Effective field theories for collective excitations of atomic nuclei

Let us start from a microscopic Hamiltonian $H_{\rm mic}$ whose degrees of freedom are the positions, spins, and isospin projections of nucleons. For simplicity, we think of an even-even nucleus and further assume that neither proton nor neutron numbers are magic numbers. Thus, we deal with an open-shell nucleus. It is profitable to break down the computation of its ground state $|\psi\rangle$ into several steps, each of which gives increasingly better approximations. We denote the states, energies, and angular-momentum expectation values at the $n^{\rm th}$ step as $\ket{\psi_{(n)}}$, $E_{(n)}$, and $J_{(n)}^{2}$.

The first step usually consists of a Hartree-Fock calculation ¹¹1We discuss the more general case of Hartree-Fock-Bogoliubov calculation in section 2.2. Let us assume that this calculation is based on single-particle states with good angular momentum projection $j_{z}$, and that we seek a state with total angular momentum projection $J_{z}=0$. In practice this is achieved by starting from a trial product states where pairs of time-reversed single-particle states are occupied. The Hartree-Fock computation then yields an axially symmetric state $\ket{\psi_{(1)}}$. The deformation results from a competition between short and long-range correlations (Lipkin, 1960). The state has zero angular momentum projection with respect to the symmetry axis (which we choose as the laboratory $z$ axis) but not good angular momentum. We have $J_{(1)}^{2}\equiv\braket{\psi_{(1)}|J^{2}|\psi_{(1)}}>0$. While the energy $E_{(1)}\equiv\braket{\psi_{(1)}|H_{\rm mic}|\psi_{(1)}}$ generally is a poor approximation of the true ground-state energy, the Hartree-Fock state is a great starting point for more refined calculations.

The second step could, for instance, consist of a coupled-cluster computation where particle-hole excitations of the Hartree-Fock reference are included. Typically, such calculations are limited to up to two-particle–two-hole or three-particle-three-hole excitations (Hagen et al., 2014). If one were to include four-particle–four-hole excitations, the treatment of short-range physics would be complete: spin-isospin degrees of freedom permit up to four nucleons to be very close. This then yields the refined approximation $|\psi_{(2)}\rangle$ of the ground state. The energy $E_{(2)}$ of this state is a much lower than the Hartree-Fock energy $E_{(1)}$ and usually already close to the exact ground-state energy. However, the state $|\psi_{(2)}\rangle$ still does not have good angular momentum. Typically one finds $J_{(2)}^{2}<J_{(1)}^{2}$ but one still has $J_{(2)}^{2}>0$. This is not surprising: It would require $A$-particle–$A$-hole excitations to get good angular momentum for a nucleus with mass-number $A$. These missing correlations are long range. So one realistically deals with a situation where rotational symmetry is broken down to axial symmetry at any step $n$ that can be achieved in practical calculations. Results that illustrate these arguments are shown in table 1.

The energy after symmetry restoration is denoted by $E_{0}$. (That the energies deviate somewhat from data is not important here; it mainly points to an inaccuracy of the employed interaction.) We see that the symmetry breaking is not very costly in energy as $E_{(2)}$ is close to $E_{0}$. Symmetry restoration is about capturing low-energy (or long wavelength) physics. The small energy gain comes from lowering the kinetic energy and not from improving the contributions from the short-ranged potential.

One can construct the corresponding low-energy or collective Hamiltonian. To do this, we follow the well known approach described, for example, by Ring and Schuck (1980). We first identify the relevant Hilbert space. The symmetry axis of the deformed nucleus is along the $z$ axis. Because of this symmetry, it is not the full group of rotations with elements $\exp{(-i\phi J_{z})}\exp{(-i\theta J_{y})}\exp{(-i\gamma J_{z})}$ and Euler angles $(\phi,\theta,\gamma)$ that we need to consider when restoring the symmetry, but rather those group elements that are in the coset SO(3)/SO(2); these are the rotations

$$R(\phi,\theta)\equiv e^{-i\phi J_{z}}e^{-i\theta J_{y}}\ ,$$

(1)

where the angles ($\phi,\theta$) parameterize the sphere. When acting onto the state $|\psi_{(n)}\rangle$ these rotations yield states $\ket{\psi_{(n)},\Omega}\equiv R(\phi,\theta)\ket{\psi_{(n)}}$. Here we combined $\Omega\equiv(\phi,\theta)$ to keep a compact notation for these states. We have $|\psi_{(n)}\rangle=|\psi_{(n)},0\rangle$. The states $\ket{\psi_{(n)},\Omega}$ all have identical energy expectation values because the Hamiltonian $H_{\rm mic}$ is invariant under rotations, i.e. $R^{-1}H_{\rm mic}R=H_{\rm mic}$. Thus, one needs to diagonalize the microscopic Hamiltonian in this basis (Peierls and Yoccoz, 1957). The basis set is not orthogonal, and one computes the Hamiltonian and norm kernels

	$\displaystyle{\cal H}(\Omega^{\prime},\Omega)$	$\displaystyle\equiv\braket{\psi_{(n)},\Omega^{\prime}|H_{\rm mic}|\psi_{(n)},\Omega}\ ,$		(2)
	$\displaystyle{\cal N}(\Omega^{\prime},\Omega)$	$\displaystyle\equiv\braket{\psi_{(n)},\Omega^{\prime}|\psi_{(n)},\Omega}\ .$

Diagonalization of the norm kernel yields an orthonormal basis. One can re-express the Hamiltonian kernel in this basis set. In practice, one computes the Hamiltonian

$$H(\Omega^{\prime},\Omega)\equiv{\cal N}^{-\frac{1}{2}}{\cal H}{\cal N}^{-\frac{1}{2}}\ ,$$

(3)

The key point is that a diagonalization of the matrix $H(\Omega^{\prime},\Omega)$ yields a rotational band, i.e. the resulting energies are approximately

$$E_{J}=E_{0}+aJ(J+1)\ ,$$

(4)

and each energy has degeneracy $2J+1$. Here, $a$ is the rotational constant (and proportional to the inverse moment of inertia).

Thus, a very simple physics picture results from a possibly quite complicated microscopic Hamiltonian. The microscopic details are all contained in the ground-state energy $E_{0}$ and the rotational constant $a$ (and vary from nucleus to nucleus), while the rotational $J(J+1)$ pattern is universal. We note that the computation of the matrix elements is only simple for product states, i.e. for the $n=1$ approximation. In general, ab initio computations of $E_{0}$ and $a$ are somewhat challenging (Qiu et al., 2017; Hagen et al., 2022; Sun et al., 2024b). An important insight gained from such calculations is that $E_{(n)}-E_{0}\ll|E_{0}|$ and that $a\ll|E_{0}|$: Both the energy gained from the calculation and the spacing of the resulting levels are small compared to the ground-state energy, and both or of the size ${\cal O}(a)$. Thus, $H(\Omega^{\prime},\Omega)$ is a low-energy Hamiltonian.

