations becomes larger and amounts to 10 − 20%. WithoutQ (2) , the deviation from the exact solution become larger with increasing 17 Fig. 6: Ratio of the inverse inertial mass B(φ) calculated without Q (2) to that with Q (2) for N = 10, χ = 1.8, ǫ = 1.0. the excitation energy, while, withQ (2) , the deviation stays relatively small. The fourth and fifth columns in Table 1 show the deviations of the calculated excitation energies from the exact solution. Although the calculation withoutQ (2) gives a better agreement with the exact solution for the first excited state, for all the other states, the calculation withQ (2) included gives a better agreement. Fig. 7 shows the excitation energies of the first three excited states as functions of χ with N = 10. Although the calculation withQ (2) does not always give a better agreement with the exact solution than that withoutQ (2) , as a whole, the agreement with the exact solution is rather good for both of the calculations with and withoutQ (2) . Fig. 8 displays the results for the next three, the fourth, fifth and sixth excited states. With-outQ (2) , as excitation energy increases, the calculated excitation energies start to deviate from the exact solution, and the excitation energies are systematically underestimated. On the other hand, the excitation energies calculated withQ (2) are still in good agreement with the exact solution. These results suggest that the role of the inertial mass becomes more important as the excitation energy increases. 18 Table 1: Excitation energies calculated with Q (2) for ǫ = 1, χ = 1.8, N = 10 in comparison with those calculated without Q (2) and the exact solution. The first, second, and third columns show the excitation energies of the exact solution, the calculation withoutQ (2) , and withQ (2) , respectively. The fourth and fifth columns show the deviation from the exact solution for the excitation energies calculated without and with Q (2) , respectively. All the energies are in units of ǫ(= 1). The rightmost column shows the percentage of the excitationenergy increase byQ (2) : We performed the calculation with a larger particle number N = 40 similarly, and the results are shown in Figs. 9 and 10. As seen in the previous section, withoutQ (2) , B(φ) is obtained analytically and is proportional to N −1 , whereas the collective potential energy is proportional to N . If the N dependence of the inertial mass in the case withQ (2) is similar to that withoutQ (2) , with increasing N , the potential energy becomes larger relative to the kinetic energy, and the difference of the inertial mass between the two calculations becomes less important. Actually, for N = 40, the two calculations give similar results and both are in good agreement with the exact solution.

Basic equations in the case without higher-order operators
In the previous section, we determinedQ (2)    Let us move to the coordinate system with B ≡ 1. Using Eq. (2.65) and multiplying both sides by Q (1) , we find Here we have used the canonical-variable condition (2.48). The above equation is easily integrated with the initial condition (RPA solution) at φ = 0 shown in Eq. (2.63) as Obviously, this is inconsistent with Q (1) which is determined from the O(p) equation of CS (2.82). This inconsistency is thought to be caused by the neglect ofQ (2) andQ ( Next, we shall adopt the moving-frame RPA equation of O(p 2 ). We have already investigated this case in Sect. 2.5. In the case of the two-level Lipkin model, (Q (1) , P ) are determined from the moving-frame RPA equation of O(p) and the canonical-variable condition only, as shown in Eq. (2.82). Therefore, there is no difference in the solution (Q (1) , P ) between the two cases where the moving-frame RPA equation of O(p 2 ) is adopted and where the q-derivative of the O(1) equation of CS is adopted. In both cases, BC plays a role of the squared eigenfrequency of the eigenvalue equations, and it is in the eigenfrequency that there appears a difference between the two cases. We have already seen the eigenfrequency squared for the moving-frame RPA equations without the higher-order operators in Sect. 2.5, and it is given by Eq. (2.84). If the set of the basic equations are self-consistent, the eigenfrequency squared ω 2 should coincide with the product of the inverse inertial mass B and the potential curvature C.
We shall calculate the product of the inverse inertial mass B and the potential curvature C in the conventional moving-frame HF & RPA equations without the higher-order operators. The inverse inertial mass B has been already obtained in Eq. (2.83). The Christoffel symbol of the second kind is given by from which we have Then, C is calculated as Thus, we obtain The first term coincides with the squared eigenfrequency of the LRPA equations ω LRPA , in which the curvature term [the third term in Eq. (2.11)] is omitted. The second term is the contribution from the connection term, and the curvature term in the moving-frame RPA equations should give this contribution. As BC is a scalar, the same result is obtained with the calculation in the q space (See Appendix B). Clearly, the squared eigenfrequency of the moving-frame RPA equation withoutQ (2) (2.84) and the product of the potential curvature and inertial mass parameter BC (4.8) are different by 9) and in this sense the self-consistency is broken. This can be attributed to the fact that the higher-order operators are neglected in the conventional moving-frame RPA equations.