Let us assume that we had a set of microscopic Hamiltonians that differ in their cutoffs and thus exhibit very different short-range physics. If these Hamiltonians are accurate, they will all yield the spectrum (4) to a good approximation. Thus, it must be possible to approach the low-energy physics of the very complicated (and non-local) Hamiltonian $H(\Omega^{\prime},\Omega)$ of equation (3) via an effective (field) theory, i.e. by constructing a Hamiltonian $H_{\rm EFT}$. Postulating locality, it is clear that $H_{\rm EFT}$ cannot depend on $\Omega$ itself because of rotational invariance. This leaves us with derivatives. As the parameter space is the two-sphere, the derivative is (Varshalovich et al., 1988)

$$\nabla_{\Omega}\equiv\mathbf{e}_{\theta}\partial_{\theta}+\frac{\mathbf{e}_{\phi}}{\sin\theta}\partial_{\phi}.$$

(5)

Here

$\displaystyle\mathbf{e}_{\theta}(\phi,\theta)\equiv\left(\begin{array}[]{c}\cos\phi\cos\theta\\ \sin\phi\cos\theta\\ -\sin\theta\end{array}\right)$

(9)

and

$\displaystyle\mathbf{e}_{\phi}(\phi,\theta)\equiv\left(\begin{array}[]{c}-\sin\phi\\ \cos\phi\\ 0\end{array}\right)$

(13)

are the usual tangential unit vectors on the sphere at $\Omega$. Together with the radial unit vector

$$\mathbf{e}_{r}(\phi,\theta)\equiv\left(\begin{array}[]{c}\cos\phi\sin\theta\\ \sin\phi\sin\theta\\ \cos\theta\end{array}\right),$$

(14)

which denotes the direction of the symmetry axis of the state $|\psi_{(n)},\Omega\rangle$, the set $(\mathbf{e}_{\theta},\mathbf{e}_{\phi},\mathbf{e}_{r})$ forms a right-handed coordinate system. The most simple Hamiltonian one can write down is then

	$\displaystyle H_{\rm EFT}$	$\displaystyle=E_{0}+a\nabla_{\Omega}\cdot\nabla_{\Omega}$
		$\displaystyle=E_{0}-a\left(\frac{1}{\sin\theta}\partial_{\theta}\sin\theta\partial_{\theta}+\frac{1}{\sin^{2}\theta}\partial^{2}_{\phi}\right)\ .$		(15)

The corresponding eigenfunctions are spherical harmonics and the energies are given by equation (4). Of course, $E_{0}$ and $a$ are low-energy constants of the effective field theory and need to be adjusted to data.

This example shows how simple and powerful the construction of an effective theory can be. The steps involved were (i) the identification of the angles parameterizing the coset space SO(3)/SO(2) dictated by the pattern of symmetry breaking as the relevant degrees of freedom, and (ii) the insight that rotational invariance only allows derivatives to appear in the effective Hamiltonian. Usually, the spectrum (4) is presented as a result of the rigid rotor model or the variable moment of inertia model (Scharff-Goldhaber et al., 1976). However, one does not need any model to arrive at the spectrum; symmetry arguments alone are sufficient.

Let us now consider a semi-magic nucleus, for example, an isotope of tin or lead. We also assume an even number of neutrons. In what follows we will use the neutron number expectation value $N_{(n)}\equiv\braket{\psi_{(n)}|N|\psi_{(n)}}$ and the variance $\Delta N^{2}_{(n)}\equiv\braket{\psi_{(n)}|N^{2}|\psi_{(n)}}-N_{(n)}^{2}$ at step $n$ of the calculation.

The starting point in this case is a Hartree-Fock-Bogoliubov computation. While the resulting product state $|\psi_{(1)}\rangle$ now exhibits good angular momentum, its neutron number, denoted as $N_{(1)}$, is not a good quantum number. Thus, the number variance fulfills $\Delta N^{2}_{(1)}>0$. A second step could be a Bogoliubov coupled-cluster computation (Signoracci et al., 2015; Tichai et al., 2023) yielding the state $|\psi_{(2)}\rangle$, for which $\Delta N_{(2)}^{2}<\Delta N_{(1)}^{2}$, but $\Delta N^{2}_{(2)}>0$. As for deformed nuclei, the symmetry breaking costs only little in energy. This situation is illustrated in table 2, based on the data from (Tichai et al., 2023). We see again that the (estimated) ground-state energy $E_{0}$ is close to $E_{(2)}$, i.e. symmetry projection yields little gain in energy.

The particle-number breaking product state (Bogoljubov, 1958; Valatin, 1958) points into a definite direction of gauge space. Since the microscopic Hamiltonian $H_{\rm mic}$ preserves neutron number, the action of (global) gauge transformations

$$g(\alpha)\equiv e^{-i\alpha\hat{N}}$$

(16)

onto $|\psi_{(n)}\rangle$ introduces states $|\psi_{(n)},\alpha\rangle\equiv g(\alpha)|\psi_{(n)}\rangle$ with identical energy expectation values as $g^{-1}(\alpha)H_{\rm mic}g(\alpha)=H_{\rm mic}$.

The reader now sees where this journey is heading: One can again diagonalize the microscopic Hamiltonian in the subset of degenerate states $|\psi_{(n)},\alpha\rangle$ with $0\leq\alpha<2\pi$. This yields so-called pairing rotational bands, and ground states in nuclei that differ by pairs of neutrons are members of such a band (Bohr, 1969; Brink and Broglia, 2005). The energy scale associated with the pairing rotational band and the gain from the ensuing particle-number restoration are again small when compared to $E_{(2)}$.

Similarly as in the case of deformed nuclei, one can construct an effective field theory, and the corresponding Hamiltonian is based on the derivative $\partial_{\alpha}$ that acts on the unit circle. The effective field theory entirely rests on the fact that the approximate states $|\psi_{(n)}\rangle$ break particle number, i.e., a U(1) phase symmetry of the Hamiltonian. More details are presented in section 9.

We have seen that the effective field theories of deformed and of superfluid nuclei have a microscopic foundation. They naturally arise whenever approximations of nuclear states break a symmetry of the Hamiltonian. While we have based our arguments on microscopic Hamiltonians, we see that the universality of these phenomena holds for any nuclear model that exhibits the symmetry breakings described above. From the low-energy perspective, any such model falls into a “universality class” that is entirely determined by the pattern of the symmetry breaking. Thus, the effective field theory truly is model independent.

Given the simplicity of the parameter spaces – the unit sphere in the case of deformed, axially symmetric nuclei and the unit circle in the case of pairing – the reader might wonder about how complex the corresponding phenomena can possibly be. As we will see below, interesting phenomena will enter because of non-trivial topological effects. In the case of the unit sphere, radially symmetric “monopole-like” gauge potentials are consistent with rotational invariance, and the similar effects are possible for the unit circle. This will introduce Berry-phase physics and explain interesting phenomena.

The construction of Hamiltonians within effective field theory is based on symmetry breaking and only uses derivative (and possibly gauge) couplings. This rings familiar from quantum field theory: In the presence of spontaneous symmetry breaking of a group $G$ to a subgroup $S$, Nambu-Goldstone bosons are the relevant low-energy degrees of freedom. They parameterize the coset $G/S$ and only derivative and gauge couplings are allowed. This connection to spontaneous symmetry breaking will be discussed in the following section.

The connection between rotational bands and symmetry restoration was made soon after the collective models (Bohr, 1952; Bohr and Mottelson, 1953) arrived. The Nilsson model (Nilsson, 1955) exposed the shell structure of axially symmetric, deformed nuclei. A diagonalization of the Hamiltonian in the degenerate set of symmetry-breaking states then led to rotational bands, and this approach combined shell-model and collective aspects (Peierls and Yoccoz, 1957; Peierls and Thouless, 1962; Villars, 1965). Similarly, the understanding of superconductivity within BCS theory, and its usage in nuclear physics(Bohr et al., 1958; Migdal, 1959) introduced pairing rotations as a consequence of particle-number restoration (Bohr, 1969; Bès et al., 1970; Broglia et al., 1973).

The development of BCS theory was also most fruitful in particle physics. Nambu (1960) and Goldstone (1961) discovered that massless excitations (now referred to as Nambu-Goldstone bosons) accompany spontaneous symmetry breaking. Nambu and Jona-Lasinio (1961a, b) presented a model where pions emerged as the very light bosons of the spontaneously broken chiral symmetry, and Weinberg (1968) introduced chiral effective field theory as a model-independent approach that exploits spontaneous symmetry breaking of the strong force. Coleman et al. (1969) and Callan et al. (1969) generalized Weinberg’s approach from the spontaneous breaking of SU(2) symmetry to other continuous groups. Thus, there were parallel developments regarding symmetry restoration in nuclear physics and spontaneous symmetry breaking in particle physics.

Bohr (1975) pointed out the connection between nuclear rotation, spontaneous symmetry breaking and Goldstone bosons in his Nobel lecture. This picture has been emphasized by several authors (Ui and Takeda, 1983; Fujikawa and Ui, 1986; Nazarewicz, 1993, 1994; Frauendorf, 2001; Broglia et al., 2000; Papenbrock, 2011). However, significant differences exist: Spontaneous symmetry breaking only happens in infinite systems while nuclei are finite. Nambu-Goldstone modes are excitations with arbitrary small energies while rotational bands and pairing rotational bands have finite spacings. To emphasize the difference Koma and Tasaki (1994) and Yannouleas and Landman (2007) introduced the expressions “obscured symmetry breaking” and “emergent symmetry breaking”, respectively. We adopt the latter and want to discuss commonalities and differences between spontaneous and emergent symmetry breaking.

Let us consider the breaking of SO(3) rotational symmetry down to SO(2) axial symmetry, and take ferromagnets (where the spins point into the direction of the $z$ axis) and deformed nuclei (as discussed in section 2.1) as respective examples. For the ferromagnet the ground state $|{\rm gs}\rangle$ spontaneously breaks the symmetry while we take the correlated state $|\psi_{(2)}\rangle$ as the symmetry breaking state for the deformed nucleus.

There is a fundamental difference between the Hilbert spaces of finite and infinite systems that exhibit emergent and spontaneous symmetry breaking, respectively (Ui and Takeda, 1983). To see this, let us return to equation (4), valid for a finite system. Here, the rotational constant $a$ is proportional to the inverse moment of inertia and vanishes in the limit of infinite particle number. As the ground state of the infinite system cannot be infinitely degenerate, one must introduce inequivalent Hilbert spaces and exclude rotations of the whole system.

Let us also present an alternative argument, and this time start from the symmetry-breaking state. In the case of nuclei the symmetry-breaking state $|\psi_{(2)}\rangle$ and its rotated kin have a nonzero overlap, i.e. $\langle\psi_{(2)}|R(\phi,\theta)|\psi_{(2)}\rangle\neq 0$ for almost all angles. In the case of the ferromagnet’s ground state $|{\rm gs}\rangle$, however, we have $\langle{\rm gs}|R(\phi,\theta)|{\rm gs}\rangle=0$ for all finite rotation angles. The latter is so because the overlap is an infinite product of single-spin overlaps that all have magnitudes smaller than one. Thus, for infinite systems a global rotation yields a state that is orthogonal to the symmetry breaking state. The rotated state belongs to an inequivalent Hilbert space, and there is no symmetry restoration.

For ferreomagnets, Nambu-Goldstone modes $|\phi(\mathbf{x},t),\theta(\mathbf{x},t)\rangle$ are generated by acting with the space- and time-dependent rotation operator $R(\phi(\mathbf{x},t),\theta(\mathbf{x},t))$ of equation (1) onto the ground state, i.e.

$$|\phi(\mathbf{x},t),\theta(\mathbf{x},t)\rangle=R\left(\phi(\mathbf{x},t),\theta(\mathbf{x},t)\right)\,|{\rm gs}\rangle\ .$$

(17)

The quantum fields $\phi(\mathbf{x},t)$ and $\theta(\mathbf{x},t)$ generate spin waves. They can have arbitrarily long wave length and arbitrarily low energy. Spatially constant, i.e. $\mathbf{x}$-independent, fields are forbidden because a rotation of the infinite system is not allowed (because it leads to an inequivalent Hilbert space). The Nambu-Goldstone states $|\phi(\mathbf{x},t),\theta(\mathbf{x},t)\rangle$ are orthogonal to the ground state because they involve infinite products of individual overlaps that are almost all smaller than unity.

In the case of nuclei the Nambu-Goldstone modes are symmetry restoring and can be purely time-dependent. They are generated by acting with $R(\phi(t),\theta(t))$ onto the symmetry-breaking state $|\psi_{(2)}\rangle$ which gives

$$|\phi(t),\theta(t)\rangle=R\left(\phi(t),\theta(t)\right)\,|\psi_{(2)}\rangle\ .$$

(18)

These states generally are not orthogonal to the state $|\psi_{(2)}\rangle$.

This discussion shows how rotational excitations differ from Nambu-Goldstone modes. However, rotational excitations and Nambu-Goldstone modes both arise from the action of rotation operators whose angles are time-dependent variables and fields, respectively. In both cases, the angles parameterize the coset SO(3)/SO(2) that reflects the pattern of the symmetry breaking. (We note that the coset space SO(3)/SO(2) is isomorph to the surface of the unit sphere.) This common technical aspect allows one to use the coset approach (Coleman et al., 1969; Callan et al., 1969) to develop effective Lagrangians for collective excitations in finite systems (Leutwyler, 1987; Chandrasekharan et al., 2008; Papenbrock, 2011). We briefly discuss this approach next.