22/30
The third term vanishes when we employ the coordinate system with B ≡ 1. By rewriting ∂ q V and ∂ q Q (1) with use of Eqs. (2.52) and (2.65), respectively, BC can be rewritten in terms of (Q (1) , P ). Then, with Eq. (2.82), we readily obtain Unlike the conventional moving-frame RPA equations withoutQ (2) , the eigenfrequency squared (4.11) coincides with BC (4.8) obtained from the potential curvature and the inverse inertial mass, so they are self-consistent in this case. While the higher-order operators are involved in the moving-frame RPA equation of O(p 2 ), which were neglected in the previous subsection, the q-derivative of the O(1) equation of CS does not contain the higher-order operators, and no approximation is made. In this sense, the q-derivative of the O(1) equation of CS is better than the moving-frame RPA equation of O(p 2 ) with the higher-order operators ignored. It is noteworthy thatQ (2) is omitted in the O(p) equation of CS in both of the two cases. 23/30

Discussion
So far we have considered a method of determiningQ (2) in the case without the pairing correlation. As shown in Refs. [2,3], the higher-order operators have much to do with the gauge symmetry. In this section, we briefly discuss the gauge symmetry of the basic equations when the pairing correlation is included. First we shall reconsider the gauge symmetry that Hinohara et al found [11]. As mentioned in Introduction, Hinohara et al encountered the numerical instability caused by the gauge symmetry of the basic equations, and needed to introduce a gauge-fixing prescription for successful calculation. This is a "numerical" problem in the sense explained below. The moving-frame HFB & QRPA equations including up to the first-order operatorQ (1) are given by, (1) . (Assume thatQ (1) contains only A-terms.) When ∂ q V = 0, these equations are invariant under the transformation (1.2)-(1.5). When ∂ q V = 0, the basic equations are not invariant under this transformation. Nevertheless, there occurred the numerical instability. This may be understood as follows.
As shown in Ref. [2], for ∂ q V = 0, the moving-frame HFB equation ( breaks the gauge symmetry as easily confirmed. Hinohara et al started the calculation from the HFB equilibrium point (q = 0) where ∂ q V (q = 0) = 0, and at the next step q = ±δq, ∂ q V (δq) is still small. Although there are gauge-symmetry-breaking terms in the movingframe QRPA equations as seen above, they are proportional to ∂ q V , and their contributions are small in the vicinity of the HFB equilibrium point ∂ q V (q) = 0. Thus, the gauge symmetry is approximately retained, which leads to the numerical instability. If the gauge-symmetrybreaking terms gave sufficiently large contributions, the numerical instability would not occur. Then, once the gauge is fixed at the HFB equilibrium point, the gauge-fixing prescription would not be necessary to solve the moving-frame equations at non-equilibrium points.
is gauge invariant, then what we have to check is the gauge symmetry of ∂ q V D qQ (1) . It is sufficient to investigate the gauge symmetry in the vicinity of the HFB equilibrium point ∂ q V = 0. We choose the coordinate system with B(q) = 1, and then D qQ (1) = ∂ qQ (1) . In the previous section, due to the simplicity of the Lipkin model, the basic equations reduced to one differential equation. In general cases, however, ∂ qQ (1) should be approximated by finite difference. With the simplest scheme, It is not gauge invariant except for the zero modes.  (2) , so the consideration above holds as it is. Unless C(0) = 0, the gauge symmetry is significantly broken even in the vicinity of HFB equilibrium point ∂ q V = 0, and the gauge is fixed. The term [Q (1) ,Q (2) ] in Eq. (2.21) is a B-term and does not contribute, so Eq. (2.21) with this term omitted, so the gauge-symmetry-breaking term is proportional to ∂ q V . The rest transforms as

Concluding remarks
In this paper, we have considered the ASCC theory including the second-order collectivecoordinate operatorQ (2) which consists of only A-terms, proposed a new set of basic equations to determine the collective operators includingQ (2) , and applied it to the two-level Lipkin model. We have compared the ASCC calculations with and withoutQ (2) and found that, for a first few low-energy states, the difference between the results of the two calculations is not so significant and that both of the calculations reproduce the exact solution well. However, with increasing the excitation energy, the deviation from the exact solution becomes larger in the case withoutQ (2) , while, withQ (2) , the agreement with the exact solution is good even for the higher excited states. As discussed in Refs. [2,3] and this paper, Q (2) does contribute to the inertial mass. As the excitation energy increases, the kinetic energy becomes important relatively to the collective potential energy. The results shown in this paper illustrates the importance of the correct evaluation of the inertial mass.