The collective degrees of freedom involved in symmetry projection and the Nambu-Goldstone bosons in spontaneous symmetry breaking parameterize the coset $G/S$ when the symmetry is broken from a group $G$ to a subgroup $S$. This allows one to construct nonlinear realizations of the symmetry group $G$ (Coleman et al., 1969; Callan et al., 1969). In the example considered so far, $G={\rm SO}(3)$ and $S={\rm SO}(2)$, and the coset $G/S$ is the two-sphere. Each point on the sphere can be parameterized by the usual azimuth and polar angles $(\phi,\theta)$. A rotation maps the point with coordinates $(\phi,\theta)$ to a new point $(\phi^{\prime},\theta^{\prime})$, and the new angles are nonlinear functions of the old ones. This then constitutes the nonlinear (or Nambu-Goldstone) realization of the symmetry group $G$. To nuclear physicists, these may be somewhat less familiar than the usual linear (Wigner-Weyl) realizations. Nonlinear realizations apply in cases of spontaneous and emergent symmetry breaking. The nonlinear transformation properties also allow one to introduce quantities that are invariant under symmetry operations and thereby to construct effective Lagrangians. As usual the Noether theorem allows one to identify the corresponding conserved quantities. The original arguments and derivations of this approach are by Coleman et al. (1969); Callan et al. (1969). Excellent expositions can be found in references (Weinberg, 1996; Brauner, 2010). In section 4.4 we briefly display the main arguments for deformed nuclei (Papenbrock, 2011).

Here we follow the more geometric approach by Papenbrock and Weidenmüller (2020) as it simplifies steps (i) and (ii) above, as well as the construction of invariants via Noether’s theorem. This approach combines the ($\phi$, $\theta$) angles into the radial unit vector (14), oriented along the symmetry axis of the nucleus ²²2Ground states of axially symmetric nuclei are often invariant under rotations by $\pi$ around an axis perpendicular to the symmetry axis, i.e. they exhibit ${\cal R}$ invariance (Bohr and Mottelson, 1975) and thus are nematics (Mermin, 1979). Then, one only needs a preferred axis but no direction. This identifies opposite points on the sphere and reduces the coset space to half of the sphere.. The velocity of this vector,

$$\mathbf{v}\equiv\frac{d}{dt}\mathbf{e}_{r}(\phi,\theta)=\dot{\theta}\mathbf{e}_{\theta}(\phi,\theta)+\dot{\phi}\sin\theta\mathbf{e}_{\phi}(\phi,\theta)$$

(19)

is the building block of the effective theory. Here and in what follows the dot denotes the time derivative. The polar and azimuthal unit vectors were introduced in equations (9) and (13).

The leading contribution to the effective Lagrangian is the simplest term built from the above velocity that is invariant under rotations

$$L_{\rm LO}=\frac{C_{0}}{2}\mathbf{v}^{2}=\frac{C_{0}}{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2}\sin^{2}\theta\right).$$

(20)

Here, $C_{0}$ is a low-energy constant that must be fit to data. The Legendre transformation of this Lagrangian yields the leading-order effective Hamiltonian

$$H_{\rm LO}=\frac{1}{2C_{0}}\left(p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin^{2}\theta}\right)\ .$$

(21)

Here we used the usual canonical momenta

	$\displaystyle p_{\theta}$	$\displaystyle=\frac{\partial L_{\rm LO}}{\partial\dot{\theta}}\ ,$		(22)
	$\displaystyle p_{\phi}$	$\displaystyle=\frac{\partial L_{\rm LO}}{\partial\dot{\phi}}\ .$

Application of Noether’s theorem yields the total angular momentum $\mathbf{I}$ with components

	$\displaystyle I_{x}$	$\displaystyle=-\sin\phi p_{\theta}-\cos\phi\cot\theta p_{\phi}\ ,$		(23)
	$\displaystyle I_{y}$	$\displaystyle=\cos\phi p_{\theta}-\sin\phi\cot\theta p_{\phi}\ ,$
	$\displaystyle I_{z}$	$\displaystyle=p_{\phi}\ ,$

as the conserved quantity. One can combine these expressions into

$$\mathbf{I}=-\frac{p_{\phi}}{\sin\theta}\mathbf{e}_{\theta}+p_{\theta}\mathbf{e}_{\phi}\ ,$$

(24)

and see that the angular momentum has no component in direction of the symmetry axis. One can now rewrite the Hamiltonian as

$$H_{\rm LO}=\frac{\mathbf{I}^{2}}{2C_{0}}\ .$$

(25)

Its quantization yields the energy spectrum

$$E_{\rm LO}(I)=\frac{I(I+1)}{2C_{0}}\ .$$

(26)

Thus, the well-known rigid-rotor spectrum results from the assumption of emergent symmetry breaking from SO(3) to SO(2).

Several comments are in order. First, we note that the SO(3) symmetry is realized nonlinearly, as rotations transform the angles ($\phi$, $\theta$) nonlinearly. Transformation laws were presented in (Papenbrock and Weidenmüller, 2020). Second, one quantizes the canonical momentum according to

$$\mathbf{p}\equiv p_{\theta}\mathbf{e}_{\theta}+\frac{p_{\phi}}{\sin\theta}\mathbf{e}_{\phi}=-i\nabla_{\Omega}\ ,$$

(27)

where $\nabla_{\Omega}$ is the angular derivative (5). Writing the angular momentum as $\mathbf{I}=\mathbf{e}_{r}\times\mathbf{p}$, and noticing that $\mathbf{I}^{2}=-\nabla_{\Omega}^{2}$ yields

$$H_{\rm LO}=-\frac{\nabla_{\Omega}^{2}}{2C_{0}}\ .$$

(28)

The eigenfunctions of this Hamiltonian are the usual spherical harmonics $Y_{IM}(\theta,\phi)$ and $M\in\{-I,-I+1,\ldots I\}$ is the eigenvalue of $I_{z}$. Finally, ground-state bands in even-even nuclei only contain states with even spins. This pattern arises from the ${\cal R}$ invariance (Bohr and Mottelson, 1975).

We can now consider more general Lagrangians. The rotational invariance permits only powers of $\mathbf{v}^{2}$ and this yields Hamiltonians in powers or $\mathbf{I}^{2}$. Thus, the most general spectrum is a polynomial in $I(I+1)$ with coefficients that must be adjusted to data for a given rotational band. As we will now discuss, this is indeed the power counting of the effective theory.

We introduce the low-energy (or small-frequency) scale $\xi$ that is typical for nuclear rotations. Then, the angular velocity (19) is slow, i.e.

	$\displaystyle\dot{\theta}$	$\displaystyle\sim\xi$		(29)
	$\displaystyle\dot{\phi}$	$\displaystyle\sim\xi\ .$

We also have $E(2^{+}_{1})\sim\xi$ in the leading-order spectrum (26) and this yields the estimate

$$C_{0}\sim\xi^{-1}\ .$$

(30)

Next we introduce a high-energy scale $\Lambda$ at which the effective theory breaks down. This scale is due to neglected degrees of freedom that appear at an energy $\Lambda$. An example is shown in figure 2. The ground-state band closely follows the leading spectrum (26), setting the scale $\xi$ as the first level spacing. Clearly, the description of ²³⁸Pu as a single rotational band breaks down at the energy $\Lambda$ where the second rotational band starts. We have $\xi\ll\Lambda$, and this separation of scales allows us to introduce a power counting.

One can, of course, introduce additional degrees of freedom to describe the second rotational band depicted in figure 2, but this is not what we want to consider here. Instead, the mere existence of other degrees of freedom impacts the low-energy theory. As there is a separation of scales between the excluded degrees of freedoms and the low-energy ones, the effect of the former on the latter can be captured by effective Lagrangians.

Let us return to the most general Lagrangian

$$L=\sum_{n=1}^{\infty}\frac{C_{n-1}}{2n}\mathbf{v}^{2n}.$$

(31)

Here, the factors $1/(2n)$ are introduced out of convenience. If we set the breakdown energy as $\Lambda$, we must have $C_{0}\mathbf{v}^{2}\sim\Lambda$ and thus find $\mathbf{v}^{2}\sim\xi\Lambda$. This establishes the breakdown velocity. The key idea is now that each term in the Lagrangian (31) yields an equal contribution at the breakdown scale. Thus $C_{n-1}\mathbf{v}^{2n}\sim\Lambda$ and our estimate for the size of the low-energy coefficients is

$$C_{n-1}\sim\xi^{-n}\Lambda^{1-n}\ .$$

(32)

This now establishes a power counting. At low energies, where $\mathbf{v}^{2}\sim\xi^{2}$, the contribution of each term in the Lagrangian (31) is then

$$C_{n-1}\mathbf{v}^{2n}\sim\xi\left(\frac{\xi}{\Lambda}\right)^{n}\ .$$

(33)

These clearly are increasingly smaller corrections ³³3One can also use breakdown velocity rather than a breakdown energy to establish a power counting, and that approach was taken in (Papenbrock, 2011; Zhang and Papenbrock, 2013; Coello Pérez and Papenbrock, 2015b; Papenbrock and Weidenmüller, 2020).. Based on this counting scheme, terms containing higher powers of $\mathbf{v}^{2}$ are higher orders of the effective theory.

When higher orders are included in the effective Lagrangian, the exact Legendre transformation to the Hamiltonian is not anymore possible because one cannot easily solve for the velocities in terms of the conjugate momenta. Instead, one pursues a perturbative inversion where one expands around the leading-order result (Fukuda, 1988). This then yields a Hamiltonian expansion (Zhang and Papenbrock, 2013)

$$H=\sum_{n=1}^{\infty}g_{n}\mathbf{I}^{2n}\ ,$$

(34)

where the couplings $g_{n}$ are given in terms of the low-energy coefficients $C_{k-1}$. The resulting energy spectrum is then

$$E(I)=\sum_{n=1}^{\infty}g_{n}\left[I(I+1)\right]^{n}\ .$$

(35)

From the power counting one finds that $g_{n}\sim\xi(\xi/\Lambda)^{n-1}$. Coello Pérez and Papenbrock (2015b) confirmed that low-energy coefficients for molecules and deformed nuclei scale as estimated by the power counting.

We also want to discuss the key concepts behind the nonlinear realization of spontaneously broken symmetries (Weinberg, 1968; Coleman et al., 1969; Callan et al., 1969). This topic is presented in detail in Weinberg’s textbook (1996) and Brauner’s review (2010), and for deformed nuclei in references (Papenbrock, 2011; Papenbrock and Weidenmüller, 2014). We let the angles $(\phi,\theta)$ set the orientation of the nucleus’s symmetry axis. Under a rotation $R(\alpha,\beta,\gamma)$ with Euler angles $(\alpha,\beta,\gamma)$, the $(\phi,\theta)$ angles transform into $(\phi^{\prime},\theta^{\prime})$, which are complicated nonlinear functions of the original angles. This nonlinear representation of SO(3) is in contrast to the usual linear representation where spherical tensor transforms linearly via multiplication by a Wigner-$D$ matrix (Varshalovich et al., 1988).

The nonlinear realization has important consequences. Under a a rotation

	$\displaystyle(\phi,\theta)$	$\displaystyle\to(\phi^{\prime},\theta^{\prime})\ ,$		(36)
	$\displaystyle\mathbf{e}_{r}(\theta,\phi)$	$\displaystyle\to\mathbf{e}_{r}(\theta^{\prime},\phi^{\prime})\ ,$

and

	$\displaystyle\mathbf{e}_{\theta}(\phi,\theta)$	$\displaystyle\to+\mathbf{e}_{\theta}(\phi^{\prime},\theta^{\prime})\cos\eta+\mathbf{e}_{\phi}(\phi^{\prime},\theta^{\prime})\sin\eta\ ,$		(37)
	$\displaystyle\mathbf{e}_{\phi}(\phi,\theta)$	$\displaystyle\to-\mathbf{e}_{\theta}(\phi^{\prime},\theta^{\prime})\sin\eta+\mathbf{e}_{\phi}(\phi^{\prime},\theta^{\prime})\cos\eta\ ,$

where $\eta$ is an angle that depends on the Euler angles of the rotation (and the original angles $(\phi,\theta)$ of the nuclear symmetry axis). Thus, the rotated body-fixed coordinate system differs from the basis vectors $(\mathbf{e}_{\theta},\mathbf{e}_{\phi},\mathbf{e}_{r})$ at the rotated point $(\phi^{\prime},\theta^{\prime})$ by a rotation around the symmetry axis $\mathbf{e}_{r}(\phi^{\prime},\theta^{\prime})$ with the angle $\eta$. This has two important consequences. First, under a rotation any degrees of freedom defined in the body-fixed coordinate system transform linearly by a SO(2) rotation around the nucleus’s symmetry axis. Second, any terms constructed from such degrees of freedom that are invariant under SO(2) rotations around the nucleus’s symmetry axis are in fact invariant under full rotations of the system.

The coset approach starts from the rotation operator (1) with time-dependent Euler angles. One computes the expression

$$R^{-1}\partial_{t}R=a_{x}J_{x}+a_{y}J_{y}+a_{z}J_{z}$$

(38)

via the Baker-Campbell-Hausdorff expansion. This defines functions

	$\displaystyle a_{x}$	$\displaystyle=-\dot{\phi}\sin\theta\ ,$		(39)
	$\displaystyle a_{y}$	$\displaystyle=\dot{\theta}\,$
	$\displaystyle a_{z}$	$\displaystyle=\dot{\phi}\cos\theta\ .$

These quantities are the building blocks for effective Lagrangians because they exhibit definite transformation properties under rotations. The components $a_{x}$ and $a_{y}$ transform linearly, i.e. by a rotation around the $z^{\prime}$ axis with a rotation angle $\eta$ of equation (37) in the body-fixed coordinate system. They are readily identified with the components of the velocity vector, see equation (19). The quantity $a_{z}$ is part of the covariant derivative

$$D_{t}=\partial_{t}-ia_{z}J_{z}$$

(40)

and comes into play when other degrees of freedom are coupled to the rotor.