We have also reconsidered the basic equations to adopt in the case where no higherorder operator is included. It has been shown that, in the conventional moving-frame RPA equations, the self-consistency is broken in the sense that the eigenfrequency squared ω 2 does not coincide with the product of the potential curvature and the inverse inertial mass BC. In contrast, when we employ the q-derivative of the O(1) equation of CS, in which no approximation is made for the higher-order operators, the relation ω 2 = BC holds, and the self-consistency is kept. In the case of the two-level Lipkin model we have used, (Q (1) ,P ) are determined from the O(p) equation of CS and the canonical-variable condition, and the difference between the moving-frame RPA equation of O(p 2 ) and the q-derivative of the O(1) equation of CS appears only in ω 2 . It would be interesting to investigate how (Q (1) ,P ) are affected depending on which of the two equations to adopt, using the three-level Lipkin or more realistic models. The observation above may lead to an intuitive understanding as follows. The O(1) equation of CS is an equation for the state vector |φ(q) and the Lagrange multiplier, onceQ (1) is given. The O(p) equation of CS and the q-derivative of the O(1) equation of CS can be viewed as equations to determineQ (1) andP , respectively, whileQ (2) is determined from the O(p 2 ) equation of CS. Note that all the basic equations should be solved self-consistently.
Although we have mainly studied the ASCC theory without the pairing correlation in this paper, the basic equations with the pairing correlation are also derived in a straightforward way, and we have briefly discussed the gauge symmetry of the basic equations. The gauge transformation changes the gauge angle and the chemical potential as well as the collective operators, and it plays an important role in the treatment of superfluid systems.
As shown in Refs. [1,2], the equation of CS before the adiabatic expansion is gauge invariant, but the gauge symmetry is (partially) broken by the adiabatic expansion. In Refs. [1,2], we analyzed the gauge symmetry under the general gauge transformation, and found that four examples or types of the gauge transformations play an essential role in the analysis. One of the four is the gauge transformation Eqs.(1.2) -(1.5) (We call it Example 1 in Refs. [1,2]). The gauge symmetry of Example 1 can be retained by including the higher-order collective-coordinate operators. However, the symmetry under the gauge transformation of Example 3 in Ref. [1,2], which mixesP withÑ , cannot be retained even if the higher-order operators are introduced. This can be regarded as a gauge fixing by the adiabatic expansion. 26/30 In Hinohara's prescription, they require only the gauge symmetry under the transformation which mixesQ (1) withÑ (Example 1), and the gauge symmetry under the transformation which mixesP withÑ (Example 3) is left broken. In this sense, Hinohara's prescription attaches more weight to Example 1 than Example 3. If the two transformations are of equal weight, the basic equations breaking the symmetry under the gauge transformation of Example 1 can be adopted. Then one may regarded it as a gauge fixing as in Example 3. In general, if there is gauge symmetry, one can choose a convenient gauge, so a set of the basic equations which are not gauge invariant may be possible.
As mentioned in Introduction, the extension of the ATDHF to the ATDHFB theory is not straightforward, because one has to decouple the number-fluctuating mode from the collective mode of interest. For the collision of two nuclei, the chemical potentials of the two nuclei are different unless it is a collision between the same nuclides. To describe such a phenomenon, it is necessary to construct a theory with which one can treat the gauge degrees of freedom correctly. In Ref. [23], Wen and Nakatsukasa could not find a collective path connecting the superdeformed state and the ground state in 32 S, and pointed out a possible improvement of the problem by including the pairing correlation. For that purpose, a theory is required which can treat both cases with and without the pairing correlation on an equal footing. The basic equations we have proposed in this paper has such an advantage and treat the higher-order collective operator in both of the cases on an equal footing. It would be very interesting to apply the set of the basic equations proposed here to systems with the pairing correlation, and it will be reported in a future publication.