The early work (Papenbrock, 2011) followed Bohr (1952) and employed quadrupole degrees of freedom $\Phi_{\mu}$ with $\mu=-2,-1,\ldots,2$ modeling the shape of the nuclear surface. In the presence of emergent symmetry breaking one works in the co-rotating coordinate system. The component $\Phi_{0}$ acquires a vacuum expectation value and small oscillations around this expectation value introduce the $\beta$ vibrations. The modes $\Phi_{\pm 1}$ become replaced by the angles $(\phi,\theta)$ that determine the orientation of the symmetry axis. Finally, the modes $\Phi_{\pm 2}$ are the $\gamma$ vibrations. The rotational excitations are assumed to be at lowest energy and well separated from the $\beta$ and $\gamma$ vibrations. This allows one to set up a power counting.

Papenbrock (2011) derived the theory up to next-to-next-to-leading order. At leading order one only deals with (harmonic) vibrations; at next-to-leading order, the ground-state rotational band and the bands on top of the $\beta$ and $\gamma$ vibrational band heads appear and add small corrections. All three bands have identical moments of inertia. At next-to-next-to-leading order, couplings between the different rotational bands appear, adding finer details. At even higher order, the different bands become non-rigid, i.e. they deviate from the $I(I+1)$ pattern (Zhang and Papenbrock, 2013). We briefly review these developments in what follows.

The spherical components of the velocity (19)

$$v_{\pm 1}=(v_{\theta}\pm iv_{\phi})\ ,$$

(41)

are the remnants of the components $\Phi_{\pm 1}$ in the case of emergent symmetry breaking of the quadrupole oscillator. We have ${v}_{\pm 1}\sim\xi$. The remaining components of the quadrupole field can be parameterized as

	$\displaystyle\Phi_{0}(t)$	$\displaystyle\equiv\upsilon_{0}+\varphi_{0}(t)\ ,$		(42)
	$\displaystyle\Phi_{\pm 2}(t)$	$\displaystyle\equiv\varphi_{2}(t)e^{\pm i2\gamma(t)}\ .$

Here $\phi_{2}$ and $\gamma$ are real functions and $\Phi_{2}=\Phi_{-2}^{*}$ (because the nuclear surface must be a real function), $\upsilon_{0}$ is the constant vacuum expectation value, and (being related to the emergent symmetry breaking) scales as an inverse power of the rotational energy scale

$$\upsilon_{0}\sim\xi^{-1/2}\ .$$

(43)

The other quantities in equation (42) scale as the vibrational energy scale $\Omega$. We have $\Omega\gg\xi$, and

	$\displaystyle\varphi_{0}$	$\displaystyle\sim\varphi_{2}\sim\omega^{-1/2}\ ,$
	$\displaystyle\dot{\varphi}_{0}$	$\displaystyle\sim\dot{\varphi}_{2}\sim\omega^{1/2}\ ,$		(44)
	$\displaystyle\dot{\gamma}$	$\displaystyle\sim\omega\ .$

Under a general SO(3) rotation, the components of the quadrupole field transform as

$$\Phi_{\mu}\rightarrow\exp(i\mu\eta)\Phi_{\mu}\ ,$$

(45)

where $\eta$ is a complicated angle of the rotation angles and the angles that define the orientation of the body-fixed symmetry axis (Papenbrock, 2011).

This simple transformation allows for the construction of effective Lagrangians. Terms that appear to be invariant under SO(2) are in fact invariant under SO(3). The effective Lagrangian at the high-energy vibrational scale is

	$\displaystyle L_{\rm LO}$	$\displaystyle=\frac{1}{2}\left(D_{t}\Phi_{0}\right)^{2}+\frac{1}{2}D_{t}\Phi_{+2}D_{t}\Phi_{-2}-\frac{\omega_{0}^{2}}{2}\Phi_{0}^{2}-\frac{\omega_{2}^{2}}{2}\Phi_{+2}\Phi_{-2}$
		$\displaystyle\approx\frac{1}{2}\dot{\varphi}_{0}^{2}+\frac{1}{2}\dot{\varphi}_{2}^{2}+2\varphi_{2}^{2}\dot{\gamma}^{2}-\frac{\omega_{0}^{2}}{2}\varphi_{0}^{2}-\frac{\omega_{2}^{2}}{2}\varphi_{2}^{2}\ .$		(46)

Here $\omega_{0}$ and $\omega_{2}$ are low-energy constants. The approximation in the second line neglects terms of the order $\xi\ll\omega$ coming from the time derivatives of the rotational angles or the expectation value $\upsilon_{0}$. This yield the Lagrangian of uncoupled harmonic oscillators. The low-energy constants scale as the energies of the excited bandheads, i.e.

$$\displaystyle\omega_{0}\sim\omega_{2}\sim\Omega\ .$$

(47)

At next-to-leading order, the smaller details of the rotational scale $\xi$ enter via the Lagrangian

	$\displaystyle L_{\rm NLO}$	$\displaystyle=\frac{C_{0}}{2}v_{+1}v_{-1}+4\varphi_{2}^{2}\dot{\gamma}\dot{\phi}\cos{\theta}$
		$\displaystyle=\frac{C_{0}}{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2}\sin^{2}{\theta}\right)+4\varphi_{2}^{2}\dot{\gamma}\dot{\phi}\cos{\theta}\ .$		(48)

Here the low-energy constant $C_{0}$ scales as $C_{0}\sim\xi^{-1}$, see equation (30).

A Legendre transformation of the effective Lagrangian at this order yields a Hamiltonian of the form $H=H_{\rm LO}+H_{\rm NLO}$, where

	$\displaystyle H_{\rm LO}$	$\displaystyle=\frac{p_{0}^{2}}{2}+\frac{\omega_{0}}{2}\varphi_{0}^{2}+\frac{p_{2}^{2}}{2}+\frac{p_{\gamma}^{2}}{8\varphi_{2}^{2}}+\frac{\omega_{2}^{2}}{2}\varphi_{2}^{2},$
	$\displaystyle H_{\rm NLO}$	$\displaystyle=\frac{\mathbf{p}_{\Omega\gamma}^{2}}{2C_{0}}=\frac{\left(\mathbf{I}^{2}-p_{\gamma}^{2}\right)}{2C_{0}},$		(49)

with $p_{i}\equiv-i\partial_{i}$ and

$\displaystyle\mathbf{p}_{\Omega\gamma}\equiv\mathbf{e}_{\theta}p_{\theta}+\mathbf{e}_{\phi}\frac{p_{\phi}-\cos{\theta}p_{\gamma}}{\sin{\theta}}.$

(50)

The eigenstates $\ket{n_{0}n_{2}IMK}$ of this Hamiltonian, with $n_{0}$, $n_{2}$ and $K/2$ the number of quanta for the different oscillation modes, have eigenenergies $E=E_{\rm LO}+E_{\rm NLO}$ with

	$\displaystyle E_{\rm LO}$	$\displaystyle=\omega_{0}n_{0}+\omega_{2}\left(2n_{2}+\frac{K}{2}\right)\ ,$		(51)
	$\displaystyle E_{\rm NLO}$	$\displaystyle=\frac{1}{2C_{0}}\left[I(I+1)-K^{2}\right]\ ,$

where the next-to-leading contribution is the rotational energy of the system.

Thus, spectra consists of rigid rotational bands on top of harmonic excitations. Deviations from the harmonic behavior of the band heads can be accounted for by terms containing only vibrational degrees of freedom yielding a correction that can be expanded as a series in powers of $\Omega/\Lambda$ (where $\Lambda$ is the breakdown scale). Since the theory focuses only in the lowest $0^{+}$ and $2^{+}$ bands, traditionally known as the $\beta$ and $\gamma$ bands, those contributions are neglected in what follows. The next-to-next-to-leading order Lagrangian

	$\displaystyle L_{\rm N^{2}LO}$	$\displaystyle=\frac{C_{\beta}}{2}\Phi_{0}v_{+1}v_{-1}+\frac{C_{\gamma}}{4}\left(\Phi_{+2}v_{-1}v_{-1}+\Phi_{-2}v_{+1}v_{+1}\right)$
		$\displaystyle=\frac{C_{\beta}}{2}\varphi_{0}\left(\dot{\theta}^{2}+\dot{\phi}^{2}\sin^{2}{\theta}\right)+\frac{C_{\gamma}}{2}\varphi_{2}\left[\left(\dot{\theta}^{2}-\dot{\phi}^{2}\sin^{2}{\theta}\right)\cos{2\gamma}+2\dot{\theta}\dot{\phi}\sin\theta\sin{2\gamma}\right]$		(52)

is off-diagonal and does not impact energies at that order. It will, however, play an important role in describing electromagnetic transitions between different rotational bands, see section 7.2. Thus, one has to go to one higher order to see dynamical modifications of the rotational moment of inertia.

Zhang and Papenbrock (2013) showed that the next-to-next-to-next-to-leading order contributions are

$$L_{\rm N^{3}LO}=D_{0}\Phi_{0}^{2}v_{+1}v_{-1}+D_{2}\Phi_{+2}\Phi_{-2}v_{+1}v_{-1}+D_{02}\Phi_{0}\left(\Phi_{+2}v_{-1}v_{-1}+{\rm h.c.}\right)+\ldots\ .$$

(53)

Here, the dots denote terms that are not coupled to any rotations (and not of interest to us as they only model vibrational interactions). The Lagrangian (53) corrects the rotational bands via

$$E_{\rm N^{3}LO}=E_{\rm NLO}\times\left[1+a_{0}n_{0}+a_{2}\left(2n_{2}+\frac{K}{2}\right)\right]\ .$$

(54)

Thus, one has a small shift in each band’s moment of inertia that is linear in the number of excited phonons $n_{0}$ and $n_{2}$ of the modes $\varphi_{0}$ and $\varphi_{2}$, respectively. The parameters $a_{0}$ and $a_{2}$ in equation (54) are functions of the low-energy constants of the Lagrangian up to and including equation (53).

Figure 3 compares predicted energies for states in the ground, $\beta$ and $\gamma$ bands in ${}^{166}{\rm Er}$ (blue lines) to experimental data below $2$ MeV (black lines).

The low-energy constants describing the spectra at this order were fitted to the energies of the first excited $2^{+}$ state, and the energies of the band heads and first excited states in the $\beta$ and $\gamma$ bands. The errors shown as blue boxes were estimated as to be $\varepsilon^{2}$ times smaller than the next-to-leading energies, with $\varepsilon\equiv E_{\rm NLO}/\Lambda$ defined in terms of the rotational energy and the breakdown scale, the latter lifted due to the explicit inclusion of quadrupole modes to $\Lambda\approx 3\Omega$, with $\Omega\equiv\max(\omega_{0},\omega_{2})$. These results show that the assumptions about the energy scales and the power counting are consistent.

The effective theory for quadrupole degrees of freedom in the presence of emergent symmetry breaking essentially casts the Bohr Hamiltonian (in the deformed limit or SU(3) limit) into an effective theory. While the leading couplings between vibrational and rotational degrees of freedom are model independent, the resulting effective theory is – of course – ultimately based on a model of quadrupole degrees of freedom. It is unable, for example, to account for any low-lying negative parity bands (see figure 3). On the plus side, however, it makes clear how one could make systematic corrections to the Bohr Hamiltonian.

Papenbrock and Weidenmüller (2014, 2015) considered the nucleus as a deformed liquid drop with axial symmetry. The construction of an effective field theory for this finite system differs from the one used for infinite systems such as (anti)ferromagnets (Leutwyler, 1994; Román and Soto, 1999; Hofmann, 1999; Bär et al., 2004; Kämpfer et al., 2005). In infinite systems Nambu-Goldstone bosons are based on the fields $\phi(\mathbf{x},t)$ and $\theta(\mathbf{x},t)$ that describe the local rotations of spins via the rotation operator (1). The building blocks of the effective field theory are $a_{k,\mu}(\phi,\theta)$ with $k=x,y,z$ and $\mu=t,x,y,z$ that are derived from

$$R^{-1}\partial_{\mu}R=a_{x,\mu}(\phi,\theta)J_{x}+a_{y,\mu}(\phi,\theta)J_{y}+a_{z,\mu}(\phi,\theta)J_{z}\ .$$

(55)

It is particularly important that the angles $\phi(\mathbf{x},t)$ and $\theta(\mathbf{x},t)$ and the quantities $a_{k,\mu}(\mathbf{x},t)$ really depend on position. Purely time-dependent angles that lack position dependence would induce an overall rotation of the (anti)ferromagnet. In an infinite system, the rotated state has zero overlap with the state one started from. Thus, the rotated state is really in an inequivalent Hilbert space. For this reason, overall rotations of the system must be excluded and purely time-dependent fields are forbidden.

In contrast, purely time-dependent fields are allowed in finite systems. Then one must single out the purely time-dependent angles (Leutwyler, 1987); we denote them as $\alpha(t)$ and $\beta(t)$. The key rotation operator then becomes the product

$$U(\alpha,\beta;\phi,\theta)\equiv R(\alpha,\beta)R(\phi,\theta)\ ,$$

(56)

where $\phi(\mathbf{x},t)$ and $\theta(\mathbf{x},t)$ are fields that depend on both time and position. Clearly, this operator induces local distortions in the liquid drop via the fields $\phi(\mathbf{x},t)$ and $\theta(\mathbf{x},t)$ followed by overall rotations of the whole drop via $\alpha(t)$ and $\beta(t)$. The building blocks for the effective field theory, i.e. combinations $a_{k,\mu}(\alpha,\beta;\phi,\theta)$ of degrees of freedom that transform properly under rotations appear on the right-hand-side of

$$U^{-1}\partial_{\mu}U=a_{x,\mu}(\alpha,\beta;\phi,\theta)J_{x}+a_{y,\mu}(\alpha,\beta;\phi,\theta)J_{y}+a_{z,\mu}(\alpha,\beta;\phi,\theta)J_{z}\ .$$

(57)

The derivation of that effective field theory was presented by Papenbrock and Weidenmüller (2014) and followed the coset approach. For simplicity, the position was expressed in polar coordinates, i.e. as $\mathbf{x}=(r,\varphi,\vartheta)$, and the dependence on $r$ was dropped (because radial compression modes were assumed to be high in energy). Finally, the fields $\phi(\varphi,\vartheta,t)$ and $\theta(\varphi,\vartheta,t)$ were decomposed in terms of normal modes and a cutoff was imposed. The resulting effective field theory consisted of vibrational modes (from the decomposition of the fields $\phi$ and $\theta$) and the rotation angles $\alpha,\beta$.

That theory naturally explained how a large number of vibrations arise (these quickly become anharmonic as the energy increases), and how these are coupled to the rotations of the whole droplet. This latter point is very important because the coupling of internal degrees of freedom to the rotations is universal. It consists of a Lagrangian term

$$L_{\rm coup}=K_{z^{\prime}}\dot{\alpha}\cos\beta$$

(58)

that originates from a covariant derivative and couples the angular-momentum projection $K_{z^{\prime}}$ of an internal degree of freedom to the rotation angles. As we will see below, this term can be re-written as a gauge potential.

An important assumption in the construction of the effective field theory (Papenbrock and Weidenmüller, 2014, 2015) was that energies from vibrations are much larger than those from rotations. This is certainly so for sufficiently large droplets. Here, the radius scales as $A^{1/3}$ for a droplet with mass number $A$. Thus, vibrational energies scale as $A^{-2/3}$ while the rotational energies scale as $A^{-5/3}$. This shows that internal degrees of freedom are always “fast” when compared to rotations in sufficiently heavy nuclei.

As a concrete model for a given nucleus, the effective field theory of the liquid drop is probably less useful because a considerable number of low-energy coefficients need to be adjusted to the many vibrational states of the system. This corresponds to a modeling of the internal dynamics. Another concern is that modeling the nucleus as a liquid drop is too simple because it neglects superfluidity.

Instead of modeling the physics of a liquid droplet, it is simpler to introduce internal degrees of freedom as spins $\chi_{S}$ of rank $S$. We allow $S$ to be half integer or integer. A key point is the assumption that the internal degrees of freedom are fast compared to the velocity (19) of the nucleus’s symmetry axis. (Otherwise there would be no separation of scales between rotations and internal motions.) This means that one can work in an adiabatic approximation where the fast spin instantly follows the slow motion of the symmetry axis $\mathbf{e}_{r}(\phi,\theta)$. (This approximation is also known as the “strong coupling” regime in the physics of deformed nuclei.) It is then natural to expand $\chi_{S}$ in terms of body-fixed helicity basis functions $\chi_{S\lambda}(\theta,\phi)$ that fulfill (Varshalovich et al., 1988)

	$\displaystyle\hat{S}^{2}\chi_{S\lambda}(\theta,\phi)$	$\displaystyle=S(S+1)\chi_{S\lambda}(\theta,\phi)\ ,$		(59)
	$\displaystyle\hat{S}_{z^{\prime}}\chi_{S\lambda}(\theta,\phi)$	$\displaystyle=\lambda\chi_{S\lambda}(\theta,\phi)\ .$

Here $\hat{S}_{z^{\prime}}\equiv\mathbf{e}_{r}\cdot\hat{S}$ and it is clear that the helicity basis functions are quantized with respect to their projection $\lambda$ onto the nucleus’s symmetry axis.

In the body-fixed system, the interaction between the spins and that between the spins and the deformed rotor is only invariant under rotations around the nucleus’s symmetry axis. While we do not want to model these interactions, the adiabatic approximation allows us to compute the eigenstates of the Hamiltonian that governs the spins at a given orientation of the nucleus’s symmetry axis. The eigenfunctions are superpositions

$$\psi_{Kq}(\theta,\phi)\equiv\sum_{S}U_{qS}^{(K)}\chi_{SK}(\theta,\phi)\ .$$

(60)

Here, $K$ is the spin projection of the eigenfunction $\psi_{Kq}(\theta,\phi)$ onto the nucleus’s symmetry axis (and we followed nuclear-physics conventions in using this label instead of $\lambda$), and $U_{qS}^{(K)}$ are admixture coefficients with $\sum_{S}|U^{(K)}_{qS}|^{2}=1$ The label $q$ is for other good quantum numbers such as parity and possibly isospin. Clearly, the eigenfunctions (60) are superpositions of spherical tensors with different spin $S$ and not eigenstates of the operator $\hat{S}^{2}$. We only have

$$\hat{S}_{z^{\prime}}\psi_{Kq}(\theta,\phi)=K\psi_{Kq}(\theta,\phi)\ ,$$

(61)

and

$$\psi^{\dagger}_{Lp}(\theta,\phi)\hat{S}_{\pm 1^{\prime}}\psi_{Kq}(\theta,\phi)\propto\delta_{L}^{K\pm 1}\delta_{p}^{q}\ .$$

(62)

Here, the proportionality constant can be computed from knowledge of the matrix elements $U_{qS}^{(K)}$.

Time reversal invariance demands that the eigenfunctions (60) come in pairs $\pm K$ for $K>0$. For half integer $K$, this is Kramer’s degeneracy. For integer $K$, all states with $K\neq 0$ come in doublets and a state with $K=0$ is a singlet. Thus, we can denote the eigenenergies of the internal degrees of freedom as $E_{|K|q}$. We note that the eigenfunctions are orthogonal

$$\psi_{K^{\prime}q^{\prime}}^{\dagger}(\theta,\phi)\psi_{Kq}(\theta,\phi)=\delta_{K^{\prime}}^{K}\delta_{q^{\prime}}^{q}$$

(63)

and complete

$$\sum_{Kq}\psi_{Kq}(\theta,\phi)\psi_{Kq}^{\dagger}(\theta,\phi)=1\ ,$$

(64)

because the matrix $U_{qS}^{(K)}$ in equation (60) is unitary for each $K$.

The energies and quantum numbers $(K,q)$ and energies $E_{|K|q}$ could be taken from data. Alternatively, they could also come from computations that start from deformed reference states (and do not perform symmetry projections).

We now introduce time-dependent basis functions (and drop the angular dependence for simplicity), i.e.

$$\psi_{Kq}(t)\equiv\psi_{Kq}(\theta,\phi;t)$$

(65)

describes the time dependence of an internal state. Then, the Lagrangian for the internal degrees of freedom simply is

$$L_{\psi}=\sum_{Kq}\psi_{Kq}^{\dagger}(t)\left(i\partial_{t}-E_{|K|,q}\right)\psi_{Kq}(t)\ .$$

(66)

The arguments we made in section 4.3 imply that under a rotation the coefficient functions transform linearly via

$$\psi_{Kq}(t)\to e^{-iK\eta}\psi_{Kq}(t)\ ,$$

(67)

with the angle $\eta$ introduced in equations (37). Thus, terms such as $\psi_{Kq}(t)\psi_{-Kq}(t)$ or $\psi_{Kq}^{\dagger}(t)\psi_{Kq}(t)$ (which are scalars under rotations around the body-fixed symmetry axis) are indeed scalars under full rotations thanks to the non-linear realization of the SO(3) symmetry. It is then clear that any axially-symmetric Lagrangian or Hamiltonian in the helicity components is admissible to construct a rotationally invariant effective theory.